NON-RESTRICTED SPECIAL RELATIVITY: Special Relativity Extended (to Anti-particles and) to Superluminal motions: AN EXTENDED REVIEW

RIVISTA I)EL NIJOVO CIMF,I~TO

VOL. 9, N. 6

1986

Classical Tachyons and Possible Applications (*). F~. ]%ECA.MI D i p a r t i m e n l o di lr dell' Universit~ Statale - Catania, I t a l i a Istitceto Nazionale di .Fisiea N ~ l e a r e - Sezione di Catania, I t a l i a

(ricevuto il 26 Aprile 1985) Q uone vide.q cilium de2mre et longius ire ~lfultlplexque loci spati'~m h'anscurrere eodem Tempore q u o Solis pervolgant l u m i n a coeIum? ,> (**).

LUCRETIa'S (50 ]~.C., ca.) ......................... should be thoughts, W h i c h ten times /aster glide than the S u n ' s beams Driving back shadows over low'ring hills. *

SSAKESPEARE (1597)

4 4

4 5 5 6 6

7 9

10 lI 11 11 13 15 16 16 17 21 21 2'2 22 23

1.

Introduction. 1"1. Foreword. 1"2. Plan of the review. 1"3. Previous reviews. 1"4. Lists of references. Meetings. Books. PAISa: i. - Particles and antiparticles in special r e l a t i v i t y (SR). 2. Special r e l a t i v i t y with ortho- and anti-chronous Lorcntz tr~msformations. 2"1. The Stiickelberg-Feynman (( switching principle )) in SI2. 2"2. Matter and a n t i m a t t e r from SR. 2"3. F u r t h e r remarks. l ' A l ~ I I . - Br~dyons and tachyons in SI~. 3. Historical remarks and preliminaries. 3"1. IIistorieal rem~rks. 3"2. Preliminaries t~bout tachyons 4. The postulates of SR revisited. 4"1. Existence of an invariaut speed. 4"2. The problem of Lorentz transformations. 4"3. 0rthogonal and autiorthogonal transformations: digression. 5. A m o d o l t h e o r y for tachyons: an ). In particular, we discuss bow tachyons would look like, i.e. their apparent (~shape ~>. Last but not least, all the common causality problems are thoroughly solved, on the basis of the previously reviewed tachyon kinematics. Part I I I deals with tachyons in general relativity; in particular the question of the apparent Superluminal expansions in astrophysics is reviewed. Part IV shows the interesting, possible role of tachyons in elementary-particle physics and in quantum theory. In part V, the last one, we face the (still open) problem of the Superluminal Lorentz in four dimensions, by introducing, for instance, an auxiliary six-dimensional space-time, and finally present the electromagnetic theory of tachyons: a theory that can be relevant also from the point of view. 1"3. Previous reviews. - In the past years other works were devoted to review some aspects of our subject. As far as we know, besides review I (ICEOAm and MIG~A.~I, 1974a), the following papers may be mentioned: CALI)IROL), and ]~ECA:~ (1980); ~ECA~xr (:1979a, 1978a); K~C]I (1977); BARASn:E~KOV (1975); Kii~Zl~iTs and SAI~OI~OV (1974); RECA.-~I (1973); BOLOTOVSKu and GI~ZBUICG (1972); CAME~ZI~D (1970); FEINBEI~G (1970), aS well as the short but interesting glimpse given at tachyons by GOL])I~BEI~ and SmTH (1975) in their review of all the hypothetical particles. At a simpler (or more concise) level, let us further list: GI:ASl~ (1983); VOULGAI~IS (1976); KICEISLEI~ (1973, :1969); VELA~DE (1972); GONDI~AND (1971); NEWT0~ (1970); BILANILm: and SU])ARSHA~ (1969a) and relative discussions (BILA~IIJ~ et al. 1969, 1970); and a nice talk by SIIDARSV_A~ (1968). On the experimental side, besides GOLX)~A]3EI~ and Smelt (1975), let us mention: BOICATXV(1980); J0xES (1977); BERLEu et al. (1975); CAlClC0L et al. (1975); I~A~NA MUXT~IY (1972); GIAC0MELLI (1970). 1"4. Zists o] re]erences. Meetings. Books. - As to the existing bibliographies about taehyons, let us quote i) the references at pages 285-290 of review I; at pages 592-597 in RECA~I (1979a); at pages 295-298 in CALDn~0LA and I~EOA~iI (1980); as well as in ICEOA~tI and M_tG~A~I (1972) and in MIGNAM and I~ECA~I (1973); ii) the large bibliographies b y P~EPEI,tTS~ (1980a, b); iii) the list b y FELD~AN (1974). However, the last one, a librarian's compilation, lists some references (e.g, under the numbers 87 9, 13, 14, 18, 21-23) seemingly having not much to do with tachyons; while ref. 38 therein (PEI~.S, 1969), e.g., should be associated with the comments it received from BA~DO and I~ECA:gI (1969). In connection with the experiments only, also the referenccs in :BAr~TImTT et al. (1978) ~nd BHA~ et al. (1979) may be consulted. As to meetings on the subject, to our knowledge: i) a two-day meeting was held in India.; ii) a meeting (First Session of the Interdisciplinary Seminars)

E . R]~CAMI

on ~ Tachyons and Related Topics ~ was held at Eriee (Italy) in September 1976; iii) a (~Seminar sur le Tachyons ~ exists at the Facultd des Sciences de Tours et de Poitiers (France), which organizes seminars on the subject. W i t h regards to books, we should mention i) the book b y TE~LETSKY (1968), devoted in part to tachyons; ii) the book Tachyons, Monopoles, and ttelated Topivs (North-Holland, Amsterdam), with the proceedings of the Erice meeting cited above (see RECA,~, Editor, 1978b).

PART I

Particles and Antiparticles in Special Relativity (SR). 2. - Special relativity with ortho- and anti-chronous Lorentz transformations. In this p a r t I we shall ]orget about taehyons. F r o m the ordinary postulates of special relativity (SR) it follows t h a t in such a t h e o r y - - w h i c h refers to the class of mechanical and electromagnetic p h e n o m e n a - - t h e class of reference frames equivalent to a given inertial frame is obtained b y means of transformations ~ (Lorentz transformations, LT) which satisfy the following sufficient requirements: i) to be linear x 'v = I2',x',

(1)

ii) to preserve space isotropy (with respect to electromagnetic and mechanical phenomena), iii) to form a group, iv) to leave the quadratic form invariant:

(2)

~ , dx ~ dx" ~ ~

dx'" dx '~ .

F r o m condition i), if we confine ourselves to subluminal speeds, it follows t h a t in eq. (2) (3)

~'p =

diag ( + 1, - ] , - ] , -

1) =

~,,,.

Equations (1)-(3) imply t h a t d e t J 5 2 = l , ( L ~ The set of all subluminal (Lorentz) transformations satisfying all our conditions consists--as is ~ve]l k n o w n - - o f four pieces, which form a noneompaet, nonconneeted group (the full Lorentz group). Wishing to confine ourselves to space-time (~rotations ~ only, i.e. to the case detJ5 ---- 9- 1, we are left with the two pieces (da)

{Lt+}: Z~

+1,

(db)

(L~+}: L00< -- 1,

detJ5 ~ 9 - 1 , detJ5 = 9- 1,

CLASSICAL T A C H Y O N S

7

which give origin to the group of the proper (orthochronous and antichronous) transformations

and to the subgroup of the (ordinary) proper ortochronous transformations (6)

~ft+ ~

{.Lt+},

both of which being, incidentally, iuvariant subgroups of the full Lorentz group. For reasons to be seen later on, let us rewrite ~f+ as follows:

z(2)-{#/+1}-{+1,-1}.

