Classical confined particles
Descripción
P_
_rence
op on armonic i ators
(NASA-CP-319?) WORKSHOP ON HARMONIC OSCILLATORS (NASA)
515
N93-27311 --THRU-N93-27359
p
Unclas
HI174
0160334
2_
E=....
.
J
.--
.
NASA
Conference
Publication
3197
Workshop on Harmonic Oscillators Edited by D. Han NASA
Goddard
Space Flight Center Greenbelt, Maryland
Y. S. Kim University of Maryland College Park, Maryland
W. W. Zachary Howard University Washington, D.C.
Proceedings
of a conference held at the University of Maryland College Park, Maryland March 25-28, 1992
National Aeronautics and Space Administration Office of Management Scientific and Technical Information Program 1993
ORGANIZING
D.
Han
(NASA/Goddard
Y. S. Kim W. W.
(University
Zachary
This workshop of the National
Space
Flight
of Maryland
(COMSERC,
COMMITTEE
Howard
Center,
at College University,
Greenbelt, Park,
MD)
Washington,
was supported in part by the Goddard Aeronautics and Space Administration.
ii
MD)
Space
DC)
Flight
Center
PREFACE
The Workshop on Harmonic Oscillators was held at the College Park Campus of the University of Maryland on March 25 28, 1992. This Workshop was mostly supported by the Goddard Space Flight Center of the National Aeronautics and Space Administration. The harmonic oscillator formalism has been and still is playing an important role in many branches of physics. This is the simplest mathematical device which can connect the basic principle of physics with what we observe in the real world. The oscillator formalism is, therefore, a very useful language in establishing communications among (1)
The physicists interested in fundamental interested in describing what we observe
principles in laboratories.
and
those
(2)
Researchers in different branches of physics, such as atomic, nuclear and particle physics, quantum optics, statistical and thermal physics, foundations of quantum mechanics and quantum field theory, and group representations for possible future theories.
The Workshop brought together active researchers in harmonic oscillators in many different fields, and provided the opportunity for them to learn new ideas and techniques from their colleagues in the fields of specialization different from their own. The Second International Workshop on Harmonic Oscillators will be held in Mexico in 1993. The Principal Organizer for this important meeting will be Kurt Bernardo Wolf of the Universidad Nacional Autonoma de Mexico.
iii
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Research,
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• Jin, Guo-Xiong, Dept. of Physics,Univ. of Maryland, CollegePark, MD 20742 • Kapuscik, Edward, Dept. of Physics,Univ. of Georgia,Athens, GA 30602 • Karr, Thomas J., LawrenceLivermore National Laboratory, Livermore, CA 94550 • Katriel, Jacob, Dept. of Chemistry,Technion- IsraelInst. of Technology,Haifa 3200,Israel • Kauderer, Mark, National ResearchCouncil, RomeLaoboratory, RL-OCPA, Griffith AFB, NY 13441 • Ketov, SergayV., Inst. for Theoretical Physics,Univ. of Hanover, D-3000 Hanover, Germany • Kiess, Thomas E., Dept. of Physics,Univ. of Maryland, CollegePark, MD 20742 • Kim, Do-Young, Dept. of Physicsand Astronomy,Univ. of Georgia,Athens, GA 30602 • Kim, Y. S., Dept. of Physics,Univ. of Maryland, CollegePark, MD 20742 • Kim, Yurng Dae, Dept. of Physics, Collegeof Education, Chungbuk National University, Cheongju 360-763, S. Korea • Kirson, Michael, Dept. of Nuclear Physics, Weizmann Inst. of Sciences,Rehovot 76100, Israel. • Klauder, John R., Dept. of Physics,Univ. of Florida, Gainesville,FL 32612 • Klink, William H., Dept. of Physicsand Astronomy,Univ. of Iowa, Iowa City, IA 52242 • Kobayashi, Takeshi,12-104Kounan-dai 5-1, Kounan-ku, Yokohama,Kanagawa233, Japan • Kostelecky,V. Alan, Dept. of Physics,Indiana University, Bloomington, IN 47405 • Kretzschmar, Martin, Inst. for Physics, Johannes-GutenbergUniversity, D-6500 Mainz, Germany • Lai, ShengNan, Dept. of Physics,Univ. of Maryland, CollegePark, MD 20742 • Loyola, Gerardo, Inst. de Fisica Teorica, Univ. Nacional Autonoma de Mexico, Apdo. Postal 20-364, 01000Mexico, DF, Mexico • Maguire, Gerald Q., Dept. of Computer Science,Columbia University, New York, NY 10027 • Man'ko, Margarita A., LebedevPhysical Institute, 53 Leninsky Prospect, 117924Moscow, Russia • Man'ko, Olga, Inst. for Nuclear Research,60th October Anniversary Prospect 7A, 117317 Moscow,Russia • Man'ko, Vladimir I., Lebedev Physical Institute, 53 Leninsky Prospect, 117924Moscow, Russia vii
• Mohanty, Pritiraz, Dept. of Physics,Univ. of Maryland, CollegePark, MD 20742 • Morris, Randall, GESD 137-227,GeneralElectric Co., Moorestown,NJ 08057 • Moshinsky, Marcos, Inst. de Fisica Teorica, Univ. Naeional Autonoma de Mexico, Apdo. Postal 20-364, 01000Mexico, DF, Mexico • Mun, Bong Chin, Dept. of Physics,Univ. of Maryland, CollegePark, MD 20742 • Nassar,Antonio B., Harvard-WestlakeSchool,3700Coldwater Canyon, N. Hollywood, CA 91604 • Nelson, Charles, Dept. of Physics,State Univ. of New York, Binghamton, NY 13902 • Nieto, Luis M., Center for Mathematical Research,Univ. of Montreal, Montreal,
Quebec,
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Dept.
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Canada.
Massachusetts
County,
Baltimore,
Washington,
University,
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Research
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and
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Branch,
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DC 20:375
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Postal • Uzes,
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Luc,
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Giuseppe,
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de Ciencias
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E. Baltimore
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of Georgia,
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Univ.
Univ. Street,
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Dept.
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and
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TABLE OF CONTENTS PAGE INTRODUCTION
SECTION
....
I
QUANTUM
,°°°°o°,,°°
....
°°o°°°Do°°,,,°°°
....
MECHANICS
°.°°oo°
....
°,,°-°
....
° .....
° ........................
°°°,.°°,
The
1
......................................................................
Quons, An Interpolation Between Bose and Fermi Oscillators O. W. Greenberg
.......
3
,--
.............................................................................................................
Universal Propagator John R. Klauder ..................................................................................................................
...'_..5
19
Phase-Space Quantum Mechanics Study of Two Identical Particles in an External Oscillatory Potential Luis M. Nieto and Manuel Wave and
SU(2)
Packet
Motion
in Harmonic
Computer Visualization Hideo Tsuru and Takeshi Action-Angle Demosthenes
Gadella ..................................................................................
29
Potential Kobayashi
.............................................................................
35
Variables Ellinas .........................................................................................................
41
Alternative Descriptions of Wave and Particle Aspects of the Harmonic Oscillator Dieter Schuch .....................................................................................................................
47
Symmetries of Coupled Harmonic Oscillators D. Hart and Y. S. Kim .........................................................................................................
53
SECTION
67
II
QUANTUM
GROUPS
...........................................................................
q-Harmonic Oscillators, q-Coherent States and the q-Symplecton L. C. Biedenharn, M. A. Lohre and Masao Nomura ...........................................................................................................
xi
69
Which Q-Analogue of the Squeezed Oscillator? Allan 1.Solomon ................................................................................................................. 85 Deformation of Supersymmetric and Conformal Quantum Mechanics Through Affine Transformations Vyacheslav Spiridonov ..................................................................................................... 93 Phase of the Quantum Harmonic Oscillator with Applications to Optical Polarization Scott R. Shepard ............................................................................................................. 109 Condition for Equivalence of q-Deformed and Anharmonic Oscillators M. Artoni, Jun Zang and Joseph L. Birman............................................................................................................. 115 Novel Properties of the q-Analogue Quantized Radiation Field Charles A. Nelson ........................................................................................................... 121 SECTION III
QUANTUM OPTICS
.......................................................................... 127
Distribution of Photons in "Squeezed" Polymode Light, V. I. Man'ko ....................................................................................................................... 129 Two Photon Annihilation Operators and Squeezed Vacuum Anil K. Roy, C. L. Mehta and G. M. Saxena............................................................................................................ 141 Boundary Conditions in Tunneling via Quantum Hydrodynamics Antonio B. Nassar and Physics Department, Harvard-WestlakeSchool.............................................................................................. 149 Using Harmonic Oscillators to Determine the Spot Size of Hermite-Gaussian Laser Beams Sidney L. Steely .............................................................................................................. 155 Squeezed States of Damped Oscillator Chain O. V. Man'ko ..................................................................................................................... 171
xii
The Uncertainty Principle In Resonant Gravitational Wave Antennae and Quantum Non-Demolition Measurement Schemes Pierluigi Fortini, Roberto Onofrio and Alessandro Rioli....................................................................................................... 179 Harmonic Oscillator States in Aberration Kurt
Bernardo
SECTION
Point
IV
Form
Optics Wolf ..........................................................................................................
SPECIAL
Relativistic
RELATIVITY
......................................................................
Phase Space Localization for Anti-De Sitter Quantum Mechanics and Its Zero Curvature Amine M. El Gradechi and
About
of Mathematics University ......................................................................................................
Massive
219
Limit
Proper-Time Relativistic Dynamics Tepper L. Gill, W. W. Zachary and John Lindesay .......................................................................................................... Remarks
217
Quantum
Mechanics and Relativistic SU(6) W. H. Klink .........................................................................................................................
Department Concordia
195
235
241
and Massless
Particles in Supersymmetry S. V. Ketov and Y.-S.
Kim .........................................
:....................................................
255
On Harmonic Oscillators and Their Kemmer Relativistic Forms Nathalie
Debergh
and Jules
Beckers
..........................................................................
The Algebra of Supertraces for 2+1 Super De Sitter Gravity L. F. Urrutia, H. Waelbroeck and F. Zertuche ................................................................................................................ Quantum
Wormholes
and
Harmonic
261
267
Oscillators
Luis J. Garay .....................................................................................................................
xiii
273
On Inducing Finite Dimensional Physical Field Representations for Massless Particles in Even Dimensions Vineer Bhansali ............................................................................................................... Non Linear
Modes
of the Tensor
Dirac
Equation and CPT Violation Frank J. Reifler and Randall Galilean
Covariant
Andrezej
SECTION
Harmonic
Horzela
V
and
281
D. Morris
.........................................................................
289
Oscillator Edward
Kapuscik
THERMODYNAMICS STATISTICAL
....................................................................
295
AND
MECHANICS
...........................................................
307
Double Simple-Harmonic-Oscillator Formulation of the Thermal Equilibrium of A Fluid Interacting with A Coherent Source of Phonons B. DeFacio, Alan Van Nevel and O. Brander ................................................................................................................. Wigner Expansions for Partition of Nonrelativistic and Relativistic Oscillator Systems Christian Zylka Decoherence Squeezed
and Gunter
of Multimode Coherent
Functions
Vojta ..................................................................................
Yuhong
Zhang
323
Thermal
States
Leehwa Yeh and Physics Division, Lawrence Berkeley Laboratory ..................................................................................... Quantum Harmonic In A Thermal Bath
309
337
Oscillator .................................................................................................................
A Study of Electron Transfer Using A Three-Level System Coupled to An Ohmic Bath Masako Takasu and David Chandler
..........................................................................
xiv
353
365
SECTION
Vl
GROUP
REPRESENTATIONS
Symmetry Algebra of A Generalized Anisotropic Harmonic Oscillator O. Castanos and R. Lopez-Pena
........................................................
..................................................................................
371
373
Fermion Realization of Exceptional Lie Algebras from Maximal Unitary Subalgebras A. Sciarrino .......................................................................................................................
379
Quantized Discrete Space Oscillators C. A. Uzes and E. Kapuscik ..........................................................................................
385
Orbit-Orbit Branching Rules F. Gingras, J. Patera and R. T. Sharp .........................................................................
397
SECTION
VII
ATOMIC, PARTICLE
NUCLEAR, PHYSICS
AND ....................................................................
403
.........................................................................................
405
Mass Spectra of the Particle-Antiparticle System with A Dirac Oscillator Interaction M. Moshinsky From
and G. Loyola
Harmonic to Anharmonic Oscillators F. lachello ..........................................................................................................................
Atomic Supersymmetry V. Alan Kostelecky The
429
...........................................................................................................
443
Harmonic Oscillator and Nuclear Physics D. J. Rowe .........................................................................................................................
453
On the Spring and Mass of the Dirac Oscillator James P. Crawford .......................................................................................................... Classical
Confined
Andrzej
Horzela
469
Particles and
Edward
Kapuscik
......................................................................
Second Quantization in Bit-String Physics H. Pierre Noyes ................................................................................................................
](V
475
483
Covariant Harmonic Oscillators -- 1973 Revisited M. E. Noz ...........................................................................................................................
495
Calculation of the Nucleon Structure Function from the Nucleon Wave Function Paul E. Hussar .................................................................................................................
511
xvi
INTRODUCTION The harmonic oscillator is the basic scientific language for physics. It was Einstein who predicted the existence of quantized energy levels using the harmonic oscillator model
for
the
specific
head
of
solids.
The
role
of
harmonic
oscillators
in
the
development of quantum mechanics and quantum field theory is well known. Indeed, because of its mathematical simplicity, the harmonic oscillator model often precedes new physical theories. It also acts as an approximation in many of the existing theories. Among the most respected physicists of our century, Paul A. M. Dirac and Richard P. Feynman were quite fond of harmonic oscillators. It was Dirac who started using harmonic oscillators for representing the Lorentz group. It was Feynman who said that we should try an oscillator formalism, instead of Feynman diagrams, to understand relativistic bound-state problems. Feynman's path integral formulation of quantum mechanics is also based on harmonic oscillators. These two physicists have left a profound
influence
on what
we are doing
now.
In spite of its past role, it is important to realize that we do not study harmonic oscillators to learn the history of physics. Our major concern is the future of physics. Let us look at one of the cases of what we are doing today. Since the development of lasers in the late 1950's and early 1960's, the theory of coherent radiation has become a major branch of modern physics. It is generally agreed that this new theory is more or less the physics of harmonic oscillators or the study of the Lorentz group using harmonic oscillators which Dirac pioneered (J. Math. Phys. Vol. 4, page 901, 1963). Let us look at another example. The question of thermal excitations and the lack of coherence is of current interest. Here also the basic theoretical tool is the harmonic oscillator 1972).
as Feynman
noted
in his book
on statistical
mechanics
(Benjamin/Cummings,
In view of the past and present roles of harmonic oscillators in physics, it is fully justified to develop new oscillator formalisms for possible new physical theories in the future, even though their immediate physical applications are not clear. This typically takes the form of constructing representations of groups using harmonic oscillators. Developing a mathematical formalism before the birth of a new physical theory is the most sacred role of mathematical physics. The theory of squeezed states of light is a case in point. It was possible to construct this theory very quickly because all relevant mathematical techniques were available when its physical idea was conceived. The harmonic oscillator indeed occupies a very important place in mathematical physics. The
Workshop
on Harmonic
Oscillators
The Workshop was attended by many including those in atomic, nuclear, and physics, as well as mathematical physics. are potential developers of new theories.
was
the
first
scientific
meeting
of its
kind.
of the researchers in harmonic oscillators, particle physics, quantum optics, statistical It was also attended by many students who Many interesting papers were presented.
There were many lively informal discussions. However, the Workshop was by no means a perfect meeting. Many potential participants did not attend the Workshop because the purpose of the meeting was not clear enough to them. Yet, those who came to the Workshop have set the tone for future meetings in the same series. It is the participants, not the organizers, who decide the success or failure of any given scientific meeting. Indeed, the participants of the Workshop on Harmonic Oscillators did very well, and well enough to generate the second meeting in the same series.
I.
QUANTUM
MECHANICS
N93" 7812 QUONS AN INTERPOLATION BOSE AND FERMI O.W. Center
for
BETWEEN OSCILLATORS
Greenberg Theoretical
Physics
Department of Physics and Astronomy University of Maryland College
Park,
MD
20742-_111
Abstract After
a brief
mention
of Bose and
Fermi oscillators
and of particles
which
obey
other
types of statistics, including intermediate statistics, parastatistics, paronic statistics, anyon statistics and infinite statistics, I discuss the statistics of "quons" (pronounced to rhyme with muons), particles whose annihilation and creation operators obey the q-deformed commutation relation (the quon algebra or q-mutator) which interpolates between fermions and bosons. I emphasize that the operator for interaction with an external source must be an effective Bose operator in all cases. To accomplish this for parabose, parafermi and quon operators, I introduce also discuss interactions
parabose, parafermi and quon Grassmann numbers, respectively. I of non-relativistic quons, quantization of quon fields with antiparti-
cles, calculation of vacuum matrix elements of relativistic quon fields, demonstration of the TCP theorem, cluster decomposition, and Wick's theorem for relativistic quon fields, and the failure of local commutativity of observables for relativistic quon fields. I conclude with the bound
1
on the parameter
q for electrons
due to the Ramberg-Snow
experiment.
Introduction
I start
by reviewing
the
(Bose)
harmonic
oscillator.
I want
to emphasize
that
the
commutation
relation, [ai, a_]._ _ a,a_ - a_a, = 5,j, and
the
vacuum
condition
which
(1) characterizes
the
Fock representation
a 10) =0 suffice
to calculate
operators. contraction the vacuum
(2) all vacuum
matrix
elements
of polynomials
in the
annihilation
The strategy is to move annihilation operators to the right, picking of an annihilation and a creation operator. When the annihilation on the right,
(Olaflai2...ai,,a_m
it annihilates
up terms operator
creation with a gets to
it. For example,
" .. aj2 t ajlt lO) = 6_,jm (Olai, ai2 ..... a,,__l a__l +(Olai, a,_'"ai,.__aj,.a.,t
.
PAGE BLAI'_K
NOT
uJ2-ta tjl IO)
a tj__,'"aj2aj_t
5
PRE6F.£)ii_i_
and
FILMED
t IO).
(:_;)
-Continuing
rion,
thag Yeduc
it is clear
that
this
vacuum
matrix
element
vanishes,
unless
the
set
{il,i2,...,i,_} is a permutation of the set {jl,j2,'",jm} (this includes n = m). In particular, relation is needed between two a's or between two a t 's. As we know, it turns out that [a;, aj]_ = 0 = [ai,aj]-, t t but
these
relations
are
(one-dimensional)
redundant
observables
nk = nkk = at ak, The
(4) in the
representations
To construct operator, nkt,
commutation
Fock
of the symmetric
in the free theory
nk,
representation.
Also, only
(i.e., permutation)
we can use the number
the
group
totally
symmetric
S,_ occur.
operator,
nk, or the transition
(5)
=atat.
relation,
[nkt, at] - =6tm at, follows
Using
from
Eq.(1).
(6)
The
number
nk(at)_rl0)
= A/'(at)arl0).
nk and
nkl we can construct
Y = _
no
operator
has integer
eigenvalues, (7)
the Hamiltonian,
eknk,
(8)
k
and
other
observables
[H,a]]_ Analogous
for the free theory.
The
Hamiltonian
obeys
= e,a].
formulas
(9) of higher
degree
I want to pay special attention ration for that I write the external H_xt = __(j;ak
in the a's and to couplings Hamiltonian
at's give interaction
to external sources in the Bose case,
terms.
in the
quon
theory;
in prepa-
+ atjk),
(10)
k
where
jk is a c-number;
i.e.,
[jk, a_]_ = [jk, j[]This satisfies
the
[H, xt,a_]_ Equations free theory
= 0, etc.
commutation
(11)
relation
= j;.
(9) and
(12)
(12) state
with an external
[n, at at,, t_ "" "at]_
=
that
H and
source.
H¢,t are
In particular,
"effective Eq.
E e,ala], • • •at
(9) and
Bose operators"
in the
context
of a
(12) imply (13)
i
and [H,_:t,a tat I_
12 ""
.aLl_
=Eat i
ll
a 12 t a l,_a.]l, t ..
t ali+l
"
..at,
(14)
so the energy operator
is additive
is that
by a large
spacelike
definition
for all cases,
I stated
with obvious relation;
commutes
The
general
with the
definition
field when
the
of an effective points
Bose
are separated
---, 0, Ix - Yl ---' _.
holds
Everything
of free particles.
density
distance,
[7"/(x), ¢(y)]_ This
for a system
the Hamiltonian
for the
modifications.
(15)
including
quons.
Bose harmonic
The
oscillator
commutation
relation
can be repeated
for the
Eq. (1) is replaced
Fermi
oscillator,
by the anticommutation
[ai, a_]+ = aiaJ + a_ai = 6ij that,
together ail0)
again
with
condition
which
to calculate
operators.
Continuing
this
{il,i2,...,in}
all vacuum
between
these
reduction,
matrix
it is clear
two a's
that
of the
or between
elements
of multinomials
in the
annihilation
relations
again
antisymmetric
....."ai,, _ ,atmai.at _ this vacuum
set
ai_ai ,t * 10)
, m_ ,
matrix
{jl,j2,'",jm}.
t 't lO). a_aj,
element
it turns
(18)
vanishes,
In particular,
two a t 's. As we know,
unless
again
no
commutation
from
are redundant
in the
in the
Fock
representation.
representations free
theory
of the
we again
use
Also,
symmetric the
as we know, group
number
only
the
operator,
nk,
or the
(20)
relation
(2_) relation
Eq.(16).
The
number
operator
ues; now, however, the number of particles in a single quantum state since Eq.(19), which holds in the Fock representation, implies a t2 i = 0,
n_(a_)_rl0)-- g_,ar(aIFrlO),ar-- o, 1
n = _
is
occur.
_k,= 4a,.
the commutation
nk and
set
out that
[n_,,at]_ = 6,m_L
Using
the
relation
(19)
(one-dimensional)
_k = nkk= 4a_,
follows
and
= 0 = [a_,aJ]+,
To construct observables transition operator, nkt,
The
representation,
t t •. %%10) = _,o,_(01(01a_,a_...... ai,,_,aJ,,,_,
is a permutation
[ai,aj]+
totally
Fock
For example,
-(0la,%
but
the
(17)
(0lai, a, 2 .... ai.a_
needed
characterizes
=0,
suffices
creation
the vacuum
(16)
nkt we again
can construct
again can
has only
integer be zero
eigenvalor one,
(22)
the Hamiltonian,
eknk
(23)
k
7
and other
observables
for the
free theory.
The
Hamiltonian
obeys
[H, a]]_ = eraS. Analogous
formulas
(24)
of higher
I again pay special Fermi case is
degree
attention
in the a's and
to couplings
at's give interaction
to external
sources;
terms.
the external
Hamiltonian
in the
(25) k
where
fk is an anticommuting [fk,ft]+
The
The
= [fk, ft*]+ = [fk,a,]+
external [H,,,,
Hamiltonian
number, (26)
= [fk, a/]+ = 0.
satisfies
the commutation
relation,
a[]_ = ft*.
commutation
tors"
(Grassmann)
in the
(27)
relations
context
Eq.(24)
and
of a free theory
Eq.(27)
with
state
that
an external
H and
source.
H_t
are
"effective
In particular,
Bose
Eq.(24)
and
operaEq.(27)
imply [H,a_a
t12"'"
al, ]_
eiall tat 12"'"
-" _ 2._ i
at
(28)
and [Hezt,a_la_2
...aL]_
= y_ a_a_2a_,_,ft:a_,+,
...aL,
(29)
i
so that the energy is additive Notice that Eq.(2,5,6,8,9,13) Fermi
case.
Eq.(7,10,12,14)
for the
case. Finally, Eq.(1,3,4,11) by minus signs.
2
for a system of free particles. for the Bose case are identical
Generalizations the first attempt
[1]. He suggested
"intermediate
Clearly
Fermi
case
statistics
are
case and
of Bose
As far as I know, state.
Bose
for the Bose
statistics,"
Fermi
for the
for the
case differ
statistics
up to n particles
for n = 1 and
Bose
dubbed operator,
his invention Eq.(5,
a generalization
"parastatistics"
20), have
the same
which
[3]. Green form
is invariant
noticed
for both
that
bosons
Fermi only
by G. Gentile
a given
is recovered
quantum
in the
limit
a proper quantum statistics, state is not invariant under
under
the number
and
was made
can occupy
statistics
n --+ c_. As formulated by Gentile, intermediate statistics is not because the condition of having at most n particles in a quantum change of basis. H.S. Green [2] invented
Fermi
Statistics
Bose and Fermi
in which
for the
to Eq.(22,25,27,29)
Eq.(16,18,19,26)
and
to go beyond
is recovered
analogous
to Eq.(17,20,21,23,24,28)
fermions,
change
of basis.
operator as do the
and
I later transition
commutation
relations
between
Green
the
generalized
transition
the
operator
transition
_nd the creation
operator
and
annihilation
operators,
Eq.(6,
21).
by writing
= where
the
(30)
upper
signs
for the generalization states
of fermions
are necessary akt0)
the new
are for the generalization ("parafermions").
to fix the Focklike
Since
representation:
("parabosons") Eq.(30)
the usual
and
is trilinear, vacuum
the lower
signs are
two conditions
condition
the
is
= 0;
(31)
condition
aka[[O) = p 5k,,p contains obeying
of bosons
integer,
the parameter parastatistics
(32)
p which is the order of the parastatistics. The Hamiltonian for free particles has the same form, in terms of the number operators, as for Bose and Fermi
statistics, g = _
e_n_,
where,
as usual
[H,a[]_
= e,a[.
(33)
k For interactions
with an external
source,
we must introduce
the interaction Hamiltonian an effective Bose operator. Bose and Fermi sources discussed above. We want
para-Grassmann
This is in analogy
numbers
which
make
with the cases of external
= c;. We accomplish Hi? where
the
this
the
"etc." retains the
by choosing
Hext = Ekl rz,_t J_ kl
-- (1/2)([c_,a,]+
+ [a_,c,]+),
para-Grassmann
numbers
[[ck, cl]+,em]_ and
(34)
upper
-
(lower)
'
with (35)
ck and c_ obey
O, [[c*k,a,]_,a_]_
= 25track,
etc.,
sign is for parabose-Grassmann
(36) (parafermi-Grassmann)
numbers.
right-hand-side. It is worthwhile
to make
explicit
the fact that
in theories
with
parastatistics
states
many-dimensional representations of the symmetric group. This contrasts with and Fermi statistics in which only the one dimensional representations occur. I emphasize that parastatistics is a perfectly consistent local quantum ables, such as the current, are local provided the proper symmetrization used;
belong
the cases
to
of Bose
field theory. The observor antisymmetrization is
for example, ff(x.)
= (1/2)[(k(x),7_'f(z)]_
for the current are
The
in Eq.(36) means that when some of the c's or ct's is replaced by an a or an a t, the relation its form, except when the a and a t can contract, in which case the contraction appears on
no negative
of a spin-l/2 probabilities.
(37) field. On
Further, the
all norms
other
hand,
in a parastatistics parastatistics
theory
of order
are positive;
p >
1 gives
there a gross
violation of statistics; for example,for a parafermitheory of order p be occupied
p times.
Within
the
dimensions) (i.e.,
last
have
Fermi
are
subject
can be traced
Bose
norms
first appear
from
(i.e.,
state
with
those
with
there
manifestly
Young positive
covariant
statistics
theories
can
statistics
in which
be violated
quantum
a gross (in three
the
Pauli
by a small
state
can
violation. or more
exclusion
amount.
space
principle
One
of these
commutation
relations
[14]. (Deformations
in particular,
quantum
interest
and
[15].) The which states:
particles,
(3,1).
squared
norms
called
probabilities
The
"parons,"
but some
states
extensive
which
of such
this type
The
the
negative
theories
with
of statistics negative
squared
norm
norms
norm
states
states
parons
on this
have
negative
the corresponding
Thus
must
negative
in [5] the first
seem that
(as, in contrast,
obey
literature
theories
in the theory). considered
It does not
electrodynamics).
and groups,
at present.
in the model
tableau
quantum
of algebras
activity
is local,
are negative
in many-particle
to particle
to provide
> 1 each
to rule out such
[4, 5, 6, 7, 8, 9, 10, 11, 12, 13] of the trilinear
of great
field theory
is not needed
two new approaches in order
and/or
a subject
a quantum
from
years
studied
[2] and Volkov.
groups,
in the
experiment
uses deformations
of H.S. Green
squared
three
been
statistics)
approaches
have
A precise
occurs
decouple
do decouple
seem
to have
in
a fatal
flaw.
The
other
approach
uses deformations
[16, 17, 18, 19, 20, 21, 23]. The positive-definite squared norms of such the
theories
validity
problem
fail
to have
of relativistic
with
hold for free relativistic Still other
the
quon
non-relativistic
desired
quon
locality
theories, quon
approaches
of the
but,
theories.
theories,
so even
to violations
another
type
Bose
of statistics,
properties.
in contrast
anyon
relativistic
quon
statistics,
were given
been
In the
form
framework references
3
usually
The
In their included
Quon
questions does
theorem may
extensively
not cause
and
a
clustering
be interesting.)
in [24, 25, 26]. order
about
An interpolation
was studied discussed
superconductivity.
in [27]; this recently,
and
For anyons,
1).
the
(38)
considered,
I am considering. [29, 30].
The
3.1
which
= _'%(2,
theories,
theories
relations
called "quons," have but the observables raises
the TCP
of increasing has
commutation
failure
to the paron below,
parastatistics
Fermi
of statistics, parameter, This
applied to the fractional Hall effect and to high-temperature transposition of two particles can give any phase, ¢(1,2)
and
(As I prove
of statistics
between Bose and Fermi statistics using also does not give a small violation. Yet
bilinear
particles which obey this type for a range of the deformation
anyons
only exist
I will not
discuss
in two space them
further
dimensions,
and
here;
I give
rather
are
outside two
the
relevant
Algebra
q = 0 case
general bosons,
classification of possible particle statistics, Doplicher, Haag and fermions, parabosons, parafermions and one other case, infinite
all representations
of the symmetric
group
10
could
occur,
but
Roberts statistics,
did not give an algebra
[31] in which
led to this last the
Bose
and
case.
Roger
Fermi
aka_ -- _k,,
to Hegstrom
Hegstrom's
permission,
statistics. (ark1
vanishes
that
one can choose
me, this algebra
I followed
Consider
This norm
at Wake
Forest
University,
suggested
averaging
to get (39)
and
atk.lO), d p_lkl
• . .
a chemist
relations
ak]0) = 0.
(Unknown infinite
Hegstrom,
commutation
unless
had
up his idea and
a general
scalar
been
considered
showed
that
earlier
by Cuntz
[28].)
gives
an example
this algebra
With
(4o)
"" "';p-,kmlO)).
n = m and coefficients
P is the c(P)
of
product,
identity,
to project
and
then
into states
it equals
in each
one.
From
irreducible
this
of S,
it follows
and
that
the
will be positive,
•.. atp_, .lo)ll > o;
[[ y_ c(P)atp_,k,
(41)
P
thus every representation of S,_ occurs. Note as before, the Fock vacuum condition makes To construct
Once
observables,
[nk,a_]_
= 5_,a_,
Eq.(42)
holds,
we want
[nkt, at]_
that there is no relation between such relations unnecessary.
a number
operator
and
a transition
two a's or two a s;
operator
which
= 5t,_a t.
the Hamiltonian
obey (42)
and other
observables
can be constructed
in the
usual
way;
for example, H = _eknk,
etc.
(43)
k
The
obvious
thing
is to try
nk = atkak.
(44)
[nk, a_]_ = atkaka_ -- a_a tak.
(45)
Then
The first term This
in Eq.(45)
can be done
is 6kta t as desired;
by adding
the
term
however
the second
Z, a_at_akat to the term
term is extra in Eq.(44).
term, but adds a new extra term, which must be canceled by another term. an infinite series for the number operator and for the transition operator,
nkt = a a,+ E ala a,a,+ E d
As in the Bose
case, this infinite
operator whose domain includes the vacuum. (As far as I know, etc.
for a free field are of infinite
, t ,
at2atlakalaqat2
and must This
be canceled.
cancels
This
the extra
procedure
yields
(46)
+,..
tl ,t2
series for the transition
or number
states made by polynomials this is the first case in which degree.)
11
operator
defines
an unbounded
in the creation operators acting on the number operator, Hamiltonian,
For nonrelativistic theories,the x-space pl(x;
y) = ¢'t(x)¢(y)
+ f
if- / d3 zld3Z2C(z2)_bt(zl which
obeys
The apparent significance.
Qj_(y)]_
= o,
where Q = f d'xjO(x). the Q factors; however,
= O,
then
Eq.(49)
holds,
despite
tion,
not
representation
the
electrodynamics In a similar
Then
the
in the way,
and
p(x;y)]0)
associated consider
= 0.
with
(48)
the
space
integrals
has
no
physical
x ... y,
(49)
(49)
seems
to have
nonlocality
because
of the space
integral
x .._ y, the
Coulomb
in
(50) apparent
in terms
y', x'), ¢t(z)]_
Hamiltonian
(47)
requirement
w)¢t(x),
Equation if
_,(x),j_(y)l_
is
-t-...,
)_(z2)
nonlocality of this formula To support this last statement,
[Qj_,(x),
[p2(x,y;
locality
= _(y-
operator
d3z_t(z)¢t(x)_(y)¢(z)
)g't(x)c(y)¢(zz
the nonrelativistic
[p,(x;y),¢t(w)]_
form of the transition
nonlocality.
of a space
gauge
= _(x'-
is another
is relevant
(The
is the
apparent
commutation
nonlocality
rela-
of quantum
such example.)
z)_t(x)pl(y,
of a nonrelativistic
What
integral.
y ') + _(y'-
theory
z)¢t(y)pl(x,
with two-body
x').
interactions
(51)
has the
form
1
g=(2m)-'
[H, _t(zl)...
f d3xV,'Vx,pl(x,x')Ix=x,+ f
_bt(z,_)]_
= -(2m)
-1 _
d3xd3yV(lx-yl)p2(x,y;bfy,
j=l
+
v(Iz,
Vz2, + _
- zjl)]¢t(z_)...
x).
(52)
_pt(z.)
i0, and
¢ +
2 +ikxo(X-Xo)-ri(y-yo)
_ are complex
the
h:v: 2m
equation
as
= _0+ i_1, which
Potential
N is a normalization
rio>O,
4_orio-A_
>0,
(19)
constant
(20) Using evolution
the same techniques of the wave packet
and procedure in the one-dimensional in terms of an infinite series. O0
_b(x,y,t)
we obtain
the
= _[un(y)
_
Cm,nu,n(x)exp(-iw(m
+ n + 1)t)],
(21)
m=O
expansion coefficients Cm,,, are also calculated For an uncorrelated initial condition
explicitly.
_(x,y,0)= ¢ 7¢_°0°exp(-_(_- Xo) _+ __o(x- _o)- ,(y- yo)_+ _kyo(y - yo)), we can
evaluate
the
infinite
¢(_,u,t) = ¢ X
exp(i(1-
time
oo
n----O
The
case,
(22)
series
1 27raxt_ryt
exp(_
4a_) sin(2wt + 2%)(x16a_a_t
(x - x,) z 4a_t
x,) 2 + i (1 - 4a_)sin(2wt
x exp(-i
.k_oxo
2
.1 arctan(
y,)2)
tan(wt))
+ ion)
tan(cot))
k2Yt
38
+ ikyty)
+ 2%)(y-
16o_o_,
kxt3_t
× exp(-i
(yy,)_ 4a_t )exp(ikxtx
(23)
where (1 - 4(((_ + (_))_o
2 O':r
2__ Gy
(-16(,7o 2 + r/_) 2 + 4(r]02 1 %=_arctan(l_4(_02+_)),
2
sin2( wt + 72,) + 4_
cr_,=
r?_)) cos2(%)
4_1
cos2(wt
0x and
0r are
time
+ %:)
2 '
independent
_(
We obtain
the
phase
time
dependence
x, = XoCOs(wt) k,, The
trajectory
around
4
the
= kxocos(wt)
of the
origin
with
Uniform
The
SchrSdinger
two dimensional
center
equation flat
plane
(27)
),
(28)
+ rh)Y°2 +2
1 arc_an( , , tan(Tv) -2a_
)'
(29)
following
parameters
y, = yocos(wt)
-
,
ku, = l%ocos(wt)-
magnetic
'
. tan(7_) _2or
+ _ arctan(
,
probability
frequency
(26)
+ %)
4cr2 u
+ k,osin(wt)
of the
),
sin2( wt + %) + 4a_ cos2(tot
) + _l)xg
x0sin(wt)
(25)
'
4,71 1_4(,7o2+,./2)
1
of the
of mass
an angular
1 arctan(
r/_)) sin2(Tv)
factors
_(1 - _4_! sin(2"Tv! 4(sin2(Tv) + 4a_ cos2(%))
explicit
4(7702-
cut=
(1 - 4a_) sin(2%) = 0. -(4(sin:(%) + 4a_ cos2(%)
Or=
+ (1 7y=_
4cr_
Here
(24)
(-16(_o2 + _1_):+ 4(_o 2- _)) cos2(%)+ (1- 4(_o 2 - _)) sin2(%:)' (1-4(70: + ,7,_))_o
density
(30)
+ kvosin(wt), yosin(wt) function
.
is an elliptic
(31) motion
w.
field
for a single
electron
in a uniform
magnetic
field perpendicular
to the
is
(32) where
the
vector
potential
A in Landau
gauge
is (33)
a = (-By, O) We
separate a special solutionof the wave equation as ¢(x,
The
wave
equation
for f(k,
y, t) = exp(ikx)f(k,
y, t)
(34)
y, t) becomes
(35)
39
where
eB c_ -
This
is the
one
k/o_.
Thus
above
of the are
wave
dimensional techniques
packet
presented
[6].
in the
following
equation
and procedures
We choose
literatutre
harmonic potential and is the half of the latter. For the
SchrSdinger
magnetic This fact
initial
the
[6].
-ch for the
harmonic
can be applied
in order
initial
The
(36)
wave
major
packet
difference
field is the period is also interpreted
potential
centered
to obtain
eq.(17).
The
between
the time
comlete
at y = evolution
descriptions
two dimensional
isotropic
of the change of the variance. The former by the pass integration technique [6], [8].
condition i
_=_' the
shape
of the contour
lines
of the
_=2
probability
-
(37)
density
function
remains
circular
during
the
motion. For the
following
initial
condition 1
i
= _ = _ ' _ = 2' the
shape
5
of the
probability
density
remains
(38)
unchanged.
Conclusion
Using
a frame
buffer
which
can
us an intuitive
The the
give
potentials
wave
NVS2000
are simple
packet
motions
and
video
recorder
understanding but
are very
due
BVW-75,
of the wave
to the
quantum
we have
packet
mecanical
made
CG
animations
motions. property
the
analytic
form
complicated.
References [1] E.
SchrSdinger:
[2] L.
I. Schiff:
[3] I.
Fujiwara
[4] S.
Brandt
Berlin,
Naturwiss. Quantum
and and
14 664 (1926).
Mechanics
K. Miyoshi
3rd ed. (McGraw-Hill,
: Prog.
H. D. Dahmen
Theor.
: Quantum
Phys. Mechanics
York,
1968).
715 (1980). on the Personal
Computer
(Springer,
1989).
[5] H.
Tsuru
: J. Phys.
Soc.
Japan.
60
3657
(1991).
[6] H.
Tsuru
: J. Phys.
Soc.
Japan.
61
2246
(1991).
[7] H.
Tsuru
: J. Phys.
Soc.
Japan.
61
2595
(1991).
[8] R.
P. Feynmann
New
64
New
York,
and
A. R. Hibbs:
Quantum
Mechanics
1965).
40
and Path
Integrals
(McGraw-Hill,
of
N93-27316
SU(2)
ACTION-ANGLE
VARIABLES
Demosthenes
Ellinas
Department of Theoretical Physics, Siltavuorenpenger _0, SF-00170
University Helsinki,
of Helsinki Finland
Abstract Operator aaagle-action variables are studied in the fra_me of the SU(2) algebra, and their eigenstates and coherent states are discussed. The quantum mechanical addition of actionangle variables is shown to lead to a novel non commutative Hopf algebra. The group contraction is used to make the connection with the harmonic oscillator.
1
Introduction
Action-angle variables in quantum mechanics one known to lack, in the operator level, properties of their classical analogues [1,2]. Especially the exponential phase operators harmonic
oscillator,
occuring
in the polar
operators (an operator analogon and satify the weaker condition
decomposition
of the bosonic
of the polar decomposition of a complex of one side-unitary or isometry operator.
creation
some of for the
and annihilation
number), lack the unitary Based on the mathemat-
ical fact that, unlike in finite dimensional Hilbert spaces as the Fock space of harmonic in finite spaces an isometry is equivalent to a unitary operator, we have in recent works,
oscillator, suggested
a group theoretical construction of a unitary phase operator by introducing action-angle for the SU(2) algebra and going over to their oscillator counterparts via the InSn_-Wigner
variables method
of group contraction [3-6]. In this report we will briefly review and then expand this work with respect to two aspects: first, a set of coherent states will be introduced along the lines of the displacement operator creating the usual coherent states from the vacuum state and second, we will show that aAdition of spins in terms of their action-angles addition in terms of the step (cartesian) operators, involves commutative Hopf algebra structure and relates interestingly subject
2
of quantum
(polar) operators, unlike the usual a genuine no commutative, no cothe phase operators subject to the
groups.
Action-angle
Variables
and
Let us start with the SU(2)
action-angle
operators
J_=ei_
J_+J_=
41
States
J_/r)__J+ei¢
(1)
fi-+j_,-,.
(2)
%
_r_ ¸
,
ir
where 2j J+ = _.,
Cm(2j
- m + 1)]J;rn
+ 1 >< J;m[
,
J-
= J_.
(3)
rnmO
2j & = __,(m-j)lJ;m
><
J;m I
(4)
11 ,
(5)
rn_0
and _j
e_¢ = _--_IJ;g >< J;l+ /=0
mod(2j
+ 1), and
hh + = h+h = 1 with
h = e i¢, h + = e -i'_ the
from the fact that h, generates the cyclic group Z2_+1 acting space of the algebra we can construct phase states 1 ]¢;k
>=
FlJ;k
>=
unitary
angle
as a cyclic permutation
operator.
Then
in the weight
2j
__, _k'_lJ;n
>
(6)
through the finite Fourier transform FF + = F+F = 1, which maps action eigenstates to angle eigenstates and conjugates the respective variables, where w = expi(2rr/2j + 1). Indeed, if g := w Ja+jl then FgF + = h, FhF + = g-1 and g(h) acts as step operator in the angle (action) state basis, i.e,
hlJ;n>=lJ;n+l>
,
hl¢;m>=w'_l_;m>
(7)
glJ;m>=w_lJ;m>
(s)
while
g-_l¢;n>=l@;n+l>
,
mod(2j + 1) and h 2i+1 = g_j+l = 1, (notice that the state lJ; n > and 1_; m > where denoted as tn > and I_,_ > respectively, in Refs. 3-6). The noncommutativity between the action and the angle variables is best expressed by the formula wgh = hg
(9)
which resembles the exponential form of the Heisenberg canonical commutation relations (CR) as were originally written by Weyl with the association that here the action operator ,/3 is a finite version of the position operator and the angle virtue of this analogy we may interpret eqs. directions of the phase space of our problem, torus, parametrized by the discrete action noncommutative character of two succesive effect of group contraction which is discussed until the continous limit j _ oo. Furthermore
operator stands for the momentum operator. By (7-8) ks the translations along the two different which due to the module condition is a lattice
and angle values. Also eq. translations along different
(9), exhibits the unusual directions. Moreover, the
below, is to increase the density of the lattice points this association to position and momentum suggests 42
that we should by diagonalizing IN; m >, related
look for the "number
states"
IN; m >, m = 0, 1, ...,2j
the finite Fourier transform FIN; m >= i"]N; e.g. with the orthonormal action states as:
in our finite system.
Indeed
m >, we find the number
states
2j IN;k
>=
_
]J;m
><
J;mlN;k>,
(10)
tn----O
with expansion
coefficients
< J;mlN;k This situation
given >=
in terms
_
is akin to that
of the Hermite
polynomial,
e- ,r_r+_('Oi+l)+')'Hk
of the harmonic
H_ with discrete
(p(23 + 1)+
oscillator
number'trtates
states of the usual Fourier transform operator which conjugates position a fact that stems from the property of the oscillator eigenstates exp(-{ Fourier transforms. Especially the vacum or lowest number state is,
which
argument,
m)
(11)
are similarly
eigen-
and momentum operators, X2)H_(x), to be their own
2.i
IN;0 >=
_
w_"2Os(imli(2j
+ 1))[J;m
>
(12)
m=O
where
Os is the theta-Jacobi
function
[7]: oo
0s(zl ) =
(13)
Having the action [d;m >, the angle [¢;n > and the number states IN; k > as were given above, we can further built, as have been outlined in Ref. 4, the quantum theory of action_agle variables by introducing the corresponding coherent states acting on the vacum IN; 0 >, with a displacement operator. Such an operator is furnished by the unitary traceless elements J,,,_.=2 := w=_'_/2g =_h=2 , where J+Wil ,l'r;,2 - J-,-_,-=2 = J2j+l-,na.2j+l-,n2, with belonging to the square index-lattice 0 < m],rn2 < 2j with boundary conditions pair excluded.
(rnl,m_) and the
The following interesting properties of these operators suggest them as the Glauber ment operator of our case; first they constitute an orthonormal set of (2j + 1)2 _ 1 elements the relation < J,,a,J_ >:= where
Tr J,,aJn = (2j + 1)/5,,a+_,5 ,
pairs (0,0)
displaceobeying
(14)
e.g. J,,a = J,n_,,2, and further,
J,,a
=
J,a+
(15)
and
J J,a = jax
(16)
and finally [J,,a,J_]
= -2i
sin[_rfi t2j + 1 43
(17)
mod(2j
+ 1), while
coherent
states
[[>,
r_ x _ = rnln_ - m_nl. for the action-angle
With
system
the
aid of these
by acting
operators
we now introduce
on the vacum:
2j
I/>:=
J_N;
0 >= w] 6t2 _
+ 1))l J; rn + 6 >
w6"+l_O3(imli(2j
(18)
rtl_0
These are now coherent states defined on the lattice phase space of the quantum action-angle variables. They involve
space which is the appropriate phase the Jacobi theta functions which are
also appearing in the case of the ordinary coherent states when, looking for a complete subset out of the over complete set of coherent states we lattice the phase space. Elsewhere, the normalization and minimum uncertainty properties of the states will be studied in detail.
3
Quantum
Angles
Let us now turn
to the
Addition
case where there
are several
for the way we combine them quantum mechanically. generators J_, with [Ji, Jj] = 2ieijkJk is the fundamental is solved by tensoring the generators,
action-angle
degrees
of freedom
and
AJ_ := J_ ® l + l ® J_ which again satisfy the commutation generators g = w (J3+jl) and h = w
relations, F(J3+jl)F+
(coproduct in the jargon of Hopf algebras), such coproducts we have found, ,
(19)
[AJ_, AJj] = 2ie_j_AJ k. In our case, for the "polar" with wgh = hg we must find an appropriate tensoring
which
Ag=g®g
search
The similar problem for the "cartesian" theme of addition of spins and customanily
provides
such Ag and Ah that wag
= Ah.
Two
(20)
Ah=h®l+g®h
and Ag=g®g which both
have
the remarkable
,
property
ponents involved in the tensor products. where there is no sence of order in the
Ah=h®g+g-l®h of not been
(21)
the same under
permutation
of their
com-
This is distingly different to the usual addition of spins, tensoring the spins. Technically speaking we have here
a natural case of no co-commutativity unlike in eq. (19), where the product is co-commutative [8-11]. We end here this discussion, as we intent to expand it elsewhere, by saying that it is also possible to show the Hopf and quasi triangular Hopf algebra structure then to find the R-matrix and to verify the Yang-Baxter equation.
4
Contraction
to
Before we came to conclusions that the SU(2) aspects
action-angle
of this procces
the
tensoring
and
Oscillator
let us mention variables
of the above
that
as was shown in Ref. 3 via the group
can be contructed
could be exemplified
by studing
44
to those
of the oscillator
the Jaynes-Cummings
contraction
and the dynamical model.
We illustrate
now thisidea be contractingthe SU(2) generatorsto the oscillator generatorsin the Bargmann analyticrealization. In the space of analyticpolynomialsof degree2j the SU(2) algebraisrealized aS,
J+=-
z2 d _z +z2j
d ,/3 = Z_zz - j
d
J-=d"_
(22)
where z is the complex label of the spin coherent states, and geometrically stands for the projective coordinate of the coset sphere SU(2)/U(1) .._ S 1. Transforming now the generators like J, --_ J±/v_ and J3 -'* .13+jl we find in the large j limit, the oscillator generators in their Bargmann form as follows:
=
2j
d(x/'Tjz)
J_
+
d
_
=a +
(23)
d
w3-- = a(v z)
=
(24)
and J3 + j = y/_z
d(v dz)
d
=N
(25)
where v_z _ o_ is the complex variable of the Glauber coherent states which is now becoming the coordinate of the tangent phase plane of the harmonic oscillator. One can further show that the overlap, the completeness relation and all other notions of the spin coherent states can be contracted q-deformed
to their respective oscillator oscillator with q deformation
counterparts. parameter
Moreover in Ref. 5 has been shown how a to be root of unity can be employed to define
action-angles variables in a finite Fock Hilbert space and a number of their properties have been worked out. In such an approach we have shown [5], that the contraction method is substituted by the limit procedure of undeforming the q-oscillator to the usual ocillators.
5
Conclusion
In conclusion,
we have
shown
that
the
quantization
of action-angle
classical
variables
can be
developed in the framework of the SU(2) algebra in a manner which allows for the classical properties of these variables to find well defined operator analogues. Interesting relations to the quantum groups and Hopf algebras are naturally emerge from the present method of angle quantization which will be pursued further, together with the introduction of the Wigner function for the action-angles variables and the star and Moyal product defined between functions of the phase space of our problem.
References [I]P.A.M. Dirac,Proc. R. Soc. Lond. A, 114,243 (1927).
45
[2] P. Carruthers
and M.M. Nieto,
[3] D. Ellinas,
J. Math.
Phys.
[4] D. Ellinas,
J. Mod.
Optics.
[5] D. Ellinas,
Phys.
Rev.
38, 2393 (1991).
A. 45, 3358 (1992).
and D. Ellinas,
[7] D. Mumford,
Tara Lectures
J. Phys. on Theta
[8] M.E.
Sweedler,
Hopf algebras,
[9] V.G.
Drienfeld,
Sov. Math.
Lett.
Math.
[11] N. Yu. Reshetikhin,
40, 411 (1968).
32, 135 (1991).
[6] M. Chaichian
[10] M. Jimbo,
Rev. Mod. Phys.
Phys.
(W.A.
Pokl.
A 23, L291 (1990). (Birkhauser, Benjamin,
1984).
Inc. New York 1969).
32, 254 (1985).
10, 63 (1985);
L.A. Takhtajan
Boston,
and
Commun.
L.D. Faddeev,
48
Math.
Phys.
Leningrad
102, Math.
537 (1986). J. 1,193
(1990).
N93"27317. ALTERNATIVE ASPECTS
DESCRIPTIONS OF WAVE AND PARTICLE OF THE HARMONIC OSCILLATOR
Dieter Schuch Institut fiir Theoretische Physik J. W. Goethe- Universit_t Robert-Mayer-Sir. 8-10, W-6000 Frankfurt am Main,
FRG
Abstract The dynamical properties of the wave and particle aspects of the harmonic oscillator can be studied with the help of the time-dependent SchrSdinger equation (SE). Especially the time-dependence of maximum and width of Gaussian wave packet solutions allow to show the evolution and connections of those two complementary aspects. The investigation of the relations between the equations describing wave and particle aspects leads to an alternative description of the considered systems. This can be achieved by means of a Newtonian equation for a complex variable in connection with a conservation law for a nonclassical angular momentum-type quantity. With the help of this complex variable it is also possible to develop a Hamiltonian formalism for the wave aspect contained in the SE, which allows to de: ::ribe the dynamics of the position and momentum uncertainties. In this case the Hamiltonian function is equivalent to the difference between the mean value of the Hazniltonian operator and the classical Hamiltonia_ function.
1
Introduction
In wave mechanics a complex equation, the SchrSdinger equation (SE), is used to describe the dynamics and energetics of the particle and wave aspects of a material system under the influence of conservative forces, e.g., the harmonic force of an undamped oscillator. In classical mechanics Newton's equation of motion is a real equation which is only capable of describing the particle aspect. It will be shown that it is possible to also take into account the wave aspect by changing to a complez Newtonian equation. However, real and imaginary parts of the new complex variable are not independent of each other, but are coupled by a well-defined relation which is connected with a conservation law for a nonclassical angular momentum-type quantity. With the help of this new complex variable it is also possible to express the groundstate energy ]_ in a way that it can serve as a Hamiltonian function for the position and momentum uncertainties.
2 The
Dynamics wave mechanical
of Particle equation
and
Wave
(SE) for the harmonic
0 _(x,t)=
iJl -_
{-
_2
C_2
2m Ox 2
47
Aspects
oscillator
+
772
(HO)
_z 2} _(x,t),
(1)
possessesexact analyticsolutionsof the form of Gaussian wave packets (WP). The dynamics of the p_rticleaspect is reflectedby the fact that the maximum of the WP followsthe classical trajectoryof the correspondingparticle. The wave aspect isexpressed by the finitewidth of the WP. This width can alsobe time-dependent. This time-dependence iscloselyconnected with a contributionto the convectivecurrent densityin the continuityequation for the (real)density functioncorrespondingto the (complex) WP. Insertingthe Gaussian WP given in the form
• L(z,t)=
NL(t)e=p
{i [y(t)_2+
lIp)_+
K(t)]}
,
(2)
(where $ - _ - (=) = = - r/(t)and (p)- rn_(z) denotes the mean value of momentum p, the explicitform of N(t) and K(t) isnot relevantforthe followingdiscussion), into the SE(1) shows that the maximum at position(z) = T/(t) fulfills the classical Newtonian equationof motion
,_+ _,2,z= 0.