(5')

We shall skip in the following, for simplicity's sake, the subscript ~ in the transformations JSt+, ~+. Given a transformation D , another transformation L'l'e ,.s always exists such t h a t

(7)

/-?' = (- 1)./2",

V~r

,.~+,

and vice versa. Such a one-to-one correspondence allows us to write formally (7')

ff

= -

I t follows in particular t h a t the central elements of ~ + are C ~ ( + 1, -- 1). Usually, even the piece (4b) is discarded. Our present aim is to show--on the c o n t r a r y - - t h a t a physical meaning can be attributed also to the transformations (4b). Confining ourselves here to the active point of view (cf. I~EOA~ and I~O])I~GVES (1982) and references therein), we wish precisely to show that the theory of SlY, once based on the whole proper Lorentz group (5) and not only on its orthochronous part, will describe a Minkowski space-time populated by both matter and antimatter. 2"1. The St#vkelberg-~eynman () in SlY. - Besides the usual postulates of SI~ (prineiple of relativity, and light speed invariance), let us assume--us commonly admitted, e.g. for the reasons in GA~vccIo et al. (1980), MIGNA~ and ~EdA~W (1976a)--the following

Assumption: c negative-energy objects travelling forward in time do not exist )). We shall give this assumption, later on, the status of a fundamental postulate. Let us, therefore, start from a positive-energy particle P travelling forward in time. As is well known, any orthoehronous LT (4a) transforms it into another particle still endowed with positive energy and motion forward in time, On

8

E. R~CAMI

the contrary, any antichronous (~-nonorthochronous) LT (db) will change s i g n - - a m o n g the o t h e r s ~ t o the time components of all the Jour-vectors associated with _P. A n y L ~ will transform P into a particle P ' endowed in particular with negative energy and motion backwards in time (fig. 1). t

I

"x

futuPe --" "*

,'dJ~,

X!

s

7

] --I--\

/ ' - po.st--",, s(

I -. I

"x

Fig. 1. In other words, SR together with the natural assumption above implies t h a t a particle going backwards in time (G6DEL, 1973) (fig. 1) corresponds in the four-momentum space (fig. 2) to a particle carrying negative energy; and, vice versa, t h a t changing the energy sign in one space corresponds to changing the sign of time in the dual space. I t is then easy to see t h a t these two paradoxical occurrences (~ negative energy ~ and ~ motion backwards in time ~) give rise to a phenomenon t h a t a n y observer will describe in a quite orthodox way, when t h e y are--as t h e y actually are--simultaneous (1-r 1978e, 1979a and references therein).

rnetrfc (+ - - - ) ; c=1 E = • ~/p; + m~ \

/

/)" " \ / //

p,

\ \

Fig. 2. Notice, namely, t h a t i) every observer (a macro-object) explores spacetime (fig. 1) in the positive t-direction, so t h a t we shall meet B as the first and A as the last event; ii) emission of positive quantity is equivalent to absorption of negative quantity, as ( - - ) - ( - - ) = ( - F ) ' ( + ) ; and so on. Let us now suppose (fig. 3) t h a t a particle P ' with negative energT (and, e.g., charge -- e) moving backwards in time is emitted b y A at time tl and

CLASSICAL TACtIYONS

[t negative energy an(t charge, i.e. loses positive energy and ch'~rge. The physical phenomenon here described is nothing b u t the exchange ]rom B to A of a particle Q with positive energy, charge + e, and going ]orward lit time. h~otice t h a t Q has, however, charges opposite to P ' ; this means in a sense t h a t the present (in the past ca.lled ~ I~IP ~) effects a operations P~ T to be equivalent to the space, time reflections acting on the space-time both external and internal to the particle world-tube (see subsect. 11"3 in the following). Once accepted eq. (10), t h e n eq. (7') can be written

(70

f,+ = (p~) fit+ ~ (cP~) f t + .

In particular, the total inversion Lr ~ "l transforms the process a + b--> --> e + g into the process d + ~ -> b d- a without a n y change in the velocities. d) All the ordinary relativistic laws (of mechanics and electromagnetism) are actually already covariant under the whole proper group s eq. (5), since t h e y are CPT symmetric besides being covariant under f~+. e) A few quantities, t h a t happened (cf. subsect. 5"17 in the following) to be Lorentz-invari~nt under the transformations L r ~ f~+, ~re no longer invariant under the transformations J5 e f + . We have already seen this to be true for the sign of the additive charges, e.g. for the sign of the electric charge e of a particle/9. The ordinary derivation of the electric-charge invariance is obtained b y evaltmting the integral flux of a current through ~ surface which, under J~+, does move, changing the angle formed with the current. Under L ~ s f~+ the surface (( rotates >>so much with respect to the current (cf. also fig. 6, 12 in the following) t h a t the current enters it through the opposite face; as a consequence, the integrated flux (i.e. the charge) changes sign.

I~A~ II

Bradyons and Tachyons in $1~. 3. - Historical remarks and preliminaries. 3"1. Historical remarks. - l~et as now t a k e on the issue of tachyons. To our knowledge (Col~BEi~, 1975; I~CAMI, 1978a), the first scientist mentioning objects (~faster t h a n the Sun's light >> was LVCl~ETIUS (50 B.C., ca.), in his De R e r u m Natura. Still remaining in lore-relativistic times, after having

12

~. ~xcxm

recalled, e.g., Z~eLAOE (1845), let us only melltion the recent progTess represented by the noticeable papers by Tt[o~soN (1889), H~AVISlDE (1892), DES COD-D~ES (1900) and mainly SO~[ERF~mD (1904, 1905). In 1905, however, together with St~ (EInSTEIn, 1905; POI~CAm~, 1906) the conviction t h a t the light speed c in vacuum was the upper limit of any speed started to spread over the scientific community, the early-century physicists being led by the evidence that ordinary bodies cannot overtake t h a t speed. They beh~ved in a sense like Sudarshan's (1972) imaginary demographer studying the poplflation patterns of the Indian subcontinent: ~ Suppose a demographer calmly asserts that there are no people North o/ the Himalayas, since none could climb over the mountain ranges/ That would be an absurd conclusion. People o/central A s i a are born there and live there: They did not have to be born in I n d i a and cross the mountain range. So with /aster-than-light particles ~> (el. fig. 4). Notice that photons are born, live and die just ~(on the top of the mountain )>, i.e. always at the speed of light, without any need to violate S1%,t h a t is to say to accelerate from rest to the light speed.

A\

,\ -c

0

a)

c

v

c

v

b)

Fig. 4. iV[oreover, ToL~_~ (1917) believed to have shown in his antitelephone (( paradox ~> (based on the already well-known fact that the chronological order along a spacelike path is not Lorentz invari,mt) that the existence of Superluminal (v~> c~) particles allowed information transmission into the past. In recent times that (~paradox )) has been proposed again and agMn by authors apparently unaware of the existing literature (for instance, some of l~olnick's (1972; see also 1969) arguments had been already (~answered ~> by C s o N ~ (1970) before they appeared). Incidentally, we shall solve it in subsect. 9"1. Therefore, except for the pioneering paper by SOmG~A~A (1922; recently rediscovered by CMmI~OLA et al., 1980), after the mathematical considerations by MAJO~A~A (1932) and Wm~E~ (1939) on the spacelike particles one had to wait lmtil the fifties to see our problem tackled again in the works by AZCZEL~S (1955, 1957, 1958), S C ~ D T (1958), TAXGm~LL'~I (1959); and a little later by TANAKA (1960), TERLETSKY (1960) and GICEGO]~Y (1961, 1962). I t started to be fully reconsidered in the sixties: In 1962 the first article

13

CLASSICAL TACI:[YONS

by SUDA/~SttA]N and co-workers ( B ~ A ~ K et al., 1962) appeared, and after t h a t paper a number of physicists took up studying the subject--among whom, for instance, JONES (1963) and F ~ ] 3 E ~ G (1967) in the USA and I~ECAlW (1968, 1969a) and colleagues (RECAp, 1968, 1969; OLKH0VSKu and t~ECA~ 1970a, b, 1971) in Europe. The first experimental searches for SuperluminM particles were carried out by A~vXG]~ et al. (1963, 1965, 1966). As is well known, SuperluminM particles have been given the name (( taehyons ~) (T) by F v . i ~ a (1967) from the Greek word Tagi~ = fast. (~ Une particule qui a u n nora poss~de ddj~ u n ddbut d'existence )) (A particle bearing a name has already taken on some existence)~ was later commented on by A~zELI]~S (1974). We shall call (( luxons ~) (l), following B ~ L A ~ K et aI. (1962), the objects travelling exactly at the speed of light, like photons. At last, we shall call ~ oZ);

if) in St~ the speed of light v = o plays a role similar to the one played by the infinite speed v = oo in the Galileian relativity (G~II~I, 1632, 1953). Two of the aims of this review will just be to show how objection i)--which touches a really difficult problem--has been answered, and to illnstrate the meaning of point if). With regard to eq. (12), notice that a priori ~ - ~ - - 1 = = i i ~ 2, since ( - 4 - i ) 2 = - 1. Moreover, we shall alway~ mlderstand that v / 1 - - f i ~ for fl~> 1 represents the upper half-plane solution. Since a priori we know nothing about Ts, the safest w~y to build up a theory for them is trying to generalize the ordinary theories (starting with the classical relativistic one, only later on passing to the quantum field theory) through (~minimal exteltsions ~), i.e. by performing modifications as small as possible. Only after possessing a theoretical model we shall be able to start experiments: Let us remember that, not only good experiments are required before getting sensible ideas (GAL~LEI,1632), but also a good theoretical background is required before sensible experiments can be performed. The first step consists, therefore, in facing the problem of extending SR to tachyons. In so doing, some ~uthors limited themseIves to considering objects both subluminal and Super]umina], a.ll referred, bowcvcr, to sublumin~l observers ((~weak approach ~)). Other authors attempted on the contrary to generalize SI~ by introducing both subluminal observers (s) and Superluminal observers (S), and then by extending the principle of relativity ((( strong approach ~)). This second approach is theoretically more worth of consideration (tachyons, e.g., get real proper masses), but it meets, of course, the greatest obstacles. In fact, the extension of the relativity principle to Superluminal inertial frames seems to be straightforward only in the pseudo-Euclidean space-times M(n, n) having the same number n of space axes and of time axes. l~or instance, when facing the problem of generalizing the Zorentz transformations to SuperluminaI frames in four dimensions one meets no-go theorems as Gorini et aL's ( G o ~ I , 1971 and references therein), stating no such extensions exist which satisfy all

CLASSICAL TACHYONS

15

the following properties: i) to refer to the four-dimensional Minkowski spacetime M4 ---- M(1, 3), ii) to be real, iii) to be linear, iv) to preserve the space isotropy, v) to preserve the light speed invarianee, vi) to possess the prescribed group-theoretical properties. We shall, therefore, start b y sketching the simple, instructive and very promising (, model theory )~ in two dimensions (n ---- 1). Let us first revisit, however~ the postulates of the ordinary SlY.

4. - The postulates o f SR revisited.

L e t us adhere to the ordinary postulates of S1% A suitable choice of postulates is the following one (review I; ~ A t a ~ O ~ ' ~ E and ICECA~ 1982a, a n d references therein) : 1) First postulate: principle o] relativity: ~ The physical laws of electromagnetism and mechanics are covariant (----invariant in form) when going from an inertial frame ] to another frame ]' moving with constant velocity u relative to ] ~). 2) Second postulate: (~Space and time are homogeneous and space is isotropic ~. For future col~vcnience, let us give this postulate the form: (SLT) from s to S, or from S to s, must t r a n s f o r m timelike into spacelike quantities, and vice versa. W i t h assumption (25) it follows t h a t in eq. (15) the plus sign has to hold for LTs and the minus sign for SLTs: (15)

ds ' 2 = ~: ds ~

(u~l);

therefore, in (~e x t e n d e d relativity ~>(ER), with assumption (25), the quadratic form d s 2 ~-- d x . d x .

is a scalar u n d e r LTs, b u t is a pseudoscalar u n d e r SLTs. I n t h e present case, we shall write t h a t LTs are such t h a t (279)

dt ' ~ - d x ' 2 = -~ (dr 2 - dx-')

( 9 2 < 1),

dt '* -- dx '2 = -- (dr ~-- dx 2)

(u: > 1).

whilc for SLTs it m u s t be (27b)

5"3. Energy-momentum space. - Since tachyons are just usual particles w.r.t, t h e i r own rest frames ], where t h e ]'s are Superluminal w.r.t, us, t h e y will possess real rest masses mo ( l ~ c A ~ x and I'VlXG~A~I, 1972; L ~ T ~ , 1971a; PA~R, 1969). F r o m eq. (27b) applied to the e n e r g y - m o m e n t u m vector p , , one derives for free tachyons the relation (28)

E 2 -- p~ = -- ml < 0

(too real),

provided t h a t p~ is so defined to be ~ G-vector (see the following); so t h a t one

has (cf. fig. 5) (29a)

(29b) (290)

PUP"=-

+ m o 2> O

for bradyons (timelike case),

0

for luxons

--m] 1),

which can be assumed t~s the canonical ]orm of t h e SLTs in t w o dimensions. L e t us observe t h a t eqs. (39') or (39") yield for t h e speed of so w.r.t. S' dx' x--0-~----

(42)

~: --

===F U

(u~~l)'

and all of t h e m preserve the quadratic form, its sign included: d~or d~p' -~ d~ d V. The point to be emphasized is t h a t eqs. (48) in the Superha-ninal case yield directly eq. (39"), i.e. t h e y automatically include the ((reinterpretation ~) of eqs. (39). Moreover, eqs. (48) yield --

6--1

U--~-

(u2r

(49) =

qa,

o2~

O c. I n such a particular conformal mapping (inversion) the speed e is the one, and the speeds zero, infinite correst)ond to each other. This clarifies the meaning of observation ii), subsect. 3"2, b y EInSTEIn. Of. also fig. 9, which illustrates the i m p o r t a n t eq. (32). I n fact (review I), the relative speed of two (( dual ,) frames s, S (frames dual one to the other are characterized in fig. 9 b y 2~B being orthogonal to the u-~xis) is infinite; t h e figure geometrically del)icts, therefore, the circumsta.nce t h a t (so--~ S) = (so -> s). (s -> S), i.e. the f m l d a m e n t a l t h e o r e m of t h e (bidimensional) (~extended rel~tivity ~): (( The SLT: So -* S(U) is the p r o d u c t P

B1 -- C

-)

so

Fig. 9.

tt r

32

~. R ~ C A M ]

of the LT: so-->s(u), where u------1/U, by the transcenden$ SLT ~>: of. subseet. 5"5, eq. (32) (MIGNAm and RECAMX, 1973a). Even in more dimensions, we shall call > two objects (or frames) moving along the same line with speeds satisfying eq. (51): (51')

v]7 = c =,

i.e. with infinite relative speed. Let us notice that, if p , a n d / ~ , are the energym o m e n t u m vectors of the two objects, t h e n the condition of in/inite relative speed can be written in G-invariant way as

(51")

p , P , = O.

5"12. The /or tachyons. - The problem of the double sign in eq. (50) has been already taken care of in sect. 2 for the ease of bradyons (eq. (9)). Inspection of fig. 50) shows that, in the case of tachyons, it is enough a (suitable) ordinary subluminal orthochronous Lorentz transformation JLt to transform a positive-energy t a c h y o n T into a negative-energy t a c h y o n T'. For simplicity let us here confine ourselves, therefore, to transformations 15 --/~+ e~'~+, acting on free taehyons (see also, e.g., MARx, 1970). On the other hand, it is well known in SR t h a t the chronological order along a spaeelike p a t h is not ~t+-invariant. However, in the ease of Ts it is even clearer t h a n in the b r a d y o n case t h a t the same transformation /~ which inverts the energy sign will also reverse the motion direction in time (review I; R~,CxMI, 1973, 1975, 1979a; C~])n~OL~t and R~,CXM~, 1978; see also G ~ u c c I o et al., 1980). In fact, from fig. ]0 we can see t h a t for going from a positive-energy state T~ to a negative-energy state T: it is necessary to by-pass the state T~ (with V----oo). F r o m fig. 11a) we see moreover that, given in the initial frame so a t a c h y o n T travelling, e.g., along the positive x-axis with speed Vo, the c ~ .

l~emember once more that, if u. V is nega.tive, the --is t h a t during the process C lowers its rest mass (invariant statement!) in such a w a y t h a t - - M 2 < A < - m 2,, where M, m, A are defined above.

L e t us anticipate t h a t , in the case of (~intrinsic absorption )), relation (62') will hold instead of relation (57); and let us observe the following. Since the (inva~iant) q u a n t i t y A in relation (62 ~) can assume also positive values (coI~trary to the case of eqs. (56), (57)), if an observer ] sees b o d y C to increase its rest mass in the process, t h e n the (~proper d e s c r i p t i o n , of the process can be nothing b u t an intrinsic absorption. L e t us stress once agMn t h a t the b o d y C, when in flight, can appear to emit suitable tachyons without lowering (or even cha~ging) its rest mass : in particular, a particle in ]light can a priori emit a suitable t a c h y o n t transforming

CLASSICAL TACI'KYONS

4~

into itself. But. in such cases, if we pass to the rest, frame of the initial part.icle, the (~e m i t t e d ~) t'tchyon appears t h e n as an absorbed a u t i t a c h y o n t. At last~ when A in eqs. (56)-(59) can assume only known discrete values (so as in elcmentary-I3article physics), them--once M is fixed---cq. (56) imposes a link between m and ET, i.e. between m and ]p[. 6"5. (( I n t r i n s i c a b s o r p t i o n ~ o] a tachyon. - Secondly, let us consider (MACCA~UO~E and I~J~CA_M~, ]980a, b) oltr b r a d y o n C, with rest mass M, to absorb now i n its rest / t a m e a t a c b y o n (or antitachyon) T' which is endowed with (real) rest mass m and 4 - m o m e n t u m p---- ( E T , p ) , in e m i t t e d b y a second b r a d y o n D, and travels with speed V (e.g., along the x-direction). The 4 - m o m e n t u m col~servation requires t h a t (60)

M -t V/p ~ - m ~ = V/p '-' -~ M '~

(rest frame),

wherefrom it follows t h a t a. body (or particle) (3 at rest can a p r i o r i absorb (suitabh~) l,achyons both when increasing or lowering its rest mass ~nd wher~ conserving it. Precisely~ eq. (60) gives (61.)