The WP width, _V/_ (where (5:2) = (z2) _ (x)2), complex coefficient of 5:2 in the exponent, y(t), via
To determine the time-dependence of the WP tion of Ricatti-type
2a #+
is connected
(3)
with the imaginary
paxt of the
width, the complex (quadratically) nonlinearequa-
2h (g
has to be solved.
48
+ 2=o
(5)
With the aid of the variable a(t) as defined in Eq. (4) (which is apart identical with the WP width), the corresponding real part turns into 2h
&
-yR= -o_ rrt
from
a constant
factor
(s)
and Eq. (5)yieldsthe (real)nonlinearNewtonian equation
1
a+
=
(7)
The only differencebetween thisequation,determining the dynamics of the WP width, and Eq. (3)for the dynamics of the WP maximum isthe inversecubic term on the rhs of Eq. (7). In order to elucidatethemeaning ofthisadditionalterm, the RicattiEq. (5)has to be reconsidered. Using the substitution
m y=
(s)
with the new complex variableA = fi+ i_,Eq. (5) can be linearizedto yieldthe complex linear Newtonian equation
+ J
_ = 0.
(9)
This equation is formally indentical with the Newtonian Eq. (3) for the WP maximum. It can be shown (e.g. by expressing the WP(2) in terms of A or with the help of a Green-function, see [1-3]) that the imaginary part of A is directly proportional to the classical trajectory, i.e. _ctoP0 m = (x) = r/(t)
(10)
(where CZoand Po are the initial values of a(t) and (p)(t), respectively). Furthermore, in the same way it can be shown (see e.g. [1-3]) that real and imaginary parts of A are uniquely connected via the relation zu
- uz
=I
.
(11)
Equation t_ ) for the time evolutionof A was obtained from the Ricatti Equation (5),which describes e evolutionof the WP width, as shown in Eq. (7)fora(t). In order to show how the wave aspect is contained in A, itshallbe writtenin polar coordinates, A = c_ e'_ =
a cos _0 + i a sin_.
49
(12)
Inserting this form into Eq. (8), comparison with the definitions given in Eqs. (4) and (6) shows that the quantity a in Eq. (12) denoting the absolute value of ,_ is identical with the quantity a denoting the WP width in Eq. (7), if the relation ¢
=
1 _Z
(13)
is fulfilled. However, the validity of Eq. (13) can easily be proven by inserting (12) into Eq. (11). The physical meaning of Eq. (13)becomes more transparent, when the motion of _(t) in the complez plane is compared with the motion of a two-dimensional harmonic oscillator in the real physical space, written in polar coordinates (see e.g. [1-3]). This comparison shows that relation (13) (and thus the equivalent relation (11) in cartesian coordinates) corresponds to the conservation of angular momentum in real space. Furthermore, it shows that the inverse cubic term on the rhs of Eq. (7) corresponds to a centrifugal force in real space. So, it can be stated that the complex quantity _(t) fulfilling the Newtonian Eq. (9) contains the information about the dynamics of both particle and wave aspects of the system. Written in cartesian coordinates, the imaginary part of )_ directly provides the information about the dynamics of the particle aspect, the WP maximum, written in polar coordinates, the absolute value of ), directly provides the information about the dynamics of the wave aspect, the WP width.
3
Energetics
of Particle
and
Wave
Aspects
It shall be mentioned only briefly here (for further details see e.g. [2,3]) that this new complex variable A can also provide new information contained in the groundstate energy of the harmonic oscillator, usually only given in the form/_ = _hw. The notation/_ is used to already indicate that this energy contribution is just the difference between the mean value of the Hamiltonian operator (calculated with the WP-solution (2)) and the classical energy Ecru,,, respectively
O, By making to make
with
scale changes a canonical
m =
making
(rnlm2)
4A'B'-
of xl and z2 to (mx/m2)l/4xx
transformation
1/2. This
the coordinate
B' > O,
of the above
transformation
and
C a > 0. (m2/rnl)l/4x2
Hamiltonian
is generated
to the form
We can
: (cos(_/2) sin(a/2)
- cos(a/2) sin(c,/2) ) (::)"
Under this rotation, the kinetic energy portion of the Hamiltonian Thus we can achieve the decoupling by diagonalizing the potential diagonal
respectively,
decouple
it is possible
[14, 15]
this
Hamiltonian
by
transformation:
(_:)
becomes
(30)
if the
angle
(32) in Eq.(31) remains invariant. energy. Indeed, the system
a becomes
c tan a = B-Z--A" This
diagonalization
procedure
is well known.
6O
(33)
We now introduce
the new
parameters
K and
r/defined
as
A + B + _/(A - B) _ + C 2 K = k/AB in addition form
the
to the
Hamiltonian
rotation
- C2/4,
exp (-27/) a.
In terms
A
=
K
B
=
K
A
=
K(e-2'7-e>7)
angle
can be written
=
_/4AB
of this
yl and
y2 are defined
o
variables,
the
point.
the coordinate
Hamiltonian
o2 = _l-K/rn.
(35)
and
-sin(a/2) cos(a/2)/
The above rotation
variable
takes
together
in units of (inK)
first
If 7/= 0, the system
coordinate
with that
1/4, and use (ruff)
of Eq.(32)
is generated
-1/4 for the momentum
the form
becomes
-rl
decoupled,
In Sec. 8, we will be dealing with the problem of what on the second coordinate. If the system is decoupled, in the
(37)
-e"/"
and
= o2_[? +x,j ?'_ + o2_-(p_ + physics
(36)
_+e_2Èy_},
O2
where
the
,
sina.
(qq;) = (cos( /2) \ sin(a/2) by J0. If we measure
C take
as
in Eq.(32),
This form will be our starting
A, B and
,
21
e 2" sin 2 _ + e -_" cos 2
K H=___. ml {q_+q_}+2_{e=,y where
sin _
e-2_
(34)
'
new set of variables,
e _, cos 2 __ +
(
- C2
is solely
dictated
the Hamiltonian
becomes
x_) .
(39)
happens when no observations are made as the above Hamiltonian indicates, the
by the Hamiltonian
It is important to note that the Hamiltonian of Eq.(39) canonical transformation. For this reason, the Hamiltonian
cannot be obtained of the form
from
Eq.(38)
by
O2
H'= may play a useful form of Eq.(39)
o2 (e-nq_
role in our discussion. through
a canonical
+e'Ty_)
+
This Hamiltonian
transformation.
61
(enq 2 +e-'Ty_) can be transformed
(41) into the decoupled
6
Quantum
It is remarkable SchrSd]nger function
Mechanics that
wave
both
the
function.
of Coupled
Hamiltonian
If Yl and
for this oscillator
system
Oscillators
H of Eq.(38)
y_ are measured
and
in units
H' of Eq.(41)
lead
to the
of (mA') 1/4, the ground-state
same wave
is 1
_o(x,,x_) = _exp The
wave
the story
function is quite
is separable different.
in the Yl and y2 variables.
If we write
_exp
-2
en(XlC°S-2
function
values
a unitary
transformation.
of 7?, the wave function
- x2sin
for the variables of Xl and
xl
and
x2,
x2, then
)_
sin _- + x2cos
= _exp
is the m th excited
-
of Eq.(43)
A,,,,,_(a, rrtl
where ¢,n(x) condition
(42)
)2
.
(43)
becomes
¢o(Xt,X2) For other
in terms
o
+e-n(xl If rt = O, this wave
e-'y_)}.
However,
this wave function
i {l[
_b(x,,x2)=
y, + { -_t_1__,._
(x_ + x_)
.
can be obtained
(44) from the
above
expression
by
rl)¢m,(xl)Vr_2(x2),
(45)
m2
state
wave function.
The coefficients
Am,m2(r/)
satisfy
the unitarity
IAm,m_(a,,)l_ = 1. rn
It is possible
to carry
As for unitary canonical
out
transformations
are also applicable
a similar
transformations
expansion applicable
in classical
to the Wigner
functions
[6, 4]. They
in the
case of excited
to wave
mechanics
phase-space
of the Wigner function is translated There are therefore ten generators
(46)
lYt_ 2
functions,
in Eq.(12) distribution
states
and Eq.(13). function.
into a unitary transformation of unitary transformations
[16].
let us go back
As was stated
The canonical
4= [_'1
=
_
i1
a a2 + a a 1
,
J_ = y,1
(a!a2
a_ al + a2a
(a!at
a_a2),
Jo = -_
'(!! a a
+ alal
a_a_ - a2a2),
4
62
before,
of they
transformation
of the Schr6dinger wave function. applicable to Schr6dinger wave
are
J1
the generators
-
a_al),
,
i fatat _\i l-ram+
/_ =
a_a!-a_a2) ,
_= Q_ -
_
-
_-
+_@
+
+
+
i \rata t 63 = E ' _a,a_) . where
a_" and
functions.
7
a are
The
the
above
Wigner
The Wigner
and
operators
step-down
also satisfy
Functions
phase-space
uncertainty
relations.
canonical
step-up
Unitary
transformations
operators
mechanics
transformations space.
given
is often
more
In his book
on statistical with
measurements
mechanics
the density
in the
first
[18], Feynman
matrix.
part,
but
Feynman
known
are not
raises
divides
able
picture that
the
issue
the universe
to measure
In the present
case of coupled
to study
the
It is often
density
matrix,
harmonic
more
convenient
especially
when
oscillators,
of the
we want
to study
function
is defined
that
in the
are
of the parts.
second
[17]. universe We make
part.
The
plays the essential [19, 20].
role
we are not able to measure
phase-space the
rest
into two
the
through
transformations transformations
anything
we assume
to use the Wigner
for studying
can be achieved
canonical
second part is Feynman's rest of the universe. Indeed, the density matrix when we are not able to measure all the variables in quantum mechanics the x2 coordinate.
wave
in Eq.(6).
convenient
uncertainty-preserving transformations. They are also entropy-preserving Are there then non-canonical transformations in quantum mechanics? in connection
oscillator
Relations
in the Schr6dinger It has been
to harmonic
relations
Uncertainty
of quantum
in phase
applicable
the commutation
and
picture
(47)
uncertainty
distribution products
function in detail
[15, 18]. For two coordinate
variables,
W(Xl,X2;pl,P2)
the Wigner
=
exp{-2i(p_y,
as [15]
+ P2Y2)}
x _'(xl + yl, x_ + y2)¢(xl - yl, x2 - u_)au_du2. The
Wigner
function
corresponding
W(xl,
x2; pl, P2) =
to the oscillator
exp
63
wave function
-e'(xl
of Eq.(43)
cos 7 - x2 sin
(48) is
-e-_(x, sin _-+
x2cos
)2 _ e-,(p,
-e_(plsin If we do not make
observations
in the z2p2 coordinates, W(xl,p1)
The
evaluation
of the integral
leads
= /
This Wigner gives
function
-
product
becomes
is coupled
with
z2, our
rest
universe,
of the
system
in which
8
ignorance
In addition
distribution
and those
to the
about
the
of the
oscillator
state,
of the Wigner
ten generators
second and
coordinate.
space
becomes
of xl and pl. This distribution
(52)
uncoupled which
Xl world
with
a = 0. Because
in this case
which,
both
given the
coordinate
function
What
space
acts
x_
as Feynman's
in Feynman's
words,
is the
The
of the
Hamiltonian
first
given
we can consider coordinate in Eq.(38)
the
are
suggests
significance
such a
by
(53,
in phase space.
to an increase
scale
expanded
I 00i).
first coordinate
64
are also determined
Space
also in Eq.(ll),
can be generated
physical leads
and
and entropy
space.
and momentum
contracted.
of the
is the
of phase space
phase
in Eq.(10)
position
scale transformations
phase
the temperature
in Phase
the transformation
generates of the
7, the expansion
cosh r/+ sin r/cos
in the phase
in the
i
second
(50)
} 1/2
x2 coordinate,
So=_(
expansion
z2; pl,pe)dx2dp2.
system
the uncertainty
excited
in which
transformation,
This matrix
becomes
2 = 1(1 + sinh 2 r/sin s a).
Transformations
transformation
function
(49)
phase-space picture, the uncertainty is measured in terms of the area in phase Wigner function is sufficiently different from zero. According to the Wigner
of the spread
Scale
the Wigner
77cos a
1/4 if the oscillator
increases
for a thermally
by the degree
.
we are interested.
In the Wigner space where the function
)2
of
(Ax)2(Ap) This expression
_ +p2cos
1 sin +sinh2
cosh 77-sm
gives an elliptic
the uncertainty
)2
to
W(x ,x2;pl,p ) = x exp
W(xl,
cos -_ - p2sin
The
transformation
[21] and .contracts of this operation? in uncertainty
and
the
leads phase
to a radial
space
As we discussed entropy.
of the in Sec.
Mathematically
speaking,the contraction of the secondcoordinateshould causea decreasein uncertainty and entropy. Can this happen? The answeris clearly No, becauseit will violate the uncertainty principle. This questionwill be addressedin future publications. In the meantime,let us study what happenswhen the matrix So is introduced into the set of matrices given in Eq.(10) and Eq.(ll). It commutes commutators with the rest of the matrices produce
[S0,J1]=
,
[S0, K3]=-_ If we take
into account
there
are
fifteen
SL(4,
r).
This
the
generators. SL(4,
with J0, J3, Ka, h'2, Q1, and four more generators:
-o.1
0
-2
five generators
They
form
r) symmetry
in addition
the closed
of the
coupled
its
,
'
above
Q2. However,
o'3
0
to the sixteen
set of commutation oscillator
"
system
generators
relations may
have
of Sp(4),
for the
the group
interesting
physical
implications.
References [1] S. S. Schweber, ford,
New
An
York,
Introduction
to Relativistic
Quantum
Field
Theory
(Row-Peterson,
Elms-
1961).
[2] A. L. Fetter and J. D. Walecka, Quantum New York, 1971); M. Tinkham, Introduction
Theory of Many Particle Systems (McGraw-Hill, to Superconductivity (Krieger, Malabar, Florida,
1975). [3] S. K. Kim and
J. L. Birman,
[4] P. A. M. Dirac,
J. Math.
Phys.
Phys.
Rev.
B 38, 4291
(1988).
4, 901 (1963).
[5] C. M. Caves and B. L. Schumaker, Phys. Rev. A 31, 3068 M. Caves, Phys. Rev. A 31, 3093 (1985)', B. L. Schumaker, also Fan, [6] D. Hun,
Hong-Yi Y. S. Kim,
[7] Y. S. Kim,
Phys.
[8] F. Iachello
and
[9] H. Umezawa,
and
[11] A. K. Ekert Barnett
and
J. Vander
and
Rev. S. Oss,
Lett.
Phys.
Phys.
M. Potasek,
Rev.
Rev.
Lett.
A 39, 2987
A 41, 6233
(1989).
(1990).
66, 2976 (1991).
and M. Tachiki,
Thermo
Field
Dynamics
and Condensed
States
1982). Phys.
P. L. Knight,
P. L. Knight,
Rev.
63, 348 (1989).
Amsterdam,
and
Linder,
M. E. Noz, Phys.
H. Matsumoto,
(North-Holland, [10] B. Yurke
and
(1985); B. L. Schumaker and C. Phys. Rep. 135,317 (1986). See
Rev.
A 36, 3464 (1987).
Am. J. Phys.
J. Opt.
57, 692 (1989).
Soc. of Amer.
65
For an earlier
B 2 467 (1985).
paper,
see S. M.
[12] D. Han, Y. S. Kim, and M. E. Noz, Phys.Lett. A 144, 111(1990). [13] H. Goldstein, Classical [14] P. K. Aravind, [15] Y. S. Kim Singapore,
Mechanics,
Am. J. Phys.
and
Second
Edition
(Addison-Wesley,
Reading,
MA,
1980).
57, 309 (1989).
M. E. Noz,
Phase
Space
Picture
of Quantum
Mechanics
(World
Scientific,
1991).
[16] Y. S. Kim and
M. E. Noz,
Theory
and Applications
of the Poincard
Group
(Reidel,
Dordrecht,
1986). [17] D. Han,
Y. S. Kim,
[18] R. P. Feynman,
Statistical
[19] J. von Neumann, Princeton, [20] E. P. Wigner [21] Y. S. Kim
M. E. Noz, and Mechanics
Mathematical
L. Yeh, Univ.
of Maryland
(Benjamin/Cummings,
Foundation
Physics Reading,
Paper MA,
of Quantum
Mechanics
(Princeton
Academy
of Sciences
(U.S.A.)
93-23
(1992).
1972). Univ.
Press,
1955). and M. M. Yanase, and
M. Li, Phys.
Lett.
Proc.
National
A 139
445 (1989).
66
49,910
(1963).
II.
QUANTUM
GROUPS
67
N93-27319
q-HARMONIC
OSCILLATORS, AND
THE
q-COHERENT
q-SYMPLECTON
STATES
t $
L. C. BIEDENHARN Department Durham,
of Physics, Duke North Carolina,
University U.S.A.
M. A. LOHE Northern
Territory
Casuarina,
NT
University
0811,
A_tralia
MASAO NOMURA University Komaba,
of Tokyo
Meguro-ku,
Tokyo,
155,
Japan
Abstract The recently introduced notion of a quantum group is discussed conceptually and then related to deformed harmonic oscillators ("q-harmonic oscillators"). Two developments in applying q-harmonic oscillators are reviewed: q-coherent states and the q-symplecton.
1 Introduction It is not
unfamiliar
in physics
that
a new
theory
appears
in the
form
of a 'deformation'
of a previous 'classical' theory; thus, for example, quantum mechanics can be considered to be a deformation of classical mechanics (which is recovered in the limit that the 'deformation parameter' ti --* 0), and Einsteinian relativity to be a deformation of Newtonian relativity (which is recovered when the 'deformation parameter' c --* o0). Recently this notion of deformation has been applied [1,2] to symmetry itself, leading to the concept of a 'quantum group' as a deformation of a classical (Lie) group with a deformation parameter denoted by q. This new development has had numerous important applications in both physics and mathematics [3,4]. Since harmonic oscillators of symmetry in quantum t Supported $ Invited (College
in part paper
Park),
have played a fundamental--and physics, it is not surprising that
by the National
presented
25-28
March
at the
Science Harmonic
Foundation, Oscillator
pervasive!--r61e in the applications the concepts of quantum groups, and grant
No. PHY-9008007.
Conference,
1992.
69 PRIE_D!NG
P,-_GE P.LAr_K _!OT _q._IF'D
University
of Maryland
deformations,
are
important
here
also,
and
hence
ingly, it is our purpose to discuss here deformed deformed coherent states ("q-coherent states") harmonic
oscillators)
We will begin tum groups--SUq(2), deformed harmonic
2 The
group
called
the
relevant
to the
present
conference.
Accord-
harmonic oscillators ("q-harmonic oscillators" and the deformed algebraic structure (based
), on
"q-symplecton".
by discussing, in conceptual and motivational terms, the simplest of quanthe q-deformed quantal rotation group--to set the stage for introducing oscillators, and then the remaining topics mentioned above.
Quantum
Group
The commutation relations SUq(2) ate given by:
SUq(2)
for
the
three
generators
{J_,
J[,
dq}
defining
the
quantum
[JLJ ]=+JL
(2.1)
qJ_ _ q-J; [J_,J__]=
q½_q-½
qc
'
(2.2)
These defining relations for SUq(2) differ from those of ordinary angular momentum (SU(2)) in two ways: (a) the commutator in (2.2) is not 2J, as usual, but an infinite series (for generic q) involving all odd powers: (jq)l,(jq)3, .... Each such power is a linearly independent operator in the enveloping algebra; (b) For q _ 1, usual commutation The differences
accordingly, the Lie algebra of SU,(2) is not of finite dimension. the right hand side of (2.2) becomes 2J,. Thus we recover in the limit the relations for angular momentum. noted in (a) and (b) are expressed by saying that the quantum group SUq(2)
is a deformation of the enveloping algebra of SU(2). The deformation parameter q occurs in SUq(2) in a characteristic by [n]q such
way, as q-integers
denoted
that: q_ -q-_
[n]q= q½ _ q_½, =ql_i'D"
+q_
+...q
-_,
nC_.
(2.3)
These q-integers, [nlq obey the rule: [-n]q = (-1)In]q, with [0]q = 0 and [1]q = 1. Note that [n]q = [n]q-,, so that the defining relations (2.1) and (2.2) are invariant to q _ q-1. The quantum group concept involves much more than just deforming the commutation relations of the classical group generators. Actually an interesting new algebraic structure is also imposed, that of a ttopf algebra [5]. Let us first define this new structure and then discuss its meaning. Consider algebra
involves
an associative
algebra
A, with
a unit
element,
1, over
a field say, _T. Then
the
the operations: multiplication: unit:
rn : A ® A ---*A, 1 :g' ---* A,
7O
and,
(2.4) (2.5)
subject tion.
to the familiar
We can (2.4)
and
extend
axioms this
(2.5) above,
of associativity
algebra
that
to become
is, if we can define co.multiplication:
the algebra
the compatibility
a Hopf
algebra
of addition
if we can
"reverse
and
multiplica-
the
arrows"
A ® A,
and,
(2.6)
• : A _tT.
A is a group 3':
in
two new operations: A :A _
co-unit: Since for a quantum group have a third new operation:
and
A _
(2.7)
algebra,
it is reasonable
to require
that
A,
one (2.8)
called
"antipode", (the analog to the inverse in the group). These three new operations must satisfy the requirement that A and e are homomorphisms of the algebra A and that 7 is an anti-homomorphism. In addition, the operations must satisfy the compatibility
axioms:
Associativity
of co-multiplication:
(id ® A )A(a)
= (A ® id)A(a),
aEA
m(id ®.r)A(.) = m('r ® ia)A(a) = Co-unit axiom: (e ® id)A(a) = (id ® = a. Antipode
axiom:
• For a physicist, the introduction of such complicated of the blue" is very disconcerting. Certainly it requires "why a Hopf algebra"? Let us try to answer this.
and heavy motivation.
(2.9) (2.10) ,(2.11)
algebraic machinery "out The obvious question is:
Physicists are already very familiar with the algebraic approach to symmetry in quantum mechanics; what is needed is a physical reason for "reversing the arrows". What this really means, in effect, is that all one needs is a simple motivating physical example. Here is that example. Consider angular momentum: there is a natural, classical, concept for adding angular momenta, which is taken over in quantum mechanics. Consider Jtotal as the total angular momentum operator which is to be the sum of two independent constituent angular momenta Jx and .]2. Writing the total angular momentum operator "]total as an action on the two constituent state vectors we have: JtotaI¢)total
=
Jl1_)1
where we have been careful to use a precise notation for the tensor product independent systems. Writing this same result in an abstract formal manner, we discover that what done
by "adding
angular
momentum"
is to define A(J)
where
J denotes
a generic
angular
(2.12)
® llX)2 + 11_)1 ® J2[_)2,
® of the
we have really
a co-multiplication: (2.13)
= .] ® 1 + 1®.I,
momentum
(defined
as obeying
In other words: The vector addition of angular momenta in a ttopf algebra. One sees accordingly that a (commutative)
71
two
the commutation
defineJ a commutative Hopf algebra structure
relations). co-product is not only
very natural in quantum physics, but actually implicit, and in fact essential--unfamiliar only because unrecognized. The remaining Hopf algebra axioms are required to make the structures compatible and well-defined, and in a sense analogous to group concepts. What we wish to emphasize is that the deformation of the algebraic structure in a quantum group is only part of the basic concept--requiring the additional Hopf algebra structure, which is natural to quantum mechanics, provides an important constraint on the freedom to deform the commutation relations. One quantum
can now understand groups for physics:
intuitively from our example one now has the new possibility
the fundamental significance of defining a non-commutative
co-multiplication, as actually occurs for the quantum group SUq(2). This means that: (i) the fundamental commutation relations are changed ("deformed"); that is, one has matic symmetry breaking. (Recall that Hamiltonian perturbation theory is dynamical leaves commutation relations (which are kinematical) invariant); (ii)
the
"addition
of q-angular
momentum"
depends
on the
There is one other feature of the commutation relations the relations (2.1) and (2.2) single out J_ and thus appear
of
kineand
order of addition. for SUq(2) that deserves to break the rotational
comment: symmetry.
For generic values of q this seeming result is incorrect: the degeneracy structure of q-group irreps is in fact preserved, a consequence of the Rosso-Lusztig theorem. (We take this opportunity to note that ref. [6] is misleading on this particular point.) For completeness, since we have emphasized the importance of the complete Hopf algebra structure, let us give explicitly the remaining Hopf algebra operations for the quantum group
svq(2): =
1 + 1 J.,, Jq
(2.14)
Jq
A(j ) =
(2.15)
e(1)=l, 7(J_)
3 q-Boson
e(J_)=e(J_)=O,
= -q
7(J_)
= -J_-
(2.17)
operators
In order to understand the meaning it is natural to look for representations For the usual angular momentum group, Schwinger
'J_,
(2.16)
map [7], which
maps
of the deformed commutation relations (2.1) and (2.2) of the operators J_, J_ as finite-dimensional matrices. there is a standard way to do this: one uses the Jordan-
the 2 x 2 matrices
{J+, Jz} of the fundamental
irrep
into boson
operators.
Let us recall how this works. One begins with a realization of a pair of commuting boson creation operators (al, a2) and and defines the Jordan-Schwinger map: J+--' This map preserves
a1_2,
the angular
ger map is a homomorphism) of SU(2).
J-
--, a2_1,
momentum and from
J,
commutation
of the operators J:t:, Jz in terms annihilation operators, (_1,a2),
1 _(alal
- a_'d2).
relations
this map one can explicitly
72
(that
(3.1a, b,c) is, the Jordan-Schwin-
construct
all unitary
irreps
Is there
a q-analog
to the
basic idea is to construct operator aq, its Hermitian vector
[0) defined
Jordan-Schwinger
q-analogs conjugate
map?
There
is indeed!
(Refs.
[8,9,10]).
to the boson operator_. To do so introduce the q-creation the q-destruction operator a-q, and the q-boson vacuum ket
by the equation _10) =0.
Instead
of the Heisenberg
relation,
N q is the Hermitian
number
[Nq,a q]=a Thi_
algebra
i_ a deformation
(3.2)
[_, a] = 1, let us postulate a-qaq
where
The
__ q_aq'_q
operator q,
the
algebraic
relation:
-. q-_-,
(3.3)
satisfying
[Nq,a -q]---a
_,
with
of the tteisenberg-Weft
q -, 1. (Note that the q-number operator Heisenberg case.) Orthonormal ket vectors corresponding
Nq[O)-O.
algebra,
N ¢ is now no to states
which
longer
of n q-quanta
(3.4a, b, c) is recovered
the
operator
in the a_
as in the
are given by:
tn)q - ([n],!)-½(a')"lO>, with:
limit
(3.5)
Nqln)q =
(3.6)
It is now easy to define a q-analog for the algebra of the generators of the quantum group SUq(2). In the language of q-boson operators, one defines a pair of mutually commuting q-bosons a_ for i = 1, 2. That relations:
is, for each,
i, a iq and a-_ obey
for i # j: The
generators
{ J._, Jq_, JT} of SU,(2)
--. ala2, q_ The construction straightforward
of all unitary irreps [6] but will be omitted.
Remarks: (1) We have particularly co-multiplication--is
q
equations
q
[a,,a,] = are then
--' a2al, q---q
St
of the
q
= realized
(3.3),
(3.4)
and, in addition,
= 0.
the
(3.7)
by
Jg
½(N q - Nq).
quantum
group
SUq(2)--for
generic
(3.8a,
b, c)
q--is
now
emphasized in Section 2 that the Hopf algebra structure---more an important constraint on possible deformations. Let us note
that the deformation of q-bosons given by eq. (3.3) does allow a (non-commutative) co-product to be defined. However, as shown by Prof. T. Palev (private communication), a complete Hopf algebra structure is not possible. (2) The deformation given equivalent, forms. For example, becomes:
in eq. (3.3) can be put into many if we define Aq = aqq_ N" and -_ AqA
a form often found
q =
qAq"A
in the literature.
73
q +
1,
differently appearing, = q__N'_a', then eq.
but (3.3) (3.9)
4 The We
q-Harmonic have
motivated
Oscillator the
introduction
concept of a quantum group. Let us now examine the q-harmonic a, _ we can define q-momentum operators. That is, we define:
(P)
of q-deformed oscillator
O = t2-_ [P, Q] is then
(using
as a way
on its own merits.
and q-position
(O) operators
P ---z v T(a
The commutator
bosons
to implement
From the q-boson in the same
operators
way as for boson
q -a-q),
(4.1)
(aa + am).
(4.2)
(3.3)):
i[p, Q] = h[_',aq] = _([N + 1], - IN],). The eigenvalues
(N --* n) of the right
hand
h(In + 1], -In]q) One sees that dent
the
Heisenberg
uncertainty
(4.3)
side are therefore
= h
cosh(¼(2n + 1)log q) cosh(¼ log q)
in the q-harmonic
oscillator
of q) only in the limit q --* 1; the uncertainty increases The q-harmonic oscillator Hamiltonian is defined from p2 rn_2 7"/= 2rn + _ hw-m = -_-(a
From
(4.4)
is minimal
(and
indepen-
with n for q # 1. P, Q according to
Q2,
q a + a'a_).
(4.5)
+ 1]q + IN],),
(4.6)
(3.3) we find = ::_-([N
showing
that
the eigenvalues
of 7"/are
Z(n) The normMized
eigenstates
= _([n
+ 1], + [hi,).
(4.7)
In) are: ]n) = ([n]!)-½(a')n]0).
The the
the
energy
undeformed
spectrum case.
for the
For q laxge,
q-harmonic one
oscillator
sees that
the
h_q_(1 + o(_)).
74
(4.8) is uniformly
spectrum
becomes
spaced
only
exponential:
for q = E(n)
1,
5 Coherent
exist
States
It is natural to ask, once one has defined q-deformed bosons, whether or not coherent states for this new harmonic oscillator structure. The answer is yes [11], as one might expect.
Let us review this structure briefly here. There are two key characteristics of the and and
Skagerstam [12]: (a) continuity of the coherent (b) the resolution of unity:
state
(usual)
coherent
[z} as a function
states,
as identified
by Klauder
of z.
1 = J I=)(=1du(z),
(5.1)
where
the integration takes place with respect to a positive measure The best known examples of coherent states, which certainly teristics, are the canonical coherent states generated by the (usual) operators
a and
_. These
canonical
coherent
states
are defined
dl_(Z). satisfy these creation and
two characannihilation
by [8]
Iz) = e-1_l'/2eZ"lO) o_
zn
= _-izl:/__ _., In),
(5.2)
n-_---O
where
In} denotes the orthonormal vectors generated by the creation operator a. We can immediately write down q-coherent states [z}q by replacing the boson
(5.2) by its q-boson
analog,
and replacing
the exponential
operator
in (5.2) by the q-ezponen*ial
of
function
expq:
Iz>, = (expq(lzl=))-½ expq(zaq)lO>q oo
These
states
zn
= (exp, (Iz12))-½ _ _ln),. n_O
(5.3)
_qlz)_= zlz)q,
(5.4)
satisfy:
showing that the eigenvalue z and,
q-coherent state [z)q is an eigenstate of the annihilation operator since z = ,(z[a-q[z}, (assuming the states [z)q are normalized), the
a--q with label z is
the mean of a-q in the state [z)q. The definition (5.3) is not a unique q-extension of (5.2), for we could have chosen any one of the family e_ of exponential functions in [13]; this would introduce explicit q-factors in equations such as (5.4). We outline below how the particular q-harmonic oscillator model of Section 4 (above) leads naturally to these q-coherent states. (The states (5.3) were first considered in Ref. [8] and subsequently also in Refs. [14-17]. In fact, as with many q-analogs of classical and quantum concepts, some q-generalizations were obtained before the appearance of quantum groups [18]). Let us now consider the two characteristic properties of coherent states, continuity and completeness. the continuity
(a) The continuity properties of the deformed exponential
of [z)q, as a function of z, follow immediately function, expq in (5.3).
75
from
(b) The resolution of unity within the Hilbert space, in terms of the states Iz)q, has been considered by Gray and Nelson [15] and also Bracken et al [17]. The q-analog of Euler's formula for P(x)
is required,
and
is expressed
in terms
of the q-integration
fo¢ eXpq(-X)xn
dqx = [n]q!
defined
in [13]:
(5.5)
where ( is the largest zero of expq(x) (note that, unlike e =, expq(x) alternates in sign as z --* -oo). A natural restriction is ]z] 2 < ( and then, with the help of (3.5), the resolution of unity can be derived [17],
1 = [ Iz), ,(zl d_,(z)
(5.6)
d where
the measure
d_(z)
is given d_(z)
by
where 0 = arg(z). It follows from the states Iz)q. (In fact, q-coherent
(5.6) that an arbitrary states are overcomplete,
non-orthogonal Coherent
within
to Iz)q, for any z.) states arise naturally
by defining boson factors to unity:
operators
from
position
the framework
we can use these formulas
state can be expanded in terms for an arbitrary q-coherent state of the harmonic
and momentum
a, = _(Q - iF), Conversely,
(5.7)
= _ expq(Izl2)expq(-lzl2)dqlzl2dO,
operators,
oscillator
Q, P, putting
of Section
4,
dimensional
1 _ = 7_(Q + iP).
to define momentum
of is
(5.8)
and position
operators
and so, given
q-boson operators, these formulas also provide convenient q-analog definitions of q-momentum and position operators [8]. Alternatively, one can define a q-harmonic oscillator by starting with SchrSdinger's equation and replacing the derivative by a finite difference operator which provides an alternative form for the deformation. We use the following q-derivative,
V,f(x) = and
the q-harmonic
oscillator
states
f(xq) - f(x) x(q- 1) '
are now deterrained
(5.9)
by the equation
(5.10)
1 2 _(-v, + qx2)¢(x)= E¢(_).
Effectively,
we have
chosen
q-momentum
and q-position
operators
Qq, Pq satisfying (5.11)
qQqPq - P_Qq = i, with the realization the deformation.)
Qq = x,
Pq = iVq.
(This
is yet another
76
realization
different
from
(3.9) for
Solutions involve
of the difference
q-extensions
of the
equation
Hermite
(5.10)
have
polynomials.
¢0(x) =
.=0
been
The
given
ground
by several
state
authors
¢0 is given
[19,20],
and
by
(_).q-4x2. [2n]4!! '
(5.12)
where [2nlq![ = [2n]q[2n - 214... [2]4. Upon using the identity [2n]4 = [214[n]42 we can identify the function (5.12) as one of the family of q-exponential functions given by Exton [13]. The eigenstates
¢,, of the
deformed
SchrSdinger
equation
(5.10)
are labelled
by an integer
n, and the energy levels are E, = ½12n + 114. (For comparison, note that in the model defined in Section 4, the energy levels are different: E, = ½(In + 114 + [nl4 ) = ½[2n + 114,,,). The eigenstates of (5.10), ¢,,, take the form Cn(x) where ¢0 is given by (5.12) with the explicit formula:
and H_(z)
= H_(x)¢o
denotes
= where
the coefficients
Cr are given
(5.13)
(xq-_),
a q-extension
C,.x"q-
[rb!
r--_0
of the classical
,
(5.14)
'
_ 4]4...
[2n - 4m + 4]4
C2,-,,+1 = (_)m q(2,,+1),,,/212 n _ 21412n _ 6]q... From the explicit eigenstates one can identify q-boson states Cn(x), from which one can form the q-coherent oscillator [20].
The
polynomial,
(for even or odd r) by
C2m = (_),n q(2,,+l),,,/212n1412n
6 The
Hermite
(5.15a)
(5.15b)
[2n - 4m + 2]4.
operators which step between the eigenstates of this model of the q-harmonic
q-Symplecton idea
behind
the
symplecton
construction
has a close relationship
to harmonic
oscil-
lators. In the Jordan-Schwinger realization of angular momentum one obtains uniformly all unitary irreps in terms of two independent harmonic oscillators. This naturally suggests the question: can one do better and realize all irreps uniformly in terms of one harmonic oscillator? The
answer
is (of course)
yes--this
is the symplecton
realization
[7,21], which
uses the creation
operator (a) as the spin-½ "up" state and the destruction operator (_) as the "down" state. This implies that there is no longer a vacuum ket [0) annihilated by _. Instead we define a formal ket 1) and seek to interpret both a[) and _[) as non-vanishing vectors. Operators in this symplecton calculus will be defined as polynomials over (a, _) with complex numbers as scalars. State vectors will be defined as operators multiplied on the right by the basic formal ket, i.e.,
Iv) - o.I),
77
(6.1)
where Iv / is a vector and O_ the operator creating this vector. The action of the generators state vectors will be defined as commutation on the relevant operator O_, that is,
.z,(Iv))-=[J,,o,,11). To be completely explicit we are considering operator a and its conjugate _ obeying:
(for the
(6.2)
undeformed
symplecton)
a single
[_, a] -- 1, all other
commutators J+
It is easily
verified
zero. _
that
-la2,
The
generators
(note
the
this realization
J_
obeys
Note that the tion relations,
action of these generators succeeds precisely because
are realized 1_2,
_
J0
the desired
[Jo, J±] = +J±,
by: "*
commutation
relations:
(6.5)
[J+, J-] = 2,/o.
on symplecton of the Jacobi
=
(6.4)
1(a_+_a).
state vectors, identity. Using
verifying the commutation
generators, the labels J and M can be assigned to define characteristic polynomials angular momentum irrep eigenvectors are then given by the set of vectors T'jMI). The adjoint polynomial (T'jM) ad5 is defined by: (,p)adj
boson
(6.3)
of SU(2)
sign!)
on
commutaunder the 79jM. The
(6.6)
( _ I ) J- M ,_:); M ,
with _ taken to be adjoint to a. The adjoint (dual space) vector to MI) is defined as (I(T_) _j. The crucial problem in this (undeformed) symplecton construction is the proper definition of an inner product for the Hilbert space of the irreps. Omitting details [7], the answer is obtained from the multiplication law for symplec_on eigen-polynomials. TUEOREM [21]: these polynomials
Let _ and P_ be normalized obey the product law:
eigen-polynomiads
of the generators
Ji.
Then
a+b
c, bo, p_,+_
(6.7a)
c=la-bl
wh ere
(clalb) = (2c +
A(abc) and
,,-,b°c is the
=
[ (a
(6.7b)
+ b - c)!(a - b + c)!(-a
usua/Wigner-Clebsch-Gordan
coe_cient
78
+ b + c)!
for SU(2).
(6.7c)
Using this theoremit is now easyto understandthe inner product (#Iv): one appliesthe product law to the polynomials O_ j and O,, and then projects onto the J = 0 part. The Wigner-Clebsch-Gordan Remark:
coefficient
It is clear also that
(for J = 0) quite one can extend
literally
defines
this structure
here
a metric!
by adjoining
additional
symplec-
tons. That is, one considers a symplecton having n "internal" states: ax, a2,..., a,, and their conjugates al, a_,... ,_,,. Just as the adjunction of a boson with n "internal" states suffices to realize SU(n), so does an n state symplecton suffice to realize the structure Sp(2n). An important consequence of the symplecton construction is the definition of a new invariant angular momentum function: the triangle coefficient A(abc), eq. (6.7c). This triangle function, A(abc), has gratifyingly simple properties. It is a function defined symmetrically on three "lengths" or "sides" a, b, c, which (from the properties of the factorial function) vanishes unless the triangle conditions (that the sum of any two sides equals or exceeds the third side) are fulfilled. The symplecton addition in a particularly The triangle function
realization of angular graphic way. is clearly a rotationally
momentum invariant
yields
the
function
triangle
defined
rule
of vector
on three
angular
momenta; as such, it fits very nicely into the series of invariant functions defined on 3n angular momenta: (6j) [Racah coefficient] and (9j) [Fano coefficient]. The Wigner coefficients are often called "(3j) symbols", but in view of the fact--emphasized by Wigner--that these coefficients are coordinate frame dependent (i.e., involve the triangle function as the more appropriate The triangle function obeys the following A(acf)A(bdf)
magnetic quantum numbers) one might to designate as the (3j) symbol. transformation law, Ref. [21]:
= (2f + 1) Z
A(abe)A(cde)W(abcd;
It is quite remarkable that the Racah function appears four triangles by pairs. Having reviewed now the symplecton construction can one define cillator? deformed formation
a deformed
symplecton
ef).
(6.8)
here as a tetrahedral it is time
("q-symplecton")
using
consider
to return
a single
function
coupling
to our main
deformed
theme:
harmonic
os-
The answer (of course) is yes, but there are some surprises [22]. We will develop the structure using finite q-transformations, which provides further insights into the deprocess [23]. (The infinitesimal approach--which obtains the q-generators {J_ } using
a single q-boson, the q-boson analogs to eqs. (6.4)--was Let aq and _q be q-boson creation and annihilation
developed operators
earlier in ref. [24].) obeying:
_qaq - q½aq-dq = 1. This q-commutation
relation
is invariant
(a,
under
the transformation
= (a,n)
(6.9) of q-spaces
[23]:
(; u '
where: ux = q½zu,
vx = q½xv,
yu=q½uy,
uv=vu, 1
xy - q-_vu
yv = q½vy,
(6.11a,
b,c)
(6.lid, 1
= yx - q_vu
79
= 1.
(6.11f)
e)
The
adjoint
to (a, _) is:
_,-q-_a)
and obeys:
u.) u" '
(6.12)
with: z" = y,
order
u* =-q-½v,
Let us denote the q-sympleeton j + m in a and j - m in _ and
v" =-q½u,
y* = x.
(6.13a,
eigenpolynomials by: Q_. defined to transform as:
Then
b,c,d)
Q_" is a polynomial
07(,',_) = _ d,L,.(_, u,,,, _)QT(,,, n).
of
(6.14)
n
din,re(x, u,
Here
is the q-rotation
V, y)
matrix
which
obeys:
_-,_,(_," ,,,,,, y)d_',,,.,,(_. _,v._,) = _ ,c_;:';,,. J
×,c_i(,'4 dL(_...v.u). where
qC)_( are
irreducible
q-WCG
tensor
coefficients.
of rank j.
It follows
Moreover
that
the set {Q_n, m = -j,-j
Q}" is a q-symmetric
+ q-u+_)4u-')
+½aJ+m-l.ffa_J-m-I
Here _ is the least
number
of transpositions
Example: As is clear from
[4]½Q_ = q-]a3_
our review
(of the usual
law for the deformed
q-eigenpolynomials,
THEOREM
Q j,
[23]:
Let
fl,t t
and
_rt It
Q j,,
q
C j'y''j mJff;tt_
of q dependent
symplectons),
is an
•
4
•
"5'-ma '+m.
f(a, il) in normal-ordered
the major
task
(6.16) form. (6.17)
is to prove a product
Q_. q-eigenpolynomials.
Z N(j'j"j). 1
is the q-Wigner-Clebsch-Gordan only
to put
+ q
+ q-¼a2"_a + q_a_a 2 + qS"ffa_.
be normalized
,,' m" Q.j, (a,_)Qf, (a,'6)=
where:
needed
,j}
+... (/- ,,,)C/+,,,)
+...
+ 1,...
function:
•
+ q-(t+')(4J")+½y(a,i_)
(6._5)
j' j" j qCJm,m,,,n • Qr_(a,'6),
coe_cient,
on j',j",3.
8O
Then:
and
N(jtj"j)
(6.18)
is a scalar
function
N(j'ffj)
obeys
the recursion
([2j"][2j
relation:
+ 1])_N(j'j"j)=([i'-j"
+j
+ 1],D" +j"-j]q)½
x N (j',j"-
½,j + ½) g (j + ½, !,j_2/
+ ([j' + j" + j + l],[-j'
+ j" + j],)½
xg(j',j"-},j-]).
cases.
The determination We find:
of the coefficient
N(j N(j,,j2,jl N(j,
N(fj"j)
is very
difficult.
It helps
to see a few special
0 j) = 1,
(6.20)
+ j2) = 1
(6.21)
½,j-½)
with:
=
F(n)
-q-¼F(2j) ([2j][2j + 1])½'
=- [11 + [2] +...
We remark that the appearance of the function F(n) q-symplecton [23]. One can prove the further property, at this stage,
(6.22)
+ In],
F(0)
= 0.
is characteristic that
(6.23)
of relations
the function
N(j',
involving
the
j", j) is symmetric
in the first two indices. One of the surprising properties [23] is that the (q-rotationally invariant) function N(j',j",j) is not symmetric under q _ q-1. These results show that N(j',j",j) is not the proper q-analog to the triangle function A(a, b, c), despite the fact that the q-symplecton N(j',j",j) in the proper form. It has been shown is via the definition:
product law seemingly appears to define in Ref. [22], that the proper way to proceed
/F(2c)![2a
+ 1]![2b + 1]!
= This q-triangle
coefficient
(6.24)
has the desired
symmetry.
symmetric in its arguments jl ,j_, ja--precisely triangle coefficient A(jlj2ja)in (6.7c). Moreover,
it is now possible A,(acf)A(hdf)
[22] to obtain --[2f
+ 1] Z
As shown
in Ref.
the same property the
proper
[22], Aq(jlj_ja)
possessed
q-analog
A,(abe)A,(cde)Wq(abcd;
Let us conclude by citing the product law for q-eigenpolynomials to show the desired q-analog structure [22]:
is totally
by the (undeformed)
of (6.8): (6.25)
el).
in the
proper
form
now
a+b
eZQf -
[2c+ 1]-½nq(ak)(b Z lc c=l.-bl
81
+
(6.26)
Note
the surprising
appearance
of the
q-WCG
coefficient
involving
q-a
as the
proper
form
to
show
the analogy. Space is lacking for more than this brief survey of the q-symplecton and the associated subtleties of q-analysis. More detail can be found in [22], and related discussions--from the aspect of Weyl-ordered boson polynomials--is given in [25] and [26].
References [1] M. Jimbo,
Lett.
Math.
[2] V. G. Drinfeld,
Phys.
Quantum
[3] C. N. Yang and Advanced Series
1, 63 (1985).
Groups.
Proc.
Int. Congr.
Math.
1,798
(1986).
M. L. Ge (Eds.), Braid Group, Knot Theory and Statistical in Mathematical Physics, Vol. 9, World Scientific, Singapore
[4] H.-D. Doebner, J.-D. Hennig (Eds.), Proceedings of the 8th Int. Workshop [5] E. Abe, HopfA1gebras,
Cambridge
Quantum on Math.
Groups, Lecture Phys., Clausthal,
Tracts in Math:,
74, Cambridge
Notes in Physics FRG (1989). University
[6] L. C. Biedenharn, An Overview of Quantum Groups, in Lecture Notes V. Dodonov, V. I. Man'ko, Eds.), Springer Verlag, Berlin (1991).
Press
in Physics
[7] L. C. Biedenharn and 3. D. Louck, Angular Momentum in Quantum Physics, dia of Mathematics and Its Applications, Vol. 8, Addison-Wesley, Reading, reprinted Cambridge University Press (1989). [8] L. C. Biedenharn,
J. Phys.
[9] A. J. Macfarlane, [10] C. P. Sun and
J. Phys.
A: Math.
Gen. 22, L873-878
(1989).
A: Math.
Gen. 22, 4581-4588
(1989).
H. C. Fu, J. Phys.
A: Math.
Gen. 22, L983-L986
[11] M. A. Lohe and L. C. Biedenharn, Klauder Festschrift (World Scientific,
On q-Analogs Singapore).
[12] J. R. Klauder
Coherent
matical
and
Physics,
[13] H. Exton, (1983). [14] P. Kulish
B. S. Skagerstam,
World
Scientific,
q-Hypergeometric
and
[15] R. W. Gray [16] C. Quesne,
Phys.
[17] A. J. Bracken,
Functions
E. Damashinsky,
and
C. A. Nelson, Lett.
Singapore
153A,
D. S. McAnally,
J. Phys.
D. D. Coon,
States,
Applications
Applications,
A. Math.
382,
(V.
EncycIopeMA (1981),
to appear
in Physics
A. Math.
E11is Horwood
Gen. 23, L415
and
in the
Mathe-
Series,
Wiley
(1990).
Gen. 23, L945 (1990).
303 (1991). R. B. Zhang
and
M. D. Gould,
1379 (1991). [18] M. Arik and
(1980).
(1985).
and
J. Phys.
370,
(1989).
of Coherent
States.
Mechanics, (1989).
J. Math.
Phys.
17, 524 (1976).
82
J. Phys.
A. Math.
Gen. 24,
[19] 3.
A. Minahan,
The
q-SchrSdinger
Equation,
[20] N. M. Atakishiev and S. K. Suslov, Theor. ibid. The'or. i. Mat. Fiz. 87, 154 (1991). [21] L. C. Biedenharn [22] M. Nomura SUq(2),
and
and L. C. Biedenharn,
to be published
[23] M. Nomura,
J. D. Louck,
J. Phys.
and Math.
Phys.
of Florida Phys.
(N.Y.)
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[26] M. Gel'fand and D. B. Fairlie, The Algebra Quantum Extension HUTMP 90/B226, DTP
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43D,
of Weyl Symmetrised 90/27 (1990).
83
D. Fairlie
and
C.
and
its
417 (1991). Polynomials
N93"27320
WHICH
Q-ANALOGUE
OF
THE
Allan Faculty
1
of Mathematics,
The Open
Introduction
The
noise
tum
oscillator
-in the
and
the mass
and
state
(variance
determined
same
which is that
defines
mode 1).
the
associated (and
algebra,
ratio.
Thus
states.
its hermitian
coherent However,
Coherent
more
• Eigenstate
original,
productive
from
effect.
• Automorphism
• M and
Definitions Group
Definition
Signal-to-Noise
• q-Coherent
States
Definitions
(Group)
• Example:
Squeezed
and
q-Squeezed
Ratio States
P q-bosons
• Eigenstate
Definitions
85
other
states
conventional
creation
operator) generates
the calculational
to define they
the
state the
the algebra stance,
Needless In this
are
states
coherent
Heisenberg-Weyl
operator
generates
is
are not squeezed,
In fact, squeezed
constant
of squeezing
difficult than
as a quanof a squeezed
amount
as the annihilation
a s and its conjugate
Definitions
• Exponential • Algebra
and
Kingdom
Planck's
definition
the
it is not
to groups
1). Just
a t, the
(taking
(Glauber)
this noise-reducing
The
conjugate
- and
automorphism group of the Heisenberg-Weyl algebra is SU(1, 1). viewpoints generalizes differently to the quantum group context. both. The structure of the talk is as follows:
• Conventional
United
field - considered units
value
vacuum.
SU(1,
operator
viewpoint,
6AA,
to 1). A practical
vacuum
the usual
do have
the group
so the pair-photon
Another
all equal the
sense but with respect
Glauber with
MK7
of the electromagnetic
as the
which
group
Keynes,
to one half, in appropriate
is less than
variance
states
Milton
Content
is equal
noise
general
University,
of the oscillator
the
to coherent
Group
SU(1,
the
in the more
optics Weyl
frequency
OSCILLATOR?
I. Solomon
of a component
vacuum
by the appropriate produce
analogous
photon
squared)
is one for which
as they states
and
SQUEEZED
in quantum a of a single Heisenbergof the group
is to note
that
the
to say, each of these talk we shall discuss
• Exponential Definitions • Algebra (q-Group)Definitions • Example: Signal-to-NoiseRatio • Automorphism q-Group
2
Conventional
The
elementary
the
identification
in suitably
Definition
Coherent
treatment
of (a single
chosen
then
where
N
units.
photons;
satisfy
= the
frequency)
Squeezed of the
We may
the
btb.
B,,_p,
introduce
boson
(b + bt)/v/2
Heisenberg-Weyl
The
vacuum
E-,_x,
quantum
quantized
a similar The
field leads
to
noise
result
conventional
b, bt by
p=
(b-bt)/v/2i
[b, bt]
=
1
[N,b t]
=
bt
Algebra
interpretation state
operators
of these
(1)
operators
is that
they
annihilate
(resp.
create)
]0 > satisfies
of the x-component
(E-field)
0. in the vacuum
(Ax):=-2 with
electromagnetic
[x,p]=i
b[0 >= The
States
of its components
x= which
and
for the B-component. coherent
states
is given
by
=1/2
The vacuum
(Glauber
state
[1] states)
signal
( < x >2) vanishes.
are defined
as eigenstates
of the operator
b,
blA>= For these
states
one readily
>.
(2)
evaluates (Ax) 2 = 112
< x >2=
(A +X)2/2.
An alternative, suggestive definition of the coherent states which readily lends itself to generalization, is that they are obtained by the action of the realizations of the group corresponding to the Heisenberg-Weyl
Algebra
generated
by {b, bt, 1} on the vacuum, ]A >-- exp(Abt)10 86
>.
thus; (3)
It is an important
practical
here we are of course p vanishes where
for the
N, is the
In a classic
problem
to maximize
only considering
vacuum;
number paper,
the quantum
it attains
of photons
the
value
in the
Yuen [2] showed
coherent
He further light
states
only mathematical (1).