I P [ : 2M1 V/i.m.) ,_ A) 2 . 4m.2 Mo"

(rest frame)

which corresponds to (62)

zJ = - - m ~-+ 2 M E T ,

so t h a t (62')

--

m ~< A < ~

(absorption).

Equal~ion (6~) tells us t h a t b o d y C in its rest: frame can absorb T' only when the l a c h y o n speed is (63)

V ----- V l - - l - . i ~ ; ~ M o ' / ( ' m ~! A { .

Nol,ice th.~t eq. (62) differs from eq. (56), such a difference being in agreement with the fact t h a t , if bradyort C varies its velocity w.r.t.t.acbyon T', then--i~l the C rest frame---cq. (60) c a n t r a n s f o r m into eq. (55) ((ft. subsect. 5"]2-5"]4). Equatiolls (61), (63) formally coincide, on the contrary, with eqs. (55'), (55"), respectively; but t h e y refer to different domains of A: in eq. (55") we have A < - - m ~, whih, in eq.(63) we have A > - - m " . I n particular, eq. (63) yields t h a t C can absorb (in its rest frame) infinitespeed tachyons only when m 2 + A ~ O, i.e. (64)

V ~ oo A ---- -- m 2

in agreement with eq. (58), as expected.

(rest frame)

~

E. ~ E C A M I

Q u a n t i t y A, of course, is again invariant. can be w r i t t e n

(65)

A --

--

I n a generic f r a m e / eq. (62)

m ~ ~- 2p~/~",

1 ' , being now the initial C-foatr-momentum in /. Still A ~>-- m ~. Notice also here t h a t t h e word absorption in eq. (62') m e a n s (~intrinsic absorption ~, since it refers to ((absorption (as seen) in the rest f r a m e of the absorbing b o d y or particle >>. This m e a n s t h a t , if a m o v i n g observer / sees relation (62) being satisfied, the (~intrinsic ~> description of the process, in the C rest f r a m e , is a t a c h y o n absorption, b o t h when / observes an actual absorption and when J observes on the c o n t r a r y an emission. L e t us s t a t e the following t h e o r e m : Theorem 2. (r Necessary and sufficient condition for a process, observed either as the emission or as the absorption of a t a c h y o n T ' b y a b r a d y o n C, to be a t a c h y o n absoI:ption in the C re,st f r a m c - - / . e , to be an "intrinsic absorpt i o n " - - i s t h a t -- m~"~ c 2. Case oJ ~ intrinsic emission ~ at A. Still in the A resl, frame, let us now consider A, B to exchange a t a c h y o n T w h e u u. V>~c '~. Again we can h a v e either or (( intrinsic absorption ~) at A. The present cases di]]er f r o m the previous ones (subsect. 6"8~ 6"9) in the fact t h a t n o w - - d u e to the (( switching procedure ~) (cf. the third postul a t e ) - - a n y process described b y A as a T emission at A a n d a T "rbsorptiou at B is described in the B rest f r a m e as a T absorption at A and a T emission a~, B, respectively. L e t us analyse the ease of (( intrinsic e m i s s i o n , b y b o d y A. D u e to the condition u . V > c'- (el. eq. (52')) and to t h e consequent ~ - - m s. I n conclusion, the p r e s e n t t a c h y o n exchange is k i n e m a t i c a l l y allowed when eqs. (70) are satisfied, b u t now w i t h (76)

A A > - - m s,

A~--

m S.

I n the p a r t i c u l a r case in which P~ a n d p are collinear, we can h a v e only (-- P~)lip (approaching phase), a n d we get (77)

2M2]pI--~ E~ V ( m s ~- A~) 2 + 4m~M~ - (m ~ Q- AB)]PB[

(PBi[(--P)),

with A~ ~ - - ms. Finally, let us recall t h a t in t h e p r e s e n t case ((( intrinsic a b s o r p t i o n s , at B a n d at A) b o t h quantities AA, A~ can vanish. W h e n AA ---- 0, we simply get 2 M A E v ---- ms. I n t h e p a r t i c u l a r case w h e n A~ ~ 0 one gets 2 E ~ E B ( u . V - 1) ---= m s, and t h e n [p] = (m/2M~) [EB(m s § 4M~) t - mlP~l]. 6"12. Conclusion about the tachyon exchange. - W i t h regard to t h e process a t B, t h e k i n e m a t i c a l results of subsect. 6"8-6"11 yield w h a t follows (M),CCAR~o~v, a n d R E c k , 1980b): ---- - (78a)

u. V--

mS,

AB ~ ~-- _ mS _ 2p~ P~ < -- m s ; _-- - As ~

~

m S -~

_ mS _

2p~ P C

at A : u . V < c" > An > -- m 2 ~ intrinsic a b s o r p t i o n at B ,

(79)

u.V>

c~ >

l1 n <

--

m 2 ~ intrinsic emission a t B ;

b) in t h e case of ((intrinsic a b s o r p t i o n ~ at A :

(s0) 6"13.

u. V <

c ~ ~/|~3

<

--

m "~~ intrinsic emission at B ,

u . V > c" =~ -/]B > -- m2 ::~ intrinsic a b s o r p t i o n a t B . Applications

to

elementary-particle

physics:

examples.

1'aehyons as

(( i n t e r n a l lines ~. - L e t us recall t h a t , w h e n e l e m e n t a r y i n t e r a c t i o n s arc considered

t o be m e d i a t e d b y e x c h a n g e d objects, no o r d i n a r y (bradyonic) particles c a n be t h e classical, (( realistic )) carriers of t h e t r a n s f e r r e d e n e r g y - m o m e n t u m . On t h e c o n t r a r y , c]assical l a c h y o n s - - i n place of t h e so-called v i r t u a l p a r t i c l e s - c a n a p r i o r i a c t as t h e a c t u a l carriers of t h e f m t d a m e n t a l s n b n u c l c a r interactions. F o r instance, a n y elastic s c a t t e r i n g can be r e g a r d e d as classically (((realistically )>) m e d i a t e d b y a suitable t a c h y o n e x c h a n g e dlu'ing t h e a p p r o a c h i n g p h a s e of t h e t w o bodies (cf. subsect. 6"7). I n such a case, eqs. (70), (76) read, a l w a y s in t h e A rest f r a m e (A A = A R = 0),

(s~)

E T ---- m~

,

E~ :

MA/(u'V--

1),

w h e r e t h e a n g u l a r - m o m e n t u m c o n s e r v a t i o n is n o t considered. we w o u l d h a v e Ip~[ = ]Pn] ~ ]P] a n d (82)

cosO ..... := :1

21P] ~

I n t h e c.m.s.

(elastie scattering),

so t h a t (once !P] is fixed) for each t a c h y o n mass m we g e t a particlllar 0.... ; if m assumes only discrete v a l u e s - - a s e x p e c t e d f r o m t h e d u a l i t y principle, subscct. S ' I - - , the~.l 0 reslflts to be classically (( q u a n t i z e d ~>, a p a r t f r o m t h e cylindrical s y m m e t r y . More in genera], for cac,h discrete v a l u e of t h e Cachyon m a s s m, t h e q u a n t i t y 0 .... assumes a. discrete value too, which is m e r e l y a f u n c t i o n of [PI. These e l e m e n t a r y c o n s i d e r a t i o n s neglect t h e possible m a s s w i d t h of t h e t a c h y o n i c (( resonances ~) (e.g., of t h e t a c h y o n mesons). L e t us recall f r o m subsect. 5"3, 6"7 t h a t in t h e c.m.s, a n y elastic s c a t t e r i n g a p p e a r s classically as mediate(t b y an infinite-speed t a c h y o n h a v i n g p , :~= (0, p ) , wit.h [p] = m. Moreover~ eqs. (81) i m p o s e a link b e t w e e n m a n d t h e direction of p , or r a t h e r b e t w e e n m a n d a ~pP (where we can choose P = P s ; r e m e m b e r t h a t P~ = -- PA): m

(83)

c o s ~ .... - - 2 ] P I

;

50

~.