The
term
attain values experimentally. generally,
showed
that
generated input
below the vacuum These squeezed
on Glauber
coherent
states;
state,
the preceding
is that
taking
field the maximum
is attained of the
the fact
(or coherent) states may states)
p for radiation
we see from
for a coherent
by the
of the operator
consists
from
What
ratio
p,,,ox = 4Ns(N, + 1), ( effectively a maximum
this value
result
derives
noise. 4N,
for any radiation
as eigenstates
to this
"squeezed"
signal-to-noise
(real))_2
= Ns
signal.
that
noise ratio p for fixed energy has the value limit on the number of photons in the signal constraint).
the
where N, gives the upper power per unit frequency
squeezed
states
in these
commutation states
the
relations
quantum
state value of 1/2. Such states have also be defined by the action on the
of the group
corresponding
to the
[3], two-photon
I 12- lul2 =
#b + vb t where
canonical
that
signal-to-quantum
algebra
1. The
Equation
dispersion
may
been produced vacuum (more generated
by
{b_, (bt)_, (bb t + b_b)}. Thus
a typical
squeezed
state
(up to normalization)
I(,z The state
1_, z > is an eigenstate
(4)
may be written
>= exp(_((bt)2)exp(zbt)[o
of (b-(b
>.
t) with eigenvalue
(5)
z, in agreement
squeezed state above (# = 1, u = -_ and for convergence we require in Equation (4) satisfy the commutation relations of SU(1, 1)
that
with
the
definition
1(I < 1.) The
of
operators
= -2I,o [K0, K+]
An alternative definition which results ized) eigenstates of the of the lowering
=
in states operator
-Fife.
(6)
exhibiting squeezing is to define them as (normalK_ - b2. These states have the form i _1
16 > =
>
i=0 oo
[_> An appropriate A more group
sum of these
basic
of the
H-W
definition algebra
=
squeezed
_i 1) '12i+1>" ,=o/(2i+
states
of squeezed is SU(1,
Y_
1); thus b_
where
Ipl 2 -
eigenstate
lul 2 = 1.
of the
The
transformed
recovers
states
conventional
a Glauber from
a unitary
coherent
the observation
transformation
state. that
the
automorphism
U on b gives
U b U _ = #b + vb _ squeezed
bose destruction (#b+
arises
(7)
state
is then
(8) defined,
exactly
as above,
as an
operator
ubt)l_
>= 87
_l_ >.
(9)
More generally, a conventional squeezed U(#,v) on a coherent state ]z >= D(z)tO ]_ >=
state is defined >, thus:
U(_,v)D(z)lO
>=
as the
action
of the
(10) enables
calculations
using V(_,
one may
operator
U(p, v)]z >
This definition is not only elegant but, by applying the inverse transformation, in squeezed states to be made as readily as in the coherent states. For example;
unitary
readily
evaluate
v)-'
the dispersion
b V(_,
v) = _b -
of x in the squeezed
vb t state
]_ > to be
(_x) _ = _1_ - _12 and
the
signal
to be
< x >5= {(u _ _)_+ (_ _ _)z)}2. For real values
of the
parameters,
the maximum p-<
may
3
readily
be seen
to be attained
q-Coherent
A deformation Coon [4]. Their
and
of the deformed
aM
x >2/(Ax)2
+ 1) as cited
q-Squeezed
q-boson
?-
apapt
qaptap
[N, ap t] has
been
introduced
matician's
classical
more recently both
forms.
There
( I have
q-analysis,
introduced
In principle, definition
[5, 6].
is no need
either
of q-coherent
a study
physicist's Equation states
used
the
which form,
to subscript
subscript
the
[2].
ago by Arik
__
q-N
=
ap t
Heisenberg-Weyl
and
M to denote
the
relation
as far as Gauss,
second
equation
N for the reason bosons.
Algebra
(12)
(3) can be used as a starting
of deformed
years
(11)
the Quantum
P. The
operator
some
1.
goes back at least
subscript
(2) or Equation for both types
=
ap satisfying
_
above
b was introduced
qaM'taM
operator
ratio
States
operator
aMaM
noise
= 42
at pm_x = 4N,(N,
standard boson bosons satisfy
More recently, the deformed (HWq Algebra)
of the signal-to-quantum
It is easily
to the
in contrast
of (12)
given
matheto the
is satisfied
by
below.)
point shown
for an eigenstate that
an attempt
to use Equation (3) does not lead to a normalizable state (for q _ 1) in either case. Starting from Equation (2), q-coherent states for the deformed boson operator of Arik and Coon were constructed 88
by these Both
authors
forms
[4]; the
of q-boson
same
lead
equation
was used
to the q-coherent
[6] for the q-bosons
I_ >q = AZ-lexpq(Bat)] where
defined
in Equation
(12).
state (13)
0 >
a = aM or ap and
N' = expq(l l'). The
q-exponential
is defined
in both
cases
(14)
by oo
X v
(15)
expq( =) E _=0
The symbol we define
[r]q! is defined
by [r]q! = [r]q[r-
1]q[r-
2]q...
[l]q where,
[x]q = (q_ - 1)/(qand
in the
case of Equation
in the case of Equation
(11), (16)
1)
(12), we define [X]q=(q_-q-_)/(q-q-_).
Equation which
(11)
converges
gives
rise
to the
Iql > 1,
for
classical
or for
form
(17)
of the
Ixl < Ii1_--;I when
q-exponential
Iq[ < 1.
The
usually form
written
as Eq(x),
of q-exponential
sponding to Equation (12) is convergent for all z and q. In both cases, limq._._ expq(x) and when q = 1 the q-boson operators reduce to standard boson operators. The
q-bosons
are related
to the conventional
bosons
corre= exp(x),
b as follows:
a=b where N = b_b, using the appropriate definition [7] or "physical" bosons Equation (17)[8]. The in the
q-coherent
states
defined
case of the conventional
gives rise to squeezing
above coherent
(18) of [N]q for "mathematical"
do not give rise to (time-independent) states.
In fact,
it may be shown
It is not immediately
squeezing, [7} that
the
term
(16) just
as
which
< a >2
which is zero for eigenstates of a. However, Buzek [9] has shown squeezing, by choice of a suitable analogue of the usual Hamiltonian; Rasetti
Equation
is, in general, < a2 > -
by Celeghini,
bosons
and
Vitiello
that there is time-dependent and this has also been found
[10].
clear how conventional
squeezed
states
can be generalized
group context. The most direct approach is to use a q-boson realization algebra; one may then attempt to define the analogous q-squeezed states
89
to the quantum
of the analogous suq(1, l) by the exponential action
of the resulting given by Kulish
operators on the vacuum and Damaskinsky [8] is
(or on the q-coherent
states).
1 K+ with p = (q + q-')-' of this algebra
and
= p(at)2 K_
= p(a)2
[K+, K_] = -[2K0]q2.
fail to give a normalizable
conventional
ones,
give normalizable
(19)
the exponential
action
not only for the conventional
We may
alternatively
carry
over
q-boson
case.
For the
choice
of the operators
exponential
(which
also for expq_(x) the
"box"
facto-
[11].
the definition
(ato the
states
1)
8Uq(1,
1
was to be expected) but also for the q-exponential expq(z) defined above (and which one would have thought to be the appropriate function here). The eigenstates of K_ corresponding to Equations (7), obtained by substituting rials for the
of
Ko = -_(n+ -_)
However,
state
A realization
(at)lC z >: zl ,z >
(20)
z = 0 we obtain
I(,z>=.N'-a}--_(
i i=0
[2i-1]q!!12 [2i]q!!
i >
(21)
with normalization
i----0
The symbol is 1. The and
[r]q!! has the expected
squeezing
properties
q were calculated A more
basic
of squeezed
group
property
(8)
(9)..
One
Equation
automorphism
[r]q!! = [r]q[r-
defined
quantum
given
may
group
states
seek
of the
in the quantum
in the previous by analogy quantum
section
and
the first term
values
in (22)
of the parameters
a conjugation
1. c-numbers
c _
2. q-numbers
(quantum
A _
c* , (complex plane)
q-squeezed
_tTa
o_7"
=
/17*a
77*
=
7*7
-- ")'*7
=
1
=
1.
by its effect
,4 defined
algebra
d* = a
9O
=
from
generalizing
case,
Equation
in terms
of q-bosons.
of the Consider
by Woronowicz
[12],
(23) on
conjugation) 6 = a*
states
a and 7 as defined
=
cm* -/t_7"7
case arises
for the conventional
Heisenberg-Weyl
a7
a*a
group
to define
the quantum plane d la Manin generated by two elements satisfying the following commutation relations:
We now introduce
2]q[r-4]q.-.
in this way, for various
in [11].
definition
the automorphism and
meaning
of states
=
1
3. operators Under
A
=
(q real ).
q½(N2-N)Atq-½(N2-N)
this transformation,
A = A,
A_B =/_A
;
and
the boson
a satisfying
aa _ _ qata = q-N maps
to fi , with
the
pair
a, fi satisfying ha-
with # = q2. The
two-dimensional
fundamental
U
and
u satisfies
uJfi
#ha
[a 7
_
(24)
representation
of SU,(1,
1) is given
by
]
(25)
[a, alu
(26)
Or*
= J where
[10] 0 -p
J= The
= 1.
transformation
[a, al is an automorphism
which
may
as the eigenstates
now be defined Finally,
optimal may
we note
that
preserves one
signal-to-Quantum
be shown
may
Noise
Equation
ratio
an analogue
q-photons
is, for a radiation in the
states
a, thus generalizing of Yuen's
[13]; the
field in terms
group
version
conventional
[1] R. J. Glauber, [2] H. P. Yuen,
[4] M. Arik
group
the results
result
[2] cited
corresponding
bound
+ 1]q/([Ns
of photons
satisfying
of the Heisenberg-Weyl
case,
attained
Rev.
131, 2766
+ lie -INs]q)
for the SU(1,
the
Algebra. 1) squeezed
References
[3] D. Stoler,
in the quantum
of [11].
above for
context on the
q-photons
to be
of the quantum value
Squeezed
of the transformed derive
pq = 4[N,]q[N, that
(24).
Phys. Physics
Phys. and
Letters
Rev.
56A,
(1963)
105 (1976)
D l, 3217 (1970)
D. D. Coon,
J. Math.
Phys.
17, 524 (1976)
[5] A. J. MacFarlane,
J. Phys.
A: Math.
Gen.
22, 4581
[6] L. C. Biedenharn,
J. Phys.
A: Math.
Gen. 22,L873(1989)
91
(1989)
2.
(27)
modified This
ratio
states.
commutation is always
relations tess than
the
[7] J. Katriel
and
[8] P. P. Kulish [9] V. Buzek,
A. I. Solomon,
and J. Mod.
[10] E. Celeghini, [11] A. I. Solomon
A: Math.
E. V. Damaskinsky, Opt.
M. Rasetti,
J. Phys.
Gen.
24, 2093 (1991)
A: Math.Gem
23, L415
(1990)
38, 801(1991) and
G. Vitiello,
and J. Katriel,
[12] S. L. Woronowicz,Commun. [13] A. I. Solomon
J.Phys.
J. Phys. Math.
Phys.
Rev.
A: Math.Gem
Phys.
136,
, to be published.
92
Lett.
66, 2056
23, L1209
399(1991)
(1991)
(1990)
1193-27321
DEFORMATION
OF
CONFORMAL THROUGH
SUPERSYMMETRIC
QUANTUM AFFINE
Laboratoire
MECHANICS
TRANSFORMATIONS
Vyacheslav Spiridonov 1 de Physique Nucldaire, Universitd
C.P. 61_8, succ.
AND
A, Montreal,
de Montrdal,
Qudbec, H3C 3J7,
Canada
Abstract Affme transformations (dilatations and translations) areused to definea deformationof one-dimensional N = 2 supersymmetricquantum mechanics.Resultingphysicalsystemsdo not have conservedchargesand degeneracies in thespectra.Instead,super'partner Hamiltoniansareq-isospectral, i.e.thespectrumofone can be obtainedfrom another(withpossible exceptionofthelowestlevel) by q2-factor scaling. This construction allowseasilyto rederive a special serf-similar potential found by Shabat and to show thatforthelattera q-deformed harmonic oscillator algebraofBiedenharnand Macfarlaneservesas thespectrumgenerating algebra.A generalclassofpotentials relatedto the quantum conformalalgebraJuq(1,1) is described. Furtherpossibilities forq-deformation ofknown solvable potentials areoutlined.
1.
Introduction
Standard Lie theory is known to provide very usefultoolsfor descriptionof physicalsystems. Elegant applicationswere found in quantum mechanics withinthe concept of spectrum generating, or,dynamical (super)symmetry algebras[I].The most famous example isgiven by the harmonic oscillator problem (sothe name of thisworkshop) where spectrum isgenerated by the HeisenbergWeyl algebra. Some time ago a wide attentionwas drawn to the deformations of Lie algebras which nowdays arelooselycalled"quantum algebras",or,"quantum groups" [2](below we do not use the second term because Hopf algebrastructureisnot relevantin the presentcontext).Spinchain models were found [3]where Hamiltonian commutes with generatorsof the quantum algebra 8uq(2),deformation parameter q being relatedto a coupling constant. Thus, an equivalenceof a particularperturbationof the interactionbetween "particles"to the deformation of symmetry algebragoverning the dynamics was demonstrated. Biedenharn and Macfarlane introduced q-deformed harmonic oscillator as a buildingblock of the quantum algebras [4, 5]. Various applications of q-oscillators appeared since that time [6-13] (an overview of the algebraic aspects of q-analysis is given in Ref.[7]). Physical models refering to q-oscillators can be conditionally divided into three classes. The first one is related to systems on lattices
[8]. In the second
class dynamical
quantities
are defined
on "quantum
'On leave of absence from the Institute forNuclear Research, Moscow, Russia
93
planes"
- the spaces
with non-commutative similar to the standard
coordinates [9]. one, all suggested
Although SchrSdinger equation in this approach looks explicit realizations of it in terms of the normal calculus
result in purely finite-difference equations. Parameter q responsible for the non-commutativity of quantum space coordinates serves as some non-local scale on the continuous manifolds and, therefore, the basic physical principles are drastically changed in this type of deformation. We shall not pursue
here the routes
of these two groups
of models.
The third - dynamical symmetry realization class - is purely phenomenological: one deforms already known spectra by postulating the form of a Hamiltonian as some combination of formal quantum algebra generators [10], or, as an anticommutator of q-oscillator creation and annihilation operators [4, 8]. This application, in fact, does not have straightforward physical meaning because of the non-uniqueness of deformation procedure. Even exact knowledge of a spectrum is not enough for precise reconstruction of an interaction. For a given potential with some number of bound states
one can
spectrum
associate
[14]. Therefore
another
potential
the physics
containing
behind
new
parameters
such deformations
and exhibiting
is not completely
fixed.
the same Moreover,
for a rich class of spectral problems there are powerful restrictions on the asymptotic growth of discrete eigenvalues [15] so that not any ordered set of numbers can represent a spectrum. All this means that one should more rigorously define physical interaction responsible for a prescribed deformation of a given simple spectrum, q-Analogs of the harmonic oscillators were also used for the description of small violation of statistics of identical particles [13] (general idea on the treatment of this problem on the basis of a parametric deformation of commutation relations was suggested in Ref.[16]). The papers listed above represent only a small fraction of works devoted to quantum algebras and q-analysis. For an account of unmentioned here applications we refer to reviews [17, 18]. Recently Shabat have found one-dimensional refiectionless potential showing peculiar selfsimilar behavior and describing an infinite number soliton system [19]. Following this development the author proposed [20] to take known exactly solvable Schr_dinger potentials and try to deform their shape in such acquires complicated
a way that the problem remains to be exactly solvable but the spectrum functional character. So, the Shabat's potential was identified in Ref.[20] as a
q-deformation of conformally invariant harmonic and particular forms of Rosen-Morse and P6schiTeller potentials. The hidden q-deformed Heisenberg-Weyl algebra was found to be responsible for purely exponential character of the spectrum. In comparison with the discussed above third group of models present approach to "quantum" symmetries is the direct is fixed first and the question on quantum algebra behind prescribed secondary. In accordance with this guiding principle mechanics [21, 22] was proposed in Ref.[23].
one - physical interaction rule of q-deformation is
a deformation of supersymmetric This talk is devoted to description
(SUSY) quantum of the results of
Refs.[19, 20, 23] and subsequent developments. We start by giving in Sect.2 a brief account of the properties of simplest (0 + 1)-dimensional SUSY models. In Sect.3 we describe a deformation of these models on the basis of pure scaling transformation of a superpartner potential, namely, we find q-SUSY algebra following from this rule and analyze its properties. Sect.4 outlines possible extensions of the simplest potential deformation. In Sect.5 we show that mentioned above selfsimilar potential of Hamiltonian. the
Hilbert
space
naturally appears within In this case factorization of square
integrable
q-SUSY as that characterized by the simplest structure operators entering the supercharges are well defined on
functions
and
94
generate
q-oscillator
algebra.
As a result,
a representationof
q-deformed
conformal
algebra
suq(1,1)
is obtained.
In Sect.6
we give short
description of further generalizations of the Shabat's potential which correspond to general qdeformed conformal quantum mechanics and q-deformation of (hyper)elliptic potentials. Sect.7 contains some conclusions. We would like to stress once more that suggested realizations of qalgebras
are continuous
the standard
2.
physical
SUSY
(i.e.
they are not purely finite-difference
ones)
and they are used within
concepts.
quantum
The simplestN = 2 SUSY
mechanics
quantum mechanics isfixedby thefollowingalgebraicrelations between
the Hamiltonian of a system H and superchargesQt, Q [21]
{Qt,Q}=H,
Q2=(Qt)2=0,
[H,Q]=[H,
Qt]=0.
(1)
All operators are supposed to be well defined on the relevant Hilbert space. Then, dently on explicit realizations the spectrum is two-fold degenerate and the ground state
indepenenergy is
semipositive, E.ac :> 0. Let us consider a particle moving in one-dimensional space. Below, the coordinate z is tacitly assumed to cover the whole line, z E R, if it is not explicitly stated that it belongs to some cut. Standard
representation
of the algebra
(1) contains
one free superpotential
0(o :/ 0, (0.:) ._i._ 0) 0) '
0
=
H+
0
,
=
A=(p-iW(z))/V_,
AA?
It describes
a particle
with
two-dimensional
internal
H_A t = At H+, only
sponds
possible
difference
to the harmonic
annihilation
operators
concerns
oscillator
problem
at, a which
satisfy
[a, at] = 1, where
N is the number
correspond
the
operator,
to the conformally
space
W'(z)aa),
level.
and then At,
the
basis
vectors
of the intertwining
AH_
lowest
(3)
of which
can be
relations
= H+A. Note
(4)
that
A coincide
the
choice
W(z)
with the bosonic
= x correcreation
and
the algebra [N,a t] = a t,
N = ata.
invariant
(2)
(lo _°1) .
identified with the spin "up" and "down" states. The subhamiltonians H+ are isospectral as a result
The
[22]:
[z,p]=i,
= _(p_ + W2(z)-
d w(z),
W(z)
This,
dynamics
and [24].
95
[N,a] another
= -a, particular
(5) choice,
W(z)
--- A/x,
3. Now
q-Deformed
SUSY
we shall introduce
construction.
the tools
Let Tq be smooth
quantum needed
q-scaling
mechanics
for the
quantum
operator
defined
algebraic
deformation
on the continuous
of the above
functions
(6)
Tqf(z) = f(qz), where q is a realnon-negativeparameter. Evident propertiesof thisoperator are listedbelow
TJ(.)g(.) = [YJ(.)][Ta(.)], T' ddx TqT.=T.,,, T;I= Y,-,, On the Hilbert
space
of square integrable
functions
= q-l d
TI
Tq'q
(7)
-- 1,
£_ one has
/:_'(_)_(q_)dx = q-' /:_'(q-'.)_,(_)d., oo
where from the hermitian
conjugate
of Tq can be found
T: = q-iTS", As a result, tiahle,
_
Tq is a unitary
an explicit
re_llzatlon
operator.
(10) into the formal
(T:)
Because
of Tq is provided Yq
Expanding
=
we take wave functions
--
where formed
W(z)
is arbitrary
case (3).
A and
(10)
using integration
by parts
AA t
We define q-deformed
.:(.
0
H+
conjugates
_q'-"(p _ + W2(x)
½q(p2 + q-2W2(q-,w)
SUSY Hamiltonian
of each other on £2.
'
Q=
96
Now one has
= q-'H_,
(12)
+ q-'W'(q-'z))
o)
as in the unde-
W'(_,))rq
and supercharges
q-aAA?
(11)
we use the same notations
- W'(z))
1 _-1,_0-1 = _q *_ (p_+w_(,)+
=
(9)
cut considerations
A = -_ T;'(p- iW(,)),
T.,
and for convinience
At are hermitian =
one can prove relations
be taken for finite
q-I
(p + iW(.))
function
A ?A
differen-
qZd/dx.
1 -_
to be infinitely
by the operator
e lnqxd/d*
series and
(9)
t = T,.
on the infinite line and semillne [0, oo]. A special care should since Tq moves boundary point(s). Let us define the q-deformed factorization operators
At=
(8)
O_
(13)
- qH+.
to be
o) 0
'
=
(o 0
"
(14)
Theseoperatorssatisfy
the following
{Qt,Q}q where
we introduced
=//,
version
{Q, Q}q = {Qt,
Qt}q
of the N = 2 SUSY
= 0,
algebra
(15)
[//, Q]q = [Qt, H]q = 0,
q-brackets [X,Y]q
- qXY
{X,Y}q Note that
q-deformed
the supercharges
- q-'YX,
= qXY
[Y,X]q
÷ q-1YX,
are not conserved
= -[X,Y]q-,,
{Y,X}q
because
(lfi)
= {X,Y)q-l.
(17)
they do not commute
with the Hamiltonian
(in this respect our algebra principally differs from the construction of Ref.[11]). An interesting property of the algebra (15) is that it shares with (1) the semipositiveness of the ground state energy
which
follows
SUSY algebra mechanics.
(1).
from
the observation
Evidently,
For the subhamiltonians
in the
that
limit
Qt
Q and
q -,
H_: the intertwining
1 one recovers relations
H_A "_= q2A? H+, Hence, H± are not isospectral from the spectrum of H+ just
AH_
but rather q-isospectral, by the q2-factor scaling:
H+ ¢(+) = E(+)¢ (+), E(-
Possible
exception
SUSY
quantum
dence
between
for it Ev,c
) =
concerns
the
spectra.
We name
= and describes
+ 1 2 _p
a spin-l/2
1
+_q1
2
+
i.e.
the spectrum
2o'a z 2
particle
-_
]_eraO"
I
quantum
of H_ can be obtained
_b(+) oc A lb (-).
zero modes
this fituation
+
SUSY
(18)
level in the same spirit
has zero mode
ordinary
= q2H+A.
-1
there
is one-to-one
as a spontaneously
then
q-SUSY
(19)
as it was in the undeformed
then is exact,
has one level less than its superpartner H_ (or, H+). As a simplest physical example let us consider the case W(z) the form
H =
conventional
satisfy
H_ ¢(-) = E(-)¢ (-),
If At, A do not have
> 0. If A (or, At)
q-_'H
look as follows
_2(-) 0¢ At_b (+),
q2 E(+)
only the lowest
mechanics.
the operator
broken
correspon-
q-SUSY
because
E_,c = 0, and H+ (or,
= qz.
_ q) + ((q2 _
The Hamiltonian
takes
_ q_ (20)
3,
in the harmonic
H_)
potential
and
related
magnetic
field along
the
third axis. The physical meaning of the deformation paramete r q is analogous to that in the XXZmodel [31 - it is a specific interaction constant in the standard physical sense. This model has exact
q-SUSY
and
if q2 is a rational
numbex then
the spectrum
97
exhibits
accidental
degeneracies.
4.
General
deformation
of superpartner
Hamiltonians
Described above q-deformation of the SUSY quantum mechanics is by no means unique. If one chooses in the formulas (11) Tq to be not q-scaling operator but, instead, the shift operator
Tqf(=) =
+ q),
Tq=
(21)
then SUSY algebra will not be deformed at all. The superpartner Hamiltonians will be isospectral and the presence of Tq-operator results in the very simple deformation of old superpartner potential U+(a:) -. U+(z - q) (kinetic term is invariant). Evidently such deformation does not change the spectrum of U+(z) and that is why SUSY algebra remains intact. Nevertheless it creates new physically
relevant
SUSY quantum
was the implication A more general
mechanical
models.
The crucial
point
in generating
of essentially infinite order differential operators as the intertwining Tq is given by the shift operator in arbitrary coordinate system Tqf(z(z))
= f(z(z)
+ q),
Yq
---- e qd/dz(x)
d
'
1
dz -
d
dz"
of them operators.
(22)
The effects of choices z = In a: and z = z were already discussed above. In general, operator Tq will not preserve the form of kinetic term in H+-Hamiltonian. Physically, such change would correspond to the transition from motion of a particle on flat space to the curved space dynamics. Below we shall assume the definition (6) but full afflne transformation on the line Tqf(z) = f(qx + a) may be used in all formulas
without
changes.
An interesting question is whether inversion transformation can be joined to the a_ne part so that a complete SL(2) group element z --. (az + b)/(cz + d) will enter the formalism in a meaningful way? Application of the described construction to the higher dimensional problems is not so straightforward. If variables separate (spherically symmetric or other special potentials) then it may work in a parallel with the non-deformed models. In the many-body case one can perform independent affine transformations for each of the superselected by fermionic number subhamiltonians and thus to "deform" these SUSY models as well.
5.
q-Deformed
conformal
Particular form of the su(1, 1) algebra creation and annihilation operators (5) K+=½(at)
_,
quantum generators
K_=_a,'
can
that harmonic potential split into two irreducible
the potential introduced in Ref.[19] complete parallel with (23),(24).
be realized
2
[Ku, K+] = -4-K_, This means states being
mechanics
[K+,K_]
via
the
the quantum
98
oscillator
Ko=_*(N+_),
(23)
= -2Ko.
(24)
has su(1, 1) as the dynamical symmetry representalons according to their parity. obeys
harmonic
conformal
symmetry
algebra, physical Let us show that algebra
su_(1, 1) in
First, we
shall rederive
this potential
Hamiltonian of a spin-l/2 along the third axis
particle
within
in an external H=1
and impose
two conditions:
q-SUSY
2
we take magnetic
physical
potential
situation.
{U(z)
Let us consider
and a magnetic
field
U(_)
require the presence
of q-SUSY
(15).
field to be homogeneous (26)
(25) and (14) we arrive at the potential
Equating
v(z) = w2(z) - w'(z) + ;32q-2, where
W(z)
satisfies
the following W'(t)
mixed
finite-difference
+ qW'(qx)
{B(x) (25)
B = -;32q -2 = constant and
the
+ W2(z)
(27)
and differential
- q2W2(qz)
equation
(28)
= 2;32.
This is the condition of a self-similarity [19] which bootstraps the potential in different points (in Ref.[20] ;32 = 72(1 + q2)/2 parametrization was used). Smooth solution of (28) for symmetric potentials
V(,x)
= U(z)
is given
by the following
OC
= "-" In different
limits
q2i_ 1 1 q2;+12i_1
ci=
i=l
of the parameters
power
series i-1 _c'-mcm' m=l
several well known exactly
2;32 cl =--. 1 +q2
solvable
Morse - at q ---, 0; 2) PSschl-Teller - at ;3 o¢ q ---, oo; 3) harmonic 4) 1/z2-potential - at q ---* 0 and ;3 ---* 0. However, strictly speaking valid one has to prove their smoothness, procedures do not commute, etc. Note
problems
(29) arise:
1) Rosen-
potential - at q _ 1; for all these limits to be
e.g., for 4) there may be solutions for which two limiting also that for the case 2) the coordinate range should be
restricted to finite cut because of the presence of singularities. Infinite soliton solution of Shabat corresponds to the range 0 < q < 1 at fixed ;3. If q # 0, 1, c_, there is no tmalytical expression for W(x) but some general properties of this function The spectrum can be derived by pure algebraic H± subhamiltonians are related via the q2-scaling
may be found along means. We already
the analysis of Ref.[19]. know that the spectra of
=
(30)
where the number n numerates levels from below for both this model the lowest level of H_ corresponds to the first restriction
(26)
the spectra
differ only by a constant, E(-)
Conditions
(30) and
= ;32
__ q2., 1 - q2
q-Z,n
(31)
= EL+) _ ;32q-2,
(31) give us the spectrum E.,,.,
spectra. Because q-SUSY is exact in excited state of H+. But due to the
of H m=O,
99
1; n=O,
1,...,cx_.
(32)
At q < i there are two finite At q > I energy eigenvalues
accumulation points, i.e. (32) looks similar to two-band seem to grow exponential]y to the infinity but there
spectrum. is a catch
which does not allow to identify (32) in this case with real physical spectrum. In Ref.[19] it was proven that for 0 < q < I the superpotential is smooth and positive at L = +or. But then
= exp(-f
is a norm zable wavefunctiondefiningthe groundstate of H_-
subhamiltonian and all other states are generated from it without violation of the normalizabi]Jty condition. Therefore relation (32) at 0 < q < i defines real physical spectrum. At q > i the series inequalities
defining
W(L) q_-
p2:-q2-_l we have
0 < c! I) < c, < c! _), where
converges 1
only on a finite
q2i_
interval
[z I < r < oc.
From
1
< q2i+
1 < I,
c! I'2) are defined
i>1
by the rule (29) when
q-factor
on the right
hand side is replaced by p2 and 1 respectively (cl J'2) = c_). As a result, 1 < 2v/'_r/_" < p-_, which means that W(z) is smooth only on a cut at the ends of which it has some singularities. From the basic relation (28) it follows that these are simple poles with negative unit residues. In fact there should be an infinite number of simple "primary" and "secondary" poles. The former ones are characterized by negative unit residues and location points z,, tending to _r(m + 1/2)/v/_, rn E Z, at q _ oo (cl is fixed). "Secondary" poles axe situated at x = q"z,,, n E Z +, with corresponding residues defined by some algebraic equations. Unfortunately, general analytical structure of the function W(L) is not known yet, presented above hypothesis needs rigorous proof with exact identification of aLl singularities and this is quite challenging problem. On the other hand, existence of singularities in superpotential does not allow to take formal consequences of SUSY as granted. Namely, isospectr_ty (or, q-isospectrallty) the whole line problem is broken at this point. Hence one is forced to consider (25) on a cut [-r,r]
with boundary
conditions
¢,,(+r)
= 0. Pole character
leads to ¢o(-)(+r) = 0, i.e. ¢0(-) is true ground state of H_. It also gaxantees on the physical boundaries, U_(+r) < oo. Note, however, that the spectrum problems can not grow faster than n 2 at n _ oo [15] in apparent discrepancy is resolved by observation that action of Tq-operator
of H+ and H_ for ShrSdinger operator of W(L)
singularities
that U_(L) is finite E, for such type of
contradiction with creates singularities
(32). This inside the
interval [--r,r] so that U+(L) and q2U+(qz) are not isospectral potentials (in ordinary sense) as it was at q < 1. Hence, the q > 1 case of (32) does not correspond to real physical spectrum of the model. The number by the presented
of deformations above q-curling
of a given function is not limited. is the property of exact solvability
The crucial property preserved of "undeformed" Rosen-Morse,
harmonic oscillator, and PSschl-TeUer potentials. It is well known that potentials at infinitely small and exact zero values of a parameter may obey completely different spectra. In our case, deformation with q < 1 converts one-level Rosen-Morse problem into the infinite-level one with exponentially small energy eigenvalues. Whether one gets exactly solvable potential at q > 1 is an open question but this is quite plausible because at q = oo a problem with known spectrum arises. Derivation
of the dynamical
symmetry
algebra
is not difficult.
100
To find that we rewrite
relations
(12),(13) for the superpotenti_'(28) AtA
where H is the Hamiltonian
= q-lH
_2q-1 + -1 - q2'
2q-I
AA t =qH+_
with purely exponential
(33)
1 - q2'
spectrum 82
H = _(p2 + W2(x)_
w'Cx))
En-
1 -q2'
1 --q2 q2..
(34)
Evidently, AA t _ q2At A = _2q-1. Normalization
of the right
hand
side of (35)
to unity
definition of q-deformed Heisenberg-Wey] algebra. The shifted Hamiltonian (34) and At, A operators
(35)
results satisfy
[At, H]g = [H,A]q
in the first relation braid-type
entering
commutation
the
relations
= 0,
or, H At = q2A_H, Energy
eigenfunctions
In} can be uniquely 1
It is convinient
to introduce
A H = q2H A.
determined
from the ladder operators
_I-1-- q_" - I,,- 1) A In)= t_q-_/_v 1 - q_ "
In+ a),
the formal
only on the eigenstates
gq
satisfy
original
q-deformed
=
harmonic
number
-_ q A q-N/2,
quantum
conformal
algebra
NIn) = nln>,
a_q
oscillator
.._
algebra [N,a_]
8uq(1, 1) is realized K-
(3_)
operator
of the Hamiltonian.
aqa_ - qa!aq : q-N, The
action
f
q2(.+l) 1 - q;
N = ha[(q2 - 1)H//32], In q2 which is defined
(36)
:
(38)
Now one can check that operators
(39)
-_q q-N/2At
of Biedenharn a_,
and
Macfarlane
[N, aq] = -aq.
[4, 5] (40)
a_ follows,
=(K+)
?,
]
Ko = _(lv + +),
q q4K0
[Ko, K :_] = +K +,
[K+,K
_
q2 _
101
q-4Kn
-] = q-2
(41)
Since H (x q4K0, the dynamical symmetry algebra of the model is SUq(1, 1). Generators K :_ are parity invariant and therefore even and odd wave functions belong to different irreducible representations of this algebra. We conclude that quantum algebras have useful applications even within the continuous dynamics described by ordinary differential equations. A different approach to q-deformation of conformal quantum mechanics on the basis of pure finite difference realizations was suggested in Ref.[25]. Let us compare presented model with the construction of Ref.[26]. Kalnins, Levine, and Miller called as the conformal symmetry generator any differential operator L(t) which maps solutions of the time-dependent Schr6dinger equation to the solutions, i.e. which satisfies the relation
0 L-[H,L] = R(i 0 _ H),
(42)
where R is some operator. On the shell of Schr6dinger equation solutions L(t) is conserved and all higher powers of space derivative, entering the definition of L(t), can be replaced by the powers of O/Ot and linear in 0/0r term. But any analytical function of O/Ot is replaced by the function of energy when applied to stationary states. This trick allows to simulate any infinite order differential operator by the one linear in space derivative and to prove that a solution with energy E can always be mapped to the not-necessarily normalizable solution with the energy E +/(E) where y(E) is arbitrary analytical function. "On-shell" raising and lowering operators always can be found if one knows the basis solutions of the SchrSdinger equation but sometimes it is easier to find symmetry generators and use them in search of the spectrum. In our construction we have "off-shetl" symmetry generators, which map physical states onto each other and satisfy quantum algebraic relations in the rigorous operator sense. In this respect our results are complimentary to those of the Ref.[26]. It is clear that affine transformations provide a particular example of possible potential deformations leading just to scaling of spectra. In general one can try to find a map of a given potential with spectrum E, to a particular related potential with the spectrum/(E,) for any analytical function y(E). A problem of arbitrary non-linear deformation of Lie algebras was treated in Ref.[12] using the symbols of operators which were not well defined on proper Hilbert space. Certainly, the method of Ref.[26] should be helpful in the analysis of this interesting problem in a more rigorous fashion and the model presented above shows that sometimes an "off-shell" realization of symmetry generators can be found.
6.
Factorization
method
and
new
potentials
mechanics is related to the factorization method of solving of SchrSdinger equation [27-29]. Within the latter approach one has to find solutions of the following nonlinear chain of coupled differential equations for superpotentials Wj(z) SUSY quantum
W_+W_+ I+Wf-W2+1 where kj, Aj are some constants.
=kj+l
=A_+1-Aj,
j=0,1,2...
The Hamiltonians associated
to (43) are
2Hi = p_ + Ui(:c ) = p_ + W_(x)-
102
14_(_:) + Ai,
(43)
(44)
u0(_) = w_ - w_ + _0, where _u is an arbitrary SUSY Hamiltonians
uj+,(_) = uj(_) + 2w;(_),
energy shift parameter. are obtained by unification
of any two successive
pairs
Hi, Hj+i
in a
diagonal 2 x 2 matrix. Analogous construction for a piece of the chain (44) with more entries was called an order N parasupersymmetric quantum mechanics [30, 31]. In the latter case relations (43) naturally arise as the diagonality parasupersymmetric Hamiltonian.
conditions
If Wj(z)'s do not have severe singularities differ only by a finite number of lowest levels. ¢(J)(z) are square normalizable
of a general
then the spectra of two operators from (44) may Under the additional condition that the functions = e- f" Wj(u)du
function
(45)
one finds the spectrum
Hj¢(2)(_)= E(Z)¢(.°)(_), where subscript
(N + 1) x (N ÷ 1)-dimensional
n numerates
of Hj from which
levels
from
below.
one can determine
E.(_)= '_j+.,
In this case (45)
lowest
excited
states
(46) represents of Hf,
ground
state
wave
j_ < j,
¢(j)(_) = (p + iw;)(p + iw,+,)... (p + iw;+._,) ¢,(j+"). Any exactly solvable discrete spectrum times it is easier to solve SchrSdinger Hamiltonians
(44).
Cu(N) normalizable.
If Uo(z)
problem equation
can be represented in the form (43)-(47). Someby direct construction of the chain of associated
has only N bound
If WN(Z)
= 0, then
(47)
states
Hj (j < N)
then
there
has exactly
does
not exist
g - j levels,
WN(z)
making
the potential
Uj(z)
is refiectionless and corresponds to (N -j)-soliton In order to solve evidently underdetermined conditions. At this stage it is an art of a researcher
solution of the KdV-equation. system (43) one has to impose some closure to find such an Ansatz which allows to generate
infinite number of Wj and kj from fewer entries. the choice Wj(z) = a(z)j + b(z) + c(z)/j where
Most a,b,c
of old known examples are generated are some functions determined from
by the
recurrence relations [27, 28] (see also [19]). New look on the equations (43) was expressed in Ref.[32]. It was suggested to consider that chain as some infinite dimensional dynamical system and to analyze general constraints reducing it to the finite-dimensional integrable cases. In particular, it was shown that
very
simple
periodic
closure
conditions
w,+_,(=)= w,(=), for N odd lead to all known hyperelliptic
potentials
A,+N= _,, describing
(48)
finite-gap
spectra
(i.e.
those
with
finite number of permitted bands). In this case parameters ,_j do not, of cause, coincide with the spectrum. The first non-trivial example appears at N = 3 and corresponds to Lame equation with one finite gap in the spectrum. Equivalently one can consider arising Schrodinger equation in the Weierstrass
form (then
periodic
potential
has singular
points
where
wave functions
are required
to
be equal to zero) and again parameters ,_j do not coincide with (purely discrete) spectrum. Note that in the analysis of parasupersymmetric models [30, 31] constants kj were naturally treated as arbitrary parameters only occasionally giving the energy levels.
103
The self-similar
potential
of Sect.5
was found
in Ref.[19]
by the following
Ansatz
in the chain
(43) (49)
Wj(z) = qiW(qJx), which gives a solution each other as follows
provided
W(x)
satisfies
the equation
(28)
and constants
k s are related
ks o(q:;, j > 0. As it was already and
therefore
discussed,
closure
the parameters
(49) seems
to be completely
different
w,+,(=) = wA=), which leads
to harmonic
oscillator
(50)
_j o¢ q2j give the spectrum
q-SUSY quantum mechanics and subsequent derivation q-deformation of the following closure condition
potential.
to
from (48).
of (49),(50)
of problem However,
shows
that
at 0 < q < 1
described in fact
k_+,= k,,
Indeed,
one may write
Wj+,(r) = qWj(qz),
kj+, = q2kj
above (49) is a
(51)
(52)
and check that (49), (50) follow from these conditions. As it was announced in Ref.[23] one can easily generalize deformation of SUSY quantum mechanical models to the parasupersymmetric ones. In the paxticular case defined by (N ÷ 1)member piece of the chain (44) one simply has to act on the successive Hamiltonians by different affine transformation group elements. This would lead to multiparameter deformation of the parasupersymmetric analogous physical accepting general
these
algebraic restrictions
relations. Following on the Hamiltonians
constraints.
q-periodic
closure
Analyzing of the chain Wj+N(Z)
These
conditions
describe
q-deformation
Let us find a symmetry algebra First we write out explicitly
the consideration of Ref.[30] one may impose and look for the explicit form of potentials
such possibilities
the
author
have
found
the
following
(43) = qWj(qz),
(53)
kj+N = q2kj.
of the finite-gap
and related
potentials
appearing
at q = 1.
behind (53) at N = 2. the system of arising equations
w:(_) + w_(_) + w}(_)- w_(_) = 2_, w_(x) ÷ qW_(qz)+ w_(z) - q2W}(q=)= 2_.
(54)
One can check that the operators K + = _(p+ satisfy
iW,)(p+
iW2)v_Tq,
K- = (g+)t
(55)
the relations K+K - = H(H-
a),
K-K
+ = (q2H ÷ B)(q_H
H =_(p'2 + _7(_) - w;(_)).
104
+ a + _),
(56)
The operator
H obeys
the following
commutation
HK+-q_K+H=(a+_)K Note
that by adding
+,
to H of some constant
+ - q4K+ K-
The formal map onto the relations particular form of the "quantization" Described q-deformation sented in Sect.5. Indeed, potential corresponding soliton potential arises
with K +
K-H-q2HK
equations
On the basis of (56) one may define various The simplest one would be the following K-K
relations
(57) may be rewritten
q-commutation
= q2(c_(1 + q2) + 23)H
(57)
- --(_+3)K-.
relations
+ _(a
in the form (36).
between
K + and K-.
(58)
+ 8).
(41) is also available. Therefore relations (57),(58) give a of the algebra au(1, 1) which is explicitly recovered at q = 1.
of the conforms] quantum mechanics various limits of q give the following
is more general solvable cases:
than that pre1) a two-level
to two-soliton system appears at q = 0; 2) a finite cut analog of twoat q -+ _; 3) the general conformal potential comprising both oscillator
and 1/z 2 parts is recovered spectrum of H at arbitrary
in the limit q -+ 1 when W(z) cx a/z + bz. In order to find the q it is neccessary to know general properties of the superpotential
W_. Let us suppose that there exists a sohtion for positive a and _ such that exp(- f_ W_,2) are normalizable wave functions. Then the spectrum consists of two geometric series and by shifting can be represented
in the form
En = { Elq E°q_"' 2m, with
the
reducible structures
7.
E,
< E,+1
ordering
fulfilled.
Even
for for n=2m n=2m-kl
(59)
and odd wave functions
fall into independent
ir-
representations of 8uq(1, 1). A more detailed consideration of potentials and algebraic arising from the q-periodic closure of the chain (43) will be given elsewhere.
Conclusions
To conclude, we described a deformation of the SUSY transformations. The main feature of the construction
quantum mechanics is that superpartner
non-trivial braid-type intertwining relations which remove degeneracies tra. Obtained formalism naturally leads to the Shabat's self-similar decreasing
solutions
of the KdV equation.
The latter
is shown
on the basis of sffine Hamiltonians satisfy
of the original SUSY specpotential describing slowly
to have
straightforward
mean-
ing as a q-deformation of the harmonic oscillator potential. Equivalently, one may consider it as a deformation of a one-soliton system. Corresponding raising and lowering operators satisfy q-deformed Heisenberg-Weyl algebra atop of which a quantum conformal algebra SUq(1, 1) can be built. We also outlined a generalization of the Shabat's periodic closure condition and presented q-deformation potentials. In this paper
the parameter
q was taken
potential on the basis of q-deformation of of general conformal quantum mechanics
to be real but nothing
prevents
from consideration
of
complex values as well (this changes only hermicity properties). The most interesting cases appear when q is a root of unity [331. For example, at q3 _ I eq. (28) generates a potential proportional to
105
the so-canedequianharmonic Weierstrassfunctions. generated
at higher
roots of unity.
The nontrivial
More complicated hyperelliptic potentials are Hopf algebra structure of the quantum groups
was not considered because it is not relevant in the context of quantum in one dimension. Perhaps higher dimensional and many body problems
mechanics of one particle shall elucidate this point.
In fact, there seems to be no principle obstacles for higher dimensional generalizations although resulting systems may not have direct physical meaning. Another possibility is that described self-similar systems may arise from higher dimensional ones after the similarity reductions. In order to illustrate various possibilities we rewrite the simplest out scaling (i.e. at q = 1) but with non-trivial translationary part
self-similarity
equation
with-
w'(=) + w'(= + a) + W2(=)- W2(=+ a) = co,,,tant.
(6o)
Solutions of this equation provide a realization of the ordinary undeformed Heisenberg-Weyl algebra. The full effect of the presence of the parameter a in (60) is not known to the author but solutions whose absolute values monotonically increase at z ---* 4-oo seem to be forbidden. Note also that in all formulas of SUSY and q-SUSY quantum mechanics superpotential W(z) may be replaced by a hermitian n x n matrix function. The equations (28), (35), (60) may be equally thought as being the matrix ones with the right hand sides proportional to unit matrices. We end by a speculative of the non-linear
8.
conjecture integrable
that described machinery may be useful in seeking evolution equations, like KdV, s/n-Gordon, etc.
for q-deformations
Acknowledgments
The author is indebted
to J.LeTourneax,
W.Milhr,
A.Shabat,
to Y.S.Kim for kind invitation to present this paper at the This research was supported by the NSERC of Canada.
L.Vinet
for valuable
Workshop
discussions
on Harmonic
and
OsciUators.
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R.Floreanini,
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and
T.N.Tomaras,
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B251,
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Functions,
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163 (1990).
B18,302
M.Schlieker, AS, 2635
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B.Zumino,
and S.Watamura, (1990); L.Baulieu
Mod.Phys.Lett.
Z.Phys. C49, 439 mad E.G.Floratos,
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Scientific,
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B262,
C.Dasl_loyann_s
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S.Kamefuchi,
V.A.Matveev
p.133.
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A24,
and K.Ypsilantis,
L591 (1991); preprint
D.Gangopadhyay, and 554 (1991); K.Odaka,
C.Daskaloyannis,
THES-TP-91/09,
J.Phys.
A24,
L789
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[13] O.W.Greenberg, Phys.Rev.Lett. 64, 705 (1990) and talk at this workshop; R.Mohapatra, Phys.Lett. B242, 407 (1990); V.P.Spiridonov, Dynamical Parasupersymmetry in Quantum Systems, in: Proc. of the Intern. Seminar "Quarks'90", 14-19 May 1990, Telavi, USSR. Eds. V.A.Matveev [14] M.M.Nieto,
et al (World Phys.Lett.
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B145,
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Sturm-Liouville
Operators
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D.L.Pursey,
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Generalized
rences for Heat and Schr6dinger Equations in One Spatial Dimension,/n: sis and Geometry: 200 Years after Lagrange, Ed. M.Francaviglia (Elsevier B.V., 1991) p.237
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V.P.Spiridonov,
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108
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A6,
(World
3163 (1991).
N93-27322
PHASE
OF
WITH
THE
QUANTUM
APPLICATIONS
TO Scott
Department Massachusetts
OSCILLATOR
OPTICAL
R.
of Electrical Institute
HARMONIC
POLARIZATION
Shepard
Engineering
o.f Technology,
and
Computer
Cambridge,
Science,
Massachusetts
02139
Abstract The phase of the quantum
harmonic
oscillator,
the temporal
distribution
of a particle
in
a square-wen potential, and a quantum theory of angles are derived from a general theory of complementarity. Schwhager's harmonic oscillator model of angular momenta [1] is modified for the case of photons. Angular distributions for systems of identical and distinguishable particles are discussed. Unitary and antiunitary time reversal operators are then presented and applied to optical polarization
1
General
The
fact
that
transform
linear
relations
mechanics relations oscillator, _(h (i.e.
Theory
in birefringent
media.
of Complementarity
momentum between
states
is the
the
generator
momentum
of translations
and
spatial
in space,
representations
leads
to the
Fourier
of Schrodinger's
wave-
[2]. Similarly, since energy generates translations in time, there are Fourier transform between the energy and temporal representations [3]. For the case of the harmonic the energy eigenspectrum is proportional to the integers n = 0, 1,2... (recall If/ =
+ 1/2), where h = &t(i is the photon number operator) not periodic). Therefore the temporal distribution of the
periodic.
Indeed,
the
quantum
harmonic
simplest
oscillator
way
(that
I have
found)
and this spectrum is aperiodic oscillator will be continuous and
to describe
the
phase
(¢I, = wt)
of the
is to form the wavefunction ¢1o
¢(¢)
= __, ¢,e -i'_¢
(1)
n=O
which
is the
Fourier
en
(nl¢/,
where
-
period
of ¢(¢))
associated realizable
series
bin ) = nln ).
is then
simply
n-space The
wavefunction
probability
I¢( ¢)12/2_r.
(series)
transform
(or number-ket
density
The wavefunction
with the equally correct perspective measurement of the Susskind-Glogower
Suppose we wish to study "phase of the infinite square Fourier
of the
for finding approach
expansion ¢ on any circumvents
[4] that this phase distribution (SG) [5] phase operator.
coefficients)
2r interval
corresponds
the temporal behavior of a particle in a one dimensional well"). We do not have to start all over, we can simply
of the discrete
energy
wavefunction,
which
underlies
the
(the
complications
discrete
to the box (the take the energy
eigenspecta: _2_.2
Ei-
2mL2(i)_
(i=1,2,3...)
109
(2)
=
=
:
?
where
_
=.:
--o
=!
L is the length
of the box and m the mass
of the particle.
In other
words,
labeling
eigenstates, {[E,,)}, according to the value of n - (i) 2, we'd use the ¢, _= (E,,[¢) series coefficients in ¢(4) = Z, en e-in¢, where • = t(hr2/2mLU). The temporal therefore
like that of a harmonic
For a well of finite to the squares
one weighted
since
the _(t
the bound
of integers,
each and
depth,
oscillator state
¢_ -- 0 -
eigenenergies
by e -IE_t/_,
to form
integer
be made
¢(t)
which
multiples
as small
and we would other,
being
(it can't
however
by making
for this problem, the aperiodic
from
etc.
proportional
be exactly
the difference
T large
however,
= es,
still sum over the (Ei [¢) with
is quasi-periodic
of each
as we wish
"quasi-periodic"). The unbound states in energy and for these we would form
¢3, ¢5 = 0 = ¢6 -- er
will be perturbed
but they will still be discrete
Ei are no longer + T) can
for which
the energy
as the Fourier distribution is
enough
have
--
periodic
between hence
a continuous
¢(t)
the term
distribution
¢(t) = f dE where i.e.
¢(E)
they
-
(E[¢).
Notice
can be "here
today
that
the unbound
and
(3)
states
gone tomorrow"
exhibit
as they
an aperiodic
zip past
temporal
the potential
bound states are trapped into quasi-periodic oscillations. From the general theory of complementarity we can also obtain a quantum The z component of angular momentum, ,]z, is (by definition) the generator the
angle
about
eigenvalues
the
given
z axis,
by mtt
which
where
shall
be denoted
m E {-j,-j
as ¢.
+ 1,...j
-
It is well known 1,j}
and
j is the
distribution,
well, whereas
the
theory of angles. of translations in that
Jz has
discrete
label
of the
discrete
eigenvalues of the simultaneously measurable j_ - j_ + ,]u2 + J_ which are j(j + 1)h 2. For states in which each value of m is uniquely represented (the degenerate case will be discussed in the next
section),
such
as a particle
of spin
s (i.e.
j = s = a fixed
number),
we can form
the
angle
representation ¢(¢) where emits transform
= _
is a simple
2
Harmonic
and immediate
consequence
Oscillator
of the periodicity
Models
In 1952, Schwinger [1] demonstrated a connection oscillators and the algebra of angular momenta. follows: 3_
From
a= and
ad denote
this we obtain
the
_
_atdau
annihilation
the fundamental [J+,J_]
where behave
J+ -
(4)
(j, ml¢) and the angular distribution is p(¢) = [¢(¢)[2/2_r. Since ¢(4)is ¢,,_ must be discrete, i.e. the quantization of angular momentum (projected
axis)
where
¢,,,e -''¢
(J_)t
and
like spin 1/2
J+ = j_ 4- ijy, objects
(as seen
of Angular
3z
operators
from
_
_(nu
--
that
axis).
Momenta
7"ld),
for the "up type"
commutation
relations
and
so [J_,Ju]
(about
between the algebra of two uncoupled harmonic The key points of Schwinger's model are as
and
= 2hjz
of the angle
periodic onto an
[L,J_] = ihJ_
eq.(5)), 110
yet
(5)
and
of angular
"down
type"
= ±h3_, etc. only
Since totaly
oscillators.
momentum: (6)
the
quanta
symmetrical
of these states
oscillators are con-
structed
by this method,
connection
is merely
We put
some
these
within
quanta
are not believed
the mathematics
physics
into
this
to correspond
to actual
particles
and the
[2].
connection
by
considering
a rotation
of a single
frequency
electromagnetic wave about result that a right handed
the z axis (along which the k vector lies) which leads to the well known circularly polarized photon is an eigenstate of j_/h with eigenvalue
m = +1.
handed
consider
a left
transverse
1 with sense
Similarly,
m=0 that
algebra
components
missing" its spin
of angular
space
is associated
of the vector
[6]. Since
the photon
is two
momenta
photon
potential,
the photon which
it seems
physically
fi, and
a single
hi are the
frequency,
annihilation
operators
z propagating,
[.]+,.]_] where
as before
however Notice up
J+ = (J+)t
J_ now that
and
phase
oscillators
about
than
[J_,J+l
case
of fermions
the
these
of "spin
1/2 object
to attempt
in the
to reconstruct
primitives.
Indeed,
the taking
(7) left circularly
polarized
modes
of
(8)
= -4-2h J+. etc.
is exactly
circularly
and
only
we obtain
by 1) which
between
a spin
photonic
so [L, J_] = 2ihL
shift
for the
to a rotation
and
we need
-
wave,
J± = L 4- iju,
m by 2 (rather
a differential
down
equivalent
lowers
and
= 4h,Jz
and since
is said to be a particle
reasonable
for the right and
electromagnetic
= -1
resembles
significant
£ - 2ha1 . and Jz where
m
is a boson
dimensional,
from these
with
This is the same
what
polarized
ordinary,
i.e.
we want modes
group,
for photons.
(or between
"non-photonic",
the
bosons)
is
z axis:
= Y] ns-
We can
relable
(n_ + rid)/2)
and
our two-mode
states
according
m = n, - n, (or m = (n= - rid)/2).
case of identical particles should allow for quantum than
number
(9)
ti'l I
probabilities)
and
to the
To obtain
values the
of j = n, + nt (or j =
angle
representation
for the
(e.g. all these states are photons, or they are all electrons, etc.) we interference of all these states (i.e. we should add amplitudes rather therefore
simply
use ¢_
_ _(j,
ml¢>
(10)
J
in eq.(4)
for these
degenerate
cases.
in j) we must
Since
the
Cm defined
¢(¢) --, For bosons at most
27r, and
or photons since
in eq.(10)
are no longer
normalized
(for m states
renormalize:
whe,e c = /_"
the minimal
Am,,,
non-zero
= 1 the period
value
ofp(¢)
(11)
of lml is one therefore is at most
the
27r. For fermions
period
of ¢(¢)
is
m can be 1/2 so
the period of ¢(¢) can be 47r. This indicates the rotational Berry's phase "for fermions"[7], which we now see to be more correctly stated as being "for fermions which have non-zero overlap with m = +1/2
states."
Since
Arnm_,,
for fermions
is still 111
one,
p(¢)
is still
mod
27r, indicating
that
observation of the "rood is well known. Notice
that
we could
if we allowed
have
interference radically
Am,,,i,,
with alter
might
argue
therefore
4_r Berry's
for particles
= 1/2
another
phase"
so that
state
that
it is physicaly
(not
reasonable
(each
¢(¢))
p(¢)
have
i.e.
should
be (at most)
mod
one, as spin
47r! Since
of this type
strip?). at most
no
would
Alternatively, a period
we
of 2:r and
fact that a mixed
does not occur" [2]. If however, we had a system comprised of a fermion electron and a photon) then since these distinguishable particles do not distribution
mod
particles
of the '%. empirical
another
and half-integer
be periodic
like a Mobius
that
with
of integer
would of proposed
point
to require
explanation
of this state
of mixtures
just
the existence
of space
have a theoretical
interference
comprised
p(¢)
is required,
our conceptualization
we would
requires
symmetry
and a boson (e.g. an interfere, the angular
2_r.
For the case of distinguishable particles we should add probabilities (rather than amplitudes), rather than the procedure defined by eq.s (4), (10), and (11), we should do the following. For
each
distinguishable
particle
we should
form
an angular
wavefunction,
then square
its magnitude
and divide by 27r to form each different particle's individual angular distribution, then add these individual distributions to form the angular distribution of the entire system. When these distinguishable
particles
and a spin form
have
1 photon
distinct
values
for example)
of spin (such
this proceedure
¢ 1 (else the pseudo-coherent
(A2Ex = 1), where we assume towards the vacuum).
for the linear
shot
(with
+
Both states yield polarization "noise"
(A2E:
in
for the media
x/_)e-I"f'/210),10)t , which 1'11 refer to as the pseudo-coherent state. polarization "signals" (/_=) __ -2asin(wt) and (/_t,) = 0, yet, the state
of
of putting
a reduction
These states therefore provide a foundation of devices which utilize circularly birefringent
etc.). two T,, eigenstates:
la),.la)t
which
similar of the
coherent states
state
both
of the differential
tend phase
to optical polarization states propagating ¢(¢) would describe the polarization state
since
we lose the
connection
with
the angular
as the energy eigenstates in the linear basis are not eigenstates of angular momenta. the mode exchange eigenstates in this basis correspond to an expected value of the
electric
field operator
for the
study
that
of quantum
resembles
circular
limits
on the
Atomic
Energy
polarization
performance
and
these
of quarter-wave
states plates,
provide
a foundation
etc.
References [1] J.
Schwinger,
U.S.
Report
No.
NYO-3071
(U.S.
GPO,
Washington,
D.C.,
1952). [2] J. J. Sakurai,
Modern
[3] S. R. Shepard, MA,
Quantum
Ph.D.
Thesis
Mechanics Proposal
(Benjamin/Cummings,
(Massachusetts
Institute
Menlo
Park,
CA,
of Technology,
1985).
Cambridge,
1990).
[4] S. R. Shepard
and
Washington, [5] L. Susskind
D.C., and
J. H. Shapiro,
J. Glogower,
[6] V. B. Berestetskii,
et. al., Phys.
[8] K. Gottfried,
Quantum
Washington,
Physics
et. al., Quantum
[7] S. A. Werner,
[9] S. R. Shepard,
'88 Technical
Digest
(Optical
(Permagon
Press,
Society
of America,
1, 49 (1964). Electrodynamics
Rev. Lett.
Mechanics
Workshop D.C.,
OSA
1988).
(W.
on Squeezed
35,
Elmsford,
NY, 1982).
1053 (1975).
A. Benjamin, States
1991).
114
and
New York, Uncertainty
NY, 1966). Relations
(NASA
CP-3135,
N93-$7323
CONDITION
FOR
AND
EQUIVALENCE
OF
ANHARMONIC
q-DEFORMED
OSCILLATORS
M.Artoni N.A.S.A.
Goddard
Space
Flight
Greenbelt, Jun Physics Convent
Zang
and
Department, Avenue
Center,
MD
Joseph
The City
at 138th
Str.,
Photonics
20771,
Code
715
USA
L. Birman College
New
of The C.U.N.
York,
NY
I003i,
Y
U.S.A.
Abstract We discuss the equivalence between the q-deformed harmonic oscillator and a specific anharmonic oscillator model, by which some new insight into the problem of tile physical meaning of the parameter q can be attained.
1
Introduction
Recently
there
Of particular
has
been
interest
by Biedenharn q-analogue
a great
of interest
here is the development
[2] of the
of the
deal
realization
quantum
of the
harnomic
in the
study
by Macfarlane quantum
oscillator.
group
of quantum [1] and
sg(2)q
Although
many
independently in terms
aspects
deformation of the most
of the bose harmonic oscillator algebra have been investigated, appealing issues is perhaps the physics behind the parameter
an attempt
is made
We show a specific
that
the q-deformed
harmonic
oscillator.
Thus
effective anharmonic deformation, harmonicity. The anharmonic and latter
in section
can in turn
a q-deformation forward
2 The
in section
2 and
be used during 4, and
oscillator
of the
q-
still one q. Here
their
equivalence
interesting
time-evolution
can be used to describe
can
where q is proportional the q-deformed oscillator
3 and
discussed
model
a q-deformation
to examine
the
Anharmonic anharmonic
of the
in this direction.
anharmonic
respectively
groups.
be understood
to the strength of the models are presented
is therein
non-classical
of a SU(2)
as an
coherent
discussed.
features state.
The
induced This
by
is put
in [3]
oscillator
oscillator
we wish to discuss
H_ = Ho +
has the
___Na = N + 1
_oo
hamiltonian tt N3
-2+ -O3o
115
(1)
•
,_-l,
where
tl_,
H0 is the
mental
free
frenquency
hamiltonian
is Wo. N
of the
simple
= b_b is the
harmonic
number
oscillator
operator,
whose
whereas
funder-
bt and
b are
respectively the lowering and raising bose operators. Ha is in units of w0 when H0 is in units of w0. The anharmonic term is taken proportional to N 3, and the anharmonicity small of
parameter
is positive:
anharmonic
seen
that
we take
the hamiltonian
a_ = _/-_' It is readily
specifically
deformations
72(btb
[1 +
here
# -
in Eq.(1)
+ 1) _ 2.3! ]
Ft., = 3'-'
in terms
sinh3'
(2)
+ a/2)
(3)
is indeed equivalent [4] to HA in Eq.(1). States of our anharmonic oscillator can be constructed note that
In the limit of
in this representation H_ = _(a'_a,
First
wo3`2/6.
can be discussed
the vacuum
]0)_, defined
as quantum
states
for H r.
as a_[0).y = 0, is the same as the vacuum
]0)_
for the harmonic oscillator. However, eigenstates of the number operator N_ = a_a_ substantially differ from those for the harmonic oscillator. The former can be defined as
? n
10),
In)
CV/-_n,_
while
the normalization
Cn-
condition
w(rn]n)w
= 6m,,_ determines
n 72k2 c,,w=n]ag"II(l+_.3!12=n!ag"[(l+_.3!)2l
(4)
1 ,At
the cn._'s:
72n2 I,
co,.y=l
(5)
k=l
Itere we will be concerned, 0, 1,2,...)
these
in particular,
can be expressed
with coherent
states.
In the basis
{[n)_}
(n =
as [5]
oo
c_n
]_)_ = C_ Z
_]n)._,
n=O
___
ot2n
n=O
Cn,"t
C_-2 = z_.,
_¢/Cn,"¢
(6)
Where C_ derives from the normalization condition -r(o]a). r = 1. The resemblance of the ]c_),'s with coherent states of the harmonic oscillator is resdily seen: however, we should stress that only in the limit 3` _ m the anharmonic and harmonic oscillator models
3
are exactly
q-deformed
the same.
harmonic
Let us recall
the (b, bt) bose operators
They
the Weyl-Heisenberg
satisfy
[1] and
Biedenharm
for the harmonic
oscillator
introduced
earlier.
algebra
[b,b _] = 1 Macfarlane that
oscillator
[N,b t] = b_
[2] have
aqa_ - qa;aq
discussed
= q-N
N = bib a deformation [N, a_] = a_
116
(7) of this
algebra
so (8)
and,
in particular,
parameter
its realization
q [6] characterizes
in terms
of a q-deformed
the strength
in the
previous
q-operators
where
on the section
of the harmonic
for the
can be realized
[X]q = (q= -
q-harmonic
states
q-_:)/(q
oscillator.
aq is a function oscillator turn can be defined
of b and out as
power
to be the
construct
vacuum
of b_b, IO)q and
same.
model.
bose operators
we first
q-deformed
similarly
oscillator
of the
- q-i),
The
The
between q-deformations and anharWe will first study the effect of a oscillator,
anharmonic
in terms
oscillator.
of t,he deformation.
We explore in this section the connection monic deformations of the harmonic oscillator. q-deformation
harmonic
Eigenstates
the
form
that
quantum
number
states
for the
10) of the operator
since
harmonic Nq = a_aq
=
(10) CV/-_,q
With
the choice
the
[1, 2]
as aq]0)q = 0, and
vacuum
of the
was done
By recalling
of the
is defined the
to what
Cn-
of c,,q = [X/_q. I, where
vectors {In)q} (n = 0,1,2,...) Fock space for the q-deformed can express the coherent states
l ,q
[nlq! = [nlq[n - 1]q-.-[1]q,
the
set of eigen-
is orthonormal (q(mln)q = 5re,n) and generates oscillator. On the basis {In)q} (n = 0,1,2,...) of the q-deformed harmonic oscillator as
the one
O{ n
where
the
factor
Cq is again
set by the normalization
condition
q(c_lo)q
= 1. Here
expq stands for the q-exponential, i.e. expq = _,_°°=o x/[n]q!. Again note that as q --* 1 this q-deformed model exactly reduces to that of a simple harmonic oscillator. A connection can be established between coherent states of q-deformed harmonic oscillator and coherent states of the anharmonic oscillator in the sense that there exists a condition under oscillator displacements
which o and
the ]o)q's and the 7 (or q) such that
[c_)_'s are equivalent. [3]
Namely,
o_(c_+ 8) < In -x qal4 we have Ic_)q _ Is)w, provided is beyond the aim of this paper can compare
here the
the equivalence
of the
Fig.1
of q and
for values
number
l(nl ) l = and over
states
(12)
7 = lnq. An analytic proof of this equivalence and will be reported elsewhere [3]. However, we
probability
Ic_)q's, that is, P$(_) = of probability as overlap
for
the
same
[cr).r and
cr respectitively
distribution
for the
I_)_'s
to that
for the
P_(c_) = I(nl ),l state In), equal
Owing to the definition distributions would infer
I_)q.
evaluation
A numerical
conforming
117
and
not
is reported
conforming
with
in the
condition (12). whereas in the
In this latter case P_(a2) is strongly shifted with respect former case the two distribution are nearly the same.
In conclusion, coherent
states
for appropriate
displacements
of an oscillator
with
(o)
and
anharmonicity
anharmonic
_
N 3 (N
to P_2(a2), couplings
is the
number
(p) of
particles) are correctly described in terms of coherent states of the q-deformed Lie algebra of SU(2), where q __ exp(lz/Wo) a/2. This result is particularly important because
the
parameter
to tile square
root
0.15
q can be given
of the anharmonic
I
a direct
physical
coupling
I
I
I
q - Coherent
A_
it is proportional
strength.
I
P_2 (o_2)
meaning:
I
!
State
0.12
r' (a_)
Pq' (oq) 0.09
/x
0.06
;
i
P?(a)
i tate
0.03
0.00 C
FIG.1.
I0
Probability
of a q-deformed of a quantum ticle
number.
between
the
20
30
number quantum From
with their
corresponding
a third
states,
oscillator
(al
= 10, 71 = 0.1) the condition Poission
and
satisfy
70
for coherent
states states
anharmonicity
one
which
60
for coherent order
equivalence
the
is a reference
parameters
SO "
distributions oscillator
oscillator
40
can
holds
infer
(12), respectively.
(qo = 1) distribution
118
with
I >q2) (Ic_)-,,, la)-,2) in the
the
depending
(_1 = 4, 71 = 0.05)
BO
par-
equivalence on whether
or do not satisfy
Here q = e'. a = 7.
p q0(_)
4
q-deformation
and
non-classical
harmonic
os-
cillator The
equivalence
harmonic with
oscillator
a definite
tigating
and
effects
induced
The
we have established
most
is a very
physical attaining
helpful
meaning, a sound
of these
does herent
also
alter
state,
the but
not only does it turn
the
of a SU(2) self-squeezing:
of a
q parameter
to be useful
of interesting
time-evolution
is a q-dependent
q-deformation
it provide out
interpretation
during
and
for inves-
non-calssical coherent i.e.
state.
a reduction
of the two orthogonal components (quadratures) of vacuum values that varies with q. A q-deformation
minimality not
anharmonicity
also dose
physical
effects
of the uncertainty expactations the oscillator field below their
one: but
by a q-deformation
important
between
properties
its possionian
of an
counting
initial
mimimum
statistics.
The
uncertainty
connection
co-
between
q-deformations of the harmonic oscillator and these rather interesting phenonena however beyond the purpose of this paper and will be discussed elsewhere [3].
5
is
Acknowledgments
This
work
was done
Photonics) to D. Han,
while
one
of the authors
(M.A.)
held
a N.R.C
(NASA/GSFC,
Research Associateship. We also would like to express our appreciation Y.S. Kim, and W.W. Zachary for organizing a most splendid workshop.
References [1] A.J.
Macfarlane,
[2] L.C.
Biedenharm,
[3] M. Artoni,
Jun
[4] We here retain anharmonic
J. Phys.
A 22, 4581,
J. Phys.
(1989)
A 22, L873,
Zang,
and
terms
only of the order
deformations
[5] For simplicity,
we take
[6] q is in general
complex:
Joseph
(1989)
L. Birman
at ordinary
(to be submitted
73 or lower,
as typically
energies;
a real; however,
here
q > 1 and
119
real;
for publication) done
for small
N93-27324
NOVEL
PROPERTIES
OF THE
QUANTIZED
RADIATION Charles
University
of Physics
of New
Binghamton,
FIELD
A. Nelson
Department State
q-ANALOGUE
York at Binyhamton
N. Y. I390_-6000
Abstract The "classical limit" of the q-analogue quantized radiation field is studied paralleling conventional quantum optics analyses. The q-generalizations of the phase operator of Susskind and Glogower (circa 1964) and that of Pegg and Barnett (circa 1988) are constructed. Both generalizations and their associated number-phase uncertainty relations are manifestly qindependent in the In >q number basis. However, in the q-coherent state Iz >q basis, the variance of the generic electric field, (AE) 2, is found to be increased by a factor _(z) where _(z) > I if q _ 1. At large amplitudes, the amplitude itself would be quantized if the available resolution of unity for the q-analogue coherent states is accepted in the formulation. These consequences are remarkable versus the conventional q = 1 limit.
1 On
Introduction several
occasions
during
the last
fifty
years,
new mathematical
symmetries
have
been
con-
structed in theoretical physics but only found to be relevant to nature five or more years later. If this is occurring now in the case of quantum algebras, we need to know the physical implications of these new and distinctly novel symmetry structures. If there are q-oscillators in nature which realize these new algebras, surely there must be a quantum field which has such q-oscillators as its normal modes. Until we know the physical properties of such a field, say in its "classical limit", we may not be able to glean its distinct relevance many body physics, particle physics ....
to problems
2
for the q-Analogue
A
Completeness
States The
q-analogue
( q _
Relation
and phenomena
in quantum
optics,
Coherent
by q-Integration coherent
states
[z >q satisfy
alz >q=
z[z
>q where
1, usual bosons)
121 PRE6EDtN6
PAGE 6LAPIK
NOT
FILMED
the q-oscillator
algebra
is [1]
aat _ ql/_at a __ q-N2
[N, at] = at
[N,a] =-a
It is physically very important that there remains In the !n >q basis, < rain >= g,,,, and x
The
q-analogue
the q-analogue
[_]n-
(q'/_-q-./2)/(q,/2_ q-,/_) is the where q = exps, 0 < q < 1.
coherent
quantized
states
[z >q are
radiation
(2)
the mathematically
al_ >=
atin >= _-n + 1]In + 1 > where [zlq = [z] = [z] = sinh(sz/2)/sinh(s/2)
(1)
good
field because
1 >
a resolution
[a, a] = 0.
0
(3)
of =.
for studying
exists
bosonic
a[O >=
"q-deformation"
candidates
(i) there
trivial
the
More
classical
of unity
simply limit
of
[2]
x = f I_ >< _1du(_) (ii) they indeed tation relation
are "minimum
uncertainty
UQj, with
Utlz> = 0 but
states"
2AQAP
Uh,>¢10 > = _,
(4)
for they do minimize
the fundamental
commu-
-[ < [Q, P] > I > 0 [Q,P] > I -
l<
and
(iii) the n th order
(5)
correlation
function
factorizes,
i.e.
Tr(pE-(z)E+(Y)) But, uncertainty along. In the
simultaneously, properties Iz >_ basis,
there
are intriguing
of the q-analogue from
= £-(z)g+(y),...
a]z >=
differences
quantized
z[z > it follows vo
in the
field.
of the
"q-exponential
lz >q basis
Some of these
that
for < zIz >=
for other
coherence
will be discussed
and
as we go
1
Zn
I= >_-- N(_).=o_--_._ I'_>, in terms
(6)
N(=) = _,(1_1_)-_/_
(7)
1]...
[0]! = 1
(8)
For z > 0, it's positive,
but for x < 0 it
function" Z n
eq(Z) = n--O _ _.t'
[n]!-=
_< _l 1 f0 ¢i : __o [-_],.
=
e_(-_-)
d:
(14)
1'_ ><
'_l,
_=lz[
2
(15)
n-----O oo
= _
In>< nl = x
(16)
_,_0
Several coherent actually
remarks states
are appropriate:(i)
are not
overcomplete,
orthogonal
states since
with
< a]B
]z[ 2 > _ do not contribute,(ii) >=
N(a)N(3)%(ct*_)
zlf
0 ,(iii)
the
]z >q
{z >qare
since
{_>_=f l_>< zla> d_,(z), (iv) with f(z)=<
#
arbitrary
>,
the at,a act < zlatl:
>=
z'/(z),
< zla># o, and < zlel:
>=
(17) N(z)d-_-gN(z)-'
f(z),
(v) any zero of eq(-_) = 0 can be the upper endpoint of integration provided something restricts %(z) beyond -¢i. If not, on the rhs of (12) there is also r,_ = -[n]!E_=o __kl,.(q'/2zk)'_-_%(--zk) where
zk = qk/21¢_l • This restriction I_k >_=
with
k = 0, 1,...;
occurs
Mk _ (q'/'zk) .i=0
Mk = eq(qa/2{i;kl2)-'/2;
if there
are q-discrete
j 13 + k >,
with
a discrete
123
auxiliary
akl£'k >q=
measure
states
(q'/4zk)l£'k
(l_kl_ = zk) (18)
>q
d/_k -- 2,--;_-_%(-I_k[
2) dO
3
The
q-Analogue
Quantized
Uncertainty
Radiation
Field
and
Its
Relations
In analyzing the fieldin the Iz >q classical limit,we suppress the k mode and _ polarization indicesfor the generic electricand magnetic fields, etc. There are diagonal representationsof operators,e.g. the single-mode density operator = f d/t(z)CN(Z, z')lz><
where f d/_(z)¢_v(z,z')= i as Tr(p) = I;so < (af)'a° >=
Similarly, < _-(_t).
>=
z[
(19)
Tr[pCat)'a°]= f d/_(z)(z')'z'¢N(z,z*).
fd#(z)z'(z')'¢N(z,z *) forCN(z,z*) _=<
zl_lz >,:d_(z)¢_(z,z*) = I,
and so
CN(z,z') = f d_(u)¢_(u,u')lv(y)'N(z)'_,(uz')_,(zy')
(2o)
Note that due to the use of q-lntegratlonto obtain (16), a new " q-quantizatlon" in the z complex plane has occurred, e.g.CN contributesto (19) only when
Izl' --qc,.+x)/,_,, _ - 0,I,2, .... Consequently,
for the genetic
electric
_. = i( _12_o
and magnetic
(21)
fields
V )a/'[ae'(-k" W-_,)
_ ate-'(_._--,)]
(22)
with z = tzlexp(iO),
< zl_lz
>=
-2(h,w/2eoV)
1/2
Izl sin(-_. _-_t
which indeed "looks " like a classical field but the possible squared assumes a geometric series of discrete values. With the usual uncertainties _
definitions and a/_
I[
P = -i(hw/2)'/'(a are of 0(1)
< zl[Q,P]lz where
the important
function
-
amplitudes
at),
+ a) are q-quantized;
(23) the modulus
C}= (a/2w)a/2(a + at), the fractional
for [z] ---, oo and
> = < zl[a, at]lz >=
it_,X(z) > it_
(24)
( q = exp 8 ) oo
A(z) - N(z)' _
lzt3"c°sh(8(2n + 1)/4)
,=o
[n]! cosh(s/4)
goes as (q-_/_- 1)lzt' + 1 as Izl-* o_. However,AQAP values,
=
1/2l < [Q,P] >If or lz >,
(25) expectation
per (5).
For the genetic
electric
field, in the
In >q basis
(a_)_.> : (_/2_0v)([. 124
+ 1]+ [.])
(26)
Instead,
in the[z
>q basis
(A_) G = (_/2_0V) < zl[a,at]lz >= (_/2_0V) _(z) and
so the
fractional
uncertainty
A(z) = g(z)2eq(lzl2/q
1/2) -]z12(1 (-(i/h)[Q,
which
fundamentally
( quadratic
relates
in P,Q
in amp
the
basic
is a curious
2 - ((2/hw)H
commutation
of Oil ) . operator
sinhCn/4))
relation
and
the
Note
identity
that
from
(25)
for q _ 1
2= 1
(28)
single-mode
hamiltonian
2 ,
) + aa _) = (1/2)(/5a
+ w2_)2).
(29)
small, that ,_(z) __ _1 + ((2/_/hw) 2 - 4E/hw) tanh2(s/4) where E = Ea-hw/2 > or < zl[N]lz > , so ,_ depends on the deviation from the vacuum energy.
4
q-Generalizations
Since
z's magnitude
operators.
There
P] cosh(s/4))
g = (1/2)£w(ata We get for (l-q) for n =< zlNIz
_7 ( or /3 ) is also
- ql/2).
(27)
Recall
may
of the be q-quantized
z = Iz[ exp(iS)
Phase
as in basic
and that
Operators analysis,
mathematically
we next
a hermitian
consider phase
possible
phase
operator
conjugate
[3] is defined
by [4]
to N, to [N] = ata, or to H does not exist [3]. An e'_p(i¢)q
generalization
of the phase
operator
_f ---_p(-i¢)([N + 1])'"
___ ([_ + 1])'2_p(i¢) and there
are hermitian
_-_(¢)
of Susskind-Glogower
(30)
operators
= (1/2)[_'p(i¢)
+ _-p(-i¢)]
s"/n(¢)
-- (1/2i)[g-_p(i¢)
-
e'_p(-i¢)].
(31)
These generalizations give many q-independent operator commutation relations , see [4]. So, from [N,c_(¢)] = -is'_n(¢),... the usual number-phase uncertainty relations follow for arbitrary q:
AN a_(¢) In the
In >q basis,
aN A;_n(¢) > (1/2)1 < _(¢)
> (1/2)1< ;_(¢) > 1
these
definitions
(30-31)
correspond
> I
(32)
to
oo
(33) n=O
which
is manifestly
q-independent
in In
>q,
non-unitary,
and a q-analogue
of the SG operator.
2For H, the energy is not additive for two widely separated systems, violating the usual cluster decomposition "axiom" in quantum field theory. But, for q-quanta this is not unreasonable since the fractional uncertainty in the energy based on H is also O(1) in the ]z > basis and the quanta by (1) are compelled to be always interacting,i.e. by exclusion-principle-like q-forces! An alternative hamiltonian is HN = _(N + 1/2) where N is the number operator and it has the usual free-quanta additivity, etc.. 125
Analogously,
a q-generalization
ing a complete,
orthonormal
0,_ = Oo + 2mTr/(s and
of the Pegg and Barnett
basis of (s+ 1) phase
+ 1) , with m = 0, 1,...,
states
s, . These
operator
[0,, >q=
[5] is obtained
[4] by introduc-
(s+ 1) -1/2 _,=0
are eigenstates
exp (inO,O[n
of the respectively
>q,
hermitian
unitary eq
-
_
em [O_ ><
8m[
(34)
lrrt=O
exp(i$)q which is manifestly and Ellinas' polar
--IO>q basis
]z >q coherent
do in the PB-case
states
do not
[7] both
give
minimize
and
the
minimize
N, ff'_(¢), Dirac's
s'_n(¢)
uncertainty
commutation
relation,
for Iz[ large
[Y,_,] =i Also c-_(¢)q
and s'_n(¢),
show
some
"correspondence
< _l_-'/_(¢)lz> sin(e) < zl_(C)Iz > cos(e)' and
(36) principle"
type
behavior:
< ziG(C)' + s-Tn(C)'Iz >= 1 -
(i/2)eq(lzl')
(37)
-I
proportionalityfor < zlc-_(¢)' - s-Fn(¢)'lz>. This
is based
for discussions; U.S. Dept.
on work
with S.-H.
the Argonne,
of Energy
Cornell,
Contract
No.
J.
A22,
Chiu,
m.
Fields,
and Fermilab
and R.
theory
DE-FGO$-86ER_O_91
W.
Gray.
We thank
groups for intellectual
C. K. Zachos
stimulation;
and
for support.
References 1. A. Macfarlane, C.-P.
Sun
Lett.
B254,
and
Phys.
H.-C.
Fu, J.
4581(1989);
Phys.
A22,
L. Biedenharn, L983(1989);
J. Phys.
M. Chaichian
A
22,
and
L873(1989);
P. Kulish,
Phys.
72(1990).
2. R. W. Gray R. B. Zhang
and C. A. Nelson, and M. D. Gould,
J. Phys. J. Phys.
A23, A24,
L945(1990); 1379(1991);
A. J. Bracken, B. Jurco, Lett.
D. S. McAnally, Math. Phys. 21,
51(1991). 3. L. Susskind
and J. Glogower,
4. S.-H.
R. W. Gray,
Chiu,
C. A. Nelson,
7. M. Fields
1, 49,(1964).
C. A. Nelson,
Phys.
Lett.
W. H. Louisell, A164,
Phys.
237(1992);
Lett.
S.-H.
Chiu,
7, 60(1963). M. Fields,
unpublished.
5. D. T. Pegg and S. M. Barnett, 6. M. Chaichian
Physics
and
D. Ellinas,
and C. A. Nelson,
Europhys. J. Phys. SUNY
Lett.
A23,
BING
6, 483(1988)
L291(1990). 7/27/92.
126
; J. Mod.
Opt.
36, 7(1989).
III.
QUANTUM
OPTICS
127
N93-27325 DISTRIBUTION
OF
PHOTONS
POLYMODE
IN
"SQUEEZED"
LIGHT
V. I. Man'ko Lebedev Leninsky
Physics
pr.53,
Institute,
Moscow
117924,
Russia
Abstract The distribution functions multimode cases are obtained
of photons in squeezed and correlated based on the method of integrals
light for one-mode and of motion. Correlation
coefficient and squeezing parameter are calculated. The possibility to generate squeezed light using nonstationary Casimir effect is discussed. Quantum parametric Josephson junction is proposed
1 The
as quantum
generator
of electrical
integrals
of the
vibrations.
Introduction aim
of this
work
the distribution cases.
The
is to discuss
function
of photons
distribution
by Schleich The
vacuum
and
photon
function
Wheeler
[1], by
distribution
Dodonov,
Klimov
parameter,
but
Agarwal
function
and
Man'ko
also on the
in squeezed of photons
and
and
distribution
parameter
uncertainty
correlated
Adam
for squeezed
correlation
and
in squeezed and
[6]. This
motion
light
relations
and
for one-mode
and
light
for one-mode
fields
[2], and
by Chaturvedi
and
correlated
light
function
depends
connected
with
[4] and not
SchrSdinger
multimode
was
discussed
Srinivasan
[5] was only
to obtain
[3].
discussed
by
on the
squeezing
uncertainty
relation
[7] as well, h 5qSp >_ 2 lx/-f-ZT-r_' where
the
parameter
r is the
correlation
V :
The
states
with
consider
the
For such
states
nonzero
problem instead
((_q_p)-I
parameter
how
to find
of the
coefficient
{
r we call the
states
SchrSdinger
(qP
of the
"_2 _(_}
the
(1) position
(q)(P)}
states.
minimize
inequality
the
we have
momentum
"
correlated
which
and
(2)
In the
SchrSdinger
section
below
uncertainty
we'll
relation.
the equality
h
5q@ These how
states these
describe
states
squeezed
are naturally
and created
- r2
correlated
light.
for quantum
(3)
We
will demonstrate
parametric
oscillator.
129 PRECEDING
PAGE
BLANK
NOT
F!LI_ED
The
in the
next
section
case
of the
photon
distribution Schleich
function [8].
squeezed derive
Multidimensional
light this
in terms
expression
parametric
squeezing by relativistic
tivistic
models
and
Kim
The
by the
have and
on the
The
of the
polynomials
bases
of the
was considered expression
of several
result
in quantum
wave
equations
studied
obtained
[14].
whose
properties
optics
by Caves,
for the variables
Zhu,
distribution may
in [17], [9] and
As shown
particles Lorentz
is identical
squeezed
related with
[11], by Markov
in [14], the
mathematics of such
is closely
for elementary
by Yukawa
be
Milburn
and
of photons
in
reformulated.
We
[10] for a nonstationary
mass
oscillator
boost
applied
of the
oscillators
These
and
studied
[13],
oscillator
in quantum
been
derela-
Man'ko
to relativistic
squeezing have
models
spectrum.
[12], by Ginzburg
to that
relativistic
to the
optics.
by Kim
and
[15].
To obtain the photon distribution oscillator. We shall discuss first the
2
light
oscillator.
been
Noz
squeezing
statistical
Wigner
generalization
phenomenon
scribed
squeezed
of Hermite
multidimensional
The
gives
for (fie two-mode
One-mode Hamiltonian
function one-mode
we will consider the nonstationary case ill Sec. 2.
Light for one-mode
light
is given
by the
/?/: This mode mechanical
formula 1 + 7).
hw(ht6
of the electromagnetic field in a resonator oscillator with the Hamiltonian
ft In this
case
the
multidimensional
annihilation
and
creation
+
2m
may
be
described
by
the
model
of the
(5)
2
operators
with
6 = _
(4)
+
boson
commutation
relations
,
(6)
70 '
(7)
where I 2
1
t= (_Z)- , p0= (t,_)_, connect
both
Heisenberg-Weyl satisfying
the
Hamiltonians algebra.
and
forms,
In coordinate
together
with
representation
a is any
the
identity
the
complete
operator, set
the
of coherent
basis states
of the ] a}
equation fi]c_>=a
where
(s)
complex
number,
is given
,
by the
(9)
la),
fornmla
[ q_
(q I a) = _r-¢l-½ exp [- 212 130
I_ I_ 2
_q +
l
o_ 2
(10)
The
dispersions
are given
For the
of the
by the
positions
5q and
the momentum
relation
coherent
@ do not depend
on the
parameter
a, and
1
states
the
product
of these
5q-
v'_-'
(11)
@=
_-2.
(12)
dispersions
minimizes
the
Heisenberg
inequality
h @@ The
time
evolution
parameter have
of tile
c_ in the
formula
coherent (10)
state
= -_.
] o, t} may
by the term
correlation
state.
coefficient
It is also equal
of the
position
and
to zero for stationary
be obtained
aexp(-icot)
and
state
has
the
following
momentum
is equal
Fock
state
photon
distribution
function
in the
function
H_(o_)
replacement
of the
wave
function.
to zero the
for arbitrary eigenvalue
equation (15)
representation q2 212
p
for the coherent
We
coherent
n = 0, 1,2,...
q
of the
(14)
I n, t) satisfying
coordinate
i 1 . _ = rr-_l-_2-7(n[)-_H,,(7)ex
{q In,t) The
wave
the phase
-icot exp (---_).
ata I _,t) = _ I ,_,t), This
by simple
(-icot)}
(q [ c_,t) = (q ] ctexp The
(13)
state
icot(n
1 )] + 7_
(16)
l
Ict, t} is determined
by the
overlap
integral
12-- Wn(a)
I(n, t l_,t) and
coincides
with
the
Poisson
distribution
(17)
function
_,_,_,(_) _ I _,,!I_ exp(-I _ I_). The
mode
has
the
following
time-dependent
integral /i(t)
We now will change /3. The
discuss the
how the
photon
Hamiltonian
influence
distribution
of the
mechanical
of the function
of the
motion
= exp (ia)t)a.
(19)
dependence and
the
parametric
of the dispersions
oscillator
system
has
the
linear
integral
of the motion
oscillator of the
depends
i,_ m_'(t)O _ &t) = >,--7+ 2 This
(18)
frequency conjugate
on time
and
fl(t) variables has
the
on time 0 and form
(20)
[16]
l_ ] 131
(9.1)
Here
co = fl(0),
and
the
complex
function
e(t) satisfies + fl2(t)e
The
integrals
of motion
fi,(t)
and
At(t)
satisfy
Wronskian
for the
equation
(22)
of classical
oscillator
motion (22)
the boson
is given
equation
= 0.
[/](t),At(t)] if the
the
commutation
relation
= 1,
(23)
by the equality
e_* - d_ = 2ico. The
initial
condition
for the
function
e(t) may
be taken
_(0) = 1, If the
frequency
of the
the integral of the and the relation
oscillator
motion
is constant,
(19).
The
following
wave
function
the function
normalized
in the
The
state
I a, t) which
is the
and
] 0, t) satisfying
the
the formula SchrSdinger
of the
following
wave
function
in the
( ieq2_ \2cod_ j.
integral
/
Here
a is an arbitrary
complex
number
of the
(27)
motion/l(t)
given
by formula
mode
function
I n, t} satisfy
(29) the
satisfies
the
eigenvalue
1
1_+
SchrSdinger
2
solutions
to this
equation
have
z_
2
_ ].
(29)
+ fl"a). The
equation.
Fock
(30) states
of the
parametric
equation
A,(t)A(t)ln,t)=nln,t>, The
(28)
and
\ wave
(21)
representation
_--21
(fl, t [a,t)=exp(
The
equation
representation
coordinate
(q I a, t> = (q l O, t}exp
gives
(26)
A(t) Is, t> = _ la, t> has the
(21)
I 0, t ) = 0
coordinate
eigenstate
(25)
e = exp(icot)
state
' ' t) = 7r-_(/e)-_exp
(ql0,
as follows
_(0) = ico.
A(t) has the
(24)
the following
n = 0, 1,2,... form
1
in the
*
H,_(
(q In, t) = -_nv.(_) n/2 132
coordinate
q
)
/1_1"
(31) representation
(32)
Since
the
state
[ a, t) is the
generating
state
[a,t)
the
transition
probability
from
= exp(
the
initial
o
H_m=
Here
the
transition
probability
for the
W_
-]
a ]2 / _
2
state
•
is the
Fock
states
[ m, t)
a m Ira,
t)
_
m=0
(33/
'
I n) may be calculated
2
P
o
probability
,
(34)
m>n.
to be in the
ground
state
W3o = 2(1_Is+ _-21_Is + 2)-'/_. For n > m the
formula
(34)
must
be changed m
The numbers these numbers is equal
to
0 T/-t.
W,:
(n-m_
=
(W_ )
(36)
n and m in the formulae (34) and (35) are either is even and another number is odd the transition
both even or both odd. If one of probability between such states
to zero
The formulae index.
0
P._
W_+'=W:;+I=O,
netic
(35)
field
(34)
and
(36) describe
the photon
either
moving
in a resonator
Thus,
k,p
we conclude
that
with
distribution walls
the squeezing
are connected the form
with
the
photon
function
or with
parameters
Sq =
S_ =
= 0, 1,2,...
media of the
5q =l_
(2) _mw
distribution
1/25p
function
(37) for the one-mode
with
electromag-
time-dependent
parametric
refraction
oscillator
1,
(38)
=1 _1
by the
(39) ratio
(35)
which
may
be rewritten
w° = 2(S_q + s_+ 2)-1/2. In the equal
case
of vacuum
light
Sp = Sq =
the
(40)
vacuum-vacuum
transition
probability
W 2 is
to unity.
Another be described
photon by the
distribution model
function
of the
forced
[-I(t)This
1 and
in
oscillator
has
the
integral
of the
corresponds mechanical
to the oscillator
_m + -_rnw q -f(t)dl. motion
excitation with
of light
state
which
may
Hamiltonian
(41)
[16]
A(t) = exp (iwt)a + 5(t), 133
(42)
5(t)If the ,gp are
initial
state
of a forced
time-independent.
by a Poisson has the form
They
distribution.
il riot f(r)exp(i_r)dr. v'_h
oscillator
is the
are equal
to unity.
Thus,
if the
initial
_ The
physical
is just
the
force.
The
with
meaning mean
of the
photon
photon
n photons
parameter
number
distribution
by the
state
the
is the
squeezing
parameters
distribution
vacuum
function
state
the
Sq and
is described
Poisson
distribution
_ 12) .
excitation
I,l_TM in the
the
photon
] _ 12 (43) which
after
state,
The
= m!exp(l
function
is described
coherent
(43)
(44)
determines of the
case
the
vacuum
when
the
integral state
of the
by the
initial
state
motion
external
was the
(42) linear
state
function
%m = ,,!t _ I_¢_-_>[/m-_(i 6 i_)]:. m! exp([ Here
the
Now external
We have
function
L_!
consider force
taken
is the
Laguerre
a general
is present.
(45)
¢S12) t _
polynomial.
situation The
I n}
when
the
frequency
ttamiltonian
of the
[I(t)
+
= _p
m = w = h = 1. The
linear
of an oscillator
mechanical
w_(t)gt 2-
integral
oscillator
depends model
on time
looks
and
like (46)
f(t)gt.
of motion
an
,_i(t) is equal
in this
case
to
A(t) : _(t)_ + _,(t)at + _(t),
(47)
1 _t(t) = ](_(t)
(48)
-ii(t)),
1 v(t)
= _(e(t)
(49)
+ ii(t)),
i 5(t) The
normalized
eigenstate
¢_(q,
(,_(q,t)
¢o( q, t) exp
=
t) of the
_
f
integral
(50)
f(r)_(r)dr. of motion
---?--12 +--
(47)
has
the
form
(51)
+
where
Co(q,t) _ _ The
squeezing
unforced
1
parameters
parametric
oscillator
[ iq2i
exp [ 2_ Sq and (38),
v/2q5
_
S v for the (39). r :l
The
b2C
1512
2_
states
(51)
correlation
d [-1 [(ci) 2-
134
lfot(55*-55*)dr
2 + _
are
described
coefficient 1] 1#
by
the
r is given
]
(52)
formulae by the
for
the
expression
(53)
The
Fock
states
which
are the
photon
Casimir
distribution
effect
of the
integral
¢o(q,t) _--_
g,,(q,t)
The
eigenstates
function
is expressed
for the
in terms
of motion
H,_ q +
electromagnetic
of Hernfite
ft*(t)fi(t)
are
of the
form
+
field
polynomials
(54)
created
due
to the
nonstationary
of two variables
%m _/Vo0
__
H{Rm}(Xl,X2)12,
(n!lT/[)--l[
where
6" x: = 6
X 1 --
and
(55)
the
matrix
R has
the
7/6*
(56)
elements
R= I( _1) -1
The
parameters
_ and
71 are
given
by' the
-7/*
"
relation
(57)
e(t) = @it __ r/e-it. The
photon
distribution
function
the
Hermite
polynomial
of two variables.
The effect
last
photon
on the
initially
(55)
distribution thermal
has
oscillatory
function
behavior
describes
equilibrium
the
due
to the
influence
of the
oscillatory nonstationary
behavior
of
Casimir
state
(5S) z-l= The
distribution
of photons
in the light
mode Pnn
poo
2 sinh(/3/2). is expressed
(59) by the density
matrix
diagonal
elements
1
-
= nl/2ln
-
I>,
(I.
141
I)
^+in>
(n + I)I/21n
a
+ 1>.
(1.2)
^+
One
may
introduce
the
generalized
inverses
[8] of
a and
a
:
^--I
a
in>
=
(n
^+-I a In>
The
=
+ I)
(1
^-I a
operator
-
1/21n'
an, °
+
i>,
(1.3)
) (n)-I/2
behaves
as
In
-
1>,
a creation
(1.4)
operator
^+-I a
whereas
behaves
^--1
as
an
and
On
annihilation a^+-I
is
^ ^-I aa
^+-I ^+ : a a :
the
other
the
operator. left
]0>.
Using
can
solve
^ a^+-I a,
^^+-I aa
operators be
the
obtained
eigenvalue
and
(TAO). by
Eqs.(1.1)-(1.4)
_2. The
noting we
These matrix their obtain,
n m 2
^+_i
^
a
a[n>
aa+-l[n>
^2 a in> =
=
[n/(n-I)]I/2[n-2>,
(1.7)
=
[(n
(1.8)
-
a
i.e.,
eigenvalue.
photon these
of
^+2 exhibiting two photon processes, viz., a , ^-I^+ a a do not have any normal izable right
remaining two
the
IO>,
[n(n-l)Tl/2[n-2>,
(1.9)
142
whereas
their
therefore,
action
find
the
on
In>
following
wit'h matrix
1/2
=
n
[n/(n
-
[(n-
l)/n]
=
0
elements
or
1
gives
for
these
zero.
operators:
a
i)]
(i.
In> =
(i. il) In,n-2
'
I/2
We
now
2
Elgenstates
We
write
=
[n(n
consider
the
(1.12)
13]
eigenvalue
problem
for
these
TAOs
in detail.
-+_i A a a
of
an
-
lo)
Ill, n-2' i/2
is
a
elgenvalue
_
and
Expressing
IA, I>
right obtain
in the
eigenstate
of
solution
for
a
the
first
suitable
TAO complex
a+-l_
with
number
I.
form
cO
(2.2)
IA., I> = E Cnln> n=o
we
obtain
the
following
recurrence
relation
for
C n
I/2 Cn
From II, l> states
= A
this
[(n
l)/n]
recurrence
separate or
-
odd
into number
Cn_ 2
relation two
sets
states
[(2n)!] Ik,+l>
= N+
_ n=O
(2.3)
as
it of
is
states
observed involving
that
the
either
eigenstates even
number
follows:
1/2 knI2n>
2nn!
143
(2.4)
and
® Ik,-l>
Here
N+
2n
= N_ _ n=O
and
N_
are
n! i/2Anl2n
[(2n
the
÷
+ I>.
(2.5)
I)!]
normalization
constants
given
by
N+ = (I - JX]2) I/4 ,
iX]< I,
(2.S)
IXI(I-IXI2) I/2 I/2 N_= [ -i ],
Ixl
hence
elgenstate
any
of
Eigenstates
We
write
of
the
Ik,-1>
correspond
llneaur comblnation ^÷--I
^
a
a.
equation
for
the
AA÷--
3
and
TA0
of
to
these
the
states
is
an
IX,2>
I
eigenvalue
is the
eigenvalue
followed two
sets,
odd
number
A.
right
this
operator
as
we
in
find
involving
a
that
even
of
the
manner
these
number
second strictly
eigenstates states
and
operator analogous also the
^^+-1 as to
separate
other
with that into
involving
states
® I_,+2>
(3.1)
eigenstate
Proceeding
in Sec.2, one
the
aa
_i÷-1(x,2> - klx,2>, where
same
= M+
_ n=o
2nn! I/2
(3.2)
knl2n>
[(2n)!]
and
m [(2n
+
1)!]
Ix,-2>= x Z -n=O
I12 _nIRn
2nn}
144
+
I>,
(3.3)
where
M+
and
M_
are
the
normalizatlon
constants
(i-I_,12_ 3/2 +
IXI2] 3/4
by
i/2
(I-Ix12) I/2÷Ixlsin-1
M_ = [1 -
given
Ixl < I,
(3.4)
IX] < 1.
(3.5)
Ixl
, ^^
A
general
the
4
eigenstate
states
{A,+2>
Eigenstates
and
the
TAO
(aa +-1)
is
a
linear
combination
of
{A,-2>.
_2 a
of
Coherent
of
states
are
the
right
elgenstates
of
the
annihilation
^
operator
a,
into
even
the
the case
with
and
of
so
that
and the
odd
of parts
other
elgenvalue
_2
both
TAOs.
X can
be
also.
These
separate neatly ^2 the eigenstates of a , as in ^2 normalized eigenstates of a
being
Hence
expressed
the
states
in the
co
[9]
form
An
l_,÷3> = (cosh Ixl) -w2 _
(4. i)
1/2{2n>
n:o
[(2n){]
and
Ixl
Any
llnear
eigenstate to the
be
the
value
be
less
5
Squeezed
It
of
is
Ixl
_2
of
{l{,
of
course,
a
{A,+3>
and
particular
state
{(_)I/2>.
whereas
in the
(4.2)
1/2 12n+1>.
{k,-3>
linear
Further earlier
there cases
states
combination is no IX{ was
is
an
happens
restriction
on
restricted
to
I.
Vacuum
interestln
essentially
Of
An
n=O [(2n+1){]
superposition
coherent
than
=
l]
Ix.-3> = [
1-1/2
_
slnh
the
as
E
an
to
squeezed
Eigenstate
note
of
that
vacuum
the
_+-t_
state
discussed
145
in
{X,+l> literature
[Eq.(2.4)] [3,
is
I0-12].
The
squeezed
vacuum
is
generated
by
the
action
the
of
squeeze
^
operator
S(_)
on
Io>=
vacuum
[1/2
exp
Using
the
normal
state
representation
(o_ +2
ordered of
form the
of
the
squeezed
x
(coshr)
_ n=O
(5.1)
operator vacuum
S(c)
we
find
the
number
as
I/2
[(2n)!]
-1/2 s(c) I0> =
"^2 - o" a )] I0>.
(e i8
tanhr)nI2n>,
(S.2)
2nn! ie
where we
the
find
squeeze
parameter
Comparing
Eqs.(2.4)
and
(5.2)
that
Ix,+i> =
where
v = r e
the
eigenvalue
_, = e
ie
(5.3)
Io>,
A
is related
to
the
squeeze
parameter
E by
tanhr.
(8.4)
Hence we conclude that the squeezed vacuum is an elgenstate ^+_i ^ TAO a a. In a similar manner we can show that the squeezed ^ number state S(E)In=l> is an eigenstate I_,-2> of the operator
of
our
first ^^+-i aa
References
I. R.J.
Glauber,
2.
D.F.
Wails,
3.
R. Loudon
4.
C.M.
Caves,
8.
M.S.
Kim,
Phys. Nature,
and
P.L. Phys.
F.A.M.
Rev., 306,
131, 141
Knight,
J.
2766
(1983). Mod.
D, 23,
Rev.
(1963).
Opt.,
1693
34,709
(1987).
(1981).
De Oliveira
and
P.L.
Knight,
Rev.
S6,
2176
(1986).
Opt. Commun.,
72,
99
(1992,
in
(1989). 6.
H.P.
Yuen,
7.
G.S.
Agarwal
8.
C.L.
Mehta,
Phys. and Anll
Lett.,
K. Tara, K.
Roy
Phys.
and
G.M.
press).
146
Rev.
A,
Saxena,
43,
492
Phys.
(1991). Rev.
A
9.
Many or
of
may
Quantum (Reidel,
these be
considerations
derived
using
Statistics
of
Dordrecht,
3796
(1987)].
I0.
G.J.
Milburn,
II.
Anil
K.
12.
J.N.
Hollenhorst,
Roy
and
well
C.L.
Rev.
p.
Phys.
78,
Rev.
J.
Mod.
D,
147
19,
states
Nonlinear M.
4882
^2 a are
operator
coherent
and
A, 34,
Mehta,
the
known
Linear
1984)
Phys.
for
[cf.
Optical
Hillery,
either
Phys.
J. Perina, Phenomena,
Rev.
(1987). Opt., 1669
(1992, (1979).
known
in press).
A,
36,
N93-27327 BOUNDARY CONDITIONS IN TUNNELING VIA QUANTUM HYDRODYNAMICS Ant6nio Physics University
B. Nassar Department
of California,
Los Angeles
Los Angeles, CA 900_4 and Physics Department Harvard- Westlake School 3700
Coldwater
Canyon,
North
Hollywood,
CA 91604
Abstract Via the hydrodynamical formulation of quantum mechanics, a novel approach to the problem of tunneling through sharp-edged potential barriers is developed. Above all, it is shown how more general boundary conditions follow from the continuity of mass, momentum, and energy.
1
Introduction
A commonly used assumption in quantum on a surface _ where the potential undergoes the the
wave function hydrodynamical
2
conditions that both
(¢) and its derivative (O¢/Ox) be continuous on _r. We show below through formulation of quantum mechanics how more general boundary conditions
follow from the continuity conditions,
mechanics [1,2,3,4] is that the boundary a finite jump reduce to the requirement
a novel
of mass,
approach
momentum,
to tunneling
and energy
through
densities.
sharp-edged
With
potential
these
new boundary
barriers
is presented.
Formulation
Let us consider equations
the dynamics
of a quantum
Op -_+
particle
described
by the coupled
hydrodynamical
O(pv) _ O, Ox
(1)
o_ oo 1 o(v + y_ ) -_ + v_ + m ox _ o, where tion
Equation (2) describes
external
potential
(1) represents trajectories V and
the
the mass
conservation
of a particle quantum
with
potential
law with
(2) mass
v =
149 FRE_EOING
PAGE
BLAN_
density
(a/m)(OS/Oz), Vq_ = -(a2/2rn¢)(02¢/0=2), velocity
i_O3- F.li._tED
p = ¢2 and subject which
Equato an
accounts
J
for quantum-wave features, been expressed in the polar
such as interference form ¢ = eexp(iS).
and diffraction [5,6]. The wave Equations (1) and (2) yield
OS rnv 2 h-_- + (--_- + _ and
the corresponding
Schr_Sdinger
+ V)=
function
0,
has
(3)
equation
ir_°¢ = - r_ 02¢ + v¢. oat From densities
Equations as follows:
(1) and (2), we obtain
OJ
(4)
2m Ox 2 the conservation
OP
laws for the momentum
and energy
p OV
(5)
_- + _; +--0,m0_ = OU
OQ
-_-+-g;=0,
(6)
where
j = _,
(7)
P = Pv2 4m: [_z_ u=p(-_-+
p _
momentum,
mentum
density
of a more density
general
momentum
pv appearing quantum
flux,
energy, local
(9)
aea¢) Ot
energy
flux densities,
equations
momentum
(10) respectively.
can be shown
field 7_ defined
from
The
mo-
to be the real part the
momentum-
operator
P=_where v = (li/m)(OS/Ox) It follows now that energy
and
in the hydrodynamical mechanical
(s)
v_,+v),
q = vU + 2rn--'t_'_ (,02¢ ¢P_x_ are the
,
the
li g,. O¢
-g-_x=
mp(v
+ iu),
and u = -(h/2mp)(Op/Ox). boundary conditions for the
continuity
(11)
of mass,
momentum,
and
axe:
p, pv, pu, In terms
of the wave function
¢'¢, ¢.(0¢/a_),
andp(-_-+Vq_,+V).
and from Equation
and (as/&).