R~CA~I

again we find (once IPI is given, and if the intermediate-tachyon masses are discrete) t h a t also the exchanged 3 - m o m e n t u m results to be (classically) (( q u a n tized ~ in b o t h its magnitude and direction. I n particular, for each discrete value of m, also the exchanged 3 - m o m e n t u m assumes one discrete direction (except, again, for the cylindrical s y m m e t r y ) , which is a function only of [P]. I t is essential to notice t h a t such results c a n n o t be obtained at the classical level when confining ourselves only to ordinary particles, for the mere fact t h a t b r a d y o n s are not allowed b y kinematics to be the interaction carriers. Of course, also the nonelastic scattering can be regarded us mediated b y suitable t a c h y o n exchanges. We shall come back to this in the following (subsect. 13"2). 6"14. O n the v a r i a t i o n a l p r i n c i p l e : a tentative digression. - After h a v i n g e x p o u n d e d some t a c h y o n mechanics in subsect. 6"2-6"12, let us t u r n a bit our b

a t t e n t i o n to the action S for a free object. I n the ordinary case, it is S = for a free t a c h y o n let us, rather, write

(8~)

~fds;

s = fldsl 9

B y analogy with the b r a d y o n i c case, we might assume for a free t a c h y o n the Lagrangian (c = 1) (85)

L = -~- m0 V ~ -- 1

( [ - 2 ) 1),

and, therefore, evaluate, in the usual way, ~.L (86)

P -= ~ v

--

mo V

+ V'I z o :_ -_ :1

.---~ m V ,

which suggests eq. (50) to hold in the four-dimensional case too:

(50')

m -

m0

~/~-

1

If the t a c h y o n is no longer free, we can write as usual (87)

F-

d p __

dt

d [ moV dt W V : - i !

B y choosing the reference frame, at the considered time instant t, in such a

51

C L A S S I C A L T A C H YOlkS

way that V is parallel to the x-axis, i.e. IV] : V~, we get then

1 (88a)

F~ = + +no V ~ - -

V~

]

mo

1 - - V ( V ~ - ~)3 a, --

( V ~ _ 1)+ a=

and

(88b)

F~ = • ~ / m ~

a,,

mo

F~=§

The sign in eq. (88a) is consistent with the ordinary definition of work :9~ (89)

dA~ _= + F- d!

and the fact that the total energy of a tachyon increases when its speed decreases (el. fig. 4a) and 10). Notice, however, that the proportionality constant between force and acceleration does change sign when passing from the longitudinal to the transverse components. The tachyon total energy E, moreover, can still be defined as (90)

E--p'V--

L

--

m~ _

_

:

~ C

2

,

which, together with eq. (50'), extends to tachyons the relation ~----me ~. However, the following comments are in order at this point. An ordinary timelike (straight) line can be bent only in a spacelike direction; and it gets shorter. On the contrary, if you take a spacelike line and, keeping two points on it fixed, bend it slightly in between in a spacelike (time) direction, the bent line is longer (shorter) than the original straight line (see, e.g., DOl~LI~G, 1970), For simplicity, let us here skip the generic case when the bending is partly in the timelike and partly in a spacelike direction (even if such a ease looks to be b

the most interesting). Then, the action integral f Ids] of eq. (84) along the straight a

(spacelike) line is minimal w.r.t, the ((spacelike ~ bendings and maximal w.r.t. the bendings. A priori, one might then choose for a free tachyon, instead of eq. (85), the Lagrangian

(85')

L :

--moV/V~--i

,

which yields

(86')

8L

P - ~v-

mo V

~/v~_

_

inV.

52

~. R~CAMI

E q u a t i o n (86') would be r a t h e r interesting, in the light of the previous subsect. 6"13 (el. also subsect. 13"2), i.e. when tachyons are substituted for the (~virtual particles ~>as the carriers of the elementary-particle interactions. In fact, the (classical) exchange of a t a c h y o n endowed with a m o m e n t u m antiparallel to its velocity would generate an attractive interaction. F o r nonfl'ce taehyons, from eq. (86') one gets

(87')

F-

and, therefore, when ]V[ :

d. _

d t

V

V., mo

(88a')

~'* -- + ( V ~ - 1)1 a . ,

(88b')

F~ --

~no

__. __ a~

F~ --

~no

- -

a~.

Due to t h e sign in eq. (88a'), it is now necessary to define the work cf as (89')

dfLf ~ -- .F.dl ,

and analogously the t o t a l energy ~ as

(90 r)

E --=

_

(p V - - L) .

_

re~ VV~- I

:

m e

2

.

6"15. On radia$ing tachyons. - Many other results, actually independent of the v e r y existence of SLTs, will appear in the following sect. 9-13. Here, as a f u r t h e r example, let us r e p o r t the fact t h a t a. t a c h y o n - - w h e n seen b y means of its electromagnetic emissions (see the following, and review I ; BALDO et al., 1970)--will appear in general as occupying two positions at the same time (RECA~rr, 1974, 1977b, 1978a, 1979a; ]3ABUT et al., 1982; see a]so GRON, 1978). L e t us start b y considering a macro-object C emitting spherical electromagnetic waves (fig. 15c)). W h e n we see it travelling at constant Superluminal velocity V, because of the distortion due to the large relative speed, ][I > c, we shall observe the electromagnetic waves to be internally t a n g e n t to an enveloping c o n e / ~ having as its axis the m o t i o n line of C (REcAm and M_ZGNA:~I, 1972 ; review I), even ff this cone has nothing to do with ~erenkov's (MIG~A~I and Rv.cA~, ]973b). This is analogous to what happens with an airplane moving at a constant supersonic speed in the air. A first observation is the following: as we hear a sonic boom when the sonic contact with the supersonic airplane does s t a r t (BONDI, 1964), So we shall analogously see an (( optic boom ~ when we first enter in radio contact with the b o d y C, i.e. when we

CLASSICAl,

T.KCHYON$

CO

~

V

H

m-

c:-

-

-

9

c,

co

"1" 9

% --.

--~2..~

i

I ! I

0

I !

I I

--

T"

!

\

L: .

r -

I

I I

!

cz c; ( v = = )

-

"

I !

I

I

11 Ijl

I

Ill

l

0

0

a)

b)

2

"

tz

t~

S

X

/,//~2 c)

/"

........... rs ":'-x;~. . . . . . . . . . . . . . -3-~3" ' . . . . . . , . . , ~ rC

\

SuperLumino~L wor(oL-L/ne

Fig. 15. m e e t t h e / ' - c o n e surface. I n fact, w h e n C is seen b y us u n d e r t h e angle (fig. 15a))

(91)

Vcosa

= c

(V~

IVI),

all t h e ra4iations e m i t t e d b y C in a certain t i m e i n t e r v a l a r o u n d its position Co reach us simultaneously. Soon after, we shall receive a t t h e s a m e t i m e t h e light e m i t t e d f r o m suitable couples o/points, one on t h e left a n d one on the right of Co. W e shall thus see t h e initial b o d y C, at Co, split in two luminous objects C1, C2 which will t h e n be observed to recede f r o m each other with the Superhuninal ~transverse~> relative speed W (~ECAMI et al., 1976; B ~ U ~ et al., 1982):

I + d/bt (92)

W ---- 2b [1 -5 2d/bt]t'

V . . . . .2. -. 1 b . . . .%/V

(V~>I),

where d ----O H , a n d t = 0 is just t h e t i m e i n s t a n t w h e n the observer enters in radio c o n t a c t with C, or r a t h e r sees C at Co. I n the simple case in which C m o v e s with almost infinite speed (fig. 15b)), t h e a p p a r e n t relative speed of C1

54

E. ~XCAMI

and C 2 varies in t,he initial stage as W --~ (2ed/t)t, where now 0]5[ -~ O C o while t = 0 is still the instant at which the observer sees C~ ~ C2 ~= Co. We shall come back to this subject when dealing with astrophysics (subsect. 12"4); see also the interesting paper b y LAKE and Ir (1975). H e r e let us add the observation t h a t the radiation associated with one of the images of C (namely, the radiation e m i t t e d b y C while approaching us, in t h e simple case depicted in fig. 15c)) will be received b y us in the reversed chronological order (cf. :VlIaXANI and ~ n c h ~ I , 1973a; I~ECA?4I, 1977b). I t m a y be interesting to quote t h a t the circumstance t h a t the image of a t a c h y o n suddenly appears at a certain position Co and t h e n splits into two images was alreday m e t b y Bac~Y (1972) and BAClCY et al. (1975) while exploiting a group-theoreticM detinition of the motion of a charged particle in a homogeneous field; definition which wan valid for all kinds of particles (bradyons, luxons, tachyons). Analogous solutions, simulating a pair production, have been later on found even in the s u b l u m i n a l case b y BARUT (1978b), when exploring nonlinear evolution equations, and b y SALA (1979), b y merely t a k i n g account of the finite speed of the light which carries the image of a moving subluminal object. SALA (1979) did even rediscover--also in subluminal case---that one of the two images can display a time-reversed evolution. At this point, we might deal with the problem of causality for t a e h y o n s (since the most relevant aspects of t h a t problem do arise w.r.t, the (;lass of the s u b l n m i n a l observers). We shift such a question, however, to sect. 9, because we w a n t preliminarily to touch the problem of t a c h y o n localization.