150
(3) the above
(12) conditions
are equivalent
to:
3
Tunneling Next
consider
the
stationary
flow of particles
with
incident
energy
E striking
a potential
barrier of height V and width L: V(x) = V for 0 < z < L and zero elsewhere. The wave functions for z < 0 (incidence region 1), 0 < z < L (tunneling region 2), and z > L (transmission region
3) axe given
respectively
¢i(z,t)
by
= v_ exp(iS,) =
_/1 + _ + 2a cos(2k_ - _)
( ¢2(z,t)
The
k 2 = 2mE/h boundary
,_o
o]),
(13)
= v_ exp(is_) =
¢[c2e 2_" + d2e -2_* + 2dccos(7
×
exp
i (-cot
+ "7+ '5
2 and -_2 = 2m(V conditions
from
-6)]/q [ ce_
---5- + tan-' L_
\
¢3(x, t) = v_ where
o
exp(iS3)
= b exp i(-wt
- de-q*
(14)
+ de-_"
(15)
+ kx + 13),
- E)/li 2. (12) where
the potential
undergoes
a finite
jump
read:
p_(o) =m(o),
(16)
p:(L)
(17)
= p3(L), t
p'_(o)= p_(0),
(18)
I
p;(L)
= pa(L),
p,(O).,(o)=p=(o)._(o), p2(L)v:(L)
= p3(L)va(L),
& )o
\ & 1o'
---_] L = \---_'] L•
151
(19)
(20)
(21)
(22)
(23)
By applying
the above
boundary
conditions
1 + a 2 + 2acosa
on Equations
c s + d 2 + 2cdcos(7
=
Jr"2cdcos(7
cSe 2_L -4- d2e -_L
(13), (14), and
(15), we obtain:
- 15)
(24)
- 6) = bS'
(25)
-q 2ak sin a = (e s - d2),
C =
(27)
de -:_L,
2d2e -_r_ sin(7 k
1 - a2 =
(26)
2d2e -2_L sin(7 k
- 6)
(28)
- 6) = b2"
(29)
From Equations (25) and (27),we have bS = 2d2e-_L[1 + cos(_ --6)],
(30)
which combined with Equation (29) gives
tan
=
sin(7
(31)
q,
2k_ = _ + kS,
-/_)
(32)
_2 _ k _ cos(7- 6) - _2-7k2" Equations
(29) and
(33)
allow us to write
4_ b2 = (_s_ks) which,
in turn,
combined
l+a Equations
with
Equations
2+2acosa=b
(28) and
(29) imply
Equation
-
2_ ff
(30) as (34)
d2e-s_L,
(27) and
2k
(33)
(
,] .1+
(33), reduces
Equation
_s + k 2 _s _ k'-""_ cosh 2"_L) .
(24)
(35)
that as = 1 - b2,
which insertedinto Equation (35) gives
152
(36)
By the same
Combination
acosa=b
2
above,
Equation
procedure
of Equations
1+
["_-K-q2 jsinh_qL (26) can be rewritten
+
asina
= - \
(36),
(37), and
(38) leads
-
Using the identity sinh 2 2"_L = 4(sinh:_L the denominator in Equation (39), we arrive
as
g sinh L.
_t_q ]
[1 +_ {_ 2_" / sinh2_L]_
b-Z=
(37)
-1"
(3s)
to
+ {_.:_32 k 4_ /
sirda2 2_qL .
(39)
k2
+ sinh4_L), and after at the known result
dividing
the
numerator
by
( _2.+ k: '_' sinh 2_L. b-:=l+
4
Boundary Next
we show
assumption systems.
that
Conditions below
that
"¢ and
the
boundary
(O¢/0x)
by Equation
2k'_
for
Dissipative
conditions
for the
v is the friction dissipation.
(12) are not
new
coefficient,
the dynamics
and the term
By expressing
boundary
only more
of a quantum
particle
incorrect
general
but
the
for dissipative
in the tunneling
region
(1) and
conditions
= -vv,
(41)
on the right-hand
the wave function
ti ( O._ + vS ) + (-_The
Systems
at a" is physically
oo Ov 1 o(y + y_ ) _+v_ +m Owhere
(40)
]
axe continuous
To this end, let us consider
described
\
now are given
as before
side of Equation [see Equation
(41) accounts
(3)] we have
+ V_ + V ) = O. by Equations
(42)
(16) through
(21)
plus (43)
Ot ]o =
+ v S:I 0 ,
which cation,
shows
the
discontinuity
we will detail
the
in the phase
application
on the tunneling of a particle transmission coefficient.
through
of the
of the above a single,
153
(44)
=
+ vS2
wave function boundary
sharp-edged
at e_. In an upcoming
conditions rectangular
and
show
barrier
that
publifriction
diminishes
the
5
Acknowledgments
Assistance edged.
by A. Boutefnouchet
in the
preparation
of this manuscript
is gratefully
acknowl-
References [1] A. S. Davydov,
Quantum
Mechanics
(Pergamon
[2] E. Merzbacher,
Quantum
Mechanics
(John
[3] L. I. Schiff,
Quantum
Mechanics
[4] A. P. French and E. F. Taylor, Norton, New York, 1978). [5] D. Bohm, [6] D. Bohm
C. Dewdney,
and
Wiley,
(McGraw-Hill, An Introduction
B. J. Hiley, Nature
and B. J. Hiley, Phys.
Rev.
Left.
154
Press,
2nd edition,
2nd edition,
3rd edition,
New York,
New York,
to Quantum
(London)
Oxford,
Physics
315,294
55, 2511 (1985).
1965). 1970).
1970).
(MIT
(1985).
Series,
W. W.
N93-27328 Using
Harmonic Oscillators of Hermite-Gaussian
to Determine the Spot Laser Beams
Sidney
Size-
L. Steely
Calspan Corporation / AEDC MS 640, Arnold AFB, TN
Division 37389
Abstract This mechanical This
paper illustrates the similarity of harmonic oscillators and the modes
functional
similarity
large-order
mode
provides
Hermite-Gaussian
corresponding two-dimensional size of Hermite-Gaussian oscillator
provide
beam
modes
used
to
the
the
1.
for
beams;
beams.
photon
of photons
detected
for
The
probabilities laser
beams
in agreement
with
large-order
modes
theorem
the
classical
therein.
spot
size
of
limits
of
a
beam
modes
all
a direct
the
is to
within
approach
Correspondence of
laserand
photons
asymptotically the
of the
Mathematica
of detecting
include
provides
densities
large-order
modes, Sturm's
The
probability
densities
similarities.
to investigate
of quantum laser beams.
oscillator provide a definition of the spot The classical limits of the harmonic
for the
of Hermite-Gaussian
limits
laser
laser
fraction
probability
of large-order
classical
Gaussian
the
limits
limit
limits the
functional
classical
in the The
to determine
correlation
harmonic laser beams.
integration
integrate
illustrate
a direct
the functional forms of Hermite-Gaussian
unity
Principle.
nodes
for
Hermite-
proof.
Introduction There
are many
instances
in science where
different physical models
similar or identical functional forms. Scientists often exploit and glean other
disciplines to better understand exhibit similar
new
areas
functional
forms.
physical
models
powerful
tool for explaining and understanding
many
Since exact solutions exist for the classical andquantum tool and normal
simple modes.
system
model
to help model
quantized
The
harmonic
oscillator is a
similar disciplines of physics. harmonic
oscillator is an
oscillator,it is a motion
and
excellent pedagogical
the basic properties of quantum optics, and
ideas from
especially if the
basic principles of vibrational
harmonic
and understand
radiation fields,quantum
the harmonic
mechanics,
other disciplines of physics. Yes--
oscillatorrightfully deserves itsplace "on a pedestal" [1].
In this paper we quantum
to understand
In addition, the
of research,
have
harmonic
will exploit and use the similarity of the functional forms of
oscillators and Hermite-Gaussian
155
laser beams
to investigate the
spot size of laser-beam therein. As
modes
a result of two
[2,3],some
the fractional energy
Correspondence
we
beams,
approach
forms
should
corresponding
does not include most of the energy
the
oscillator and
probability
of
mode
modes.
harmonic
to the
quantum
results from
probability laser beam
Limits
Many
oscillator. Section integrating
motion
can
be described one
harmonic
and
systems
The
5
the
linear
of freedom
can
of
a direct
solution
massless
spring
of force
F=-kx
with
oscillator
quadratic
in the
one be
the
small
for the
oscillator'sclassical spot size for
harmonic
corresponding
potential
can
as the
momentum
p and
the
probability
2
small
systems is
Hamiltonian of mass
mass
is given The
of a kinetic
and
oscillations having
e09 = k/m The
Hamilton's
and
equations canonical
co = 2nv of
is the
motion
equations
angular for
the
[7] 156
more
or decoupled
well
suited
for
formulation m coupled
to a
by Hooke's
law
Hamiltonian a potential
for
a
energy
x .I
22 (1)
_=T+V=_m+_m_x
where
of the
equilibrium.
of coupled
oscillator
V'-kx2/2.
position p
the
on the
sum
and
formulation [7],
force
of stable
Some
by a set
Lagrangian
k. The
be written
discussion
zeros of the
a point
oscillator.
oscillations
constant
a
probability densities.
of freedom
described
simple
mode and
near
degree
harmonic
also
Although
provides
the
to illustrate
Densities
amounts
having
by a simple
theory
harmonic
by small
system
oscillators.
developing
to similarly
the classical probability
provides
the laser-beam
Probability
oscillate
of a simple degree
the
densities-, and the Correspondence modes are reviewed in Section 4 and
Sturm's theorem and its application to the peaks densities are discussed in Section 6.
Classical
within
in Section 2. Section 3 provides a discussion of the quantum
oscillator, the corresponding Principle. The Hermite-Gaussian Mathematica
are
to integrate the
of large-order mode
classical oscillator, it's classical limits, and
density are reviewed
modes.
mode
provides a direct proof that the classical limits
limits, therefore, serve to provide a good measure Hermite-Gaussian laser beams.
compared
modes
[6] is used
probability densities for small- and large-order modes theorem
oscillator
photons
laser-beam
Mathematica
also contain all of the probability density peaks. The
The
of large-
[4,5].In view of
laser-beam
detecting
classical limits of Hermite-Gaussian
unity for higher-order
laser-beam
than
polynomials
increases to unity for higher-order
of the quantum
expect
these principles. Sturm's
2.
incident
Principle, the probability of finding the quantum
the functional
similar,
photons
slightly different definitions for Hermite
within the classical limits asymptotically Since
and
references indicate that the spot size, as delimited by the peaks
order Hermite-Gaussian the
and
frequency harmonic
of oscillation. oscillator
are
obtained
from
Using
Hamilton's
equations
(2) with
the
Hamiltonian
given
(I), the
in
time
derivatives for the canonical variables x and p are obtained
= --
= --,
_p
= ---
m
Differentiating x with respect to time and standard
harmonic
= -mo_2x.
(3)
_x substituting for p in (3), we
obtain the
oscillatorequation + ¢o2x = O.
The
solution
of this
harmonic
oscillator x(t)
The motion.
total energy Using
momentum
equation = x o cos(cot
(4) can
be written
+ ¢).
(5)
E c of the classical harmonic
the oscillator Hamiltonian
as
oscillator is a constant
(1) and
the relationship
of the
between
the
and velocity, p = rex, the energy can be written as
E , + =;(4;-4t )to> oscillator
polynomials
normalize
the polyno-
using the differentiation inner product, Eq. (4.2). easily transformed to harmonic oscillator wave
231
functions e
in internal
2
momentum
and
Eq. (3.12). When
the
the
the
must
be antisymmetric.
5
creation
spatial
with
spin-flavor
variables. operators
polynomial and
In this
color
wave
in the
For a given
the
right-hand
functions
(for which
case
with
dim
column
state of Eq.
permutation
Yc = A) wave
Ya this fixes YI,
vacuum
]0) is realized (4.14)
symmetry
functions,
the
are
as
given
in
Y0 are combined
resulting
symmetry
type
namely
Y,
_
SU(6)
M
M
7O
S A
S A
56 20
Conclusion L
We have which
shown
Lorentz
how to construct
a relativistic
transformations
four-momentum
operator
quantum
are kinematic
is then
and
the product
mechanics
interactions
of the mass
using
appear
operator
Dirac's
"point
in the mass
form,"
in
operator.
and the four-velocity
The
operator.
For eigenstates of the four-velocity operator, mass operators are rotationally invariant self-adjoint operators. Mass operators corresponding to spin-orbit, spin-spin, and tensor forces are readily constructed because the internal coordinates of velocity states transform like nonrelativistic coordinates. way
Nevertheless,
under
the
four-velocity vectors,
kinematic
operator
forms
the
theory
Lorentz
replaces
a relativistic
is covariant
group.
A modified
the four-momentum SU(2)
spin
vectors
transform
Pauli-Lubanski
The
when
eigenvalue
in the
operator, dotted
of the
usual
in which
the
into appropriate spin
Casimir
four
operator
with an internal symmetry into a larger symmetry symmetry quantum numbers in a relativistically
way.
When
the internal
is a relativistic
SU(6)
(such
inspired
as QCD
symmetry theory.
is constructed
algebraically
to realize
the
harmonic
is SU(3)
In such
mass
flavor,
a theory
operators),
operator with equally spaced its nonrelativistic counterpart, ble
four
operator,
algebra.
is j(j + 1). Combining this SU(2) algebra produces mixing between spin and internal invariant
in that
but
and
there the
mass eigenvalues. with r 2 potentials using
a symplectic
oscillator
wave
the spin of the constituents are many
simplest
ways
choice
of choosing
algebra.
mass
is a harmonic
Such a mass operator between each of the
functions
is 1, 2 the result
By using
mass
is not constructed like constituents, but rather
Bargmann
as polynomials
operators
oscillator
with
spaces definite
it is possipermutation
properties. Moreover, the harmonic oscillator mass operator can be modified without changing the polynomial eigenfunctions by adding on the operator X+X -, in which case the eigenvalues N = 0, 1,2,... become (N - g)(N + 3_?+ 1), where Mass operators can also be formed out of SU(6) type
mass
it should their
formulae be possible
antiparticles
the meson
spectrum,
[11]. By adding to reproduce are combined
such
mass
the observed into a larger
as well as the
spectrum
e is the orbital angular generators, which then
operators baryon
to spin-orbit mass
spectrum.
internal
symmetry,
of some
of the low-mass
232
momentum. give Giirsey-Radicati
or tensor
mass
operators,
And if constituents
it should
also be possible
nuclei.
and to fit
Once realistic possible
relativistic
to compute
form
wave
factors,
functions
for mesons
structure
functions,
and
baryons
decays,
and
are the
available,
it should
like for hadrons
be
viewed
as bound states of spin ½ constituents. In a succeeding paper [12] we show how to formulate point form relativistic quantum mechanical impulse approximation, wherein the electromagnetic properties of the hadrons are determined by the electromagnetic It is possible to generalize the relativistic SU(6) theory Fock
space
this paper to compute old SU(6) operator;
is formed
by taking
the
direct
sum
of the
a
properties of their constituents. to a Fock space theory, where the
n-constituent
Hilbert
spaces
discussed
in
from n equals zero to infinity. Such a Fock space is the appropriate space on which decays of excited baryons, such as the A ---, 7r + N decay which was forbidden in the theory.
Finally,
for such
mass
constituents
and
we mention operators
correspond
that
mass
hadrons
to the
operators
consist
current
quarks
need
of a direct in QCD,
not sum
commute
with
the number
of an indefinite
in contrast
number
to constituent
of
quarks.
References [1] A.J.G.
Hey
[2] P. Dirac, [3] W. [4] Y.
and
Rev.
R. L. Kelly, Mod.
Phys.
Phys.
96, 71 (1983).
(1949).
n. Klink, Ann. Phys. 213, 31 (1992). S. Kim and M. E. Noz, Theory and
Publishing [5] S. Capstick
Co.,
Dordrecht,
and
N. Isgur,
[6] P. L. Chung,
F. Coester,
[7] F. M. Lev,
Fortschr.
Holland, Phys.
[10] W. H. Klink [11] F. Giirsey
and
and
University
L. A. Radicati, form
of Iowa
of _he Poinear_
Group
(D.
Reidel
D 34,
2809 (1986).
and
W. N. Polyzou,
Phys.
Rev.
C 37,
2000 (1988).
31, 75 (1983).
T. Ton-That,
Ref. [1], pp. 98ff. [12] W. H. Klink, Point
Applica_ion_
1986).
Rev.
B. D. Keister,
Phys.
[8] W. H. Klink, Ann. Phys. 213, [9] M. Hamermesh, Group Theory
ators,
Rep.
21,392
54 (1992). (Addison Wesley, J. Phys. Phys.
relativistic preprint
New York,
A 23, 2751
Rev.
Left.
quantum (1992).
233
1962).
(1990).
23,
mechanics
173 (1964); and
see also
electromagnetic
the
discussion current
in
oper-
N93-27333 PHASE
SPACE
LOCALIZATION
FOR
QUANTUM AND
ITS
ZERO
de Recherches C.P.
CURVATURE
Montrdal
Department
(Qudbec) and
of Mathematics,
Montrdal
LIMIT
M. El Gradechi
Mathdmatiques,
6128-A,
SITTER
MECHANICS
Amine Centre
ANTI-DE
Universitd
de Montrdal
H3C
Canada
3J7,
Concordia
(Qudbec)
H4B
IR6,
University Canada
Abstract Using optimal
techniques
of geometric
localization
on
phase
quantization
space
and
is defined
SO0(3,2)-coherent
for the
quantum
states,
theory
a notion
of a massive
of and
spinning particle in anti-de Sitter spacetime. We show that this notion disappears in the zero curvature limit, providing one with a concrete example of the regularizing character of the constant (nonzero) curvature characterization of masslessness The present quoted
1 It
contribution
of the anti-de is obtained.
is based
on a joint
Sitter
spacetime.
As a byproduct
work
with Stephan
De Bi_vre
a geometric (see references
below).
Introduction is a well
spacetime, SO0(3,2),
fact
can
obtained
the
constant thus
known be
kinematical
positive
nothing
but
a zero
to
theories
with
Up large The or
amount main
the
over
this
very
the
modulus as well
distribution.
Sitter
anti-de
limit.
which
known and
AdS
The
lack
theory
regularizing
can
role
the
(ADS)
one
would such
source
though
context
of quantum
the
emphasis
Indeed, to the
notion one
particle
not
effective
235 PRECEDING
PAGE
BLANK
NOT
FILMED
is
to approximate
made
equips
give the
AdS
regularizations. it has
received
field on the
it is a known
be interpreted,
of _; is thus
sought
exploited,
of a natural not
like
is the
procedure
approximations
curvature of the
group,
parameter
contraction
that
nonzero
fully
from
corresponding
massive
Sitter
anti-de contraction This
in the
theories.
cases,
functions free
arises
of those
the
fact,
the
been
implications
approaches
realizations
of Minkowski
hoping
Indeed, not
group
The
to this
is actually has
kinematical
spacetime. ones,
[1] [2].
for its potential
wave
Sitter
According
idea
from
spacetime.
SO0(3,2)-invariant
parameter,
as an AdS,
_+T(3,1), the
of a contraction
theories
Poincar_
of the
group,
of anti-de
stimulating
of the space
in both
a probability
by
of attention drawback
Poincar_,
means
relativistic
momentum
realizations,
by
curvature
a lengthlike
to now,
Poincar(_
_¢ of the
theories
regularized
the group
curvature
T'+T(3,1)-invariant rise
that
fact
spacetime that
.of 16calization. quantum in those for such
a
theories. such More-
states
of a
realizations,
as
realizations.
In this native.
short
contribution
we propose
fact,
for
of a free
In
space
is a K_ihler
discrete
series
resentation, states. this
case
S0o(3,2)
so its
can
Moreover
interpreted
flat
effectiveness
space
limit,
confirming
In section
physical
interpretations.
of geometric localized
of masslessness
The
The
phase
in AdS latter
space
the
regularizing
and
description
character
in order the
a brief
coherent localized
quantum
states
in
distribution.
theory
form
of the
to a rep-
of _.
to fix both
is obtained zero
We the
limit
For
more
as
and
the
application
of the
of a geometric
2.
proceed
notations
through
curvature
discussion
of section
the
physical in the
optimally
characterization
details
we refer
to the
theory
finds
of the
its
starts
best
(,%
of constant
classical
theory
formulation
with
manifold
spacetime
explicit
4 contains
from
description
construction
presymplectic
the
the
optimally
states and we show how their of this notion of localization
theory,
quantum
rise
[6].
classical
spacetime
of the
classical 3, the
Section
as it arises
[3], [4], [5] and
2
then
is given.
of these coherent the disappearance
the
In section
quantization, states
papers
the
2 we describe
functions
are
phase
integrable
states:
They
alterthe
gives
a square
of quantum states.
wave
regularizing
spacetime,
quantization to be
family
as a probability
form stress
AdS
(geometric)
to these
of the
as the
in
is known
a particular
Here we exhibit the explicit interpretation arises. We also follows.
whose
modulus
realization particle
latter
is attached
the
be actually
space
spinning
The
contains
of localization
space.
realization
space
phase
space,
of SO0(3,2).
Hilbert notion
in phase
the massive
homogeneous
representation
A natural
states
the
the
curvature
the
determination
is a closed _,
of a spin
within but
Mr.
of an
space,
2-form),
symmetries
mass
developped
evolution
degenerate
The
s and
scheme
with
of Mr
are
m _
0 free
particle
by Souriau [7]. ,_,, (E_ ,w_), which
a projection helpful
on
guides
the
The is a AdS
in doing
so.
50123
In
fact,
M,
is just
the
one
sheeted
y.y-_r/ooyay_ E {5,0,1,2,3}.
o,_ the
identity,
We through
in (Rs,r/),
= _(y5)__(yO)2
Clearly,
SO0(3,2),
hyperbolo'id
O(3,2)
is the
is the
so-called
+ (yl)2
isometry
AdS
whith
diag
+ (y2)2
group
77= (--+++),
+ (y3)2=
of (2.1),
its
_
-2
connected
(2.1) component
choose for E_" the SOo(3,2)-principal homogeneous space, E_" "_ S0o(3,2), the following SO0(3, 2)-invariant constraints in R 25 (five copies of (R s, 77)), Y'Y
= _
-2,
q.q
= -m
2,
y • q = 0 = all
to
group.
u.u
= 1,
the
other
v.v scalar
= 1
and
t-t
realized
= m2s 2,
(2.2a)
products
(2.2b)
rn2$
e_p_ The
physical
position Lubanski
vector.
description theory.
interpretation
on the
hyperboloid The
two
last
of the (2.1),
remaining
of E_ "s,i.e. The
y°q[3u_t,Q_'
E_ 's _
constraints
= _
coordinates
and (y,q.u,v,t)
q is its conjugate five-vectors (2.2b-c)
They are
yOq5 > 0.
is then
momentum,
u and
SO0(3,2).
ySqO_
v are shall
needed
236
as follows:
t is whatwe
introduced represent
in order
(2.2c)
in order the
spin
in (2.2a) call
the
to have part
to fix an orientation.
y is the
AdS-Paulia covariant
in the
quantum
The curve
choice of the
geodesic
ofw E is constrained completely
of M_,
i.e.
by the requirement
integrable the
distribution
dynamic
of the
that
generated
theory
the
projection
by kerw_
is obtained
on M_ of each
in E_ '_, results
from
kerw
E. Such
integral
in a time-like
an w E is provided
by, w E = dyAdq+ This
choice
space for
is not
of the _
-_ s, to
symmetry the
unique
theory, be
but
_,_ the
reasons
as the
where
Here
= g_,,,p"dz
gu_
is the
_,
metric
is the they
related
Y = _/_-:_
= g_,,,aUdx _', v.dy for
the
global
the fiat become,
guppY's _ = 0 = all the
_=
other
3
quantum
The
The methods
above
of geometric
[8].
ducible
In
other
space
SO0(3
,
H=
× SO(2)
allow
{
,
d#_"
fields
generating
is
the
one
[¢12dp_m"<
invariant kerw_.
to quantize
This
yields
on
E_ 's _ the
following
Hilbert
left
E_ 's and S0o(3,2),
regular
E {0,1,2,3}.
(2.4b) The
(2.2a-c)
space,
)'5o
representation
and
)'1_
exists
zero
translated
in
subset
(2.5b)
(p,a,b,s),
(2.5c) by their
zero
curvature
localization classical
theory
construct
the
7"(, is realized
Y121P=isw
are
the
a natural
of S0o(3,2).
left
the
SOo(3,2)
--0
as follows.
} .
invariant
action The
irre-
for which
E_ '_ _
and
described
unitary
of SO0(3,2)
structure
Y5o¢=i--¢
there
237
_'.
(2.5a)
orbit
bundle
way
g_,_sus _ = m2s 2,
to
toad joint
set
same equations,
= guvs_'dx
#,u
the
is able
for
a new
in the
K
measure Since
For
arena
(2.4a)
be confirmed
rn,j
the
and
optimal
principal
oc,
as the
p0 > 0.
one
prequantum
the
constraints
of the
can now
the
× S0(2). ,w_)
t .dy
and
and
to the the
1
phase
appears,
,_,
and
The
It
in section 4. way, we introduce
_=
(z°,i)
products
methods
associated
w:E_"----*C
Here
in L_(E_",d#_").
those
Exploiting
_v'n" .¢-J _.
_"
constraints
quantization
of SO0(3,2)
= guvbUdz
6 = m2s
and
v,,_,_ is a covering.
2)/S0(2)
above
using
and
1, g._bUb _ =
theory
words,
representation
phase
of the
+ (£)2;
metric.
scalar
eu_6pUa_b_s interpretation
and
coordinates
(E_
through
°
Minkowski
= --m 2, g._aUa
use
Interpreted
latters
Ysin_z
The
w_).
S0o(3,2)/S0(2)
,we
(x,p,a,b,s).
to the
$ E R3
requirement. of
case _ -- s is discussed limit in a meaningful
°, yo=
u.dy
space
on E_
of four-vectors
and
generating reduction
homogeneous
are
(2.3)
dynamic
of SO0(3,2)
set
Adv.
symplectic
yS = Ycos_z
of M_
The physical limits.
by
The special zero curvature
curvature limit ofgu_ is just terms of the new coordinates gu_pUp_
action
u,v,t),
_< _z ° _< _r,
-lr q.dy
(y,q,
above
symplectic
obvious
on E_ '_. This
five-vectors
the
is obtained
SO0(3,2) i.e.
forthcoming constructions. In order to carry out the
of coordinates
it fulfils
,w_j,
_du
3.1)
vector
of
SOo(3,2)
latter
restricts
to a unitary
(reducible)
character
e'(_
There
_+_'t
actually
exists
in 7"/ an invariant latter
gives
representation
of SO(2)
a positive 7-(_
to a unitary
the
Indeed,
invariant
rrt_$
subspace
rise
in 7"(, i.e.
× SO(2). . The
holds
K_ihlerian
restriction
irreducible
representation
this
provided
polarization
of the
induced both
s are
of _w"_" allowing
previous
representation
of SO0(3,2) m and unitary
of SO0(3,2).
by the integers.
one
to select
representation
to
Concretely,
et where
Z, = }_i+i};s,
fields.
The
way
mathematic
7-(_
litterature
also
.
The
well
unitary
irreducible
interpretation
The
of their
particular
states
passing
states,
which
way
optimally
are
the
L,_z's
The
determination
distribution
through
by
coherent
of the
classical
E passing
(3.3).
states
Lo_'s,
The
the
this
The
(3.3)
of the w.
minimize notion
optimally
q_w is then the
states
the
series
in the
representation
O_
C
I'r_,
7"(_
$
_o0 possess coherent
on
, of the
many states
by construction
E_ '_ is unitary
interesting of 5"O0(3,2),
they
are
labeled
@ {5,0,1,2,3}; L,_'s
are
specifies
to be
(3.3)
their
in fact
quantum
counterparts.
uniquely
the
a unique
point
reduction
localized
relations
given
vector
is known
distributions
state
In fact,
invariant
equations, Va,,_
said
disrete
orbit,
by symplectic
is then
are
space.
Laa(w)
incertainty
of localization
localized
ten
the
left
by 7-/_,'
probability
generalized
the
the
for the unitarity is m > s.) defined wave functions belonging
but
and
Thus
the
weight
= L,_z(w),
through state
to
the
observables
as
are
carried
highest
in phase
them
through
kerw
is specified the
are
the
nothing
localized
specifying
modulus
belonging
through
0, hence
formula
(11) becomes:
-- V' --
(12)
solution:
(13) We see in this case
that the mass m =- gc/2 is independent
of amplitude.
Wave packets are defined to be plane waves with slowly changing parameters (e.g., amplitude, spin, and momentum). To describe such wave packet_ we introduce "slow" coordinates y_ - ex_, where e is a small parameter, into formula (6)as follows[4]:
Ao(._)_ _o_) _',, (:e) - e_'-': _'' X',,_) (14)
p (x_)-.p_) where O (x_) - O(e x_),etc.The resultingequations governing the wave packete [4]are given
P_Ptt
n
rn 2
VQ pp -- V_ Pa V=J ffiwhere
now p_ -
Vo 8, where
m - re(p)
Jo -F To analyze
equations
the group velocity That
is v -
is given
in formula
vo, va==pJm,
(15) we now consider
(15)
0 (12), and
F-?p2+4p3
a space-time
with one space
(16) dimension.
Then
v, -
(v o, v i) and
vt/v o.
is, 1
Vo- y - V_ Vt IV'/
292
(17)
Similarly,
x a _= (t, x). Formula
(15) becomes:
--_
(FT) +
at
The equations
for the characteristic
(FvT) -- 0
(mvT) + _x
curves
(my) =" 0
(18)
for (18) are easily derived
dx
[4], and are given by:
v-J:8
E - x_+----_
(19)
where
- II/l_ where
F' and
m' denote
the derivatives
of F and
(20)
m with respect
y2 dv -, ± _
to p. On the curves
(19) we have:
dp
(21)
1
When teristic
V -
curves
_ g2 t p[4, m' =- 0 by formula
(19). Since then 5 - 0, the curves
the wave packets In general, will split
3
are identical
to bispinor
for wave packets
to exist,
into two wave packets
CPT
By the Caftan
(13) so that by (21), dv -
and map,
5, in formula
(20) must
along
lines. It is then straightforward
to show that
1 be real. If V _ _ g2 [P [4 a general curves
wave packet
(19).
Splitting
the CPT operation
for the tensor
on the charac-
the characteristic
which,
for bispinors,
¥ (x as) -.i% becomes
is, v is constant
wave packets.
that propagate
Velocity
(19) are straight
0, that
fields A K and
is given by [5]:
¥ (-xP)
(22)
p:
A_ (x_) -. A_ (_xtS) p (x _) -.-p Note CPT
that because
operation
Lagrange coupling
(23). Nevertheless
equations
commute
c, the Yang-MiUs
in the limiting
with
CPT.
Lagrangian
(23) L in formula
case that the coupling
In this section
constant
we examine
the
(1) is not covarlant g tends to infinity,
question
of CPT
violation
under
the
the Eulerfor finite
g.
One of the main to formula
of the constant
(-x p)
tests
for CPT covariance 1
(13), when V =- _ g2 Ipl
4
the masses
is the equality are equal.
293
of particle Therefore,
and antiparticle
suppose
instead
masses that
[6]. According
_.2
V-
_ 1014 + elpl'
(24)
where e is a small parameter. Then to first order in e, formula (12) gives:
m =- _2 + 3_ p Since p _ 0 for particle plane waves and p _ 0 for antiparticle
(25) plane waves, the mass difference Am is:
2e Am =- _ Jp[ On substituting (ignoring
formula (25) into (20), the velocity
splitting
(26) 28 becomes, to lowest order
in e and g-_
factors close to one; i.e., V_):
28-
_
(27)
Assuming a fractional mass difference for electrons and positrons of one part in a million, the velocity splitting would be 28 - 10-s or 3 × 105 meters per second, which should be observable in experiments that measure the spreading of low energy electron wavepackete. CPT violations of 10-6 are consistent with current observations of particle-antiparticle mass difference and suggest new experiments to observe velocity spiittinp of 3 × 105 meters per second, or less [6].
References [1] F. Reifler and R. Morris, "Unobservability of Bispinor Two-Valuedness o/Physics, Vol. 215, No. 2 (1 May 1992), pp. 264-276.
in Minkowski Space-Time," Annals
[2]F. Reifler and R. Morris,"The HamiltonianStructureof Dirac's Equationin Tensor Form and itsFermi Quantization," Workshop on SqueezedStatesand UncertaintyRelations, NASA (1992), pp,381-383. [3]J.Kessler, PolarizedElectrons, (Springer-Verlag, NY, 1976). [4]G.B. Whitham, Linearand NonlinearWaves (Wiley,NY, 1974),pp.48,5-510. [5]W. Greiner,Relativistic Quantum Mechanics,Wave Equations(Sprlnger-Verlag, NY, 1990). [6]L.B.Okun, Leptonsand Quarks (North Holland,Amsterdam, 1984).
294
93-.27341 GALILEAN
COVARIANT
HARMONIC
OSCILLATOR
Andrzej Horzela H. Niewodniczar[ski Institute of Nuclear ul. Radzikowskiego 152,31-3_2 Krakdw, Edward University Athens,
Physics, Poland
Kapu_cik of Georgia,
30602 Georgia,
USA.
Abstract A Galilean covariant approach to classical mechanics of a single particle is described. Within the proposed formalism we reject all non-covariant force laws defining acting forces which become to be defined covariantly by some differential equations. Such an approach leads out of the standard classical mechanics and gives an example of non-Newtonian mechanics. It is shown that the exactly solvable linear system of differential equations defining forces contains the Galilean covariant description of harmonic oscillator as its particular case. Additionally we demonstrate that in Galilean covariant classical mechanics the validity of the second Newton law of dynamics implies the Hooke law and vice versa. We show that the kinetic and total energies transform differently with respect to the Galilean transformations,
1
Introduction
Recently Galilean
we have covariant
proposed a new approach to classical mechanics which leads to a manifestly models of mechanics for a single interacting particle [1]. Our main goal was
to construct a self-consistent and complete scheme avoiding all relations of standard mechanics which break the Galilean covariance. It is easy to see that all such relations the class of the so-called _constitutive relations" reexamine the role of these relations in mechanics. is an example
of the
Galilean
covariant
[2] and in order to achieve our goal we had to The relation between momentum and velocity
constitutive
the mechanical forces in terms of positions covariance. Hence, in a Galilean covariant
classical belong to
relation
[3] while
all explicit
expressions
of
and velocities, called force laws, obviously break this formulation of classical mechanics of a single particle
we have to reject all known force laws. To keep the formalism as predictive as the usual one we propose to determine all mechanical quantities from the set of differential equations of the evolution type. Our program leads us to a broader than Newtonian formalism model of classical mechanics in which more than one vector-valued measure of mechanical interaction is introduced. The time evolution
of these measures
is described
by a set of differential
environment which are used to determine fully covariant way. The simplest version
equations
called the equations
of the
the interaction of the particle with its environment in a of such a scheme contains two measures of interaction:
295
r
the customary we have assume
force
F (t) measuring
called
the influence
a priori
the Galilean
I(t)
the momentum governing
covariant
non-conservation
the time evolution
Newton's
second
and a new quantity
of the
law of dynamics
acceleration. in the form
M_(0 = _(l) where
M denotes
the inertial
mass
of the particle
because
which
We do not
(1.1) this equation
is not
of the evolution
type for the acceleration and contains a physical constant. According to our general philosophy [2] we avoid to use any such constants unless we really need to introduce them as phenomenological parameters. In our case this will happen only for the equations of the environment for which without any doubt we are forced to use in the theory some information of the phenomenological character. All the remaining equations describing the particle are universal, interrelate only basic theoretical concepts and do not contain any phenomenological constant. In our theory the experimental input is used therefore only for the description of the environment and we consider this fact as a big advantage of our formalism. The relation between classical Newtonian mechanics based on the equation (1.1) and our scheme is established using (1.1) as a constraint put on the set of solutions of the differential equations. It is also a constraint put on solutions of the equation
d_e(l) 1 _i = _(0 which
in the framework
of the Newton's
mechanics
follows from
(1.2) the definition
of f(t).
tions of our model which satisfy (1.1) we shall call Newtonian solutions while solutions the relation (1.2) only will be called the generalized Newtonian solutions.
2
Linear
The solusatisfying
model
The aim of this talk is to illustrate our approach for the force and influence. We shall show that
on a simple example of linear evolution such a model includes, as its particular
equations case, the
Newtonian mechanics of the material point which motion is defined by the force provided by a linear in position and velocity force law. In the case under consideration the complete set of differential equations describing the system consists
of two purely
kinematical
equations
of motion
_d_c__,e___z, = ¢(0
(2.1)
d_(t) = _ (l)
(2.2)
dt
dt
one dynamical
equation
of motion dY(t)
dt one equation
= f(t)
(2.3)
of balance
dy(t) = £ (t) dt
296
(2.4)
and
the system
of two equations
of environment
dP(t)
= oF(t) +/3/'(t)
dt
(2.5) dI(t)
-
dt where and
£(t),
6(t),
momentum,
if(t)
and/7(t)
3'P (t) + 6Y(t)
are the trajectory
respectively.
The meaning
function
of the particle,
of ff (t) and
I(t)
its velocity,
has been
explained
acceleration
above
and the
parameters a,/3, 3' and _5represent dimensional coupling constants specifying the model. The model is covariant with respect to the Galilean transformations parametrized by a rotation R, a boost iT, and a space-time transformation rules
translation
(if, b) if all mechanical
quantities
used obey the following
_(t) _ e'(t') = RZ(t) + at + i i(t)-,
i'(t')=
(2.6) (2.7) (2.8)
Ri(t)
_(t) _ y'(t') = R_(t) + _ Z
(2.9)
P (t) _ P'(t') = RP (t)
(2.10)
i'(t) --, P(t') = Rf (t)
(2.11)
where t ---, t' = t + b and m is the Galilean mass. As we stressed coupling
constants
defines the constants
shape
mass of the particle
in the Introduction in the equation of general
[3] which we shall not assume
the only external
parameters
of the environment
solution
(2.12)
(2.5).
of (2.1) - (2.5).
to be equal
characterizing
The mutual
Denoting
relation
by A the
matrix
to the inertial the mode] between
are them
of coupling
3' and their
following
combinations
by A±
l [trA + ¢(TrA) we may form
write
down
for 4detA _(t)
> (TrA) _ the general
=
solution
= ,4 + Bt + _t 2 + 5 exp(A+t)
_7(t) =/3 if(t)
2 -4detA]
+ 2Ct + 6
A+ exp(A+t)
= 2C + if) A2+exp(_+t)
3'
(2.14)
of the equations + ff_ exp(A_t)
(2.15)
+ E A_ exp(A_t)
(2.16)
+ g A2_'exp()__t)
__ _ e_p(__t)
_+
3'
297
(2.1) - (2.5) in the
(2.17)
(2.18)
F(t)
=
6-
A+ A+3/_ exp(A+t)
_-
3
Vector-valued
constants
A,/3,
(,/9,/_
and/3
ential equations (2.1) - (2.5) and in order have to transform in the following way
)_- _3_ E exp(A_t) "r
are the integration to satisfy
constants
the transformation
X--. 2= RX- bRg + b R( _
(2.19)
of the system rules
of differ-
(2.6) - (2.11)
+i
(2.21)
B' = RB - 2bRC + ff (_ --, ("=
(2.22)
RC
(2.25)
z3-. 6'=
which
explicitly
show how their
they
(2.26)
fly _ ff_'= Rff, exp(,__b)
(2.27)
fi ---, fi' = Rfi + rnff
(2.28)
values
depend
on the choice of the reference
Here we should like to stress the difference between our approach, covariance as the most fundamental feature of the theory and standard
frame.
demanding expositions
the Galilean of mechanics
which treat it almost always as a branch of the theory of ordinary differential equations. There is no principle of relativity in the theory of differential equations and, consequently, there is no problem of transformation properties of the solutions and integration constants. In contradistinction to mathematics, this subject is of primary interest to physics and we have to realize that the integration constants take the whole responsibility for the transformation properties of all physical quantities. the whole
This means that the original information on the symmetries
preparation of physical system already contains almost of this system. The time evolution of the system has
only to preserve the original symmetries. It should not be unexpected that in our scheme which is an example of a non-Newtonian mechanics (and, as a matter of fact, its generalization) the careful analysis of the properties of integration conStantsand their relation to the initial conditions may lead out of the framework of standard classical mechanics. There is a lot of different initial conditions which may be imposed on the solution (2.13 - 2.18). For instance, we may use the values of the first four derivatives of the function av(t) at the same instant of time to to fix the values of the constants A to E. It remains in obvious contradiction to the widely the unique
spread opinion that in mechanics only the initial position and velocity are needed for determination of the trajectory. This is the property of Newtonian mechanics only in
which
relation
the
(t) are a priori
(1.1) is always independent
satisfied.
as determined
In our formalism from independent
the acceleration equations
if(t) and the
and the force relation
(1.1)
imposed on these quantities reduces the number of degrees of freedom for initial conditions. It enables us to calculate some parameters of a model in terms of the other. We shall see below that it may be used for determination of the inertial mass M in terms of the coupling constants given by elements of the matrix A. The analysis of the model depends on the mutual relation between TrA and detA. In order to concentrate the attention on the harmonic oscillator problem we shall omit the case TrA > 4detA because it does not describe oscillatory motion. The complete analysis of the problem will be found
in [4].
298
3
Oscillatory
motion
It is immediately seen from (2.14) and (2.15) that the trajectory (2.15) oscillates if the inequality (TrA) 2 < 4detA holds and the oscillations may be damped or not depending on the value of ReAi. The reality of all mechanical quantities requires that the constants/9 and/_ are complex valued and they
must
be complex
conjugated 3 = E"
which is the first condition Newtonian
condition
restricting
(1.1),
the arbitrariness
constants
A,/_,
C,/),/_.
The
of the form
= M [_7(t)-
2C]
(3.2)
provided M=
supplemented
of integration
as well as its generalization F(t)
are satisfied
(3.1)
additionally
6-_+
in the Newtonian
A+=
_5-A_
A_
case by the Galilean
(3.3) invariant
relation
d = 0 Relation criterion
(3.4)
(3.4) fixes invariantly one of the parameters of the of the Newtonian character of the solution considered.
Substituting in (3.3) the values ,_+ from (2.14) (3.3) may be satisfied only for a = 0 which gives
solution
and
we shall use
we come to the conclusion
that
it as a
the equality
M= The value of the Galilean mass m remains arbitrary because particle and has nothing to do with its possible interactions. Taking
into account
(2.15)
in a selected the criterion
it is a parameter
and (2.19) it is easy to see that F(t)
may be satisfied frames satisfying
(3.5)
the famous
which Hooke
identifies
the
force law
= -k£(t)
(3.6)
reference frame for which ,4 = /_ = C = 0 i.e. (3.3) of the Newtonian character of mechanics.
only for reference This means, be-
cause of the invariance of this relation, that the Newtonian condition (1.1) is equivalent to the requirement of the existence of the Hooke law. This fact has a far going consequences because in all treatments of the foundations of mechanics the forces are measured by dynamometers which operate on the principle of the Hooke law. Therefore any mechanics inition of forces must be Newtonian. The Newton laws of mechanics
using such an operational deffollow thus from the adopted
operational definition of force. In order to detect any violation of these laws we should first invent a new operational definition of the force not based on the Hooke law. It is indeed a very surprising conclusion which however The above conclusion mentum and velocity
uniquely follows is less surprising
from our more general approach after observing that the linear _t)
= Mi;(t)
299
to mechanics. relation between
mo(3.7)
is possible
also in the
Newtonian
mechanics
(3.7) prevent to observe any deviation fundamental assumptions of standard may be found only if the analysis is more general than the standard one. For the non-Newtonian mechanics
f(t) which
for A = g = C -
to the conclusion
that
0 reduces
only.
Therefore
the almost
always
assumed
relation
from the Newton's laws. We have to conclude that many mechanics are interrelated and their possible interrelations performed in the framework of the approach to mechanics Our method is just an example of such a scheme. we may replace the Hooke law by the relation
= -k [Z(t) - Xt.o (3.6).
gt-ddl
Using again
(3.8) may be satisfied
(3.8)
the solutions
(2.15)
and
(2.19)
we come
only if (3.9)
Together
with
(3.3) and
(2.14)
it implies
that
(TrA)_/(TrA) The square root must be different therefore we must have
2 - 4detA
from 0 due to the TrA
Since by
we already
have got c_ = 0 this condition
(3.10)
= 0
condition
4detA
> (TrA)
2 assumed
= 0
(3.11)
gives _ = 0. The frequency
of oscillations
w 2 = detA and because
and
is given (3.12)
of (3.5) and o = 8 = 0 we have J
We may therefore conclude mechanics the non-Newtonian evolution equations has the form
= -B'_ = -M7
(3.13)
that in the framework of Galilean covariant approach to classical generalization of the standard harmonic oscillator is given by linear
for the force and the influence
A =
and that
the matrix
of the coupling
(0 :) w2
constants
(3.14)
-_,
The most general Galilean covariant is the non-Newtonian generalization
which, after substitution of (2.15), parameters of the model
linear relation between the force, the position and the velocity of the superposition of Hooke and linear friction ('7 < 0) forces
(2.16) and (2.19) into it leads to the following MX__
=
r/A+-_
MX 2- =
r/A_-to
relations
between
(3.16)
300
if additionally
the generalized
the case 4detA
Newtonian
condition
(3.2) is demanded.
> (TrA) 2 which gives the only solution
We are still restricted
to
of (3.16) in the form
r/
(3.17)
K
-y = and it immediately
follows from it and (2.15)
that
A=
describes
damped
oscillatory
motion
M2
(3.18)
the matrix
0,t¢
with frequency
of coupling
constants
(3.19)
rI M) given
by (3.20)
and an amplitude
4
damping
Kinetic
and
In the standard
approach
alent
exponentially
total
according
17 exp-_t.
to the factor
energies
to classical
mechanics
the kinetic
energy
is defined
by one of the equiv-
expressions k(t)
where M is the inertial the Newtonian relation
-
"2
/72(t) 2M -
mass of the particle. between momentum
M v (t) 1 2 = 5/7(t).
if(t)
(4.1)
Relations (4.1) are a straightforward and velocity (3.7) which in Galilean
proposed should be treated as additional assumption we cannot identify the inertial mass present in second
only. Discarding (3.7) as a priori valid law of dynamics and the mass parameter
appearing in the momentum transformation rule (2.9). The general relation and velocity written down with Galilean mass introduced into it has now, form /7(t) = (m - M)g(t0) where g(t0) is an integration constant specified from initial conditions. We define the kinetic energy mental conditions put on it: i.) the balance
having
as bilinear
consequence of covariant scheme
between according
+ Mg(t)
the meaning
(4.2)
of an initial
form of momentum
momentum to [3], the
velocity
and velocity
which
satisfying
has to be two funda-
equation dk(t) dt
=/_
(t).
_7(t)
(4.3)
and ii.) the Galilean
transformation
rule 1
k(t)
k'(l') = k(t) + n (t)
301
.Z+
-.2
(4.4)
According
to these
conditions
[3] the kinetic k(t)
which, in notation down as
introduced
= m-
by (2.15)
energy
2M
_2
is given by
- (2.20) and in Newtonian
k(t) = (_mMJB) =-_) To obtain the general
the correct
formula
(4.5)
M._ (to) + 5-v (t)
for the kinetic
regime
(_ = 0, may be written
2 + --_v M-.2.. (t)
energy
(4.6)
in non-Newtonian
regime
we shall start
with
expression
k(t)
A!Y2(t) + B62(t)
+ C_(t)
. if(t)+
(4.7) +_. _(t) + st + ut2+ A The transformation rule (4.4) implies parameters in (4.7) have to obey
the following
A' = A,
conditions
and transformation
B' = B,
B = -2(1
properties
which
C' = C
- 2mA)
C = 1 - 2mA u'=
u,
(4.8)
_'=/_ I_' = Ia-
while the balance
equation A=
2ub-
R_.
ff
(4.3) gives 1
mM
B=
2(rn - M)'
C=
2(m - M)'
M m-M
= -2MC (4.9)
u-
.i 2M =-M u=
(; - M ).e
2M2 rn-m
302
(2
It is obvious that remains arbitrary we may represent
the balance equation (4.3) cannot fix the value of the constant A in (4.7) which but has to satisfy the transformation rule listed in (4.8) as the last. For example, A in the following form
= with
(£o, to) denoting
the space-time
- ,to -
coordinates
(4.10)
of an arbitrary
event.
They
may be chosen
as
coordinates of an event for which the momentum and the velocity of the particle simultaneously vanish. Such a choice guarantees that the kinetic energy also vanishes at this point which we consider the most natural condition possible to demand. Substituting
all values
of coefficients (fi-
k(t)
(4.9) into (4.7) we obtain
MB) 2
2(m - M)
M + _-G2(t)
- 2M_.
[_"(t) - £0 (t)] (4.11)
2M m - M (fi-
2M2 _ to + _--_/.C
MB).
--"2 2 to
and comparing it with the expression obtained for the Newtonian parameter which controls the Newtonian character of mechanics equivalent to vanishing of _, p, v and A in any reference frame. In contradistinction
to the kinetic
energy
system cannot be based on the above equation for the total energy
listed
the definition
case (4.5) we see that the only is _ the vanishing of which is
of the total
basic properties
of the
energy
kinetic
E for conservative
energy.
The
balance
dE d---t-= 0
(4.12)
does not give any hint on the transformation
rule of E. This rule cannot
for the kinetic
leads to a contradiction.
energy
since this immediately
1
E _ E' = E + n/7(t), the conservation
law (4.12)
implies
for free particles from the general
i.) it reduces
to the expression
=
if we suppose
-2
(4.13)
ff + _rnu
= 0
(4.14)
only. To construct the correct expression for the total energy bilinear form of 6,/7, _, t, F, f which satisfies the following two for k(t) if fi = f = 0,
ii.) it satisfies the conservation law (4.12). After straightforward but tedious calculations these two conditions is given by
E
Indeed,
shape as
that d/7(t) dt
which is true we shall start conditions:
be of the same
k(t) +
it can be shown that
7(6 2-TM)fi2+ 2(3`M) 3
1 2-_I
"2
the only form which
6 (7_/)_F.
obeys
" 1+
(4.15) +
---;-7.F. 63`M
-I 3'
303
and consequently we shall take harmonic oscillator. It is now easy to see that to the following rule
it as the definition
under
Galilean
The most momentum
important
point
associated
which is conserved
in time because
for the
Galilean
covariant
the total
energy
changes
according
,,10,
+ -_
with energy
the quantity
in the
second
E is not the momentum
1 (_ff(t)+ _--_
P =/7(t)
energy
1
:(t)
is connected
with the total
transformations
(
E --, E' = E + R
of the total
of the fundamental
term.
fi(t)
It shows
that
but
Mf(t)) equations
the
(4.17) (2.4)-
(2.5).
The
difference
in the
transformation rules for the kinetic and total energies is a new fact in mechanics which without Galilean covariant approach to mechanics could not be derived. Here we would like to remark
our this
so important fact is not specific for the non-Newtonian case only. As we have mentioned several times the Newtonian case which is equivalent to the Galilean invariant choice C = 0, $ = 0. However it must not be taken directly by putting these values into (4.15) because such a choice corresponds The correct
to the singular result is given
system by
EN=-_-
of algebraic
equations
a72+_+w.)
+)C
used to determine
(
ff_+
where w _ = -_'M according to (3.13) and X is arbitrary parameter. with our previous result obtained in [1] within less general approach the Galilean transformation rule
EN'---_ E_
= EN+
R
fi+w2
coefficients
in (4.15).
f
(4.18)
It remains in full agreement and gives for the total energy
) .ff +-_rnu
(4.19)
which means again that the total energy transforms differently from the kinetic its transformation properties are associated with a conserved quantity
energy
and that
P = fi(t) + Mr(t)
(4.20)
0.2 o
and not with ordinary
5
momentum/7(0.
Conclusions
We have demonstrated
that the requirement
of the Galilean
covariance
of classical
mechanics
leads
to a formalism broader than the standard Newtonian one. The new formalism enlarges the class of mechanical systems including those with some unusual properties. In particular, in the next talk we shall discuss the application of the formalism obtained to description of the so-called confined systems.
304
6
Acknowledgments
This work
was supported
in part
search (KBN) under grant enables him to participate for his hospitality
by funds
provided
by Polish
State
Committee
for Scientific
Re-
2 0342 91. AH is also very grateful to KBN for financial support which in the "Workshop on Harmonic Oscillators" and to Dr.W. Broniowski
at College
Park.
References [1] A. Horzela, [2] E. Kapu§cik,
E. Kapu§cik,
and J. Kempczyfiski,
in Proceedings
tistical Mechanics, Dubna, Singapore, 1990), p.423.
of the 5th International USSR,
[3] A. Horzela, in press.
E. Kapugcik,
[4] A. Horzela,
and E. Kapu_cik,
INP preprint
1989, edited
and J. Kempczyfiski,
Ga///ean
Dynamics
305
Symposium
by A.A.
Hadr.
of
1556/PH,
on Selected
Logunov
J. 14,79,(1991);
a Single
Krak6w,
Particle,
et al.,
Phys.
1991. Topics
(World
Essays
in preparation.
in Sta-
Scientific,
5,(1992),
#/
V.
THERMODYNAMICS
AND
STATISTICAL
307 PRECEDING
PAGE BLANK
N_.;T FILMED
MECHANICS
N93-27342 DOUBLE
SIMPLE-HARMONIC-OSCILLATOR OF
THE
THERMAL
INTERACTING
FORMULATION
EQUILIBRIUM
WITH
OF
A COHERENT
B. DeFacio Department
and
SOURCE
Alan
of Physics Missouri
Columbia,
A FLUID
Van and
OF
PHONONS
Nevel
Astronomy
University
Missouri
65211,
USA
and O. Brander Institute Chalmers
for
Theoretical
University
S-41_96
Physics
of Technology
Gothenburg,
Sweden
ABSTRACT
A formulation temperature thermal
which "noise"
Simple
and
uses
process
thermodynamic
"contrast" these
is given
is presented.
results the
and
the
the and
for
a collection
simple the other
observables The simple
role
of phonons
harmonic which are
oscillator generates
calculated
of "coherence"
harmonic
(sound)
oscillator
twice;
a coherent and
in a fluid
the
one
in an equilibrium is a key structure
calculations.
309 PRESEDINGI
PAGE BLANK
NCT
to give
Glauber
acoustic
FILE_ED
at a non-zero a stochastic
state two
system in both
of phonons.
point
function,
is clarified the
formulation
by
1.
Introduction
The
problem
volume and
is both
the
key
with
the
and
those
of understanding
old and
issue
walls
water
molecules
bubble
formation has
interaction
with
reservoir.
Since
harmonic for the
reservoir
have
shown The
is incoherent
the
model
"stochastic
presented
here
or chaotic"
a random
the
contrast
acoustic Gaussia.n
interaction collection
This
The are
functions
between
quantum
mechanics
L 2 or at
least
in the
will use and
density,
second
to generate model
This not
Soboler
The
energy reason
in L 2, quantum the
is why yet
partially
free
all calculated.
and
sound the
created.
space
20 -
2 x 109Hz,
nonlinear
approach Also,
finite
H 1 = L 2,1, which
310
here,
radiation
the
and
the
will field. 14,15
Noz 13
to gener-
state
it has
of scalar,
both
coherent
a standard
useage
function
which
gives
is so useful
is that
since
guarantees
that
be approximated
of Planck
2 was
energy
classical space
and
first
coherent
following
SHO
sound
Kim
twice:
a two-point
and
time-uncertainty.
coherent
is the
where
coherent.
Han,
in that
can
of
6-11
_2 and
mechanics
is a
problem
states
of noise,
the
work
coherent
states
and
radiation
4 This
or stochastic,
a Glauber
that
reversible)
the fluid is partially
universe
and
an incoherent,
phonon
squeezed
realistic
entropy
dense
the
of the
v --
the simple-harmonic-oscillator
It will be called
the fluid
was
rest
matter
will be studied
in the
with
to quantum
is a more
are
of oscillators.
idea
noise
components. optics.
interaction
to Feynman's of this
phonons.
total
describing
fluid
the full
and
role
will be studied
the
transitions.
problem radiation
in a finite
if approximately
addresses
a central
process the
relation
in quantum
their
(Poisson)
which
sound
played
waves,
temperature
linearized
the
with
(isentropic,
constant
field
in a water
radiation
of sound
to adiabatic
The
of a radiation
propagating
sound
scales
of us (AVN)
has
is analogous
longitudinal and
coherent
the reservoir
The
ate
oscillator
time
than
with
wave
of the
lead
waves.
properties
a sound
The
by one
by sound
the
Here
rather
a coherent
interaction The
of a project
thermal
interaction
container.
processes
case
be used
will be the
of the
special
1-3
of the
thermodynamic
fluid
subtle.
the
crucial
by an infinite
correct
even
solutions where
the
the
though
will lie in "function"
and
its "gradient" In
energy
Sec.
2 the
F and
partially
2.
the
coherent
The
In Fig.
square
model pair
integrable.
will
be
correlation
states.
In Sec.
presented,
and
function 3 the
the
g(2) are
Conclusions
density
calculated and
p,
the
entropy
S,
single
and
for both
Outlook
are
the
free
N-mode
presented.
Model
1, a schematic
state
of phonons)
than
So or F.
can
are
exchange
is given
a fluid
F in thermal
In general, particles
Source
which
phonons
shows contact
can
as well as heat
enter but
of
Phonons,
the
a source
of sound
So (treated
with
reservoir
R, which
the
fluid
all other
from
exchanges
So and are
the
as a coherent is much
fluid
and
larger
reservoir
negligible.
Fluid, _
Reservoir, R
F
So
Fig.
1.
Schematic
is So, F is the stationary sum
of So and
F.
The
of the system fluid wavy
volume lines
modeled. and
indicate
The
source
R is the reservoir boundaries
pass.
311
which
of a coherent
state
of phonons
which
is much
larger
than
the
allow
particles
and
energy
to
The solitons
is a modification
(here
others sical
idea
have fluids
easily
solitary shown
are
dent,
In other
many
physically
nonlinear,
coherent
analysis.
The
phonons
are
commutation
interesting
random and
their
creation
a unique,
translationally
astersik
complex with
power
number
invariant
of an operator
is its complex
no phonons.
The
The
other
and
is, Glimm
for plasmas
nonlinear
are
19, and and
formulation
of noise
destruction
bubbles)
clas-
given
and
M2
modes
indepen-
could
operators
can
easily
a*, a satisfy
relations,
Fock
a(-_,t)[O
The
solutions
medium.
some
(here
Wieland
components
[a,a] = 0 = [a*, and
cavitons
and
model
of the
solitons
so that
1_'17 that
Williamsson
exitations
Thus,
bosons
to Kaup
cases,
independent
components.
to the
canonical
coherent
to M1
be added
the
due
waves).
that
be generalized
of one
number
is its
[a,a*]= 1
vacuum
>=
adjoint
(1)
10 > exist
s.t.
O
(2)
(a*
and
a are Fock
conjugate.
Physically,
the
operator
N is defined
as
not
self-adjoint)
vacuum
and
is a quantum
a number
or Fock
states
is given
neZ+,
operator
with
7t is the
L 2 closure
space
will
be
the
positive
eigenvalues
written
integers nez+.
of the as
(a*)nl0 n!
including
The
Fock
linear
span
<
> and
.,.
(3)
by
In>=
for each
a
state
N = a*a
and
on
They
representation
of the the
zero.
>
In > states. inner-product
312
are eigenfunctions _'_F
The
of
the
quantum
inner-product
compatible
norm
of the
number
Hilbert of the
space Hilbert
is written
as
II II= [< ", >1%,For acting
on the
Fock
any
complex
vacuum
valued
yields
the
zeC 1 the
minimum
unitary
displacement
uncertainty
coherent
operator, state
U(z),
[z > given
by
gn
Iz >=
V(z)[0
>=
e -M`/2
E
_nl.
In >
(4)
n=0
The
Fock
states
vacuum
which
complex
will
number
is the
ground
be used
here.
z is written
state
of the
In terms
SHO
for the
of c-number
minimum
co45rdinate
uncertainty q and
coherent
momentum
p the
as
(5)
z=(q,p)=q+ip
so that
C 1 corresponds
coherent
states
are
to the
phase-space
of the
Two
(1 - d) system.
properties
of the
that alz >=
zlz
>
(6)
and < zl, z2 >=
From
eq.
(7) it is clear
an overcomplete states
family
]z > provides
be written
that
the
of states,
e -½(z'
coherent OFS.
a continuous
(7)
-'2)(z_-")e-_("t_'=-'_'=)
are The
continuous L2-closure
representation
of the
in the of the
label linear
physical
z and span
Hilbert
therefore of the
space
are
coherent
which
will
as 7-/c_.
A density
operator
is a positive,
self-adjoint
operator
which
satisfies
p2= p = p. The
expected
operator
value
p,p can
of an observable
be expressed
A = A* in a state
given
additive
thermal
noise
¢e7-/with
corresponding
density
as
< A >_=<
And
(s)
can
¢,A¢
be added
>=
"by
Tr(pcA)
hand."
(9)
The
entropy
of the
system,
S,
is
by S=
-kBTr[pInp]
313
(10)
where
kB is the
use eq.
(10).
Boltzman
The
constant.
entropy
In information
is obtained
from
theory,
maximizing
one
eq.
can set kB = 1/I_2
(10)
subject
to the
and
still
constraints
Tr(p)
= 1
(11)
Tr(pN)
=c
(12)
and
where noise
ceR 1 is a parameter is Gaussian,
which
its density
labels
operator
the can
strength
of the
be expressed
thermal
state.
If the
thermal
as
p(c) = --_cl/ d2ze_,=12/Clz>< zl In 7@
this
can
be re-written
as
p(c) = 17I'C Using
Fubini's
z in plane
theorem
polar
_2ze-lzt'/%-'zl"Zoo -eFlr' >< _I Z2 n
Fock
sort
representation.
others. convention
of calculation
The
nth-order
and
the
infinite
sum and
then
expressing
A similar
was
(15)
(c+ 1)(1+ 1/_).+,
S=
This
the integral
gives
p(c) = in the
(14)
n=0
to interchange
coordinates
(13)
kB[(c
given
correlation
by
calculation
of the
+ 1)ln(c
+ 1)-cln(c)]
Glauber
function,
entropy
1°, Wolf
g(")(X1,...,
gives
(16)
Sudarshan
TM,
Xn),
with
a5 and
Glauber's
probably
by
normalization
is
G(",")(X_,... g(n)(Xl,...,Xn)
X2,)
,
:-2n
YI [G(l'i)(Xk,Xk)] k---1
314
1/2
(17)
where
the
G(i'J)'s
eralization two-slit
Green's
are
of the
classical
interference
functions
coherence
pattern
or correlation degree
is related
functions.
7 of Born
and
This
is a quantum
Wolf. 2° The
visibility
genv of a
to 712 by
v = 1 + 1_,2(_,)1 which the
physically G(nm)'s
are
represents chosen
the
extremes
object
because
is proportional
of different
In the
case
For
the
two-point
function,
g(2)(.),
s.t
G(2)(p) This
in intensity.
(is)
=
Tr[pN(NTr(pN)]2
to Glauber's
g(2)
1)1 -
in ref.
1
(10)
(19)
but
is not
equal
to his function
normalizations..
of the
thermal
the
states,
Tr[]n
><
Fock
representation
nlN(N-1)]
of g(2) for n _ 0 is given
n 2 -- n - I ....
9_)(In >< _1)=
by
1 (20)
1=---
n 2
n
Tr(ln >< nlN)] 2 A coherent
state
is very
different
from
g_2)(Iz>< zl)=
i.e.
the For
Glauber the
state
thermal
is perfectly states,
eq.
Tr[[z
(20)
><
since
z]N(N-1)]
_1=0
(21)
2
coherent.
not
surprisingly,
g(2) has
the
opposite
behavior
from
eq.
(21)
because
(2)
Tr
,
I. >< nlNf"'_]
= IzJ2 + c It is straightfoward
(25)
to calculate
the
S(z;c)
entropy
of the
partially
coherent
state
from
= -kBTr[p(z;c)In(pz;c))]
=-kBTr[p(c)ln(p(c))]
= S(c) = kB[(C+ where partially
eq,
(16)
was
coherent
used state
in the
last
is given
1)ln(c+
two equalities.
coherent
state
of the
partially
coherent < n,p(z;c)m
basis.
Let
m _> n with
density
operator
> = c -Izl2/(l+c)
()(
,_! 1/2
•
The
In(c)] pair
,
(26)
correlation
function,
9 (2), of a
by
g(2)[P(Z;C)] = l-c, in the
1)--c
_
c +,zl2 Izl2)2
m, neZ+
and
'
(27)
form
the
nrnth
matrix
element
as 1 1 1 + c (1 + l/c)('+"
z V/C(l+c)
316
2)
(
Izl_
"-re\c(1
+c)
pm
)
(28)
}
where
P_(r)
is the
polynomial
given
by
m rn! = E k!(l + k)!(m
P[_(r)
rk -
k)!
k=0
The
thermal
and
is given
or noise by the
density familiar
can
be found
from
the
c _
0 limit
of the
previous
equation
expression
< ,p(z;0)m >= -Izt z"(z*)m By
calculating
found
another
Gaussian
integral,
an
overlap
(29)
of two
partially
coherent
states
is
to be exP[l+cl+c2] Tr[/9(z
1;C')tO(z
2;C
2)]
(30)
=
(1 + cl + c2) Clearly
as cl --* c2 = c, eq.
(30)
reduces
to 1
Tr[p(O;c)p(O;c)]
Remark: Tr[(p(z,c)) The
generalization
1, 2,...,
positive,
non-zero
part
2] give ameasure
(a, z) be M k =
The
to M
x 1 matrices, M
labels
covariance
of the
entropy
of the nagnitude
modes,
with
M
let (a*, z*) which
matrix
-- -1 + 2c
= S(z;
of the departure
a positive,
finite
be 1 × M matrices,
mode. and
S(c)
(31)
Let
express
B be a given the
M-mode
c) and
the
from
apure
integer 1 the
M
state
part
coherent
x M
unit
Let
matrix
Hermitian,
of
state.
is straightforward.
non-singular,
Fock
non-unit
where M
x M,
as
M 1/21'''"
where is given
nk is the
occupation
number
'/2m
>:
of the
H k=l
(a*)nk _"
kth-mode.
I0 >
The
(32)
M-mode
thermal
density
operator
by e-a°ln(l+B-l)a
prh( ) =
det(1 1 det(B)
+ B) f
H
(33)
d2x"
z_l
317
e-x"
.S-1 _lx
><
xl
'
(34)
where
x
= (xl,...,XM)
entropy
is the
of an M-mode
M-component,
phonon
packet
complex,
is given
coherent
state
amplitude.
The
by
s(13)= kBTT[(1+ S)tn(1 + 13)- 13t_s] = S(x; 13) for both
the
incoherent
for an M-mode
and
partially
the
partially
coherent
coherent
system
cases.
which
has
the
pair
correlation
function
g(2)
is Tr[13.13
g(2)(z;13)
The
(35)
=
+ 2z* • Bz]
(Tr(13)
+ Izl2F -
(36)
limits g(2)(0,13)
= 1
(37a)
g(2)(z,0)
= 0
(37b)
and
To find
the
ek for the
partition
function
kth-mode
which
for the
partially
coherent
the
which
M-mode
intensive
---.+ C_
e2
density
occupation
vector-valued, operator
mode
energy
(38)
variable.
The
(39)
number
for the
extensive
p(z; B) which
variable maximizes
mode
energies
for the
k th mode
is
= E_
k th mode
,_'= (El,...
is a real,
the
vector
is a vector-valued,
nk is the
needs
,
Ek = nkek
where
one
is real ek --e_
and
system
(40) and
,Era)
dual the
to eq. entropy
S = -kBTr[p
318
Inp]
(41)
(39).
In this
more
general
case
the
is subject
to the
constraints
Tr(p)
= I
Tr(pa)
,
= z
Tr(pa*)
(42a) ,
= z"
(42b) ,
(42c)
and Tr(pa*
Let
fl
given
be
an m-vector
covariance
with
matrix
the
7
value
for bosons
( A similar this
is not
argument needed
for here.)
= (e-
fl =
I/kBT
e--_
for
(42d)
each
component
and
express
the
....._
_' e
Fermi-Dirac The
z* ---*ez+
as --._
B = B(fl)
a)=
partition
_ 1)-1
= (e-_E
particles
would
function
Z(fl,
_ 1)-1
replace V)
-1
(43)
by
+1
in eq.
(43)
but
is now
1 Z(fi,
at thermal of the energy
equilibrium
equilibrium
and
V) = det(l-_--flE)
V is the
thermodynamic
volume
(44)
of the
quantities
can
fluid
radiation.
be calculated,
From
for example
eq.
(44)
all
the
average
g is Ere-ZE" g =
when
plus
_
Oln(Z) = -
z
(45a)
Off
classically Z(fl,
V) = _
e -_Er
(45b)
T
and
in quantum
pressure
statistical
mechanics
a Trace
over
matrix
elements
of e -fill
is taken.
The
p is
P-fl
10ln(Z)
OV
319
(45c)
and
the
(reversible)
differential
work
dW
is
dW
and
the
previous
equation
can
be used
entropy
of eq.
(10)
can
also
(45d)
to obtain
dW -
The
= -pdV
I Oln(Z) dV fl OV
be written
as
s = kB[Z_(Z)+ Z_ and
the
Helmholtz
free
energy
(45e)
is
F(_, V) = -_Zn[Z(fl, V)] The
coherent
state
density
operator
p(z)
Near
equilibrium,
deviations
from
exercise
3.
a Kubo
Conclusions
a reservoir
derive and
the
contrast
approaches.
Kubo
linear
response
where
to a future
and
A two-component with
= det [(1 - e-'E)e
equilibrium
will be left
can be given
the
results
These
future
-(a*-z*)zE(a-z,]
theory
21 can
fluctuations
(46)
be established
will be equal
was
by a coherent
with
both
studies
formulated
state
of phonons.
theorem the should
for a fluid One
for this system,
Langevin
equation
illuminate
and
(or simplify)
methods. Much
remains
to the
by studying
small
dissipations.
This
Outlook
fluctuation-dissipation
these
as
project.
thermodynamics
radiated
(45f)
to learned
from
harmonic,
SHO,
320
systems.
in thermal
future another the
project
will
be to
will be to compare
stochastic the
equilibrium
lattice
quantization Monte
Carlo
4.
Acknowledgements
This work of Physics
was supported
and
Astronomy
at Chalmers
University
part
analysis
of this
wisdom.
Brander
electromagnetic
in part
by AFOSR
at Missouri of Technology.
to one of us (BDF) and
DeFacio
grants
University
and
Professor long
(unpublished)
Karl ago have
scattering.
321
90-307 the
91-203,
Institute
Erick
in 1983-4. also
and
the Department
for Theoretical
Erickson
of Chalmers
He is thanked
applied
this
analysis
Physics suggested
for sharing
his
to polarized
References
1. G. Kirchhoff,
Ann
2. M. Planck,
Verh.
3.
A. Einstein,
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6.
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664 (1926).
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of Quantum
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(Springer,
Berlin,
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J.R.
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9. V. Bargmann,
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10.
R.J.
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11.
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1985): 12.
R.P.
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J. Opt
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17.
D.J.
Kaup,
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18.
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J. Glimm,
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M. Born
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R. Kubo,
and New
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E.L.
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Phys.
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187 (1961).
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2766
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11,
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3135
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269 (1992).
ibid A3, 20,
Reading,
76 (1986).
186 (1981).
(1988). 2063
Coherent
(1987). Nonlinear
Interactions
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1977). 33,
626 (1991).
Wolf,
Principles
Soc.
Japan
of Optics
12, 570 (1957).
322
4E (Pergammon,
New
York,
1970).
N93-27343
Wigner
Expansions
for
Partition
Relativistic
Functions Oscillator
Christian Leipzig
Unwersity,
Department
Leipzig
University,
Department
and
Systems
Zylka
of Physics,
Giinter
of Nonrelativistic
0-7010
Leipzig,
Germany
0-7010
Lespzig,
Germany
Vojta
of Physics,
Abstract The equilibrium quantum statistics of various anharmonic oscillator systems including relativistic systems is considered within the Wigner ph_e space formalism. For this purpose the Wigner series expansion for the partition function is generalised to include relativistic corrections. The new series for partition functions and ell thermodynnmic potentials yield quantum corrections in terms of powers of/t _ and relativistic corrections given by Kelvin functions (modified Hankel funktions) K_,(mc2/kT). As applications are treated the symmetric Toda oscillator, isotonic and singular anharmonic oscillators and hindered rotators, i.e. oscillators with cosine potential.
1
Introduction
In recent quantum
years, the Wigner formalism as a phase space representation of quantum mechanics, field theory and quantum statistics has found growing interest [1,2]. One of the main
fields is the theory of anharmonic oscillator systems modelling various quantum systems, e.g. solitonic systems, quantum field theories, and transport processes in more complicated systems. The purpose
of this paper is twofold,
1. to generalize the Wigner series expansion of equilibrium order to include special relativistic systems, 2. to show applicability
and
utility
of the formalism
phase
by means
space
statistics
in
of various examples.
Up to now a small number of papers on relativistic quantum theory have appeared for equilibrium as well as for nonequilibrium systems. nonequilibrium processes range [4]. In the realm of relativistic
quantum
in the Wigner formalism The topics treated for
from multiparticle production and kinetic theory [3] to cosmology phase space quantum mechanics there exist only a few papers.
Janussis et al. [5] starting from the Dirac Hamiltonian introduce a Wigner function with 4x4 spinor components. Ali [6] discusses the quantization of relativistic systems on phase space whereas Kim and Wigner [T] describe a covariant phase space representation for harmonic oscillators. A relativistic Fermi gas is treated in the frame work of the grand canonical ensemble by Greiner et al.
323
[8]. A pseudo-classical
phase
variables was given in several
space description
of the relativistic
versions [9]. There
are also papers
electron
in terms
on relativistic
of Grassmann
quantum
kinematics
[lO,11]. Over the years, fundaamental problems regarding the correct description of simple systems such as the classical relativistic harmonic oscillator and its quantization have been treated in a number
of publications,
a symplectic
see f.i.
formulation
[12-15].
of relativistic
Important quantum
papers
a Wigner formalism which is frame dependent. The i.e., we use the so-called synchronous gauge [19].
2
Relativistic
Wigner
on the covariance
mechanics
has been
problem
given in [17,18].
focus of our attention
are [7,16] ; We develop
is the comoving
frame,
Formalism
One of the advantages of the phase space methods of quantum statistics - Wigner-Weyl formalism [1], coherent state methods [20], Husimi transforms [21] or Bargmann representation [22] - is the possibility to evaluate exactly the partition function by means of a phase space integration. In practice it is convenient to expand the phase space integral into the Wigner series in powers of h 2 [23] as a basis of semiclassical quantum statistics. The (non-relativistic) Wigner function fw(q, p) can be defined as the Fourier transform off-diagonal elements of the density operator (qlP [qJ) (for systems without spin) [1]: 1 /w(q,p)
f
f
z
(2xl_) / J dzexp(ipzll_)(q-
Here f is the number of the degrees of freedom given in the coordinate representation by (q[P
z
_ l p [ q + _)
of the system
]q') = E
of the
considered.
(1) The
density
operator
¢_(q')w,_¢_(q)
is
(2)
f_
with w,_ = exp (-flE.)/Z,
where
H¢,_ = E,_¢.
Z = Tr exp (-fl/:/)=
,/3 = 1/kT, E
and the partition
function
is
exp(-flE.)
(3)
I't
Within
the frame of the Wigner
formalism Z
where
flw - (exp(-fl/:/))w
Bloch equation
for the calculation Oflw( q, P; _)
Off
function
can be expressed
as
f/dqdp w(q,p; _)
is the Wigner
aw(q, p; fl) The
=
the partition
equivalent
dz exp (ipz/h)(q of flw = -Hw(q,
of the operator
_ -- exp(-fl/:/)
- _ ] fi [ q + _)
within the Wigner p) cos(_A)_w(q,
,_
324
(4)
,
formalisms p; ,8)
defined
as
(5) reads (6)
here Hw(q, p) is the Wigner operator
or symplectic
equivalent
differential
of the Hamiltonian
_I,
and A denotes
the Poisson
Bracket
operator
A-
o_-o'_ "
c3"- _-'
Op aq
aq Op
(7)
acting in (6) to the left and right hand sides. Up to now the formalism is completely exact. The partition function (4) can be calculated by means of a phase space integration without any knowledge of the energy levels of the system infinite series of Boltzmann terms exp(-BE,,), In order to elucidate of freedom characterized
considered and without the zero-point energy
the principle of our formalism, we consider a system by a Hamiltonian H = H0(lfi) + V(_) with
H0( ) = and
the potential
the partition
energy
function
V(q)
quantum
system
(8)
2
-- Ho(p) + V(q).
The Wigner
series [17] for
by
'//
dqdp
exp(-_(H0(p)
+ V(q))}
E
h2" +,(q,p;/3)
(9)
rt----0
quantum
correction.
Solving
the Bloch equation
(6) for our relativistic
yields @t(q,P;3)
where the primes
denote
-8/_2H_ ' V" +
differentiation
with respect
ourselves to the first quantum correction. We will evaluate (9) with (10), this approximative begin
with a single degree
+oo
27ch
with @0 = 1 , @, : n-th
+ 2c2-
. One has Hw(q,p)
Z is given
g -
the necessity to sum up an beeing already included.
_4B3(H_ 'V ¢ + H_
V")
to the corresponding partition
function
(10) variables.
is denoted
We restrict by Z0z • We
with the p-integration: Zot(k,,_)
-
with
Zo(_,_) =
and
Zt(k,.) =
Substituting
p/mc
(11)
Zo(k,.) + Zt(k,.) 1 J e_aso(P)dp 2t"-h
i
+
= sinh tt and using
1/93(H_'V_+H_V'_dP
[43]
oo
(12)
e b¢°'h_ cosh utt dt_ = K_(b) 0
where
the K_ denote
the Kelvin
functions
(modified
Hankel functions)
Zo(_,n) = 2_e+O':mcAq(3mc
325
_)
one gets (13)
and after
some algebra Z,(k,_)
=
22,,rh_--i _+_"dch2r-ln2v" +t8,.,
x
[21-_sK,(_mc' )-
+
2_e+_'_C_mch_3V"c2x
x
[-'_28Kl(_mc_)
2_/93V'_]x
-_28t¢,%(_8mc')+ _28Ks(,Bmc')]+
+ _28Ks(_mc_)
(14)
- l-_ Ks(_mc')]
where the series expansions [1 + m2c p2 2l-S/2 = [1 + sinh 2 u] -s/2 ~ -- 1 -
3 5 sinh 2 u + 15 8 sinh _
(15)
and
p_
p2 1-'
(16)
m--T-c_ [I+ --m2c_ = sinh_ u[1+ sinh2 u]-I ~ sinh_u - sinh4 u have been used.
Now the integrationover the momentum space isdone, and (13),(14) represent,as a main result, the partition functionup to thefirst quantum correction, proportionalto h2 ,and relativistic correctionsup to second and fourth order in p/mc . To proceed furtherone has to take into account the potentialV(q) of the system considered. Then the fullpartitionfunction(in our approximation) isgiven by Zo1= Here we only mention
that
Zo + Z, =
(17)
dq exp(-_gV (q))Zo,(k,.)( q)
with
in Zo(,u,-,) and
Kl(z) ~ --
e-* (1 + 3_)
_2_
in Zl(k,,O the non-relativistic
3
Toda
'
K3(,) =
_Z e_,(l+_.z)_r
limit is reproduced
35
'
Ks(z)_W_z
e_,(l+_"
99) 8Z
correctly.
oscillator
As a first example
we choose
the symmetric V(q)
relativistic
quantum
= 1Io (coshaq- I)
Toda oscillator
with the potential (18)
Itseigenvaluesare not exactly known. Ifwe compare (18) with the potentialV(q) = m wg q_ / 2 of the harmonic oscillator we find a=
_/m_2olVo
326
(191
wherew0 is the (circular) frequency for harmonic (i.e. very small) oscillations. Toda molecules are discussed in [24], generalizations thereof in [25]. There exist a few papers on quantum statistics of toda chains [26] and Toda fields [27]. The non-relativistic phase space quantum statistics of the symmetric Toda oscillator with (18) is treated in [28]. The dynamics of non quantum relativistic Toda
lattices
The part
is the topic of [29-31].
Z0 of (17) becomes 1 Z0 =22- _
and the part
Zx can be expressed Zl
=
2h _xhc2 1
e_'_c, mc Kl(flmc _) e_v° 2_Ko(_Vo)
as
e _"c2 Kz(_V0)
1 206 × {-_[_g,(flmc rnc2fl 12 [__28
+
(20)
a
_2 V0 e _v°
_) - _K3(_,_c
Kt(19rnc,)
x
_)+ _K_(fl,_c_)]
+ 12 5"_68 Ka(flmc2)
-
+ (21)
l_KS(flrnc2)]}
The formulae (20) and (21) yield in the nonrelativistic limit kT ,_. rnc 2 the correct partition function of [28]. The evaluation of higher order relativistic and quantum corrections is straight forward. From the Wigner series for the partition function, corresponding series expressions for the thermodynamic potentials follow in the well-known manner. These results are published in
[32]. 4
Isotonic
Our next
or
example
Singular
is the quantum
Oscillator oscillator V(q)
(normalized anharmonic tions:
with
the potential
q = Vo I-_-
energy (22)
__]_
so that the potential minimum at q_,,_ = a gives V(q_,n) -- 0). This asymmetric oscillator appears in the literature under various headings in two different interpreta-
1. isotonic dratic 2. singular
oscillator oscillator oscillator
[33-36], radial
oscillator
(with
centripetal
oscillator
[42].
barrier)
[37-39]
or inverse
[40], [41] or nonpolynomial
Interesting subject for oscillators of this type are the energy spectrum [38], canonical mations [37], ladder operators and coherent states [34-36], phase space path integrals dynamics discussed
qua-
transforand the
and symplectic groups [37,41]. The connection with three-dimensional potentials was thoroughly [39]. Here we add the second quantum correction to the partition function: 1 Z(h i) = _
m
e -_v(q) _ O(q) 240m 2
327
dq
(23)
where Q(q)=-V(1)
+
here is v' - OV(q)/Oq, integrals to be calculated
/_(2V'V(3)')
"" _V'_)
-
V(s) - V",
and so forth. are of the type [43]:
1_16_2V_VIl+
For the isotonic
2-_'j_3V
oscillator
'4
potential
(24)
(22)
all
O0
(25)
e-t
+M-1Vt]
ko) > M < (_-
_
x0); (/_-
+constant _)
>t +constant
^
< _;p>/_olM/)o
< _;/_ >* D o' +constant
1
< _;i_>
1 + constant
o)
S < _; ig >t]Do'
0o'
(0
wO)Uo<
_;i_>'
+ constant
00-']/)o'
+constant
(. 0)
']0o'bo'+constant
0
^2 ^-1 ^-x _ +pi)]Uo D0
+constant
i=l
/)0Uo[_)-_w,_N_]/_)ol/)o
I + constant
i=1
DoUo/?/o/)o-1 where
VM-'
-- -
< £o;_
(21)
bo ' + constant, >, Do =/)(<
342
2o;/¢o >) and
Uo = O(S).
Without
loss of generality,
we can always
drop
the
constant
term
and
consider
= b000 o0o,bo1 It is easy
to see that
the normalized
ground
state
(22)
of this
Hamiltonian
is:
bo&oi0)_ Do exp[_(-mp)]lO), which the
is a SqCS
squeezed
state
Therefore since
(22)
and
(23)
any
trouble,
will not
4
those
take
(23)
because
appear
space,
density
operator
state
and
that
rotations
definition in (23),
of the the
multimode
correspondence
of S, hence/_'o,
S is unique
correspond
1) and
cases.
non-uniqueness
shown
(Uo = 1, Do _
up to some
to exp[_(-rnn)]
SqCS. between
will not cause 2-dimensional in U(S)
which
in (23).
Thermal
immersing
bath
coherent
as a unified
The
we have
Multimode
a heat
the
wi's in H0 do not appear
is many-to-one.
in phase
Consider
it contains
(U0 :_ 1, Do = 1) as two special
we can
However,
rotations
in general,
(23)
a physical
of temperature
T.
of this system
Squeezed system
This
Coherent
described
constitutes
by
the
Hamiltonian
a canonical
ensemble
States (22) and
in the
is:
=
Z-'
exp(--/3ft)
=
Z-'
exp[-#(bo/)o/T/oOo'Do')]
=
Z-'Do/Qoexp(-/3/-t/o)Uo'bo
_,
(24)
where Z = Tr[exp(-/_/:/)] This
density
operator
_ describes
a mixed
(25)
= Tr[exp(-_/:/o)]. state
unless
T = 0. In the
limit
as
T --_ 0, since lim
exp(-/3/:/o)
= [0)(0[,
(26)
/3---*oo
we have
_= bo&o{0>_ 1 for all i = 1,2,...
uncertainty
function
(59)
principle. corresponds
same
decohered
(65)
canonical)
transformation
on
0)
that
characteristic
7
1(7- 0
to a symplectic
a covariance
This
following
Noticing
the
coordinates,
is also
the
PA = Tr(s)(_AB)
non-physical
K=_a-1 Since
(64)
ri > 0, for all i = 1,2,...,rn.
a further
correspond
rO rO)a'
,00,
in another ,m,
the
canonical
otherwise
Therefore
(59)
we conclude
coordinates. will give a state
that
to an m-mode
thermal
(1) A unified
construction
the
reduced
SqCS.
Conclusion The
results
(thermal) SqCS
of this
SqCS's.
is still
paper
(2) Proof
are threefold of the
statement:
The
decohered
a (multimode)
thermal
SqCS.
(3)
characteristic
function,
which
is very
technique
via
to many
related
problems.
350
Introduction efficient
of multimode
multimode of the and
can
thermal decohering be applied
Acknowledgement I would valuable
like
to express
comments
and
my sincere
gratitude
to Professor
G. F. Chew
for his
advice.
References [1] A. O. Caldeira
and
[2] W. H. Zurek,
Phys.
[3] J.
in
B. Hartle,
Eds.
D. Han,
3135,
1992)
[4] J.
R.
Y. S. Kim
and
Mathematical and
[6] W.
M. Zhang,
[7] Y.
S. Kim
Group [8] X. Ma
Theoretical W.
Oz-Vogt,
references
and
Noz,
Rhodes,
States
Zachary
and
Conference
Opt.
34
(World
Scientific,
M. Revzen,
Relations, Publication
Applications
Singapore,
1985)
Mod.
Picture
Phys.
of Quantum
Singapore,
4625
J. Mod.
62,867
New
Groups,
York,
[11]
R. G. Littlejohn,
[12]
Y. Tikochinsky,
[13]
E. P. Wigner,
Lie
NY, Phys.
and Some
1974) Rep.
J. Math. Phys.
Algebras,
Rev.
138,
Phys.
193 (1986)
20, 406 (1979)
40,749
351
(1932)
(1990)
Mechanics:
1991)
(1990) Opt.
38,
2339
(1991),
therein. Lie
in
709 (1987)
Rev.
Space
A 41,
States:
Scientific,
Phase
Rev.
Uncertainty
(NASA
(World
R. Gilmore,
Phys.
and
Coherent
J. Mod.
Approach
A. Mann
R. Gilmore, (Wiley,
D. H. Feng
(1983)
36 (1991)
Skagerstam, Physics
E.
A 121,587
Squeezed
and W. W.
B.-S.
M.
Physica
(no.10),
on
P. L. Knight,
and
and
44
Workshop
and
[5] R. Loudon
[10]
Today
Klauder
Physics
[9] J.
A. J. Leggett,
of Their
Applications
and
[14]
M. Hillery,
R. F. O'connell,
M. O. Scully
and
121 (1984) [15]
B. R. Mollow,
[16]
L. Yeh
and
Phys.
Rev.
Y. S. Kim,
162,
1256
LBL-31657
352
(1967) (1991)
E. P. Wigner,
Phys.
Rep.
106,
N93-27345 Quantum
Harmonic
Oscillator
Yuhong
for Biologics Food
and 8800
Bethesda,
A Thermal
Bath
Zhang
Biophysics Center
In
Laboratory Evaluation
Drug
and
Research
Administration
Rockville MD
Pike
20982,
USA
ABSTRACT
In this tum
talk,
Brownian
harmonic it to the
we briefly motion.
oscillator problem
review
the influence
We report
on a newly
coupled
to a general
of loss of quantum
functional derived
environment
coherence.
353
exact
path-integral master
at arbitrary
treatment equation temperature.
of quan-
of a quantum We apply
INTRODUCTION
Recently motivated them
there by
motion
system
early
universe
back the
The
the
matrix
of the
These
is always
local
colored
are
been
It corresponds
Brownian
contribution
for the
a general
due
to system-
of quantum
play
and
Brownian
transition
in semiclassical important
galaxies
its environment
colored
only
time
evolution
derived
class
noise
studied
roles
is quite
of an
theory
formation
of
in particle [4]. In these
complicated
for the
by different
environment,
The
Brownian
extent.
equation)
a linear
classically.
in quantum
to a limited
before
of ohmic
to having
particle
reported
reduced
density
environment [6].
different
natural
frequencies.
(Brownian
particle)
from
first
In our
model,
principles
in this matrix
(with
perature
done
(the
all in the
Among
giving
noise
limiting
reduced
density
with
different
authors
force
associated
the
dissipation
proportional
with
is
In some
for which
damping
motion
the
to the
dissipation
is
at low temperature.
Our
system
been
has
dissipation
It was
systems.
coherence
application
appear
inflation
and
equation
motion
cases
of the
has
in quantum
also
and
motion.
noise.
dissipation
master
[5].
colored
which
Brownian
methods.
velocity
a system
Brownian
of quantum-to-classical
issues and
between
newest
issue
These
transition,
effects
The
the
fluctuation
and
problem,
quantum
where
phase
of nonlocal
an outstanding cases,
reaction,
in quantum
[1], loss of quantum
a few.
[3].
noise,
dissipation
effect
dissipation name
important
interaction
to nonlocal
with
cosmology,
in which
interest
of macroscopic
[2], just
is very
production,
tion
tunneling
interaction
open
considerable
observation
is in quantum
problems,
been
possible
are quantum
environment
rise
has
is the
of a Brownian
a general the
talk
thermal
environment
The is brought of statistical
harmonic bath
is a set
environment to contact and
derivation
is at with
quantum
354
of an oscillator
spectral
density)
of bath
thermal
physics
with
master
linearly at
harmonic
a thermal this
exact
to tem-
oscillators
with
The
Feynman
coupled
arbitrary
equilibrium bath.
equa-
state. derivation path-integral
The is
method can
and
Feynman-Vernon
accommodate
kernel.
all possible
It is a linear
non-Markovian
subohmic
efficients
numerically. special
Let open
and
functional
of the
differential
resides
We show
in these
nonlocal
master
previous
and
dependent
In particular
and
master
compute master
nonlocal
noise
coefficients.
The
we examine these
time
equations
equation
the
dependent
obtained
cases co-
otherwise
equation.
FUNCTIONAL
us briefly
system.
review
Consider
the
Feynman-Vernon
a Brownian
influence
particle
with
mass
_2. The
environment
is modeled
by a set of harmonic
frequency
con.
Brownian
particle
strength
This
kernel
time
coefficients.
all the
[7].
dissipation with
environment
that
of our
formalism
equation
superohmic
examples
INFLUENCE
forms
partial
character
of ohmic,
are just
influence
C,,.
The The
S[x,
total
q] = S[x]
action
of the
is coupled combined
+ SE[q] + Sin&,
2
functional M
= 1 and
oscillators linearly
system
formalism
plus
to
natural
(bare)
with
mass
each
bath
environment
of quantum frequency
rn,, and
natural
oscillator
with
is
q]
x
n
"_mnqn
-
_rn,_wnq,_}
(1)
n
where
x and
q,, are
It is well known environment
_(t)
the that
coordinates the
is governed
time
of the
particle
evolution
by the
following
of the
and
the
total
quantum
n-th
density Liouville
bath
oscillators.
matrix
of the
system
equation
. a^ ,h p(t) = [9, In the
coordinate
representation,
the
solution
of the
355
plus
(2) above
quantum
Liouville
equation
can
be written
as +o¢
+_
+co
+oc
(3) --00
--00
--0_
--00
x J(xl,qf,xrl,q},tixi,qi
,
' xi,qi
'o) p(x_, qi, • xi,' ,
'o)
qi;
where J(xf,qf,xll,q},t
Ixi,qi,x_,q[,O)
=-/Dx/Dx'/DqJDq
is the
propagator
full set of bath
of the
total
oscillator
exp_
density
matrix
coordinates
and
i {st_,q]- s[_',q']}
in path-integral
the
subscript
form.
i and
Here
f denote
the
(4)
q represents initial
and
the final
variables. We under
are
only
interested
in how environment
the
influence
of the
information
is the
reduced
density
the
(all
which
is propagated
I) =
in time
--00
If we assume
that
at t = 0 the
system The
(the
Brownian
quantity
particle)
containing
this
system
+oo
/
cIq / --00
the
clq I p(x,q;xl,q')_i(q
evolution
-
ql)
(5)
operator
-boo
I.x, I.x:
pr(Xf,x},t)
of the oscillators).
of the
--00
by the +oo
bath
matrix +oo
p_(x,x
dynamics
ix,, .:, o) ..,
studied off),
a free three-level and
the following
Re <
system
a three-level
system
correlation
functions.
>,&(t)
(i.e., coupling coupled
= Re <
results give good agreement with analytic results in these limits. For the special case of an asymmetric two-level system, we have calculated functions, and compared the results with the symmetric case[4]. Figure
>
to an
(14)
Our
for an asymmetric J13 = J23 = 0.
case We find
with from
the
parameters
Figure
1 that
E1
= 0, E2
the
coherent-incoherent
367
= 2K,
E3
=
the time-correlation 1 shows the results oc,
transition
and
J12 = -K,
occurs
around "'
r//h = 0.4, or (_ = 2T//hTr = 0.25. For a symmetric two-level system[4], the phase boundary for this temperature was at (_ = 0.4. Thus, the whole phase boundary is expected to lie at smaller for the asymmetric case than for the symmetric case. In the asymmetric case, the symmetry is already
broken,
thus
the coherence
is easier
to break
!
I
than
in the symmetric
case.
0.7
0.6
V
V.9
0.5
I
0.4 0.
1.0
!
2.0
3.0
t FIG. 2K,
1 The
correlation
X3K = 2.5, with
enumeration
function
rl/h =0.3
of spin paths
with
C1(_) for an asymmetric
(circles), p=2
0.4
0.4(squares), and
two-level
0.5 (triangles)
model
obtained
of E2 = by exact
q=7.
I
I
p
0.3 v-W
v
0.2
w
¢q TD
0.1 t'o I9
.
0.
0.5
1.0 t
FIG. gles)
2 The
correlation
for a three-level
functions
system
in the
Cl(t)
(circles),
coherent
region,
C2(t)
(squares)
and
J12 = J23 = -2K,
C3(t)
(trian-
J13 = -K,
E, = o.5K, E_ = K, E3 = 0, _K = 0.25,_,o3= 2, tlo_ = _3 = 0, (to = I:/,_c), _713/_/
:
1, T]'2/5
= _23/_
__ 0.
Figures 2 and 3 show results for a three-level system. Carlo simulation, with the Trotter numbers p=2 and q=10. 368
The calculations were done by Monte As has been found in quantum Monte
Carlo simulation of spin systems[8], to define 1 Monte Carlo step (MCS) as all the
possible
flip, 1 global whether the of the
weight
simulation,
exp(_)
of 10 6 MCS,
sign cancellation, defined
In our
we define
1 MCS
as 1 single
spin-flip,
1 double
spin-
spin-flip (i.e., flips all the spins), and 1 spin-flip of random length. We determine spins should be flipped by the standard Metropolis algorithm, using the modulus
complex
simulation
flips.
for systems having sign problems, it is usually more efficient to be a small subset of all the possible flips than to define it
for the
taking
about
we measured
transition 14 minutes
the quantity
probability.
In this
way,
on the
X-MP.
To estimate
r, the
Cray
remaining
ratio
we have
(related
carried the
to the
out
degree
negative
of
ratio
in [8]). Z+ - Z_
r Here
Z+ denotes
same
for absolute
signs
is large,
more
accurate
imaginary
the
sampled
values
leading
of the
of negative
positive
real parts
to inaccuracy
results.
parts
sum
real
parts
of the
of the weights.
in the data.
In this definition,
of the
(15)
z++z_ If r is small,
If r is large,
we are ignoring
weights,
and the
the cancellation
the
effect
Z_ denotes
the
cancellation is small,
of the
thus
of the cancellation
giving
due to the
weights. I
I
0.5
1.0
0.4
_----., 0.3
0.2 r.)
_,._ 0.1
0o
--
0.
t FIG.
3 The
correlation
functions
Cl(l),
the incoherent region. Jj, Ej and_ r/13/h = 1,771_/h = r1231h= 2. For the free three-level E_ = 0.hK, distance correlation
E2 = K,
between function
followed
by the energy
the
role
1 and
reaching
of dephasing
could
2 after of the
state
J12 is stronger the
to a system
3 is shorter
t. The
1, El. than
coherence,
Starting
preventing
than
tells the
energy
J13, then
C3(t)
the parameters,
> approximately time
and
as in Fig.
correspond
2, or 2 and
< n_(O)n2(t)
1 and
the exchange
H0, we assume
E3 = 0. This
states
at state since
Hamiltonian
C_(t)
are same
of the from
gradually the
369
the rate state
the state
for a three-level 2.
J12 = J23 = -2K, of redox-sites distance
in
of the
electron
-K,
3, where
1 and transfer
1, the electron
population
=
J13
1, 2 and
between
2, /]72, is assumed
goes to state
electron
system
tic3 = 2, t_ _ = t_ 3 = 1,
3.
The
starting
to be highest, moves
to state
3. In Fig.
3, the bath
from
back
going
2, has
to the
original state
state.
This
is a very
brief
picture
of electron
transfer
over 3 states,
strongly coupled to the initial and terminal states. As for the effect of the sign cancellations, the remaining
Fig.2 and incoherent
5
ratio
with
r defined
the intermediate
in Eq.
15 is 3% for
13% for Fig.3. The magnitude of error is about 0.1 in Fig.2 and 0.02 in Fig.3 case has less effect of the exchange K, thus leading to less sign cancellations.
The
Summary
We have
briefly
described
metric two-level system the coherent-incoherent a three-level
system,
mediate
high-energy
system.
Further
6
the
numerical
calculations
of the time-correlation
functions
of an asym-
and a three-level system. For an asymmetric two-level system, we find that transition occurs at smaller friction r/ than for the symmetric case. For
we calculated state.
the population
We observed
application
transfer
of the
a coherent-incoherent
of this model
electron
transition
will be discussed
when
there
similar
is an inter-
to the
two-level
elsewhere.
Acknowledgments
This research was supported were done on the Cray X-MP thank
John
N. Gehlen,
by the National at the University
Chi Mak,
and
Science Foundation. The numerical of California, Berkeley. The authors
Massimo
Marchi
Soc. 24,966
(1956).
for helpful
calculations would like to
discussions.
References [1] R. A. Marcus,
J. Am.
Chem.
[2] A. J. Leggett,
S. Chakravarty,
A.T.
Dorsey
et al, Rev.
[3] C. H. Mak,
D. Chandler,
Phys.
Rev.
A 41, 5709
(1990).
[4] C. H. Mak,
D. Chandler,
Phys.
Rev.
A 44, 2352
(1991).
[5] C. H. Mak,
Phys.
[6] B. Carmeli,
D. Chandler,
J. Chem.
[7] M. Takasu,
S. Miyashita,
M. Suzuki,
Prog.
[8] M. Takasu,
S. Miyashita,
M. Suzuki,
Springer
Rev.
Lett.
[9] E. Y: Loh,
J. E. Gubernatis,
[10] N. Furukawa,
M. Imada,
[11] N. Hatano,
M. Suzuki,
[12] W. H. Newman,
J. Phy.
A. Kuki,
Phys.
59,
1 (1987).
68, 899 (1992). Phys.
R. T. Scalettar,
Phys.
Mod.
Soc.
Lett.
J. Chem.
Jpn,
A 163, Phys.
82, 3400 Theor.
(1985) Phys.
Series
75,
in Solid State
S. R. White
et al, Phys.
60, 810 (1991). 246 (1992). 96, 1409
37O
1254 (1986).
(1992).
Sciences Rev.
74,
114 (1987).
B 41, 9301
(1990).
VI.
GROUP
REPRESENTATIONS
371
N93-27347 SYMMETRY ANISOTROPIC
ALGEBRA OF HARMONIC
O. Castafios and Inniitulo de Ciencian Apdo.
Po_taI
70-543,
A GENERALIZED OSCILLATOR
R. L6pez-Pefia Nuclearen, UNAM
Mdzico,
D. F.,
04510
Mdzico
Abstract Itisshown thatthesymmetry Liealgebraof a quantum system with accidental degeneracy can be obtainedby means of theNoether'stheorem. The procedureisillustrated by consideringa generalized anisotropic two dimensionalharmonic oscillator, which can have an infinite setof stateswith thesame energy characterized by an u(l, I) Lie algebra.
1
Introduction
We are going
to study
the accidental
degeneracy
[1,2] of the
Hamiltonian
1 _(p_ + x_) + AM
(1.1)
i
which is a two dimensional harmonic oscillator plus the projection of the angular momentum in the z direction, M. We use atomic units in which h = m = e = 1 and A is a constant parameter. This quantum system, for A = 1, describes the motion of an electron in a constant magnetic field [3, 4] and
its corresponding
symmetry
Lie algebra
[4]. A procedure that use the Noether's theorem [5] of the hamiltonian systems (1.1), for rational values (1.1) represents a generalization of the degeneracies harmonic oscillator [6,7]. For the purpose of the paper it is convenient to creation
77, and
annihilation
_i operators, 1 '7+ = _('71 "4z
with
with
+i'7_),
has been
discussed
by Moshinsky
et al
is established to get the symmetry algebra of the parameter lambda. We show that present in the anisotropic two dimensional introduce
appropriate
combinations
of the
i = 1, 2, i.e. 1 _+ = _(_1 x/z
(1.2)
_i_),
the properties
[_o,_b]=[V,,'Tb]=0; It is straightforward
to find
[_,,'1b]=6,_,
the expression
('7,)t =_,,
of the hamiltonian
(1.1)
"= +,--. in terms
H = (1 + _)N+ + (1 - _)N_,
PAGE
BLANK
NOT
operators
(1.4)
373 PREGEDING
of these
(1.3)
FILMED
where
a constant
term
was neglected
and N,,
denotes
the number
of quanta
in direction
a.
The
eigenstates of (1.4) are well known [4] and its eigenvalues are given by E,,m = v + Am; with ]m I = v, v - 2... 1 or 0 and v denoting the total number of quanta. From this expression, it is immediate that there is degeneracy for rational values of A, which can be defined as follows
,_ = ___Av =
v I - v,
Am Thus
the
accidental
to the strength
degeneracy
associated
of the parameter {A=+l},
For the
cases
(1.6a,
b), there
anisotropic
), the corresponding of the hamiltonian
2
(1.4),
we apply
its corresponding
Lie
can be classified
number
of ieveis
In the section
with
theorem
lagrangian
is given
three,
are responsible and remarks
Algebra
Noether's
according
(1.6a, the
of levels with the same energy. Classical Symmetry Lie algebra
oscillator.
Symmetry
(1.4)
{-1, because they form a complete set of commuting not all operators in (2.8) make sense all the time. According
we consider
three
(i) For commutation
A = 4-1, we have two sets of operators, {I,N,,,_2,$J} relations correspond to the direct sum w(1) @ u(1).
operators. This let us to the previous section
cases: and
{I, N2,,zl,z_},
whose
(ii) When A > 1 and A < -1, the set of constants of the motion (2.9), its quantum version, however this is ambiguous for the constants (2.9c,d) from them the F5 and F6 functions. It is easy to evaluate their commutators
must be replaced by and so we eliminate and get an algebra
but to identify a Lie algebra a redefinition of the constants is achieved by constructing the new operators [7]
must
-
(N,)!)
of the
motion
be done.
This
(_)k,,
(3.2a)
_' z'--(s')k' (l_ j (_:(N,)I k-')!_ ] where
[xJ denotes
the largest
integer
_< x.
From
the
Lie algebra A1
that
satisfy
---
N1
is identified N2
__
the commutation
by considering
J_'5
,
-t-t
---- Z 1 Z 2
the
were
Hilbert
generating (iii)
evaluated space
by using
of the
to check
that
(3.3)
the following
/_'6 = _q,_2
,
@perators
¢I = 1 (N1 -t-/_/2 + 1) 2
(3.4) "
relations
[_l,k0] = t-,, These
_
it is easy
= L_J.
£ = _, Then
(3.2)
(3.2b)
system,
[¢1,/h] = -/-,'0, [I_,,/_6] _" _" = -2¢_.
that and
[Z_,_] they
the invariant subalgebra. Finally for -1 < A < 1, the
= 60, are
which
is valid
the generators
symmetry
algebra
for
any
of a u(1,1) can
be found
(3.5)
state
Inx,n2
Lie algebra, by
> of
with
considering
hi the
operators
1(N1 -/_'2) Evaluating
the commutation
relations
between
these
operators
(3.6)
we have (3.7)
377
and the operator
h2 is the ideal
of the algebra.
Thus
we get for this case a u(2) symmetry
Lie
algebra.
4
Conclusions
We have established of a quantum that determine
a procedure
that
uses Noether_s
theorem
system with accidental degeneracy. First, the constants of the motion. Second, once
to find the symmetry
Lie algebra
we solve the differential equations we have chosen the minimal set of
constants of the motion that close under Poisson brackets, t ° identify the classical Lie algebra we need in general to form combinations of the selected Noether charges. And third, to find the corresponding quantum counterparts. Lie algebra can be done immediately
Afterwards, the identification of the quantum by making the standard replacement of Poisson
symmetry brackets by
commutators. However, this is true if there are not ambiguities in establishing the associated quantum operators for the constants of motion which form a Lie algebra under the Poisson bracket operation. If this is not the case, it is more convenient to choose the minimal set of constants of the motion that allows a quantum extension, and make the necessary redefinitions to build the associated Lie algebra of the system. Following this procedure we get for the generalized anisotropic two dimensional harmonic oscillator (1.4) the symmetry algebra which determine the degeneracy of the system. The symmetry Lie algebras are, depending on the value for A,w(1) Holstein-Primakoff
5
@ u(1),u(2), realization
and u(1,1). However with the generators [4] of a u(1, 1) Lie algebra can be obtained.
of the
first
one
a
Acknowledgment
This thank
work
was supported
to A. Frank,
in part
S. Hojman
by project
and
UNAM-DGAPA,
G. Rosensteel
IN10-3091.
for important
We would
comments
like to
on this work.
References [1] V. Fock, Z. Phys. V. Bargmann,
98, 145 (1935).
Z. Phys.
[2] J. M. Jauch,
and
[3] H. V. McIntosh,
E. L. Hill, Phys. Symmetry
75. (E. M. Loebl, [4] M. Moshinsky, [5] P. J. Olver,
99, 576 (1936).
Ed.,
57, 641 (1940).
and Degeneracy
Academic
C. Quesne, Application_
Rev.
and of Lie
Press
G. Loyola, Group_
in Group
(New
York,
Ann.
Theory
and
1971).
Phys.
to Differential
198,
Applications, k
103 (1990).