7. - Four-dimensional results independent of the explicit form of the SLTs." introduction. 7"1. A p r e l i m i n a r y a s s u m p t i o n . - L e t us s t a r t from ov_r three postulates (sect. 4). Also in foltr dimensions, when a t t e m p t i n g to generMize SR to Superluminal frames, t h e f u n d a m e n t a l req~irement of such an (~e x t e n d e d relativity )) (cf. subseet. 4"2, 4"3, as well as 5"1, 5"2) is t h a t the SLTs change timelike into spacelike t a n g e n t vectors, and vice versa, i.e. invert the quadratic-form sign. L e t us assume in these sect. 7, 8 t h a t such (( transformations I) exist in four dimensions (even if at the price of giving up possibly one of the properties i)-vi) listed at about the end of subsect. 3"2). Their actuM existence has been claimed, for instance, b y St~A~ (1977, 1978) within the (( quasi-catastrophe ~) theory. 7"2. G-vectors and G-tensors. - If we require also t h a t the SLTs form a new group G together with the snb]uminM (ortho- and anti-chronous) Lorentz transformations, lhe following remarks are then in order. Equations (14), (15) introduce the four-position x# an a G-vector; in other words, by definition of

CLASSICAL TACI[XONS

55

G[~Ts, q u a n t i t y x~, is a f o u r - v e c t o r not only w.r.t, the ~roup ~tf!_, b u t Mso w.r.t, lhe whole ga'OUl) G. As a consequence, tile (lx#dx~ b e h a v e s as a pseudoscalar under the SLTs. Under SLTs it is d s ' ~ - - --de2; it, folh)ws t h a t q u a n t i t y ~r dx#/ds, 'a. Lorentz veer;or, is not a. G - v e c t o r In order l)o detine the four-velocity as a G-vector, we m u s t set (93a)

ul~ -~ d x ~ / d v o ,

whez'(~ To is the 1)rot)er time. &nalogously for the fom'-acceleration: a# - dw'/dTo; and so on. We can expect t h a t also the electromagnetic quantities A~, (Lorentz vector) and 3'#,' (Lorentz tensor) do not h a v e a p r i o r i to be a n y longer a G-vector a n d a G-tensor, r e s p e c t i v e l y (cf. sect. 15). I I o w e v e r , once T~,~ is supposed to be a G-tensor, t h e n u n d e r a SLT it is (93b)

T ' ~ ~ G"~ G ~ T ~ ,

w h e r e f r o m it follows t h a t the o r d i n a r y i n v a r i a n t s

'~re still i n v a r i a n t (even u n d e r SLTs). This holds, of (.ourse, only for e v e n - r a n k tensors. As already mentioned, if we define u# b y eq. (93a), so to be a G-four-vector, then u n d e r a SLT the q u a n t i t y u 2--- u~,u, becomes ~ ' e = - - u - t T h a t is to say, a f t e r a SLT a b r a d y o n i c velocity has to be seen as a t a c h y o n i c velocity, a.nd vice versa, in a g r e e m e n t with eqs. (26). L e t us add here, a t this point, t h a t sometimes in the l i t e r a t u r e it has been avoided the explicit use of a metric tensor b y m a k i n g recourse to Einstein's notations, and b y writing the generic chronotopical v e c t o r as x - = (x0, xx, x2, xa) ~ (ct, ix, iy, iz), so th,tt g~,~: : ~ , ((( Euclidean metric ~>). Thus one does not h a v e to distinguish b e t w e e n c o v a r i a n t and c o n t r a v a r i a n t components. h i such a case, since on('. has practically to deal with a complex manifold, the quadratic f o r m which is Lorentz i n v a r i a n t is to be defined us the scalar p r o d u c t of t h e first vector b y the complex conjugate of the second vector: (93d)

quadratic form -- (dx, d~) = dx~, dy~ ;

in particular, the invaria.nt square interval would be d.~2 ~-= (dx, dg') -= d x , dxv.

8. - On the shape o f tachyons. 8"1. I n t r o d u c t i o n . - We h a v e already noticed t h a i a t a ( , h y ( m - - o b s e r v e d b y m e a n s of its light signals--will generMly appeqr '~s occupying two positions at ihe same t i m e (subscct. 6"]4 and fig. 15).

56

~. ~ECAM1

Still at ~ prehmimtry level, let us moreover recall t h a t free bradyons always admit a particular class of subluminal reference frames (their rest frames) wherefrom t h e y appear--in Minkowski space-time--as ((points ~) in space extended in time along a line. On the contrary, free tachyons always admit a particular class of subluminal (w.r.t. us) reference f r a m e s - - t h e critical f r a m e s - wherefrom t h e y appear with divergent speed V = o% i.e. as in time extended in space along a hne (of. fig. 7, 11). Considerations of this kind correspond to the fact t h a t the (( locahzation ~> groups (little groups) of the timelike and spacehke representations of the Poincar6 group are S 0 3 and S02,1, respectively (see, e.g., B A ~ V T , 1978a), so t h a t tachyons are not expected to be locahzable in our ordinary space (of. also 1)I~gEs, 1970; CAWLEY, ] 970; DVFFEY, 1975, ]980; VYw 1977a; SOU~EK, 1981). I t is, therefore, worthwhile studying the shape of tachyons in detail, following B ) ~ u T et al. (1982). 8"2. H o w would tachyons look like? - I~et us consider an ordinary b r a d y o n ]r -- t)~ which for simplicity is intrinsically spherical (in partieula.r pointlike), so t h a t when at rest its ~ world-tube ~>in Minkowski space-time is represented by 0 < x ~ -}- y~ + z 2 < r "~. When PB moves with subluminal speed v along the x-axis (fig. 16), its four-dimensional shape (i.e. its world-tube equation) becomes (94a)

0 < (x 9

vt) 2 1--L-~ v _}_ y2 _}_ z 2 < r 2 -

(v ~ < 1)

-

and, in Lorentz-invariant form, (94b)

0 < (x~, p~ ) ~ _ x~ x~, < r2

(v ~< I),

p~p~

where x~ = (t, x, y, z) and p, is the 4-momentum. t

I

I

Z/

~I/!

/

/ /J

.i/,," X

"

Fig. 16.

'

/I

CLASSICAL TACttYONS

57

Let. us now t a k e into examina.tion also t h e spacelike values of the 4-momenp**, still considering, however, only subluminal observers s: W e shall regard in these sections the SLTs, as well a.s the o r d i n a r y LTs, fl'om the active point of view only. B y an active SLT let us t r a n s f o r m t h e initial I)n into a tinal t.achyon P ~_~]3~ endowed with Superhtminal speed V along x. ])ue to subsect. 7"1, one can expect thai; eq. (94b) will t r a n s f o r m for 1)T into tltm

(95)

(xz p')'2

0 1-. Incidentally, let us r e m i n d t h a t even in EIr ~he light st)eed in v a c u u m goes on being the i n v a r i a n t speed, and can be crossed neither f r o m the left,, nor from the right. L e t us m a k e a c o m m e n t . T a c h y o n s a p p e a r e d to be m o r e similar to fields t h a n t.o particles. I t would be desirable to find out the space-time function yielding the density distribution of a t a e h y o n . F o r instance, when the t a e h y o n shape just reduces to t h e cone ~fo, it would be interesting to work out the L-function of x and t yielding the t a c h y o n density ditribution over ~o. 8"3. Critical comments on the preliminary assumption. - I n connection with subsect. 7"1 and 8"2 a critical warning is in order, since we saw at the end of subscct. 3"2 (and shall b e t t e r see in the following) t h a t real linear SLTs: dx, -+ dxl, which m e e t the r e q u i r e m e n t s ill-iv) of subsect. 4"2 do not exist in fOUl' dime~tsions. t We, therefore, expect t h a t real t r a n s f o r m a t i o n s .~-: x~,-> x~, m a p p i n g points of M~ into points of IM~ (in such a w a y t h a t ds~--~-- ds 2) do not exist as well; / otherwise reM linear SLTs: dx, -> dx~, should exist. L e t us state it differently. Equal,ion (95) was derived u n d e r the h y p o t h e s i s t h a t SLTs do exist in four dimensions which change the sign b o t h of the quadratic t f o r m dx, (tx~, a n d of the q l m n t i t y x~,x t'. This m e a n s t h a t the SLTs: dxt, --~ dx~, t r a n s f o r m i n g dx, d x ~' - § dx~ dx ~' h a v e to be linear. I n the case of SLTs linear and real~ it would exist as a consequence in M4 a point-to-point t r a n s f o r m a t i o n ! ,:V: x,-+x~,~ a n d f u r t h e r m o r e linear (]~I_~DL~:g, ]966). The results ill this sect. 8 seem to show, howc~'er, t h a t ill M, we m e e t m a p pings tha$ t r a n s f o r m manifolds into manifolds (e.g., points into surfaces). ! This seems to 1)redict 1-o us t h a t our SLTs: dxF,-> dx~ in M4 will be linear, b u t not real. F o r such nonreal SLTs we shall suggest in subsect. 14"16 an i n t e r p r e t a t i o n procedure t h a t will lead us f r o m linear nonreal SLTs to real nonlinear SLTs; of., e.g., fig. 5 in MACCA~rcO~'E and IIE(JA~ (1982a, 1984a). The latLer SLTs,