Equations
(Springer-Verlag,
1986). [6] J. D. Louck,
M. Moshinsky,
[7] G. Rosensteel, [8] R. Jackiw,
Ann.
and
and
J. P. Draayer,
Phys.
129,
K. B. Wolf, J. Phys.
J. Math. A: Math.
183 (1980).
378
Phys., Gen.,
Vol. II, p.
_
14,692
(1973).
22, 1323 (1989).
Berlin,
N93-$7348 FERMION
REALIZATION FROM
OF EXCEPTIONAL
MAXIMAL
UNITARY
LIE
ALGEBRAS
SUBALGEBRAS
A. SCIARRINO Universitd di Napoli Federico H Dipartimento di Scienze Fisiche Mostra d'Oltremare Pad.19 80125 NAPOLI ITALY
Abstract From the decomposition of the exceptional Lie algebras subalgebra a realization of the EL.ks is obtained in terms
1
are
of classical
known
$0(8)
long
since
(B SO(8)
the
author
[1].
which
chains.
Moreover,
while
the
contribution via
the
Koea
[2].
in a GUT structure realizations maximal
and
algebras
are
However
allow
very
embeddings
useful
SU(3)
c
G2 and
a realization
framework, of LAs
approach the
(roots,
fields makes
multilinear
weights,
Composition
Let us introduce
of ELAs
convenient
of bosonic physical
in terms appropriate
G=
C
SO(9)
SO(7) c
subalgebra. basis
a more Moreover
this
representations
of several different F4
c
and
the has
are
not
The
be quoted with
proposal
been
physical
a closer
connection
with
formalism
allows
to obtain
for generators
embedding =deformable",
that
4 has
of
obtained
realizations
in fermionic
SU(3)
connection
embedding been
subalgebras Ee
of multilinears and
oscillators
different
"deformable".
It should
keeps
Via
oscillators
way
SU(9)
evident
fermionic
contexts.
and
F4 are
in terms
in the
and/or
of fermionic to dispose
approach
etc.).
for all the fundamental embeddings
in several
of ELAs
unitary
in fermionic
Koea's
in terms
in a more
embeddings
of a maximal
as bilinears While
it is more
the
is to present
(LAs)
of ELAs
to describe e.g.,
embedding
of ELAs
Lie
C Es a realization
of ELAs
2
unitary
Introduction
Realizations
by
(ELAs) under a maximal of fermionic oscillators.
mad vector
of this oscillators
constructions obtained
by
applications the
algebraic multilinear
spaces
of all
[3].
law
for
a set of N fermionic
fermionic oscillators
multilinears
a_, a_ satisfying:
(i,j
= 1,2,...,N)
(1)
379
I
A fermionic
multilineat
(f, = a+,f_,
= o_,i
(f.m.)
X is defined
by the following
formula:
> O)
x= l-[ f,
i
I c z"
(2)
i
The
number
We define giving f,
the
a f.m.
"in"
of fi will be called (_'_)
X and
necessary
contraction
f_i
Y,
from
the
multiplied
all the
of X.
m.
by
fi near
X and Y of, resp., XY
by deleting
a factor
to f_i
(-1)",
in XY,
define a compolition
law
n being
and
N and N'
couples the
by a rational
as a operation
(if any)
number
(fi, f-i)
with
of transpositions
coet_cent
C(N,N',Z),
g
(X o Y) of two f.m. by the following equation (ik E I,j_E J)
1 XoY=_
1 _
x(X_-_"X)+t_L_jx
_._"_(f,,fj,-fj, k
We
the
order
of contractions.
being the number We
"in"
order
of two bilinears
obtained
to obtain
the
f,,,)x(-1)l'-x6,,j,
(3)
l
remark:
• XoY=-(YoX) • Xo We
3
Y = [X, Y]
(N,
N'E
put (N,N' = 1, 2, 3, 6; NT
• C(N,
N',
0) = 1
• C(N,
N',
1) = 6N_r.N or 6N_,,N,
• C(N,
N, N-l)
• C(N,
2N,
• C(N,
N, _)
1,2) = order of )_"Y):
2
=
N) =_
(N>I)
= -1
(N even)
Realization of Es
We consider
the
embedding
SU(9)
C Es.
The
248 =_ Introducing
a set
of 9 red,ionic
adjoint
representation
of E8 decomposes
as :
80 + 84 + 84
creation
and
annihilation
(4)
operators
and
we can
write
(ij = 1,2,..,9): # 0)} +
+
1
+
84
=
{a, a# a k + _
-84
=
{a_acak
1
In the
following
(6)
_i#la,,,,,_,ala,,,a,,%aqa,}
+ _.. _qja,,_,,tn,at
+
+
+
a_a,,
we call:
380
+
(s)
+
+
a r a_ a, }
(7)
• a_
"hermitian
• e_i**,,,m,a'{ Proposition satisfies
conjugate"
The
of bilinears
1
"dual
above
set
identity
generators
under
4
generator
In the
to the
a,
corresponding
Realization embedding
coniugate"
(d.c.)
and
of a_aiah.
trilinears
the composition
corresponding
cq --, a+ a,, The
of a_;
-+-+_+-+-+,,.,,,. % % ,,,
the Jacobi
The
(h.c.)
simple
in the fcrmionic
law (o) roots
defined
oscillators
in Sec.
closes
P,.
axe:
---, axa, as + d.c.,
"k -'* a+_lak
to the highest
is a_a_.
root
(3 < k _< 8)
(8)
of E7
SU(8)
c
E7 the
adjoint
representation
decomposes
as:
133 ==_ 63 + 70 The
SU(8)
C E7 is not contained
unitary
algebras
obtained
(ij,k
have = 1,2,.6
63
and
a common
in the
maximal
SU(9)
(9)
C Es,
subalgebra
Exploiting
SU(6),
the
the
property
following
that
realization
the two of E7 is
; r = 1,2,,5):
_-(10)
70 -
5
Realization
In the
embedding
{a_a_a7
+ d.c.,
(ij,k
SU(6)
_ SU(2)
= 1,2,.6;
c Ee the
(11)
h.c.}
adjoint
representation
decomposes
as:
+ (20,2)
(12)
h, - h,+l}
(13)
+ (1,3)
r = 1,2,..5): (35,1)
(1,3)
+ d.c.,
of Ee
78 ==_ (35,1) We have
a,a_ak
= {a'_a_a_
+ d.c.,
-
{a+a_,
(14)
b.c.,
(20, 2) - {a,a:j, + d.c.,
381
b.c.}
(is)
6
Realization
of F4
In the embedding
SU(4) (_ SU(2)' C F4 the adjoint representation decomposes
as:
52 ==# (15,1) + (1,3) + (4,2) + (4,2) + (6,3)
(16)
The most convenient way to identifythe elements of F4 is the following: i)draw
the Dynkin
ii)from
diagram
of Ee;
....
i) draw, by folding, the Dynkin
diagram
of }"4,identify the corresponding simple
roots and the highest root; iii)draw the extended Dynkin
diagram
of F4 and then, by deleting a dot, identify
SU(4) e SU(2)'. We
get for the 52
7 ..S _9
(ij,k= 1,2,.6):
(i + i = 7),
+d.c.,
+
(i # j # k #l;i
aj a# + (-1)t+l-la_a, a, aiah+d.c. a, aiat
b.c. +j
=-(-1)
I
3
S=1,
j
426
I
4
Q=j*I
5
T
J
L/2 z in units ( Gev)2 5,8 5,6 5,4 5,2
D
_ -
f_z= f_
f;
B
5,0 4,8
m
4,6 4,4
f.
4,2 4,0 3,8 3,6 3,4
B
3.2
n
3,0
B
2,8
B
2,6 2,4
'/'; .f_
oJ3
.f_'
2,2
m
2,0
-f_;
1,8
B
'1.6
-f_
fg
1,4 1,2
'f0
_'
1,0 0,8
oJ1
0,6 0,4
I
I
I
I
0 Fig,6
I=
, 13= -
1)j
427
-1,
.Q=j'I
I
L "E
J
N93-27352
FROM
Center
for
HARMONIC
TO
ANHARMONIC
F.
lachello
Theoretical
Yale
OSCILLATORS
Physics,
University,
New
Sloane
Haven,
CT
Laboratory, 06511
Abstract The
algebraic
oscillator
realizations
molecules
i
are
to
quantum
is
mechanics
discussed.
is
briefly
Applications
reviewed.
to
The
vibrations
of
role
of
complex
presented.
Introduction
In has
approach
recent
been
put
algebraic
Quantum
years,
a formulation
forward,
structure
Mechanical
in
which
following
of any
the
quantum logic
mechanics,
called
mechanical
scheme
shown
algebraic
problem in
Fig.
is
mapped
theory, onto
an
I.
System
Lie
algebras
Graded Algebraic
quantum
structure
Lie
Infinite
algebras
dimensional
?_deformed
(Kac-Moody)
algebras
(Hopf)algebras
Observables
Experiment
Fig.l.
In
Logic
scheme
implementing
of
algebraic
algebraic
theory.
theory,,
it
has
been
found
to
be
429 PREqSEDINg
PAGE BLANK
NOT
FILMED
very
useful
to
make
use
of
the
use
2
oscillator of
representations.
oscillators
Oscillators
I
in
begin
the
In
H(2)
Table
I
the
the
Heisenberg
algebra
:
a,
shows
the
known
anharmonic the
I,
eigenvalue
briefly
this
of case
the is
one-dimensional
review
described
by
harmonic
the
introduction
a#a
does
instead
Morse
theory
(2.1)
between
(SchrSdinger
and
consider
example
parallelism
case I
I will
[I]
a #,
operators
well
contribution,
theory.
(trivial)
algebraic
differential is
algebraic
this
v dimensions
with
oscillator• of
in
In
not
oscillator.
equation)
require
the
the and
further
(non-trlvial) The
usual
differential
the
treatment algebraic
in
terms
approach.
example
of
the
approach
requires
one-dimensional the
solution
H¢-E¢
h2 d2 + V(x) 2_
V(x)
dx 2
- D[I
- exp(-flx)]
2
(2.2)
J
The
solution
of
the
eigenvalue
problem
produces
wave
functions
i Cv(X)
where
Nv
-
is
Nv
z_-V
e
. £2 + _
a normalization
X_L2_-2v-I v
and
L(z)
(z)
(2.3)
denotes
430
This
explanation.
problem
H ....
of
a Laguerre
polynomial.
Also
of
Table
I.
Differential
and
algebraic
treatment
of
the
one
dimensional
harmonic
oscillator. Differential
approach
(p2
I
_n
+ x 2)
a
-
d2
" 2
H
-
En
n
-
i
+
!
I
I
-
[_
-
E
7)
-
Un(X)-
(x
[a,a # ] -
1 4
Uo(X)
d
--
x21
@n
(n +
-
at .
[" dx---2+
H
E
approach
1
I - _
H
Algebraic
--
2
2n
n!]
2
(x
i
1 x 2
n
d - _x)
In>-
e
(n!)
2
-_-ao
Inn,
The
- f
Un'(X)
f(x,d_)
z -
e -_x
eigenvalues
27
Un(X)
;
Inn,
dx
I 2_ _ - _-_
;
v-
are
431
0,I ..... _
-
- i
(2.4)
2 1 E(v)
The
mass
Eqs.
2_/_
algebraic
C
is
the
they
[2]
the
can
be
H - AC
+ 5)
were
the Lie
C -
Casimir
(v + 5)
#
D
and
deleted
algebra
(2.5)
range
in
_
Table
one-dimensional
written
;
- _
interaction
theory,
Hamiltonian
where
of
while
introducing
The
(v
strength
(2.2)-(2.5), In
by
#,
-
been
put
explicitly
can
be
oscillator
composed
of
four
elements
dealt
With
the
E(v)
v -
which can
A(m2-N
change
-
-
of
4A(Nv-v
FO 2
operator
the
written
intensities
, m
N2
of
- N,N-2
variable
F+,F.,F0,N.
,
the
(2.6)
0(2)
subalgebra
of
U(2).
The
eigenvalues
v -
..... 1 or
(N-m)/2
one
0
(N-odd
or
even).
(2.7)
has
,
2)
N _
1 - _
eigenvaiues
(N-
of
the
Morse
even
or
odd)
oscillator,
,
Eq.
(2.8)
(2.5).
The
eigenstates
as
I U(2) N
and
2)
N _ or
0,I .....
are be
-
can
D
0(2) v
be
(2.9)
>
computed
with
as
are
E(m)
in
I.
Morse
U(2),
have
by
taking
matrix
432
elements
of
operators
-
> "
are
given
by:
" 2vivj
,
" Jvj(vi+l)(Ni-vi)(Nj'vj+l)
,
" Jvi(vj+l)(Nj'vj)(Ni'vi+l)
(3.7)
As
an
example
molecule, This
C6H6,
of
has
conventional
12
is
feasible,
since
be
diagonalized.
one
can
view have
neighbor appropriate
atoms
treatments
equations
In
rather
of
these
the
couplings
and
of
models
consider
types and
benzene
(III) can
be
in
the
the
case
of
couplings: third
terms
first
436
an of
in
of
the
benzene
vibrations.
coupled algebraic
algebraic
the
neighbor
couplings.
as
of
side,
benzene,
(I)
neighbor
written
in
independent terms
other
expressed
geometry
of
36-6-30
molecule On
Hamiltonian,
hexagonal
thus
this
complicated.
the
three
to
of
(Fig.2).
molecule
easily
application
coupling couplings,
The
algebraic
A
differential treatment operators,
terms (II)
is can
[ijVij, second
Hamiltonian
Y Fig.2.
The
benzene
molecule.
6
where
and
the
r
c12
m
; Aij" A_H ; _iJ-_HH
c-EC
,
three
operators)
S (I)-
AI'A_H
s
c23
_
i
given
S (I),
S (II)ijMij
p
m
P
c45
'
S (II)
and
'' i= (Ole='l@)e-½{'{2ei'p. Dropping harmonic
the
factor
oscillator
e-{lzl2ei_,
state
[_) by the
one
obtains
the
holomorhpic
familiar
function
Bargrnann
(34)
representation
¢ of a complex
variable
[15] of a
z with
¢(z)= (Ole_{_>. For example,
a harmonic
oscillator
(35)
state 1 (36)
In) -- _-y (at)nlO) is represented
by the
function
1 ¢,_(z)=_n{Zn.
(37)
In the Bargrnann representation,the harmonic oscillator raisingand loweringoperators are simply the differential operators 0 a t=z, Coherent structed
state
in a similar
representations way.
Consider,
of the
a=_zz,
I=l.
symplectic
groups
for example,
466
(38) Sp(1,R)
the representation
and
Sp(3,R)
of Sp(3,R)
can
with lowest
be conweight
state
given by a closed-shell
state;
i.e., a state
which _atisfim
=0 mj o_#(a_a_# + a_a t)
A coherent
state representation
of a state
I0) = _10).
(39)
I_) is then given by s function
o(g)== (01 exp
O over Sp(3,R)
with
-
(40)
mj
which is proportional
to the holomorphic function of six complex _(z) = (01 exp _ _jl_) nij
variables
•
(41)
The expression of symplectic operators in this representation is simple and enables one to calculate their matrix elements in analytic form. Such a construction of coherent state representations is an explicit realization of the Borel-Weil theory
(see, for example,
7.2
Vector
The direct
ref. [16]) of the representations
coherent
application
state
of semi-simple
Lie groups.
representations
of the above construction
of a general
representation,
i.e., one whose lowest
weight state does not span a trivial one-dimensional representation of the SU(3) C Sp(3,R) subgroup, is much more complicated and, therefore, not so useful. However, it is po_ble to construct a so-called vector cohecvnt state representation which is simple. First observe, from eq. (39), that the gauge factor • l_ is a representation of a U(3) transformation;
i.e.,
Now, the lowestweight stateof a genericSp(3,R) irrepdoes not by itself span an irrepof the U(3) subgroup of Sp(3,R).However, itisone stateof a multidimensionalirrep.This suggeststhat more generalSp(3,R) it.re, ps can be constructedin which the one-dimensionalU(3) irrepof Eq. (42) is replacedby a generalmultidimensionalU(3) irrep.This iscorrectand one findsthat a state• of any discretesericmrepresentationof Sp(3,R) can be realizedM a holomorphic vector-valuedwave function_b with _b(z)ffi_
[u)(vlexp _
v
z_#a_a,#l_),
(43)
nijf
where {iu)} is a basis for a lowest weight irrep of the subgroup U(3) C Sp(3,R). The calculation of matrix dements of the sp(3,R) Lie algebra in such a representation
is a
simple task. When there are no missing quantum numbers, one obtains analytic expressions for the matrix elements. When there are missing quantum numbers, which is the generic situation, one has to do relatively The vector coherent an explicit
realization
small numerical state techniques
calculations to construct orthonormal basis states. apply to all the semi-simple Lie groul_. They are, in fact,
of the Harish-Chandra
theory
467
[17] of induced
holomorphic
representations.
8
Concluding
remarks
I hope to have shown that
the harmonic
oscillator
in three
dimensions
has a rich structure
and that
itsmany-particlerepresentations and coherentstatesprovide the framework forboth independentparticleand collective models of nuclearstates.Moreover, the coherentstateand vectorcoherent staterepresentations, which originatedin applicationsof the dynamical groups of the harmonic oscillator, have much wider applicability and are now essentialtoolsin the hands of those who use algebraicmethods in physics.
References [1] A. Perelomov, [2] E. Onofri,
Commun.
J. Math.
[3] 1_ Y. Glaubor, [4] D. Stoler,
Phys.
Phys.
Phys.
Math.
Phys.
26, 222 (1972).
16, 1087 (1975).
Rev.
131,
2766 (1963).
Rev. D1, 3217 (1970).
[5] A. O. Barut
and L. Girarddlo,
[6] H. P. Yuen,
Phys.
Rev. A13,
Commun.
Ma_h. Phys.
21, 41 (1971).
2226 (1976).
[7]D. J. Rowe, Can. J. Phys. 56, 442 (1978). [8]M. G. Mayer and J. H. D. Jensen, Elementary Theo_ N.Y., 1955).
of Nuclmr Shell Structure,(Wiley,
[9]M. Goldhaber and E. Teller,Phys. Rev. 74, 1046 (1948). [10]J. P. Elliott, Proc Roy. Soc. A245, 128,562 (1958). [11]G. Rosensteeland D. J. Rowe, Phys. Rev. Lett.38, I0 (1977)' [12] G. Rosensteel
and
D. J. Rowe, Ann.
[13] D. J. Rowe, Rep. Prog. Phys. [14] P. Park,
J. Carvalho,
Phys.,
N.Y.
128,
343 (1980).
48, 1419 (1985).
M. Vassanji,
D. J. Rowe
and
G. Rosensteel,
Nud.
Phys.
A414,
93
(1984). [15] V. Bargmann,
Commun.
[16] A. W. Knapp,
Representation
[17] Harish-Chandra,
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I, 564 (1956).
468
Groups,
(Princeton
Univ. Press,
1986).
N93'.27855 ON THE SPRING AND MASS OF THE DIRAC OSCILLATOR James
P. Crawford
Department
of Physics
Penn State - Fayette,
Uniontown,
PA 15401
Abstract The Dirac oscillator is a relativistic generalization of the quantum harmonic oscillator. In particular, the square of the hamiltonian for the Dirac oscillator yields the Klein-Gordon equation with a potential of the form: (ar 2 + bL • S), where a and b are constants. To obtain the Dirac oscillator, a "minimal substitution" is made in the Dirac equation, where tile ordinary derivative is replaced with a covariant derivative. However, an unusual feature of the covariant derivative in this case is that the potential is a non-trivial element of the Clifford algebra. A theory which naturally gives rise to gange potentials which are non-trivial elements of the Clifford algebra is that based oll local automorphism invariance. I present an exact solution of the automorphism gauge field equations which reproduces both the potential term and the mass term of the Dirac oscillator.
I
Introduction
The Dirac oscillator exhibits many interesting features. classic non-relativistic harmonic oscillator Schr6dinger sense
that the square
the spinor The
fields
equation
of the Dirac
with a potential is exactly
supersymmetry [4,5]. inter-quark interactions an interesting
version
all "usual"
multiply which
involving
physical locally
are
general
predictions
which
[11], upon
theories
where
S) where
non-relativistic
coupling
Klein-Gordon a and
case
has been
the potentials
of the algebra).
elements
of the
reproduces
Clifford
[4],
investigated
are Clifford which
algebra
If we then
equation
b are constants and
exhibits
for
[1,2,3]. a hidden
[8].
and further
is based.
469
(that
incorporates
is that based
upon
that this freedom
solutions
term and the mass discussion
scalars
naturally
is, the potentials gauge
local
potentials
automorphism
gauge theory is the observation that file algebra generators should not effect the
demand
gauge theory. a set of exact "chirar'
both the potential
details
this paper
A theory
idea behind autom_orphism chosen for the Clifford
of the theory.
Additional
which
as in the
a "scalar"
we obtain automorphism In this paper I present case.
the relativistic
feature of the Dirac oscillator is that the potential which is introduced as is a non-trivial element of the Clifford algebra. This is to be contrasted
[9,10]. The basic matrix representation
equations special
gauge
the unit element
invafiance particular
solvable
yields
(at 2 + bL.
of the in the
In addition, this particular form of potential has been used to model the in the hope of obtaining a realistic model of the hadrons [6,7]. Finally,
A highly unusual a "minimal substitution" with
hanfiltonian of the form:
It is the relativistic generalization equation to the Dirac equation
of choice
be allowed
of the automorphism
gauge
term of the Dirac
oscillator
of these topics
may be found
field as a
in reference
_L
2
The DiracOscillator
The
connection
be_'een
seen by considering
p _ This
algebra
generators
elements
substitution"
has
tile interesting
theory,
matrix),
since
coordinate
,
" these
the covariant
and suggests
of the Dirac tensor ruv
r_tv
Dirac
:
(Ut.tXv
,
equation
r0i
:
Xi
may be written
_,_,, are
the
moment
,
from
occur
file
as general
a unit fimelike vector and the
a radial
of the
center
0
(3)
as:
(4)
Clifford
as a particle
decide of the
is valid as long as we take equation
fidd.
with zero In this
electric
algebra
field.
(4) as our starling
charge
case Note
basis
the
matrix representation we
from the point of view
of local
may be developed in spaces of arbitrary to the case of four-dimensional spacefime. of the Clifford obtain
This
interacting vector
that the
point.
[12].
via
uo
may
a be
electromagnetic
However,
if we wish
derivative
for an
Invariance
the problem
Although the theory restrict our attention then
Clifford
(2)
l'i/:
elements
inter-pretation
with
four-velocity
bivector
LocalAutomorphism
in space,
the
can be derived naturally
this equation as arising from a minimal substitution of a covariant derivative, then the electromagnetic interpretation is untenable.
I now approach
[3.4]:
take the form:
an electromagnetic
dipole
interpretation to view ordinary
matrices
the
easily
-- UvXl.t)
of the
considered
potentials
upon
oscillator equation we introduce formed from the timelike unit
the
has
oscillator
dependent
that file theory
the gauge
0
magnetic
3
of being
I'yl-tt,_t - m + ½m03rrtv'/_-tv )q/:
equation
the Dirac
is most
algebra
ut_t= (1,0, 0, 0)
where
to obtain
theory
vector:
urturt = 1
Now
properly
in this case
To obtain the covariant form fourvector uo and an antisymmetric
In the "rest frame
thai is made
gauge
(l)
(1_ is the Dirac
gauge
of the Clifford
spacefime
and the automorphism
imo3[Sr
substitution"
antomorphism
oscillalor
the "minimal
p-
"minimal
the Dirac
a gauge
incorporate this local invariance covariant derivative:
algebra
generators
theory
into the theory,
based
on
the
invariance
[9,10].
dimension and signature, we will If we assume that the particular
may be chosen
the ordinary
470
automorphism
arbitrarily
at each
automorphism
group
derivative
be replaced
must
U(2,2).
point To
with the
where
the gauge
potential
is given
by [9,12]:
Art = art 1 + aPrtyp + -_ and for the field strength
tensor
rttpc_
we find:
= 0_tAv-0vArt+
ig_Art,Av] 1 fpcy
_
= ./_tv1 +fOrtvTp + -_j Making
the minimal
(6)
substitution
into the Dirac
(7)
~
rtvto_ - hOrtvYp
lagrangian
- hrtv_t
we f'md:
Zq-,
(8)
-b where
we have
(I)
made
= a pp
the det-mitions:
art
,
the basic
and bivector
lagrangian
since
1 ,.,vpo "_'Ertvpa-
_
and we see that the automorphism pseudovector,
%v
gauge
(spin)
fields
interactions.
the scalar
coupling
,
1 brtv = _Ertvpab
couple
to the fermion
Notice
that we have not included
(Yukawa
interaction)
pcy
field through
(9)
scalar, the mass
will give rise
vector, term in
to mass.
The
explicit form for the field strength tensor in terms of the gauge potentials and a more detailed derivation of the Dirac part of the lagrangian can be found in my notes on local automorphism invariance
[91.
The equations for the gauge fields in the absence of sources is found in the usual manner by demanding stationary action with respect to arbitrary variations of the fields. We find:
v = 3 ,Fr" + igEAt,,Fr'V] = 0 Notice
that
concordance as an
with
external
potentials equation
we
satisfy We now
have
not
included
the original field.
To
the
approach be entirely
equation (10). make the observation
for the Dirac
oscillator
(equation
fermion
source
to the Dirac consistent, that this
0o) term
oscillator however,
interacting
(4)) if the foUowing 471
in this
equation.
This
in that the potential we
must
lagrangian conditions
demand density are met:
is in
is introduced that
will
the
gauge
yield
the
1
gO = m a_ = 0 It is remarkable solutions
,
gbov = -_mO3rov
,
ao = 0
(llb)
that these particular
to the pure
(lla)
gauge
expressions
for the potentials
form a subset
of exact
"chiral"
field equations.
4 Chiral Solutions ansatz"
We now consider def'med as:
ao=O
a special
subset
of solutions
a o_ o=O
,
to equation
Consider
(13).
the
"chiral
b_ = 0
,
(12)
boo = + aPo and the field equations
f
_:ov =_)
There are, of course, field strength tensor
become:
.a
=
_:ov
h
,
()O f
_¢Ov =
many solutions to equations (13), but consider is constant and uniform. In this case equation
satisfy equation (13a) coordinate. Therefore
a_
K:v __)va_o
an obvious we write:
= clm(1
choice
+ dlm(u,
is to assume
x))g_
(13a,b)
the special case in which (13b) is clearly satisfied.
that the potential
+ c2m(1
0
is linear
the To
in the spacetime
+ d2m(u.x))u_:urt (14)
+ c3m2r_ where
the coefficients
~ roy The parameter be the mass
c i and d i are arbitrary
1 -_Eov_:.
c
appearing
,
= m2(Cldl
Sov
--=
oscillator.
in the spacetime
+ csm2s_
dimensionless
constants,
and we have
def'med:
(UoXv + UvX o)
with the dimension
in the Dirac linear
The field strength
f_v
r_
m is a quantity
form which is both constant vector.
+ c4m2r_
of mass
Finally
and it may be conveniently
note that equation
coordinate
tensor may be calculated
(15)
directly
and
which
from
(14)
involves
equation
(13a)
is the most only
472
one
to
general arbitrary
and we f'md:
2m2c4E_vxU
+c3-cs)(uog%-uvg_)+
chosen
"c
(16)
As
expected,
equation
the
field
(13b).
equations
strength
tensor
is constant
For the part of the gauge
potential
b_tv = -c3m2rlav
art = 0 that
(equation
,
only the
(17a)),
interaction equation
with the fermion
(see
+ 2cs)m2(ux)
+c2d2
(17a)
(17b)
(17c)
part
of the gauge
only the antis}anmetric (17b)).
This
recover
the Dirac
now
choices
(compare
potential
part
statement
we
equations
+ c2) = 1
c3 = 0
contributes
of the gauge
is generally
state
,
tnle
proof
to the
potential as may
scalar
interaction
contributes be seen
to the spin
by inspection
of
gauge
form
by a chiral
(equations
(11))
Summary I have
produce
the
shown Dirac
if we make
the
is constant
strength
(18a)
[11])
that
gauge
equations
(18b)
(18)
can
always
the potentials appearing potential) are essentially
since the potentials
may
be
in the unique
always
satisfied Dirac chiral
by
an
oscillator solutions
be brought
into this
transformation.
the mass may
be
and the potential viewed
as
introduced
a special
into the Dirac
case
of a chiral
field equations. In addition, this chiral solution can always be found which puts the potential into the physical interacting with
and
coordinate.
two spatial
considered
(17)):
c2d2 + 2c5) = 0
+
field equations,
that both oscillator
To gain insight situation of an electron spacetime
case
and Conclusions
automorphism gauge a gauge transformation Dirac oscillator.
strength
as a special
2gmc4 = 03
(see
of the automorphism
interactions
(11) with equations
,
(4cldl
without
oscillator
appropriate choice of gauge. In other words, (where we are including the mass as a constant
field
directly
satisfies
art = 0
and
g(4cl
in the
interacts
trivially
(9).
following
5
which
and therefore
+ c4m2rlav
symmetric
(equation
We may
Now
uniform,
(8) and (9)) we find:
• = (4Cl +c2)m+(4cldl
Notice
and
uniform,
in this paper
perpendicular
is actually
and a corresponding
more linear
to
to the
is essentially unique in the form displayed
in that in the
interpretation of this system consider the more familiar a constant magnetic field. In this case, since the field
the electromagnetic
As is well known
directions
equation
solution
[13],
potential
this system
to the magnetic like this situation potential.
electromagnetic interpretation of a particle with zero interacting with a linear electric field. Notice, however, 473
This
will
be a linear
exhibits
harmonic
field.
Therefore,
in that there should
oscillator the point
is a constant
be contrasted
function
of the behavior of view
and uniform with
the direct
charge and non-zero magnetic momen! that the case of a constant automorphism
field strength does not lend itself easily to the constnlction of hadrons as advocated by Moshinsky el al [5,6] since we do not view each particle as giving rise to the antomorphism field (though they certainly must contribute to the antomorphism field as does the electron to the magnetic
field,
mesons
(for
pair, but electrons electron
but this is taken
example)
the situation in a constant lmdergoes
uncorrelated.
here to be a "higher
we may
consider
with a constant magnetic field
cyclotron
These
be considered
important anticipated interactions, but these The
to arise
of
and we therefore
from
a generaliTation
dimension
oscillator and
discussed
signature.
in this
In particular,
avenue
orbits
do not necessarily
of Equivalence,
electroweak
and
naturally
[2]
P.A.
[3]
M. Moshinsky
[4]
J. Benitez,
Cook,
Phys. Rev.
Letl. Nuovo
Cimento
1,419
and A. Szczepaniak,
R. P. Martinez
the specific
cases
In addition, should be
J. Beckers
and N. Debergh,
of two,
[6]
M. Moshinsky,
G. Loyola,
1119 (1967).
A 22, L817
(1989).
H. N. Ntifiez-Ydpez,
and Phys.
Phys.
51A,
Rev.
and A. L. Salas-Brito,
Lett. 65, 2085E
(1090).
Rev. D 42, 1255 (1990).
A. Szczepaniak,
C. Villegas,
and N. Aquino,
"The Dirac Oscillator and its Contribution to the Baryon Mass Fommla" in "Relativistic Aspects of Nuclear Physics," Editors T. Kodama et al. (World
Scientific,
Singapore
(1990))
pp. 271 - 307.
[7]
M. Moshinsky, G. Loyola, and C. Villegas, "Relativistic Mass Formula for Baryons" in "Notas de Fisica 13", Proceedings of the Oaxtepec Conference of Nuclear Physics 1990, pp. 187 - 196.
[8]
V.V. Dixit, to appear.
[9]
J.P.
Crawford,
[10]
J.P.
Crawford,
[11 ]
J.P. Crawford,
[12]
submitted to Foundations J. P. Crawford, J. Math.
[131
L. D. Huff,
T. S. SanthananL
Automorphism
J. Math. "Notes
Phys.
M. H. Johnson
114]
"Local
Rev.
and W. D. Thacker,
Phys.
J. Math.
Invariance"
31, 1991
Phys.
33, (1992)
in preparation.
(1990).
on the D irac Oscillator
and Local
Automorphism
Invariance"
of Physics, April 9, 1992. Phys. 32, 576 (1991). 38, 501 (1931).
and B. A. Lippman,
J. P. Crawford, "A Generalization Theories" in preparation.
Phys.
Rev. 77, 702 (1950).
of the Principle
474
of Equivalence
and an
to three,
as this application a worthwhile and
(1971).
J. Phys.
y Romero,
Lett. 64, 1643 (1990);
[5]
Cimento
expect
generalizes
References D. Ilo, K. Mori, and E. Carrieri, Nuovo
are
gravitational
of exploration.
[ 1]
the
to a model of the hadrons. is more one of aesthetics in
paper
five, and six dimensions are likely to generate interesting results. shows, the general theory of local automorphism invariance interesting
cyclotron
of the Principle to the [ 14].
to build
the quark-antiquark
akin to pulling several between them. Each
individual
field theory to lend itself easily local automorphism invariance
to the Dirac
nl0 + n01 + nl_, or no0 > 0. It is the independence of our and permutation of the order parameter which allows the statistics program universe to differ or "random walks". The
"random
by defining
walk"
a single,
with which
shorter
a,o = 0, b_, = 1. Then h/mc
string
distribution
as describing
in space
and
the
is obtained
at velocity
(0, 0) in the x, ct plane,
their
boundary
at (0,0) and (x, ct) is x = (r - e)(h/mc),
from our more
of a "particle" -l-c along
condition
which
the light
point suggested by Feynman[5] and articulated, for example, a derivation of the Dirac equation in 1+1 dimensions. If the origin
to Bernoulli
of l's and g = n01 is the number
sequences,
general
cones.
on the trajectories
discrete
of l's changes to a sequence computed from the elements
the number classes:
of bends
RR,
in the
LL, RL and
right-moving
segments,
forward
cone.
light
k left-moving
Similarly
bends. RL and LR cannot bends. This classification is different. In order
string
For class LL has
of length
RR the
connecting
two events
We tie our model to this same
segments
r + g = nl0 + n01 These
first and and
last steps
k bends;
k + 1 left-moving
of
Schulman[6] for to start at the
of times
of O's or visa versa. As McGoveran discovered,this of e by k(e) = z,w=1"-'w-11.cw+l _ - c_) 2. We are interested
trajectory LR.
steps
This is the starting
space-time trajectory, but as noted above include an additional degree of freedom. We now classify any string c by the number of bends k(e), which counts the number sequence is simply
model
c_ = 0 if
of O's in c(r + g), We
is taking
by Jac0bson and particle is assumed
ct = (r + g)(h/mc).
that
string length generated by
nl0 + n01 by c_, = 1 if a_ = 1, b,o = 0, and
"motion"
h/rnc 2 in time
assigned
n0o and
constraint
result from both of the bit-strings
usually
we will be concerned of length
r = nlo is the number
now view this situation length
from the binomial
of both
to the
are
note
segments,
that
strings
to the
number here in
fall into four
right;
it has
k = 0 corresponds
k right-moving
a
k + 1 to the
segments
and
k
have k = 0 and have k right-moving segments, left-moving segments and is the same as in Jacobson and Schulman, but our statistical treatment
to distinguish
the
connectivity
we make
time trajectories considered by Feynman, additional two parameters nil and no0 that
between
the
two events
from
the
we call them paths. It is the interpretation allows us to extend our single particle treatment
spaceof the to an
interpretation that has features in common with second quantized relativistic field theory. In the case of a statistically causal trajectory, time ticks ahead at a constant rate. If the particle does not take a step to the right, it must take a step to the left. Although our particle follows the same trajectory
in space,
not move
in the
We interpret not directly
if we encounter
single
this affect
particle
an example
configuration
of w corresponding
space
that
Using
them.
light
cone
coordinates,
a bend
can
be specified
light cone and by any one of the g positions of the greater freedom in our string generation, There
we need
Feynman
nll
or n00 it does
approach
contains.
as representing background processes going on in program universe which do the particle. In a second quantized relativistic field theory these "disconnected
diagrams" are the first to be removed in a renormalization program. to the way we count numbers of paths, they do not enter directly forward because
to either
is all the
are r k ways we can pick a position
do is insure
that
the
restrictions
Although conceptually into our calculations.
by any
one
light
by the four
classes
484
r positions
on the backward light cone. there is no statistical correlation
on the forward
imposed
of the
cone
and
crucial
ik on the
of trajectories
on
the
However, between left.
given
All
above
are met.
Further,
of these rk/k!
factors
the
order
in the
by fk/k!
in which
relative
we make
probability
to get the (unnormalized)
the
choices
is irrelevant,
meet
our space-time
boundary
divide
Since
they
are independent
probability
that
both
will occur in an ensemble
characterized by k bends and meeting our boundary conditions. We conclude that the relative frequency of paths in the space which
so we must
by k!.
conditions
will have
of bit-strings
we must
of length
each
multiply of strings W > r+
the values (3)
rk+_
Fk
pLL(r,e)
2
Formal
Write
the
Dirac
Derivation Equation
(5/
]
Dirac
with
(4)
_k+l
= [_.l][(kI)_+
of the
in 1+1 dimensions
ek
Equation
h = 1 = c = 1 = m as
¢, = (a/at - o/ax)¢_; ¢_ = -(a/at + Olax)¢, With
z _ = t _ - x 2 = 4rg, this equation
is solved
(6)
by
01 = go(z)+ 2,j, (z); ¢_ = go(z)- 2ej_ (z) z z where
Jo and
,/1 are the
standard, go(z)
real Bessel
= Ej=o(-1)J(z/2)
functions. 2j/(j!)2
We note
(7)
that
(8)
= Ej=o(-1)J[_][_] j: --.
j.
Further Jl = -do
= E.i=lj(-1)J+l(z/2)
(9)
_j-' /(ji)2
Hence 2rg, z
= Ek=o(_l)
2g jl = Ek=o(_l)kr z
rk+l
_k
k k!
k!
(10)
k gk+, k! (k + 1)!
(11)
Since
J; = go- _J1
(12)
Z
we can now relate now demonstrate. We must
the solution
now interpret
states,
in the
in the
problem)
context that
the index
of our the
of the differential 1, 2 in the
bit-string
bends
equation
in the
model.
Dirac
to our equation,
We assume
trajectories
485
(since
correspond
relative where there to the
frequency it refers
counts, to the
is no coulomb emission
as we
two spininteraction
or absorption
of
a 7-ray,
and
projection
hence
states
to a spin and
the
flip.
four
We connect
classes
¢1 and
of trajectories
¢2 with
as follows.
the
two
(global)
laboratory
spin
¢1 correspond to correspond a to the wave function for which the laboratory spin projection is +Th. Consider first the RR trajectories with k + 1 right moving segments k left moving segments and k bends. For k even the relative frequency of such trajectories is P_n(r,g)= [_][_.,] that the particle starts moving to the right with positive spin; of spin flips,it will have at the end points spin projection the left with the same positive spin projection, it would end
up moving
to the right.
But
then
contribution
to ¢1 of trajectories
it has an odd number
which
end with
rk+l +
¢_ = Ek=°(-1)k[(k Note
that
by including
we can do because Note the
also
that
wrong helicity
frequencies
only
the
k = 0 case we have
relative
negative
frequencies
frequencies
of spin-flips.
compared
of trajectories
to that which
specified
end
to the
l) '][. .1=
that
to the
is conserved
right
is
2--'rJx(Z)z
(13)
probabilities we have
by the label.
up moving
what
the sum to the forward
no absolute mean
Since
to to
from the first to get the net number for these two classes taken together,
a step
normalized
and
simply
as we have already seen. Assume since it experiences an even number
+_ as desired. However, if it started have to take an odd number of bends
is global rather than local spin, these cases must be subtracted of relative cases with positive spin projection. Consequently, the
Let
cone; this
are involved.
a preponderance
Similarly,
left and
light
of cases
if we construct
contribute
the
with
relative
to ¢1 we find that
r k _,k (14)
eL = Zk=o(--llk[_.V][_]. ". = Jo(z) So
¢, = ¢_ + eL = Jo(s)
+ 2r j,(z).
(15)
Z
Similarly ¢2 = ¢_ + oL = Jo(s)
-
2--gJ,(z).
(16)
z
Thus, by imposing the spin projection at the same formal expression that equation necessity
3
in 1+1 dimensions. in any app|icatlon
Second
In our formal evolution components
Since, for either derivation, to laboratory data, we have
Quantized derivation,
from
conservation law on our relative frequency counts, we arrive is obtained by the series solution of the free particle Dirac
the
is a practical
Interpretation
we avoided
program
the truncation of the series achieved our formal goal.
introducing
universe
generation
_/'1, ¢2, was ad hoc. In a more
detailed
a "free particle of bit-strings. treatment,
Hamiltonian"; But
the
we would
labeling develop
we took our time of the
two spin
the spin,
angular
momentum, energy, momentum, and space-time discrete coordinates consistently from bit-strings. We will present this full discussion elsewhere. [7] Here we must content ourselves with supplying a label to each of the three strings already invoked
in our generation
process.
This can be simply
486
the first two bits in the string.
The
system
we model consists of fermions labeled these labels we concatenate bit-strings in either there
by f = (10), antifermions t = (01), and bosons b = (11). To representing the propagation of the three types of particle
space-time
or momentum-energy
is an interaction,
corresponding
space. roughly
The
general
to a vertex
connection
between
in a Feynman
Diagram,
the three
when
is
for®b=0
(17)
In this broader context, the single particle trajectory we have been following can be thought of as a particle moving forward in time or an antiparticle moving backward in time, and the two events
as space-like
rather
forward-backward minus
motion
antifermion
plane
rather
than
"wave
functions"
second
quantized
Once
the
body
was
one
to hear
cone,
the major
theory
free
particle
it is not the
"infinite
4
A New
Fundamental
bare
single
x, ct
particle
functions
in a
work
equation
with
probability
rk/k!,
atoms
in the
body
in the
spectrum. the
than
of a
Because
of the usual
Rather
walls
appearance
in a normal
of
of the
connection
fashion.
This
go on translating
you will find it of more which
background,
while
universe
contribution
Dirac
mass"
which
interest
satisfies
the Dirac
retaining
rather occurs
equation
the same mass.
than
from
in a second
The
a Hamiltonian quantized
field
interactions, there will be finite changes Our theory is "born renormalized".
a finite
of renormalization
first
approximation
of the
physical
concepts
derived
theory.
Theory
is a new, fundamental
recent
the
full
leads.
from program
"self-energy"
mass;
from
number
to the
particle
theory,
Derivation
any free particle
with the radiation comes
of the
in our
we hope
by symmetry and equate it to zero. Once we include effective mass, but no infinite mass renormalization.
primarily
fermion
with
single
of quanta
at this Workshop.
or visa versa, that
the bends
can proceed
us to remove
physics"
that
in space
for use as basis
states
theory.
theory in the
"Bit-string
shows
of the black
models
to fundamental
propagation
to the
(elsewhere)
result
we emphasis
of as interacting
in the
also
absorption
in his derivation
language,
new approach
and
oscillator
this new
this section,
mass
but
with
our derivation
we can interpret
allows
For us the
motion
we invoked
symmetry
with conventional
harmonic
unfamiliar
our space-time
left-right
extends
CPT
established
contact
and
for presenting
can be thought fact that
light
context,
by Planck
makes
into this
only
are analagous
we have
reason where
forward
replaces
conservation not
the appropriate
trajectory
invoked
results
To conclude
spin
This
as due to the emission
states
This
field theory.
in the
factor
the
this extended
above
radiation
familiar
have
to radiation
statistical
between
it to the
enclosure
connection
usual
confining
bends
separated.
and
conservation.
relativistic
we used
Thus
the
in time,
we have accepted
trajectory
time-like
number we derived
etc. black
than
in computer
theory
based
science.
This
on information theory
has
theoretic
already
achieved
considerable
conceptual clarity and quantitative success. In this section we present an outline of the underlying concepts and how they find physical application, following closely an earlier summary. [8] We start We base event O's.
from
sequential
our theory
intervals We connect
by
counter
on invariant bit-strings
our
model
firings
with space
squared-intervals [i.e.
finite
ordered
to laboratory
events
interval
c2T 2 -
L 4- AL and
L 2 between
counter
sequences
of O's and
by taking
L = (NI -
487
time interval
l's]
firings. with
No)(h/mc),
T 4- AT. We model
N1 l's
and
No
T = (N1 +
No)(h/mc2). events
Calling
a "particle",
If we now signals,
any conceptual the
consider
three
we can model
i.e. addition inequalities,
velocity
carrier
of conserved
v of the particle
counters,
the system
with
by three
quantum
is then
associated
bit-strings
clocks
synchronized
of the same
3 neutrinos,
within the space-time
W +, Z0, 3' and
colored
the
(2, 4, 16) decomposition
we obtain the cumulative cardinals combinatorial hierarchy, discovered neutrinos, of the
the
second
137 as a first
gen atom, consistent unification
charged
gluons
velocity
(using
XOR,
velocity discrete
addition rotations
and angular momentum commutation dimensions do not commute.
of the
labels
model.
Baryon onto
number,
Three
strings
lepton
22 - 1 = 3; 23 -
which
number,
add
charge
1 = 7; 2 r - 1 = 127
(3, 10, 137), which are the first three levels of the four level by A.F. Parker-Rhodes in 1961. The first level describes chiral
leptons
and
approximation
our
label
length
nucleons with baryon number separation h/mpc. Since the c, it is gravitostatically
the third
colored
to hc/e _ by correctly
quarks. modeling
We justify the
the identification
relativistic
Bohr
hydro-
stable
and
mapping
from
either from the weak or the electromagnetic from the theory are given in Table I. 16 to 256 we get the fourth
212r+ 136 _ 1.7 x 1038 _ hc/Gm_, suggesting conservation, we can consider an assemblage
against
particle
emission,
but
is unstable
[11] to particle ratios
physics. of particles
It also provides which
satisfy
us with
488
Dirac
cardinal closure. and anti-
pairs with average assemblage exceeds
to energy
loss
due
to
it ends up as a rotating, charged of bits of information lost in its This extends Wheeler's "it from
a non-perturbative
the free particle
(terminal) gravitational of nucleons
+1, charge +e, spin ½h containing N = hc/Gm_ escape velocity for a massive particle from this
Hawking radiation. Thanks to our baryon number conservation black hole with Beckenstein number hc/Grn2p [i.e. the number formation [10]] which is indistinguishable from a (stable) proton. which mass measured.
add
and improve on this result by deriving both the Sommerfeld formula and a logically correction factor: hc/e 2 = 137/(1 1 30-6_-_) = 137.0359 674. [9] Weak-electromagnetic at the "tree level" comes about by using the same geometrical argument to calcu-
of the combinatorial hierarchy: Since we have baryon number
bit"
by limiting which
in the strings satisfy the triangle the lines connecting the counters.
of the standard
late the electron mass in ratio to the proton mass interaction and equating the two results. Predictions Extending
+ N0)]c.
model we attach labels to the content strings which structure. Using 16 bits, the label gives us the 6
to the null string map onto a Feynman diagram vertex. and color are conserved; color is necessarily confined. Mapping
length
as defined above satisfy the usual relativistic theory is "Lorentz invariant" for finite and
In order to identify particles describe the (finite and discrete)
two distinct
by v = [(N1 - No)/(N1
and boosts. We prove that the usual position, momentum relations follow from the fact that finite rotations in three
quarks,
between
given
modulo 2) to the null string. The number of l's and hence can be used to define the angles between
It also follows that the velocities law; this shows that our integer
numbers
equation
mass
scale
derived
relative
above
to
can be
•_ __ E_ a
"_, 0
X .---.
_,.
c.,_
_'d
X _
oo
_..__..._t_
A
_'_
_--.
oO
0
_
i°
X
_ I_
_
_
_
"_,
"_
<
_
_
o.
II
oo
< x
m JI
o
_
I
x
_
,. = .._._
_
_
_o_
"-"
_
_E E.._
0
•_o7 _'
E
_
r,,.b
g
7 e
_ ¢ _
_ N
@
r..)
o .o
.
"_ _
. ,,..._
_
489
_
5
Fundamental
The theory that
Principles
has grown
provides
from results
a consistent
that
many
way to compute
physicists
several
rejected
as "numerological"
fundamental
constants
to a framework
of physical
interest.
It is
based on fundamental principles that we believe should appeal to physicists who are sympathetic to the operational approach of Bridgman and the early work of Heisenberg. These principles are
finiteness,
discreteness,
further information, and our procedures
finite
computability,
absolute
all members of a (necessarily must be strictly constructive.
as a science
of measurement,
in nature could result from has discovered that Galileo the
vertical
distance
through this
can expect
arc
to the
to about that
by context
that
time
it takes
= 1.108 2 .... We now compute
constant
the same "anywhere and time.
at least
bodies
some
absence
be given equal in which the
sensitive
theory
satisfying
our
when
extended
to large
principles
which
counts
can
to fall from rest as rc/2vF2
oscillate" counts
have
events
at
most
that
uncovered
7....
this
Drake [12] to swing to an equal
Thus
Galileo
constant
will be
of the units
by a single three
weight.] Book of
through
= 1.110
independent
of
bits of information.
of the structures
[13] We now believe
fall and pendulums
metric
a body
this ratio
0.3 % accuracy.
In any any
nature
In the
the way we perform experiments. For example, Stillman measured the ratio of the time it takes for a pendulum
a small
as 948/850
measured
leg.
finite) collection must For us, the mathematics
Nature is written is finite and discrete. We model In this sense we are participant observers. Physics,
non-uniqueness
sequence
homogeneous
of length
of integers, and
isotropic
dimensions in our finite and discrete sense synchronized by one universal ordering operator. [14] More complex degrees of freedom, indirectly inferred to be present at "short distance" automatically "compactify". Hence we can expect to observe at most three absolutely numbers at macroscopic distances and times. Guided by current experience, be lepton number, charge and baryon number, the extended Gell-Mann Nishijima rule. These stability
of the
proton,
electron
bitrary, since structures arise in our construction.
and
with
connected to the z-component of weak isospin by are reflected in the experimentally uncontroverted
electron-type
appropriate
conserved quantum we can take these to
neutrino.
conservation
This
laws
choice
isomorphic
is empirical with
this
but
not
interpretation
Take the chiral neutrino as specifying two states with lepton number 4-1 and no charge. couple to the neutral vector boson Z0. In the absence of additional information, these states The 4 electron states couple to two helical gamma's and the coulomb interaction. These states
can
gamma.
be generated
These
by any
3-vertex
3 + 7 = 10 states
when
which
includes
considered
two electron
together
then
states
and
the
W+.
generate
ar-
They close. seven
an appropriate This
completes
the leptonic sector in the first generation of the standard of length 6 provide a compact representation of these
model of quarks and leptons. Bit-strings states which closes under discrimination
(exclusive-or),
the
and
conserves
vertex. No unobserved missing. Two sector
flavors
which
(16 fermions level"
of quarks
generate times
description
states
the
and
both
three
inferred
a color octet of the quark
lepton
number
are predicted colored
gluons
z component
provide
127 quark-antiquark,
minus model.
and
at this level of complexity,
the state Bit-strings
490
seven
3 quark,
with no quantum of length
the
of weak
elements
8 gluon
needed
a compact
at each states
of the
3 antiquark, numbers)
8 provide
isospin
and no observed
are
baryonic ... states
for the "valence
model
using
seven
discriminately independent basis strings and again close producing this level of complexity. Combining them with the leptonic states the
vector
which
bosons
occur
model.
to be extended
in the standard
Extending
observed
have baryon makes CPT numbers,
color
scheme
(and
number invariance
the
(although
only
conserved)
one
structure
to the coulomb
probability
of this interaction
Our
basic
quantum
two events
conserved
quantum
numbers
the
about
If we model times, the
these
hydrogen
of the given four
relations
the hydrogen
in the are
a "particle"
occur
that
above.
Our constant of view.
2 and
gives
the
constants.
that
to which
the
challenge.
one
to three
dimensional
in several
structures
derived
c_, mr, and
Me
in well defined
basic
When
a quantitative
prediction
invariant
quantum
times
angular
the interval
numbers
(h/rnc)
2 and
and
that
(b)
momentum.
vector
we must
sweeps
the
angular
the
relativistic
degree
of freedom,
Sommerfeld
Fermi
square are
from and
[15] for
proper
the
account
formula
16 possible
comes
compared,
formula
take but
= 27rr. in equal
is g(g + 1)h _.
Bohr and
n_
areas
momentum
involves
root
have
out equal
formula
interaction
the
the
constants "first
relation
assert
that
assume
that
standards (which
could
principles". to those
the
the
for c_
states
of
conventional
my comes
already
length
constant know
and
be e, h and G) that But
units,
dielectric
they
of mass,
charge +6 in units of e. Otherwise their calculation We claim that within their framework, these three
from first principles. emerge from currently
leptons
non-uniqueness,
of the
1/2,
numbers
they
they must
from
numbers. and
information.
square
r from a center
the
-2 where
contact
from
the
the constants we compute with a calculation of the dielectric to how complicated the number hc/e _ must be from their point
can relate
laboratory
the
experimental
from first principles, Of course
quarks
in flavor
first
of quarks
conserved
we get
a second
not only
the
and
if the radius
1/137n_,
= (256mp)
critics sometimes compare of diamond as an analogy
be calculated
The
are
constant velocity can occur only an integer These give us relativistic kinematics and
a distance
that
we include
fact
us v_GF
We accept
quark).
of absolute
of two integers
with g = n_ -
we get
(a)
momentum
by events
probability
When
Similarly,
fermions
which
experimental
of further
carries
is the product
by noting
in counting,
interaction Lagrangian stability of the proton.
has
atom
with
spectrum.
ambiguities
of the standard
of 137 in our
model
absence
which
for position,
is supported
events
vertices
is not conserved
and
standard
by our principle
is 1/137
them,
value
universe
by the
postulates
by
AA/)_ 2 = (n_ - 1/4)(1/2rr)
Since
the
generations
top
number
the
by program
Hence,
occurring
between
why we obtain
required
interaction.
mechanical
commutation
This interpretation
of the
and generation
generated
137 states
connected
3-momentum
usual
only
of the first generation
16 we get the three
space-like correlations for particle states with the same number na of deBroglie wavelengths (_ = h/p) apart. the
and
of length
is confined,
to talk
of the
corresponds
between
vertices
1/3 and charges -I-1/3, q-2/3 as required. The 0 _ 1 bit-string symmetry automatic. As already noted, if we have only three large distance quantum
hierarchical
Empirically
all the
unification
to strings
a slot with
changing decays. We are now in a position between
14, producing
weak-electromagnetic
the whole
experimentally
to length
only the appropriate states at allows the strings representing
to get
a number
time
that
can
of physical
as measured
occur,
to diamond
as well as the fact
of diamond
in the
self-consistently,
they
will also need
the carbon
has no potential empirical numbers are too complicated
nucleus
test. to calculate
In fact, when Weinberg discusses how a finite coupling constant might acceptable theory, his errors are so large that he cannot even contemplate that
can
be confronted
by experiment.