60

x. RECAMI !

actuully, cannot be integrated, so t h a t no . ~ : x~, --> x~ cun be found in this case (SM~z, 1984). Let us explicitly mention t h a t nonlinear SLTs: dx~ -~ d x~ ' can exist which, nevertheless, i) do transform inertial motion into inertial motion (e.g., the inertial motion of a point into the inertial motion of a cone), ii) preserve space isotropy and space-time homogeneity, ii) retain the light speed invariance (cf. also subsect. 8"2, 8"4). 8"4. On the spaee extension o] tachyons. - In the limiting case when it is intrinsically pointlike, tachyon PT reduces to the cone g0 only, and we shall see Pr to be a double cone, infinitely extemled in space (I~]~CA.VlIand MACCA~~O~E, 1989; BARUT et al., 1982). B u t this happens only if the corresponding bradyon I)B exists for -- ~ < t < + ~ . On the contrary, if the lifetime (and extension) of PB ~re ]inite, the space extension (and life) of PT are ]inite too. Namely if 1)~ in its rest frame is spherical, is born at time t~ and is absorbed at time i~, t h e n the corresponding ta.chyoa I ~ possesses a finite space extension (I~ECAm and MACCAR~O~, 1980, 1983). Under the present hypotheses, in fact, one has to associate with eqs. (93), (94) suitable limiting spacelike hypersurfaces, which simply become the hyperplanes t ~ i~ and t : t2 when P~ is at rest (fig. 20). The generic Lorentz-invariant equation for a hyperplane is (98)

x# u~ :

( K ~ const).

K

X

Fig. 20. Due to subsect. 7"1 we get t h a t eq. (98) keeps its form even under an active SLT, i.e. x , u ' ~ - - - - K ' . The relevant fact is t h u t we passed from a timelike ua to a spacelike u~, , so t h a t the hyperplanes x~u'~ == K / are now to be referred

CLASSICAL TAeHYO~S

61

to two spatial and one t e m p o r a l basis vectors (fig. 21). Such hyperplanes represent ordinary planes (orthogonal to the x-axis, in our case) which m o v e parallely to themselves with the subliminal speed v ' = l / V , as follows from f their orthogonality to u~.

/,

/

Fig. 21.

I n conclusion, in the t a c h y o n case (V ~ > 1), one has to associate with eqs. (95), (96) the additional constraints

the shape of a realistic t a e h y o n PT, obtained from a finite-lifetime b r a d y o n P~, is got, therefore, b y imposing on the structure ~o @ 3r in fig. 17, 18 the following constraints :

,) I t seems to follow t h a t our realistic t a c h y o n is constituted not b y the whole structure in fig. 17, 18, b u t only b y its portion confined inside a mobile (( window ~), i.e. b o u n d b y the two planes x = xl and x = x~. As we saw, this (( window )) travels with the speed v' dual to the t a c h y o n speed V (100)

1 q/ ---- -V

(Ir~> 1, v ' 2 < 1)

and, if V is constant, its width is constant too (A--t ~ t ~ - tl): (lOl)

Ax' = AtV~--

~'~

(v' = 1 / V ) .

62

n.

arcA~

Chosen a tixed position x - - 5% such a (~window ~) l~) cross ~he plane x - will take a time independent of 9 (if V is still constanl):

(102)

At' = At

v~

:-~ AtV/V-"~ 1 .

The problem of the time extension of such (~realistic ~ tachyons does not seem to have been y e t cla.rified. I f ])~ is not intrinsically spherical, but ellipsoidal, t h e n ])T will be bound b y a double cone W and a two-sheeted hypcrboh)id ~ ' devoid, this time, of cylindrical s y m m e t r y (of. BARvT et a t , 1982). Those authors investigated also various limiting eases. Let us mention t h a t when V - + ~ (while i~, t2 and r remain ]inite) the ~ window ~) becomes ]ixed: Y~x ~ ei~ < x < c[2 ~ ~ . W e m a y conclude t h a t , if the lifetime of Ps is v e r y large (as it is usually for macroscopic and even more for cosmic objects), then the corresponding t a c h y o n description is essentially the old one given in subscct. 8"2, and I)T e'~n be associated with actual Superluminal motion. If, on the contrary, the lifetime of ]?r is small w.r.t, the observation time of the corresponding t a c h y o n (as it often happens in the microscopic domain), t.hen P.r would actually appear to travel with the s u b l u m i n a l (dual) speed v' ~- 1 / V ; even if I)~ is associated with a structnre ~ + .$f travelling with the Superluminal speed V. I n fact, the m a g n i t u d e of its (( group velocity }~ (i.e. the speed of its ~ front ~)) is given b y eq. (100). IIowever, within the ~ window )~ confining the real portion of the t a e h y o n (which possibly carries the t a c h y o n energy and m o m e n t u m , just as ]?s carried e n e r g y - m o m e n t u m only between t = tx and t----i~), there will be visible a (~structure ~ evolving a.t Supcrluminal speed, associable, t.herefor% with a taohyonic (~pha.se velocity )~. W h a t precedes is based on I~ECA_M_X and MACCARr~O~E (1983)~ b u t similar r e s u l t s - ~ v e n if got from quite different starting p o i n t s - - w e r e p u t forth b y F o x et al. (1969, ]970). See also ALA~AR I~A~A.~USA~ et al. (1983), SOU~:K (1981)~ ](OWALCZYlCISKI (1979), SCn~-~A~ (1971), Cr (1970). 8"5. Comments. - The tachyons' characteristics exploited in the previous subsect. 8"4 remind us once more (cf, e.g., subsect. 6".13) of the ordinary q u a n t u m particles with their (~de Broglie waves ~: in t h a t case too phase veh)city and g~'oup velocity obey eq. (100). To investigate this connection (RECAZ~I and MACCARR0~E, 1983) let uS recall the ordinary delinitions of Compton wave-length 20 and de Broglie wave-length )'an (f12< :1):

(103a)

~ c ~--

2dB --

oC'

-

fPl

4,1 9

-

fit

----- -

,

CLASSICAL TACHYON8

~

where we introduced the new (~wa.ve-length ~) 2

( ~ < 1)

E/c satisfying t h e rela.tion (103c)

1

1

)2

~t~B

1

E q u a t i o n s (103) suggest, of course, t h e following k i n e m a t i c a l i n t e r p r e t a tion: L e t 2c represent the intrinsic size of the considered (subluminal, q u a n t u m ) p a r t M e ; t h e n ). = 2 c V / 1 - / # is t h e particle size along its m o t i o n line in t h e fra.me where it travels with speed v -~ tic; and 2dB/V - )~/V is t h e n the t i m e spent b y the pa.rticle to cross in the sa.me fl'ame a plaane orthogona.1 to its m o t i o n line. L e t us now e x a m i n e our eqs. (101), (102). I n cq. (101) it is na.tural to identify

(10~a)

hx'---2' :~oV1

(

fl'~

c2 v'2) a proper mass mo depending on its intrinsic (proper) lifetime, na.mely such t h a t (105)

--;to-- ~ = At -> mo ---- --~. c m0 c 2 c 2At

Notice tha.t eq. (105) corresponds to the case • o ' A t - - - - E . A x ' / o = h, with Eo - - moo', E = moc~/V/i - (V'fe) ~. Notice, m o r e o v e r , tha.t t h e wa.vc-lcngth of the de Broglie waave a.ssocia.ted with a. ta.chyon ha.s an u p p e r limit (GRo~, 3979) which is essentia.lly equaal to its C o m p t o n wa.ve-length: ( dS)ma~ !

=

hlmoc =

2~.

64

9. - T h e c a u s a l i t y

E. I~E(~A~MI

problem.