491
In contrast
my values
for _,
and
me are
allow
good
to six or seven
me to predict
and
13 proton
masses.
to my physical terrestrial-type conducting
that
the
significant
common
I have
figures,
isotopes
systematic
and
I can
of carbon
ways
of improving
experiments.
of "space-time".
Somewhere
If the
of renormalized
and that
quantized
as you "squeeze"
density
"weak" if one
"quantum
second
change
an energy
something
interactions can extend
appropriate
these
along
place
vacuum"
(which
relativistic
it. The received
scheme.
It seems
from
prefer today
of the
is that
proton
the
adopting
On the other
values
hand,
from
"first
values
cannot
about
2 x 10 -l°
our
6 This
if one starts
"empty
principles"
and
be considered baryons
space"
systematically
"primordial". per
photon.
improve
After This
thanks
principles" different
concept,
--
views
plenum")
its properties
if the squeezing
"strong",
would
produces
"electromagnetic"
grand unification) and gravitation will find its however
beautiful,
possible experimental tests methodology for a physicist.
charges
and massive
particles
to and
one can measure masses and coupling -- get good approximations for these the
the universe
both
"first
"principles",
here and now with separated
or "constructed" space as the first approximation, in a well defined way. If one can -- as we claim
also--
to call a "quantum
"coupling constant" orders of magnitude,
to me that
and
12
of these two isotopes on a of th e kind in which we are
is the underlying
wisdom
that
come together (one basic the theory another three
in the
I would
principles"
of approximately
near as soon as theirs. us as coming from our
field theory
like 1016 times
my "first
estimates,
this line my calculation
that force one to go thirteen orders of magnitude beyond currently define fundamental parameters is -- to say the least -- a peculiar "empty" constants
that masses
cosmology -- of estimating the relative abundance planet with an age of 4.5 × 109 years in a solar system
find empirical supplements useful, but I believe no where I would locate the difference in point of view between
certainly
argue
will have
predictions, becomes
is in agreement
I fail to see why such optically
with
thin,
observation
and
we predict supports
philosophy.
Acknowledgements work
was supported
by Department
of Energy
contract
DE-AC03-76SF00515.
References [1] D.
O. McGoveran
(June,
1989);
[2] H. P. Noyes as DP. [3] T.Bastin, [4] Discrete Stanford,
and
hereinafter and
Studia
H. P. Noyes, referred
D. O. McGoveran,
Philosophica
"Foundations
for a Discrete
Physics",
SLAC-PUB-4526
to as FDP. Physics
Gandensia,
Essays,
2, 76-100
(1989);
hereinafter
referred
to
4, 77 (1966).
and Combinatorial Physics, H. P. Noyes, CA 94306, 1987; this includes FDP.
ed.,
[5] R. P. Feynman and A. R. Hibbs, Quantum McGraw-Hill, New York 1965, Problem 2-6, pp 34-36.
492
ANPA
WEST,
Mechanics
409 Lealand
and
Path
Avenue,
Integrals,
[6] T. Jacobson
and
L. S. Schulman,
[7] H. P. Noyes,
An Introduction
[8] H. P. Noyes,
Physics
[9] D. O. McGoveran [10] W. H. Zurek
and
[11] J. A. Wheeler, Tokyo,
March
"Information,
Physics,
Gen. 17, 375-383
J. C. van den Berg,
(1984). ed. (in preparation).
1992, pp 99-100.
H. P. Noyes,
K. S. Thorne,
A: Math.
Physics
Phys.Rev.
Physics,
Essays, Letters,
Quantum:
4, 115-120
(1991).
54, 2171-2175 the Search
(1985).
for Links",
in Proc.
3rd ISFQM,
1989, pp 334-368.
[12] Stillman
Drake,
[13] H. P. Noyes, [14] FDP,
to Discrete
Today,
and
J. Phys.
Theorem
[15] N. Bohr,
Phil.
Galileo:
"On
Pioneer
Scientist,
the Measurement
of r",
University
SLAC-PUB-5732,
13. Mag.
332,
Feb.
of Toronto
1915.
493
Feb.
Press, 1992.
1990,
p. 8, p. 237.
N93-2735 Covariant
Harmonic
Oscillators-
M.E.
1973
Revisited
Noz
Department
of Radiology
New York University New York,
NY
10016
Abstract
Using the relativistc harmonic oscillator, we give a physical basis to the phenomenological wave function of Yukawa which is covariant and normalizable. We show that this wave function can be interpreted in terms of the unitary irreducible representations of the Poincar_ group. The transformation properties of these covariant wave functions are also demonstrated.
1
Introduction
Because
wave
combining
functions
quantum
play
a central
mechanics
and
role in nonrelativistic special
relativity
takes
quantum
mechanics,
the form
of efforts
tivistic wave functions with an approrpriate probability interpretation. which has the useful property of mathematical simplicity, has served tion to many
new physical
theories.
It played
one method to construct
of rela-
The harmonic oscillator, as the first concrete solu-
a key role in the developing
stages
of nonrelativistc
quantum mechanics, statistical mechanics, theory of specific heat, molecular theory, quantum field theory, theory of superconductivity, theory of coherent light, and many others. It is, therefore, quite
natural
harmonic
to expect
oscillator
In connection
with
that
wave
the
first
function[l,
relativistic
nontrivial
relativistic
wave function
would
be a relativistic
2].
particles
with internal
space-time
structure,
Yukawa
attempted
to
construct relativistic oscillator wave functions in 195313]. Yukawa observed that an attempt to solve a relativistic oscillator wave equation in general leads to infinite-component wave functions, and
that
finite-component
four-momentum Markov,[4] The
Takabayasi,[5,
effectiveness
demonstrated
wave functions
of the particle
6] Sogami[7]
of Yukawa's by
Fujimura
may
is considered. and
oscillator et al.[9] who
be chosen This
if a subsidiary
proposal
of Yukawa
condition was further
involving developed
the by
Ishida.[8]
wave
function
showed
that
in the the
relativistic
Yukawa
wave
quark function
495 PREGEDING
PAGE
15LANK NOT
FILMED
model leads
was first to the
correct wave
high-energy function
The
behavior
was also rediscovered
oscillators instead paper Of Feynman the authors
asymptotic
of this paper
basic problem
facing
oscillator
for unitary
by Feynman
form
factor.
The
et a/.[10] who advocated
did not make
any attempt
any relativistic
harmonic
conservation.
harmonic
nucleon
It had once by eliminating
This belief
wave functions
irreducible
to hide those oscillator
without
representations
equation
out to be true.
time-like
use of relativistic
and interactions. wave equations,
The and
is the negative-energy
spec-
that any attempt to obtain would lead to a violation of
It is now possible
wave functions
of the Poincar_
oscillator
troubles.
been widely believed time-like excitations
did not turn
harmonic the
of Feynman diagrams for studying hadronic structures et al. contains all the troubles expected from relativistic
trum due to time-like excitations. finite- component wave functions probability
of the
which
form the
to construct vector
spaces
group.
In Section 2, we formulate the problem by writing down equation which leads to the covariant harmonic oscillator
the relativistically invariant differential formalism. In Section 3, we study solu-
tions of the oscillator differential equation which are normalizable in the four-dimensional x, y, z, t space. In Section 4, representations of the Poincar6 group for massive hadrons are constructed from
the
normalizable
for unitary
harmonic
irreducible
oscillator
representations
wave
of the
functions.
Poincar$
It is shown group,
little group for massive particles. In Section 5, Lorentz transformation oscillator wave functions are studied. The linear unitary representation is provided
2
for the harmonic
Covariant
oscillator
that
they
as well as that
where
x_ and
z_ are
Oscillator
to simplify
_
space-time
partial differential equation boundary conditions. In order
+
the above
+
coordinates
has many
Differential
Equations
for the first
different
differential
(:c_-x_)2+rn_
solutions
equation,
and
sures
four-vector the
X specifies
space-time
where
separation
the hadron
between
the
is located quarks.
second
quarks
(1) respectively.
496
This
on the choice of variables
new coordinate
and
variables:
(2)
in space-time, In terms
of two
$(x_,xb)=0,
depending
we introduce
consisting
X = + zb)/2, z = (zo -xb)/2. The
basis
O(3)-like
properties of the harmonic of Lorentz transformation
We first consider the differential equation of Feynman et al.[10] for a hadron quarks bound together by a harmonic oscillator potential of unit strength:
_
the
wave functions.
Harmonic
-2
form
for the
of these
while
the variable
variables,
x mea-
Eq. (1) can be
written
as
_ This equation
is separable
m0 _+ _
in the
X and
(3)
¢(x,x) = 0.
x variables.
Thus
¢(x, x)= y(x)¢(x), and
f(X)
and ¢(x)
satisfy
the following
differential
equations
,rig- (_ + 1) f(x) 1( Eq. (6) is a Klein-Gordon
°2
equation,
and
(4) respectively:
(5)
= o,
(8)
I
its solution
f(X)
takes
the
form
(7)
= exp [+ip.X"],
with _ p2 = _ p, p_, = M 2 = mo2 + ()_ + 1). where
M and
is determined eigenvalue dealing
P are the mass from the
for the
only with
momentum
and
If the
oscillator
3
states
of the
to x provided wave
equation
function, for the
Normalizable
We are using
p,
there
p,,+Ps,
q
=
(p. -pb).
given
to X.
exist
wave
any
for the operator confusion
since
and we are
combine
them
into
the
total
four-
The
(S) internal
functions
can be obtained equation
notation
cause
eigenvalue
the quarks:
=
differential
x space
pb, we can
between
conjugate
not
The
four-momentum.
P
functions
the
and
respectively.
the same
should
with a definite
separation
that
wave
of the hadron
This
quarks
four-momentum
momentum-energy
space-time
of Eq. (7).
momentum-energy
hadronic
four-momentum
four-momentum.
free hadronic
four-momenta
q is conjugate
solution
hadronic
As for the
P is the
and
momentum-
which
from
the
energy
separation
can be Fourier-transformed. Fourier
in the
q space
the
Relativistic
transformation
is identical
to the
of the harmonic
in Eq. (7)
Solutions
of
Oscillator
Equation Since tivistic
we are quite quantum
familiar mechanics,
with
the three-dimensional
we are naturally
harmonic
led to consider
497
oscillator
the separation
equation
from nonrela-
of the space
and
time
variables
and
write
the four-dimensional
(-V However,
the xt system
the above
harmonic
2+
+[x
oscillator
equation
of Eq.(1.6)
as
1)_,(x)=()_+l)¢(x).
2-t
2
is not the only coordinate
system
(9)
in which
the
differential
equation
takes
form.
If the hadron
moves
along
the Z direction
which
is also the z direction,
then
the hadronic
factor
f(X) The
of Eq. (8) is Lorentz-transformed in the same manner as the scalar particles are transformed. Lorentz transformation of the internal coordinates from the laboratory frame to the hadronic
rest
frame
takes
the form x, Z t
t#
where/3 is the velocity of the hadron coordinate variables in the hadronic differential
equation
=
y' = y,
(z-
s3t)l(1-
t32)'12,
(t-
Jz)l(1-
S72)'i_,
(lo)
moving along the z direction. The primed quantities are the rest frame. In terms of the primed variables, the oscillator
is
(-V This form is identical
'_ + _
to that
]) ¢(x)
+
of Eq. (10)
- t '2 , due
(11)
= (A + 1)¢(x).
to the fact that
the oscillator
differential
equation
is Lorentz-invariant.[1] Among
many
possible
¢_
solutions
=
(1)\._]
of the
above
differential
[ 1 _ (. is like that
the z direction, 2]
500
the
(23)
of z. Therefore, n _ excited
state
if the groundshould
behave
4
Irreducible
Unitary
Representations
of
the
Poincar
Group The Poincard group consists back to the quark coordinates tions as X.
on the quarks. However,
The
under
the
of space-time translations and Lorentz transformations. z_ and zb in Eq. (1) and consider performing Poincard
same
Lorentz
space-time
transformation translation
matrix
which
is applicable
changes
x_ and
Let us go transforma-
to za, Xb, X as well
Xb to z_ + a and
Xb + b
respectively, X
_
X+a,
x
_
x.
The quark separation coordinate x is not affected of translations for this system are
(24)
by translations.
For this reason,
the generators
0 P_, = -ZOX u, while
the generators
of Lorentz
transformations
(25)
are
Mu_ = L_
(26)
+ Lu,,,
where
It is straight-forward
to check
L,,
=
i
x, Ox..
that
the
ten
z_
.
generators
defined
commutation relations of the Poincar$ group. We are interested functions which are diagonal in the Casimir operators p2 and
=
_
-O-_x_ + x
W2=
M2(L')
in Eqs.
(26)
and
in constructing W2:
(27)
satisfy
normalizable
+too _,
the wave
(27)
2,
(28)
where L,i The mass,
eigenvalue
. = -zeqkxj
of p2 is M 2 = m02 + (A + 1), and
and e is the total
intrinsic
angular
momentum
501
, 0 Ox,k. that
for W 2 is of the
hadron
M e(e + 1).
M is the
due to internal
hadronic
motion
of the
spinless quarks.[12] the intrinsic angular the helicity. Because spherical rest frame
In aMdition, momentum
If the hadron
the spatial coordinate space
we can choose the along the direction
moves
along
solutions to be diagonal in the component of of the motion. This component is often called
the Z direction,
the helicity
part of the harmonic oscillator equation system, we can write its solution using
spanned
by z', y' and
z'. The
k_
gZz_t(z ) = Rt (r,)y_g(O,,
most
general
operator
is L3.
in Eq. (12) is separable also in the spherical variables in the hadronic form of the
solution e2
¢,)[1/([v/'_2kk!)],/2Hk(t,)e-t
is
/2,
(29)
where r'
=
[za + ya + za]l/2,
cosO I
=
zt/r t,
tan(
=
y'/x',
and A=2#+g-k. Reu(r ') is the
normalized
radial
Re,(r)
wave function
= (2(_!)/[r(g
where
e+1/2 (r 2 is the associated L_, the orthonormality condition:[14]
(30)
for the three-dimensional
harmonic
oscillator:
+ e + 312)13)'12rtL_+'/2(r2)e-'2/_,
Laguerre
function.[13]
The
above
radial
(31) wave
function
satisfies
(32) The spherical wave functions
form given in Eq. (30) can of course be expressed as a linear in flae Cartesian coordinate system given in Eq. (17).
combination
of the
The wave function of Eq. (30) is diagonal in the Casimir operators of Eqs. (28) and (29), as well as in L 3. It indeed forms a vector space for the O(3)-like little group.j15, 16] However, the system is infinitely
degenerate
due to excitations
along
the t' axis.
As we did in Section
the time-like oscillation by imposing the subsidiary condition be zero in Eq. (31). The solution then takes the form ¢_"at(x)
= Rt(r')Ytm(O
', ¢')[(1/rr)
of Eq. (16),
'/4 exp(-ta/2)],
2, we can suppress
or by restricting
k to
(33)
with A =2#+g. Thus for a given A, there are only a finite number expressed as a linear combination of the solutions coordinate
system
given
in Eq. (17).
502
of solutions. The without time-like
above spherical form can be excitations in the Cartesian
We can now write the solution
of the differential
equation
of Eq. (1) as
¢(x, z) = This
wave
function
space-time
describes
structure
which
a free hadron can
with
be described
Poincar_ group. The representation is unitary on the internal variable x is square-integrable, are Hermitian transformed.
5
operators.
We shall study
Transformation Wave Functions
If the hadronic velocity function then is
(34) a definite
by an
¢0(x)
then
irreducible
in the next
section
how these
of the
in obtaining
the wave functions hadron at rest,
its rest frame
= Rtu(r)YT(O,
the wave
for the rest frame.
an internal
representation
coincides
¢)[(1/7r)
function
of the
are Lorentz
Oscillator
with the laboratory
frame.
The
wave
'/4 exp(-t2/2)].
the boost
(35)
hadron is to replace the r, 0 and ¢ This produces Eq. (30). However,
for a moving
If we apply
wave functions
Harmonic
The simplest way to obtain the wave function for the moving variables in the above expression by their primed counterparts. we are interested
unitary
having
because the portion of the wave function depending and all the generators of Lorentz transformations
Properties
is zero,
four-momentum
hadron
operator
as a linear to the
combination
wave function
of
for the (36)
where
Ka is the boost
generator
along
the z axis,
z_-_+ O
K3=-i and ,7 is related
to velocity
parameter/3
the
Casimir These
rest-frame operators
eigenstate
coordinate invariant.
and
moving-frame
P_ and
W 2 of the
wave functions
systems. Therefore,
t _zz o ) '
(37)
by sinh rt =/3/(1
Both
its form is
wave Poincar_
are linear
- 132)'/_.
functions
have
the
same
set of eigenvalues
for the
group.
combinations
of the Cartesian
If the hadron moves along the z direction, we use the wave function of Eq. (19) with/3 _,_,o = [Xl(_r2,_n!)],/2H,_(z)expl_(li2)(z2
503
forms
the x and -- 0i + t_)].
in their y variables
respective remain
(3S)
The superscipt 0 indicates consider the transformation
that
there
are
_,_'°(z, t)
no time-like
=
[exp(-tr/K3)]g,o "
We are
kw--0.
excitations:
now led to
"'°(z,t)
= ¢_'°(z',t'), and This
ask what boost
However,
the boost
operator
operator
of course
we are interested
exp(-ir/Ka) changes
.n,0, !,z, t). does to W0
z and
in whether
t to z' and
remains
the
oscillator
invariant,
differential
and
t' respectively
the transformation
,/4'°(z,t) = E Because
(39)
equation
only the terms
can take
above.
form
A,,,w(fl)_b ,,,o o,,,,,,(z,t).
is Lorentz
which
as is indicated
the linear
satisfy
(40)
invariant,
the
eigenvalue
,k of Eq.
the condition
n=(n'-k') make
non-zero
contributions
in the
sum.
Thus
(18)
(41)
the above
expression
can be simplified
to
OO
g,_'°(z,
t)=
E
A'_(fi)_,'d+k'_(z,t).
(42)
k=O
This is indeed a linear unitary representation of the Lorentz group. The representation dimensional because the sum over k is extended from zero to infinity.[17] The remaining we can write
problem
is to determine
A'_(fl)
the
coefficient
=
f dzdt¢_+k'k
=
1(_)'* _r
A_(fl).
Using
the orthogonality
is infinte-
relation,
( z, t )¢_'°( z, t )
(2)'/2(
n!(n
1 _1,2 + k)t]
x f dzdtH,_+j,(z)Hk(t)Hn(z') x exp (-_(z In this integral,
the Hermite
polynomials
2 +zea +t 2 +tea)).
and the Gaussian
form are mixed
Lorentz transformation. However, if we use the generating function the evaluation of the integral is straightforward, and the result is A'_(fl) Thus
the linear
expansion
given
.,0, p _Z, t)
=
= (1 - fl2)O+,OI2 fiJ, ( (n + k )!'_ 1/=
\ hT_ ]
in Eq. (41) can be written
(43)
for the
with the kinematics Hermite
"
as
[1/(2",011"(1 fl){"+'_/=(exp[-(z =+ t=)/21) k=O
,
504
of
polynomial,
(44)
As wasindicatedwith respectto Eq. (20), this linear transformationhasto be unitary. Let us checkthis by calculatingthe sum CO
S = Y] I A (fl)12 •
(46)
k=O
According
to Eq. (45), this sum is oo (n + k)!(,,_,k n]-kT '/_ ) "
S = [1 - f121(,,+,)_
(47)
k=O
On the other
hand,
the binomial
expansion
of [1 - fl2]-(,,+l)
takes
the form
[1 - f12]-{,,+,) = _ (n + k)[ fl2 k k=o n[k[ " Therefore the sum transformation.
S is equal
It is also of interest which
are eigenstates
terms
of the
separately.
to one.
The
linear
transformation
to see how this transformation of the
spherical
Casimir
coordinate
If the hadron
k,ra
we have to write
three
rotation
generators
For this
purpose
(43)
directly
is indeed
in terms
we construct
for the three-dimensional
the
(x, y, z) space
a unitary
of solutions solutions and
in
treat
is at rest, m
_boxt (x ) = Rtu(r)Yt
Thus
of Eq.
can be achieved
operators.
variables
(48)
the generators
(O, ¢)[1/(
of Lorentz
can be written
v/'_2kk!)]l/2Hk(t)e
transformations
-'_/2.
in terms
(49)
of these
variables.
The
as[13] .0
L3
:
--_'_,
L:_
=
L, rI:L2 -_ 5: icotO
It is not difficult
to calculate
the three
boost
generators.
r0_ + tO) _r
=
cos0
=
KI+iK2
(o
generators
affect L
---cosOr
only the spherical .h k,m 3_O)d
"+_'o_tr .,.k,m
--
=
They
take
(50) the form
rSin00-0 t cO'
:,
e :t:i* r_+tsinO
The rotation
-
.
harmonics
4- rsinO
505
"
in the wave function
-- k,m mY20M,
e T m)(e
0-O
+ m + -Jw0_t
•
(51)
of Eq. (50).
Thus
t
The above relations mean that only change m. Eq. (53) indeed is like SO(3). On the other formulas:
hand,
i K 3_M .].km
rotations do not change the quantum corresponds to the fact that the little
if we apply
=
the
boost
generators,
we end
up
numbers A, g and group for massive
with
somewhat
k. They hadrons
complicated
(g 4- m -4-1)(g - m + 1)] l/_
[ (e+ 1)(e- m)],/2 + L(_7 _-i-)J Y_T'(°'¢)Qt+'F_t' iK+
=
[(_+(2g r. + + 1)(e 1)(2g ++r_+ 3) 2)]'/'.,Yl+, (2e + 1)(2_- 1) J
(O,¢)Q+(tF;,t
(53)
Ye_I(o'¢)Q + (e+ 1)F_t(_,t).
where
Q,=
_+
_i +_
,
and F_t(r,t
) = Re_(r)ll/(v_2kk!)]'/2Hk(t)exp(-t:/2).
1(3 does not change the value of m, while K+ and K_ change m by +1 and -1 respectively. In addition, unlike the rotation operators, the boost generators change A, g and k. This is a manifestation of the fact that the unitary representation is infinite-dlmensional as is indicated in Eq. (43). It is possible the
Qt operators.
answer
6
to finish
should
the
calculation
However, be from
this
by explicitly
does not
our experience
appear
carrying
out
necessary,
with the Cartesian
the differentiations
because
we already
coordinate
sytem.
model
withstood
contained know
what
in the
Conclusion
The
harmonic
oscillator
The
work of Karr[18,
present the experimental oscillator.
applied
to the
symmetric
19] has fully integrated present
status
quark
the field theorectic
of the
506
non-strange
has
aspects
baryon
the
test
of this work.
in relation
to the
of time. Below
we
harmonic
TABLE
I. Mas_
spectrum
of nonstrange
baryons.
The calculated
masses
based
Eqs. (9.1) and (9.2) in Kim and Noz,[2] Theory and Applications of the Group. The experimental masses are from "Physical Review D" 45, No.
on
Poincard 11, (June,
1992). The last column contains the identification code of the pion-nucleon resonance used in Particle Data Group. For N = 0 and N = 1, the quark model multiplet scheme is in excellent work
agreement
well, but
There
more
with the experimental
work
is needed
are still very few particles
on both
world. the
For N = 2, the model
theoretical
in N = 3. Baryonic
masses
CMculated N
L
SU(6)
0
0
56
1
2
1
0
70
56 70
2
2
2
56
7O
SU(3)
Spin
J
Mass
and
seems
experimental
are measured
fronts. in MeV.
Experimental Mass
PDG-ID
8
1/2
1/2
940
939
10
3/2
3/2
1240
1232
P3z****
8
1/2
1/2 3/2
1520 1520
1535 1520
Sn**** D13 ****
8
3/2
Pu****
1/2
1688
1650
$11 ****
3/2
1688
1700
D13 ***
5/2
1688
1675
Dis
1/2
1652
1620
$3_ ****
****
10
1/2
3/2
1652
1700
D_
8
1/2
1/2
1480
1440
PII
10
3/2
3/2
1780
1600
P33 **
8 8
1/2 3/2
1/2 3/2
1730 1898
1710 1900
Pll ***
10 8
1/2 1/2
1/2 3/2
1862 1660
1750 1720
5/2
1660
1680
P13 **** Fls ****
10
3/2
1/2
1960
1910
P31 ****
3/2
1960
1920
P33 ***
5/2
1960
1905
F3s ****
7/2 3/2
1960 1900
1950
F37 ****
5/2
1900
200O 2100
8 8
10
1/2 3/2
1/2
1/2
2078
3/2 5/2
2078 2078
7/2
2078
3/2
2030
5/2
2030
507
to
**** ****
Pls * P31 *
1990
FI7
**
2000
_35
*
Table
I. Mass
spectrum
SU(3)
Spin
N
L
SU(6)
3
1
70
8
1/2
8
3/2
10
1/2
8
1/2
70
8
56
2
3
70
70
J
baryons
Calculated Mass
1/2
2060
3/2
2060
1/2
2228
3/2
2228
5/2
2228
1/2 3/2
2192 2192
1/2
2060
3/2
2060
1/2 3/2
2228 2228
5/2
2228
continued.
Experimental Mass
PDG-ID
1900
$31 ***
10
1/2
1/2
2192
10
1/2
1/2
2192
3/2
2192
8
1/2
1/2
1810
3/2
1810
10
3/2
1/2 3/2
2110 2110
2150
$31 *
1940
D33 *
5/2
2110
1930
D3s ***
2190
G1_ ****
8
1/2
3/2
2180
8
3/2
5/2 1/2
2180 2348
3/2
2348
5/2
2348
7/2
2348
10
1/2
3/2
2312
8
1/2
5/2 5/2
2360 2528
7/2
2528
8
3/2
3/2
2528
5/2 7/2
2528 2528
2200
Dis
9/2 5/2
2528 2492
2250
G19 ****
7/2
2492
2200
G37 *
5/2
2110
7/2
2110
3/3
2410
5/2
2410
2350
D35 *
7/2
2410
2390
F3_ *
9/2
2410
2400
G39 **
10 56
3/2
of nonstrange
8 10
1/2 1/2 3/2
508
**
TABLE most
II. In addition,
of these
resonaces
there
are resonances
correspond
to even-
which
parity
do not fit in this table.
baryons,
they
presumably
Since belong
to N = 4 multiplet.
SU(3)
J
Mass
PDG-ID
8
3/2
1540
P13 *
8 8
9/2 11/2
2220 2600
H19 **** _,11 ***
8
13/2
2700
Klj3
10 10
1/2 9/2
1550 2300
P31 * //39 **
**
I0 10
7/2 11/2
2390 2420
F37" H3,11 ****
10
13/2
2750
I3,13 **
l0
15/2
2950
K3,ts
**
References [1] Y. S. Kim and
M. E. Noz.
[2] Y.S. Kim and M.E. Company,
Noz.
Dordrecht,
[3] H. Yukawa.
Physical
[4] M. Markov.
Suppl.
Nuovo
[6] T. Takabayasi.
Progress
[8] S. Ishida.
Progress
[9] K. Fujimura,
[11] Y. S. Kim and [12] Y. S. Kim,
edition,
in Theorectical
M. E. Noz.
M. E. Noz, and
[13] G. B. Arfken.
33:668,
Mathematical
and
D. Reidel
Publishing
1964. Physics,
Suppl.,
46:1352,
Physics,
67:1, 1979.
1969.
48:1570,1905,
M. Namiki.
and
Group.
331 pp.
1956.
Physics,
in Theorectical
1986.
1953. 3:760,
in Theorectical
1973.
of the Poincard
April,
79:416,
Cimento,
M. Kislinger,
D8:3521,
and Application
Cimento,
T. Kobayashi,
[10] R. P. Feynman,
Review,
Netherlands,
Review,
Nuovo
Progress
Theory
The
[5] T. Takabayasi.
[7] I. Sogami.
Physical
Progress
F. Ravndal.
1971.
in Theoretical
Physical
Review,
American
Journal
of Physics,
S.H. Oh.
Journal
of Mathematical
Methods
for
Physicists.
1985.
509
D3:2706,
46:480,
Academic
Physics,
43:73,
1970.
1971.
1978.
Physics, Press,
20:1341, New
1979.
York,
NY,
3rd
[14] P. Morse and H. Feschbach. York, NY, 1953. [15] D. I-Ian, Y. S. Kim, [16] D. Han,
Methods
of Theoretical
Physics,
I and
H.
McGraw-Hill,
M. E. Noz, and
D. Son.
Physical
Review,
D25:1740,
1982.
Y. S. Kim, M. E. Noz, and
D. Son.
Physical
Review,
D27:3032,
1983.
[17] Y. S. Kim,
M. E. Noz, and
S.HI Oh.
American
Journal
of Physics,
47:892,
New
1979.
[18] T. Karr. A Field Theory Wavefunctions. University
of Extended of Maryland
Particles Technical
Based on Covariant Report, 76-085:1-44,
Harmonic Oscillator January 1976.
[19] T.
of Extended
Particles
Based
Harmonic
Karr.
A Field
Wavefunctions 1976.
Theory
II: Interactions.
University
510
of Maryland
on Covariant Technical
Report,
Oscillator
76-255:1-51,
May
N93-27359 CALCULATION FROM
OF THE NUCLEON STRUCTURE THE NUCLEON WAVE FUNCTION
Paul HT
FUNCTION
E. Hussar
Research
185 Admiral
Institute
Cochrane
Annapolis,
MD
Dr.
21401
Abstract
Harmonic
oscillator
wave functions
have played
an historically
important
role in our
understanding of the structure of the nucleon, most notably by providing insight into the mass spectra of the low-lying states. High energy scattering experiments are known to give us a picture of the nucleon wave function at high-momentum transfer and in a frame in which the nucleon is travelling fast. This paper presents a simple model that crosses the twin bridges of momentum scale and Lorentz frame that separate the pictures of the nucleon wave function provided by the deep inelastic scattering data and by the oscillator model.
1
While namics
Introduction
a prediction (QCD)
for the nucleon
seems,
even
structure
now, a remote
functions prospect,
the structure of the nucleon, a clear interpretation structure function ratio (R '_p) and the polarization tion is essential. was made
A notable
by Le Yaouanc
wave functions
attempt
to relate
state
in the
cases were observed Against there
the
mixing
these features
in an attempt
nucleon
to the nucleon
non-relativistic
to formulate
predictions
chromody-
a deep understanding
rest-frame harmonic both
of
neutron-proton structure funcwave function oscillator
about
spatial
the structure
While both the large-x behavior of R np and the initial were well accounted for by the inclusion of an admixture
wave function,
the signs of the
mixing
angles
obtained
in the two
to disagree.
structure-functions
is no clear
if we are ever to claim
in quantum
of such gross properties as the asymmetry (A "_p) of the proton
et al. [1, 2, 3], who employed
and SU(6)
functions and the nucleon form factors. slope of the neutron electric form factor 70 excited
from first principles
prescription
concern that will be addressed be raised against the treatment
calculation
of Le Yaouanc
for Lorentz-transforming
et al. may be raised
a non-relativistic
wave
the objection function.
that
It is this
in this paper. Less widely recognized is an objection that can of the form factors by Le Yaouanc et at. The latter calculation
511
jnyoNes form
theassumption
factor
Lorentz
that
are related
transformation
spatial
matrix
calculation, value
the
for the
Since
possibility
mixing
(magnetization)
factors
cannot
spin
wave
the
and
relationship
is ignored.
There
be considered
be factorized
electric holds
are several
into a product
functions
play
that
structure
the
no role
(magnetic)
only when
models
functions
which
mixing[4, elements
of spin,
in the
the are
5, 6]. In involved
isospin,
structure
and
function
provide
the
correct
angle.
The spatial wave functions that shall be considered relativistic harmonic oscillator equation of Feynman used
density
Fourier
electric form factor in the absence of SU(6) relativistic spin wave functions, the matrix
the
must
The
wave functions
neutron plausible
of the form
elements.
charge
transformation.
of the spatial
known to predict a non-zero such models, which employ in the determination
the nucleon's
by Fourier
non-normalizable
"indefinite
metric"
are the "definite metric"[7] solutions et al.[8] In their original work, Feynman
solutions
of their
wave
equation.
These
to the et al.
solutions
yield divergent form factors as _q2 increases. To remedy this, they multiplied all matrix elements by an ad hoe factor. The "definite metric" solutions are normalizable and, when used to calculate nucleon
form
factors,
adjustments. patton
These
model,
In Section
yield
the
solutions
proper
q2 behavior,
a dipole
fall-off
for large
_q2
also help
to illuminate
features
of the
structure
functions
as will be seen later
2 the
relativistic
relativistic-oscillator
nucleon
oscillator
equation
and
its normMizable
solutions under Lorentz's transformation frame is exhibited. In Section 3 the
wave
function
is combined
with
QCD
2
The
state
is calculated.
In Section4,
Relativistic
the significance
Oscillator
For simplicity of discussion, first. This model describes
solutions
the
are
is discussed, and infinite-momentum-frame
momentum
rated via the valon model of Hwa.[9] The proton and neutron structure within the context of the resulting model, and a value for the mixing 70 excited
any
and
on.
harmonic
viewed. The behavior of these form in the infinite momentum
without
scaling
re-
their
incorpo-
functions are considered angle for an admixture of
of this calculation
is reviewed.
Model
the relativistic oscillator model is introduced for the two particle case the binding of a pair of quarks to from a meson via the differential
equation
{2
where
z 1 and z2 represent
convention
is defined
[012
71" (922] --
(032/16)(371-
the space-time
by -g00
the relativistic oscillator model is readily solved via separation
= 9ii
--
Z2)
coordinates 1. The
have been of variables
quark
2-
T}'/2}
I'I'#(_1,
of the two constituent spin will be ignored
formulated to include spin in terms of the coordinates
512
(1)
X2):0
1/2
quarks, here,
and the metric
though
quarks.[10,
versions
of
11] Eq.
(1)
X_, = 1/2 (xlu + x2u) xu = 1/2(Xlu
where the X u are the space-time determine the space-time separation
(2)
- x2u)
coordinates of the meson center of momentum of the quarks. The separated equations are
and
the
xu
(01- m:) ¢(x) =0
(3)
(-o_ + _2/4z2 + m_) _(z) = m2_(z)
(4)
and
where
qt(x_,x2)
Eq. (4) describes Eq.
(4) is itself
given
by a linear
In the
timelike
= ¢(X)_(x).
Eq. (3) is the Klein-Gordon
a four-dimensional separable
harmonic
in terms
combination direction,
of the
space-time
of the eigenvalues an increase
in the
equation
for a meson
components
corresponding excitation
xu,
while
This
solutions
condition
The solutions corresponding products timelike solutions
are required
suppresses
to obey
timelike
the subsidiary
excitations
of oscillator
solutions
in each
coordinate in the restframe can be written as
x exp [-w/4
of the
being
m, while
meson
rest
exp(iPuX_,) coordinates
space-time
restricted
number
corresponds
m 2 is
equations. to a more
which is not observed mass, the oscillator-
frame. where P_, is the four-momentum X_,. The solutions to Eq. (4) are
components,
to the fundamental
2
(x_ + y_ 4-z_ + t_)]
513
eigenvalue
condition
in the
to Eq. (3) have the familiar form to the meson center-of-momentum
the
to each of the component
quantum
negative contribution to the mass squared. To eliminate a degree of freedom in nature, and to eliminate, as well, the unphysical possibility of imaginary model
of mass
oscillator.
with mode
the
solution
in the
via Eq. (5).
Such
(6)
where
H denotes
a Hermite
of its components, corresponding m s = w(b+ The
solutions
in a frame the meson
y,,
z_ and
and
t. in the
is required
by Eq.
x' denotes
meson
the
rest
four
frame.
(3) to be equal
vector
The
x, represented
invariant
to m s, while
meson
Eq.
(4)
in terms square
determines
mass that
k + w + 1) + rn02.
above
arbitrary
x,,
to P"
polynomial
form
a complete
set of normalized
rest-frame
solutions.
in which the meson is not at rest is specified by the Lorentz rest frame and the frame in which it is moving. For example, frame
can be written
The
wave
transformation the ground
function
between state in an
as
+(X,x)
'P"x"
(7)
xexp{-w/4[z -2(P.z)2/P2]}.
The
construction
equally mation bound
of relativistic-oscillator
straightforward. on the rest-frame state
rapidly.
quarks
The
are seen to acquire
success
of the patton
differential
equation
takes
Separation
x3 are
of variables
functions
in arbitrary
frames
is
momenta
tells us that
in the frame this should
that a three particle version between each pair of quarks
where
be the
the meson
is moving
case.
of the relativistic is assumed, and
oscillator be the governing
the form
{3 [0_ + O_ + 032] - w2/36
x2 and
lightlike
model
of the nucleon requires A harmonic interaction
xl,
wave
Figure 1 provides a pictorial view of the effect of the Lorentz transforwave function, both in coordinate space and in momentum space. The
Modelling considered.
where
momentum-space
[(x_ - x2) 2 + (x2 - x3) 2 + (xz - x,) 2] - U °} qt(x,,x2,x3)
the space-time
coordinates
can be implemented
in terms
of three
of the coordinates
X u = 1/3 (xlu + x2_, + r. = 1/6(xiu
constituent
= 0
(8)
quarks. X,
r and
s, defined
as
X3.)
+ x2u - 2Xau)
s. =
-
514
(9)
QUARKS
_
; PARTONS t
t
Boos :iz I . #=0.8._11
.... z
__--'-
>,. I nbJ
i
( Weaker spring I " constant • I Quarks becbme
DEFORMATION SPACE-TIME
z
I
(almost)
free
>(D
qo
/3=0
BOOST
i- ,e '=o.8 //P'X
i "qz _, ,.,
///_\', _ I \ t ,/#____, _,___±_ ///
i
MOMENTUM-ENERGY .___I
I
Patton
DEFORMATION
distribution I I
FIG. space
and
1 The in
Lorentz naomentum
deformation
momentu
properties
space.
515
becomes
of tile
wider
relativistic
m_ I ) _'--I
oscillator
in coordinate
The ¢(X)
wave function satisfies
_(xa,
x2, x3) can be written
Eq. (3) while
¢(r)
and
in terms
O(s) satisfy
of these
variables
as ¢(X)_(r)O(s)
where
respectively
-1/2(0, The
square
mass
unphysical
the three-particle that
suppress
Application
is Eq.
timelike
(3) in the
degree relativistic
of the
equation
in the
nucleon
oscillator
wave function
whose rest-frame wave z-direction, the internal
case
model
function is given momentum-space
2/(M
_
1/2
transverse
the
degrees
momentum
of freedom
been
of states
functions
momentum
frame.
1/2
(Pz
with
(f
--
momentum
p+(-
p(p+)
where setting
p_
= P0 -
Pl+ = xP
A similar
procedure
P,. and
The
may
to the internal As/3
of
a meson along
the
1/2]
separation _
construction in which
1, the
(11)
- f12)]
coordinate
square
x, and
magnitude
everywhere else. in a distribution
where
of _(p,/3)
Integrating along for the internal
by
p(p+)
p(x)dx
be followed
conditions
= lim f dp_ [ _(p,/3)12
distribution
requiring
the
by Eqs. (10),
parameter/3
--/3po) 2 + (Po -/3P,)2)/(1
neglected.
P0 + p,) given
requires
In a frame
velocity
]_po_
becomes singular along the forward light cone, while vanishing the direction perpendicular to the forward light cone results light-cone
To remove
determined
by a pair of subsidiary
to the structure
[(2)
conjugate
have
by U ° + Ar + As.
frame.
by (6) is travelling wave function is
x exp [-1/w((p,
p represents
given
spectrum
is supplemented rest
in the infinite
,2-w!)
where
is then
from the nucleon
oscillator
relativistic
the momentum-space
nucleon
of freedom
such excitations
(I0)
is converted
into
(12)
a distribution
in Feynman
x by
= p(p+)dp+. in the three-particle
i=, x exp[-(m2/2w)(1
i
case.
(1/i!)(1/2)iH][(m/
- 3x) 2]
516
For three
particles
the result
is[12]
(1 - 3x)] (13)
in general,
and
po(x)
if the nucleon
is assumed
x in Eqs.
(I3)
and
structure
function
= 3m/(2_rw)
to be described
(14)
is the
F_P(z),
1/2 " exp[-(m2/2w)(1
by the oscillator
momentum
fraction
can, for example,
be based
- 3x) 2]
ground
variable.
state
(14)
wave function.
A calculation
directly
on Eq. (14).
of the The
The
variable
proton
charge
result
is
F?(x) =< 2 > rnx/(2rw)x/2.exp[(_9m2/2w)(x_ 1/3)2]
(15)
where the average of the charge ei is taken over the three valence quarks. This calculation scaling effects predicted by QCD and yields only qualitative agreement with experiment.
3
Structure
Avalon
is a bound
interactions. entiated.
Functions state
or constituent
To be completely Let
Gv/g(x)
ignores
represent
general,
quark
whose
valons
of different
a momentum-fraction
internal
structure
is probed
spin as well as flavor
probability
distribution
v (v representing spin and flavor) in the nucleon N. A nucleonic is expressed in terms of convolutions of G,4N(x ) with corresponding valo n s:
in high should
for a valon
energy
be differof type
structure function FN(x, Q2) structure functions for the
(16)
The Q2 dependence of the structure functions for the moments of the struct ure functions
appears only in FV(x, Q2). QCD evolution Eq. (13) are used to express this dependence. According to
Eq. (16),
function
the moments
of a nucleon
structure
MN(n,Q
are given
2) = __. M./N(n)M'_(n,Q
by a sum of products
2)
of moments:
(17)
_J
where
MN,"(n,Q
2) = fo'dXx"-2FN,"(x,Q_)
517
(18)
and
M,,/N(n)
The
evolution
equations
are
the
structure functions FV(x, Q2). distributions Within the vaion, behave
as singlets
correspondingly
and
factors
governing
twist-2
QCD.
The
of the quark that
in terms
for assuming
under
the singlet
moments
flavor
of singlet
and
a form
moments
are given
for the
transformation. nonsinglet
of the moments
MNs(n,
while
(19)
dxx"-'Gv/u(x).
Q2)
the evolution
nonsinglet
1
The F'(x, Q2) are understood to be determined by the v, which distributions can be broken up into components
as nonsinglets
expressed
scaling
basis
=
moments,
of such quark
moments
The
MY(n,
moments
which
MY(n,
are defined
distributions
Q2) are to be the
in lowest-order,
by
Q2) = exp(_a_NSS
(20)
)
are
M.(n,Q
_) = 1/2(1
+ p,)exp(-d+s)
+ 1/2(1
- p,_) exp(-am
(21)
s)
where
s = In
The coefficients stant, A, is the Since
valons
distributions
d_s, usual
d_., d[ scaling
of different
(22)
.
and p, come from the renormalization group analysis.[13] The conparameter while Q0 represents the "starting point" of the evolution.
helicity
will be required
ln(Q2/A2)) ) ln(Q_/A2
as well as flavor
to characterize
are
the nucleon.
to be distinguished,
four
separate
The corresponding
moments
valon
are denoted
as
Dr(n) rq( ) 'L:")
=
MuT/p(n)
= MDt/.(n)
=
MDT/p(n)
= Mut/,,(n)
=
Mul/,(n)
= Mot/.(n)
=
MDI/,(n)=
518
Mul/,(n)
(23)
where
the
nucleon,
symbol and
sponding
1" (_). denotes
where
the
that
the
identification
isospin-reversed
valon's
helicity
of valon
distributions
is par aiM
distributions
in the
proton
within
follows
from
(antiparallel) the
to that
neutron
charge
with
the
symmetry.
of the corre-
In terms
of the singlet and nonsinglet moments Eqs. (20) and (21) and the valon moments Eq. (23), moments of the nucleon structure functions F2P(x, Q2) and F2"(x, Q2) are given by
M2'(n,Q
2) = 2/912U(n)
M_n(n,Q
the
+ D(n)IM,(n,Q
_) + 1/914U(n)
- D(n)]MNs(n,Q
_)
(24)
+ D(n)IM,(n,Q
2) - 2/9[U(n)-
D(n)]MNs(n,Q
_)
(25)
2) = 2/912U(n)
with
U(n) = UT(n)+ U (n) D(n) = DT(n) + Dl(n). It is easily moments three Eqs.
verified of F 2p and
bound-state (24)
that
and
these
F 2'_ from
equations
describe
a starting
point
(25)
were
wave function
of its three
the lowest-order at which
twist-2
the nucleon
QCD
is viewed
evolution
of the
as consisting
of its
quarks. used
by Hwa[9]
in conjuncti
F 2" to obtain fitted values for the parameters therefore not be accurate for low Q2. Ideally space
(26)
from
constituent
high Q2 where
the energy
quarks
the structure
(with,
functions
on with
experimental
moments
of F 2p and
Q0 and A. These equations are first order, and will we would like to evolve the bound-state momentum-
scale Q02 at which perhaps,
the nucleon
an oscillator-like
are observed.
is describable momentum
The fitted
parameter
as a bound
distribution),
state out
to
Q02 is an approximation
for Q02 in the sense that the lowest-order evolution equations are used. This approximation is a key feature of the valon model and is discussed in detail in.[14] The goal of Hwa's fitting procedure was to obtain estimates for the functions G,/N(X). In Figure 2, an "average" valon distribution obtained in[9] by neglecting spin and flavor dependence is compared with po(x) given by Eq. (14) Let us now introduce
a 70 component
= [cos0qt0
The
¢'s represent
the
[56 >, +(sin0/v_)(¢_
spatial
wave functions;
and ¢0 are taken to b e excited zero orbital angular momentum. symmetry
which
are
of SU(6)
states The
characterized
because
it is of the
I 70 >,
+¢t_l
wave function
in the form
70 >O)]'exp(-iP"
¢0 is the harmonic
oscillator
X).
ground
state,
(27)
while
¢o
with total harmonic oscillator quantum number n = 2 and subscripts _ and/3 refer to the two possible types of mixed by the
exchange of the first and second quarks. determined in the oscillator model. The is disallowed
into the nucleon
wrong
behavior
of the
(three
quark)
wave
function
under
The form of the excited-state component is uniquely 70 state that involves n = 1 oscillator wave functions parity.
519
No other
n = 2 state
with
the
same
quantum
numbers that
as the
nucleon
are observed
dependent
valon
in the
interferes
with
structure
functions.
distributions
the
ground The
state
to produce
wavefunction
the
SU(6)
Eq. (27) leads
breaking
effects
to spin-and-flavor
of the form
Experimental
1.5
..Harmonic
p(x) 1.0 0.5 0 0
FIG.
0.2
2 A comparison
of Hwa's
the infinite-momentum-frame
Gur/p(x)
Gvt/,,(z)
=
=
0.4
"average"
valon
2 0 + sin _ 0 {5/36h(x)
x exp [-(m2/2w)(1
- 3x) 2]
(3m/_)
2 0 + sin 2 0 {1/36h(x)
[1/6cos
1.0
distribution
with
po(x)
momentum-space
(3m/2V"_-_rw) [5/6cos
defined
by
wavefunction.
+ 1/3i(x)}
- 2v/6/9cosOsinOj(x)]
+ 1/3i(x)}
+ vf6/lScosOsinOj(x)]
-
(3m/2_/_-w)
[1/3 cos20
x exp [-(m2/2w)(1
GDm,(x)
0.8
relativistic-oscillator
x exp GDr/,,(z)
0.6
2/3i(x)}
+ v/'6/9 cos 0 sin Oj(x)]
- 3x)2)]
(3m/2v'_-wrw) [2/3cos20
x
+ sin20 {1/lSh(x)+
+ 1/9sin20
+ 2vf6/9cosOsinOj(x)]
-
(28)
where
h(x) j(x) Moments (25)
U(n)
to obtain
and
D(n)
=
43/16
=
5/S + m2/Sw(1
- 3x) 2
=
I/4 - rn2/4w(1
- 3x) 2.
determined
fits for experimental
+ m2/8w(1
from
- 3x) 2 + m'/16o.,2(1
the
moments[15]
520
above
- 3x)'
(29)
distributions
of F_P(x)
and
were F2"(x)
used
in Eqs.
derived
from
(24) the
and
CHIO
muon
data[16]
was chosen
and
SLAC
to minimize
electron
target
data[17]
mass and
at Q2 = 22.5 GeV 2. A somewhat
higher
twist
effects
that
may
large
be present
value
of Q2
in the data.
The
ratios R"P(x) and A_P(z) do not app ear to show any appreciable Q2 dependence. The extension of the tails of the distributions into the unphysical regions x < 0 and x > 1 was ignored for purposes
of computing
not appear
to lead
the moments.
to noticeable
The resulting
small
deviation
from
the
Adler
sum rule
does
discrepancies.
1
1
# = 23.3°
_) =0
e
|
.1
.I
.01
.01 (n)
M2p(n)
.001
.001
M2n(n)/G
.0001
M2n(n)/"__]
P_
I,
I
1
I
I
2
4
8
8
10
I
I
I
I
I
2
4
6
8
10
.0o01
n n
FIG.
3 The
moments
of the nucleon
structure
functions
vs. n as fitted
by Eqs.
(24)
- (26) in conjunction with Eq. (29). Fitted moments at 0 = 0° and at a = 23.3 ° are represented by the solid curves and are compared with data from[15].
The fitted
moments
s defined
in Eq.
were functions (22).
this case cannot
be taken
interdependence
among
nant
of merit,
of X 2 occurs best-obtainable
so that
of two parameters
X 2 minimization as an absolute the the
moments quality
at 0 = 23.3 °, and predicted
was
- the mixing to determine
indication
of F 2p and
from
mixing Eqs.
angle 0 and the
of the quality
angle (24)
521
of 0 could
is clearly
and
(25)
best of the
F 2'_. l:/i2 was used,
of the fit as a function
a positive
moments
used
rather,
fit.
the scaling The
fit due
to the
as a relative
be evaluated.
preferred.
Figure
for 0 = 0 ° and
variable
X _ function
The
in
statistical determiminimum
3 compares
the
for O = 23.3 ° with
the experimental large
n.
moments.
With
the
At 0 = 0 °, the fitted
inclusion
of the
a simultaneous fit to the moments moments of F 2'_ remain somewhat
4
70 state
in the
of F 2'* fall outside
wave
function
of F 2p and F 2" appears large for large n.
the error
at a mixing
more
reasonable,
angle
limits
for
of 23.3 cite,
although
the
fitted
Conclusion
The simple model presented in this paper precise numerical correspondences between function and
moments
data.
The
momentum
approximate
model
scaling
does,
that
agreement
however,
must
between
falls short nucleonic address
the
be considered
of providing bound state crucial
us with properties
questions
of Lorentz
if such corespondences
pO(x) and:Hwa's
the ability to draw and the structure
are ever to be drawn.
phenomenologically-determined
tion (see Figure 2) allows us to believe that some of the essential value of the SU(6) 70 state mixing coefficien t obtained in this
transformation vai0n
The
distribu-
physics is being captured. model via a simultaneous
The fit to
proton and neutron structure function moments is very close to the original value determined by Le Yaouanc et al. This fact, together with the dependence of the form factors on the nucleon spin wavefunction, lends credence to the idea that the observed behavior of R"P and A 7p can be reliably
interpreted
as evidence
of SU(6)
mixing
in the nucleon
wavefunction.
References [1] A. Le Yaouanc, L. Oliver, 13 1519 (E) (1976).
O. Pene,
and
J. C. Raynal,
Phys.
Rev.
D 12 2137
[2] A. Le Yaouanc,
L. Oliver,
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[3] A. Le Yaouanc,
L. Oliver,
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[4] A. B. Henriques, [5] K. Fujimura,
B. H. Keiiett,
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J. C. Raynal,
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M. Kislinger,
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PAR-LPTHE
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[11] T. Takabayasi,
Prog.
[12] Y. S. Kim and
M. E. Noz, Prog.
[13] A. J. Buras,
Rev.
[14] R. C. Hwa,
Phys.
[15] D. W. Duke
and
[16] B. A. Gordon
Theor.
Mod. Rev.
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61 1235 (1979). Theor.
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60 801 (1978).
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D 22 1593 (1980).
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et al., Phys.
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D 20 2645
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[17] A. Bodek et hi., Phys. Rev. D 20 1471 (1979); M. Mestayer et hi., SLAC-Rep-214 (1979); J.S. Poucher et al., Phys. Rev. Lett. 32 118 (1974); SLAC-PUB-1309 (1973) ; W. B. Atwood et hi., Phys. Lett. 64 B 479 (1976); SLAC-185 (1975); S. Stein et hi., Phys. Rev. D 12 1884 (1975); G. Miller
523
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