As mentioned at the end of subsect. 6"15, the discussion t h a t will follow in this sect. 9 is independent oI the very existence oI the SLTs, since the most relevant causal problems arise v h e n describing tachyons (and bradyons) from the ordinary subluminal frames. We wanted, however, to face the causality problem for tachyons in relativity only after having at least clarified t h a t tachyons are not trivially localizable in the ordinary s p a ~ (cf. sabsect. 8"2-8"5; see also SHAY and Mrr,T.ER, 1977). Actually, a t a e h y o n T is more similar to a field t h a n to u particle, as we already noticed at the end of subsect. 8"2. There ~re reasons, however, to believe t h a t , in general, most of the t a c h y o n mass be concentrated near the centre C of T (fig. 17b), 18): At least, we shah refer in the following to the eases in which tachyons can be regarded as (! almost localized )) in space. I n what follows, therefore, we shall essentially m a k e recourse only to the results in subscct. 5"12-5"14 (which, incidentally, h a v e been seen to hold also in four dimensions) and to our results about t a c h y o n kinematics (sect. 6). As m e n t i o n e d above, we shall confine ourselves only to the subluminM observers (in the presence, of course, of both bradyons and tachyons) and, for simphcity, to the orthochronous Lorentz transformations only. The results in subsect. 5"]2-5"14, in particular, showed us t h a t each observer will always see only tachyons (and antitachyons) moving with positive energy 1orward in time. As e x p o u n d e d in subsect. 5"13 and 5"17, however, this success is obtained ~t the price of releasing the old conviction t h a t j u d g e m e n t a b o u t what is (, cause ~)and w h a t is (( e f f e c t , is independent of the observer; in subsect. 5":17 we concluded t h a t the assignment of the (( source ~) and (cdetector ~ labels is to be regarded as a description detail. As anticipated in subsect. 5"13, this fact led to the proposal of ~ series of seeming (( causal paradoxes ,, t h a t we are going to discuss and (at least , in mierophysics ~)) to solve.

9"]. Solution oI the Tolman-Regge paradox. - The oldest p a r a d o x is the (( antitelephone ~ one, originally proposed b y TOL~L~_~ (1917; see also B o ~ , 1965) and t h e n reproposed b y m a n y authors (cf. subsect. 3"1). L e t us refer to its most recent formulation (I~EGGE, 1981), and spend some care in solving it since it is the kernel of m a n y other paradoxes. 9"1.1. T h e p a r a d o x . In fig. 22 t h e axes t and t' are the world-lines of two devices A and B, respectively, able to exchange tachyons and moving with constant relative speed u (u 2 ~ 1) along the x-axis. According to the terms of the p a r a d o x (fig. 22a)), A sends t a c h y o n 1 to B (in other words, t a c h y o n 1 is supposed to m o v e forward in time w.r.t. A). The apparatus B is constructed so to send back a t a c h y o n 2 to A as soon as it receives a t a c h y o n 1 from A. If B has to emit (in its rest frame) t a c h y o n 2, t h e n 2 must m o v e forward in

CLASSICAL TACH ).'O_NS

It

~

1 ~B

...

t!

It

? .

""

.."" /./../

a)

b)

Fig. 22.

t i m e w.r.t. B, thal; is to say ils world-libra B A , m u s t h a v e a slop(: smaller t h a n the sh)pe B A ' of the x'-axis (where, BA']]x'); this m e a n s t h a t A~ m u s t s t a y above, A'. I f the sI)eed of t a c h y ( m 2 is such t h a t A~. falls b e t w e e n A ' and A~, it seems t h a t 2 reaches b a c k to 5_ (event A.,) be/ore the emission of 1 (event A 0. This a.ppears to realize an antitelephone. 9"1.2. T h e s o l u t i o n . First of all, since t a c h y o n 2 m o v e s b a c k w a r d s in t i m e w.r.t. &, t h e e v e n t A~ will a p p e a r to A as the emission of an a n t i t a c h y o n `2. The observer ((t }~ will see his a p p a r a t u s A (able to exchange taehyons) e m i t suc(~essively towards B the a n t i t a c h y o n `2 a n d the t a c h y o n 1. At this point, some supportexs of the p a r a d o x (overlooking t a c b y o n kinematics, as well as reb~tions (66)) would say t h a t , well, the description forwarded b y observer (~t )) can be orthodox, b u t t h e n the de.vice B is no longer working according to ~.he premises, because B ix no longer e m i t t h ) g a t a c h y o n 2 oll receipt of t a c h y o n 1. Such a s t a t e m e n t would be wrong, however, sh~ce the fact t h a t (( t ~> sees an (~intrinsic emission ~) at A2 does not mean t h a t (~t'~ will see an ((intrinsic at)sorption }> at B. On |.he contra.ry, we are just in the came of subsect. 6"10: intrinsic emission b y A, at A~, with u.V~ > c 2, where u and Vg are the velocities of B and '2 w.r.l. A, respectively; so t h a t both A and B suffer an intrinsic emission (of t a e h y o n 2 or of a n t i t a c h y o n `2) in their own rest frames. B u t the t e r m s of the p a r a d o x were cheating us even more, and ab initio. I n fact, fig. 22a) m a k e s clear t h a t , if u.Y~ > c ~, then for t a c h y o n :1 a /ortiori u. V~ :> c ~-, where u a n d F~ are the velocities of B and 1 w . r . t . A . Due to subsect. 6"10, therefore, the observer (~t'); will see B emit also t a c h y o n 1 (or, n~ther, a n t i t a c h y o n T). I n conchlsion the proposed cha.iu of events does not include ~my ta.chyon a b s o r p t i o n b y B.

66

~. R~CA:MI

(in

For body B to absorb tachyon 1 its own rest flame), the world-line of 1 ought to have a slope larger than the x'-axis slope (see fig. 22b)). Moreover, for body B to emit (~ intrinsically ~)) tachyon 2, the slope of 2 should be smaller than the x'-axis'. In other words, when the body B, programmed to emit 2 as soon as it receives 1, does actually do so, the event A, does regularly happen after A~ (of. fig. 22b)). 9"1.3. T h e m o r a l . The moral of the story is twofold: i) one should never mix together the descriptions of one phenomenon yielded by different observers, otherwise--even in ordhlary physics--one would immediately meet contradictions: in fig. 22a), e.g., the motion direction of 1 is assigned by A and the motion direction of 2 is assigned by B; this is illegal; ii) when proposing a problem about tachyous, one must comply (C~_LDIRO~ and l~V,OA~, 1980) with the rules of tachyon kinematics (MACO~RO~E and REOAMT, 1980b), just aS when formulating the text of an ordinary problem one must comply with the laws of ordinary physics (otherwise the problem in itself is ~ wrong ~>). Most of the paradoxes proposed in the literature suffered from the above, shortcomings; for a remarkable, late example, see GIRARD and MAROrrr.DON (1984). Igotice that, in the case of fig. 22a), neither A nor B regard event A~ as the cause of event A~ (or vice versa). In the case of fig. 22b), on the contrary, both A and B consider event A~ to be the cause of event A~: but in this case A~ does chronologically procede A~ according to both observers, in agreement with the relativistic covarianee of the law of retarded causality. We shall come back to such considerations. 9"2. Solution o/the Pirani paradox. - A more sophisticated paradox was proposed, as is well known, by P ~ - ~ I (1970). I t was substantially solved by P~_R~E~TOLA and YEE (1971), on the basis of the ideas initially expressed by SUDARSHA~ (1970a), Brr.ANrUK and 8~-DARSHA~ (1969b), CSONKX (1970), etc. 9"2.1. T h e p a r a d o x . Let us consider four observers A, B, C, D having given velocities in the plane (x, y) w.r.t, a fifth observer so. Let us imagine t h a t the four observers are given in advance the instruction to emit a tachyon

Y[ x i ~ "3 (c~ ~0)

A(A2)~4 Fig. 23.

2

I,,

C

CLASSICAL T A C H Y O N 8

67

as soon as they receive a taehyon from another observer, so that the following chain of events (fig. 23) takes place. Observer A initiates the experiment by sending tachyon 1 to B; observer B immediately emits tachyon 2 towards C; observer C sends taehyon 3 to D; and observer D sends tachyon 4 back to A, with the result--according to the paradox---that A receives tachyon 4 (event A1) be/ore having initiated the experiment by emitting tachyon 1 (event As). The sketch of this (( Gedankenexperiment ~) is in fig. 23, where oblique vectors represent the observer velocities w.r.t. So and lines parallel to the Cartesian axes represent the tachyon paths. 9"2.2. T h e s o l u t i o n . The above paradoxical situation arises once more from mixing together observations b y four different observers. In fact, the arrow of each tachyon line simply denotes its motion direction w.r.t, the observer which emitted it. Following the previous subseet. 9"1, it is easy to check that fig. 23 does not represent the actual description of the process b y any observer. It is necessary to investigate, on the contrary, how each observer describes the event chain. Let us pass, to this end, to the Ylinkowski space-time and study the description given, e.g., by observer A. The other observers can be replaced b y objects (nuclei, let us say) able to absorb and emit taehyons. Figure 24 shows that the

'ct ~ c

~

B 1

I

X

~

3~

Fig. 24.

absorption of 4 happens be/ore the emission of 1; it might seem that one can send signals into the past of A. However (el. subsect. 5"12-5"14 and sect. 6, as well as I~ECA~ 1973, 1978V), observer ik will actually see the sequence of events in the following way: The event at D consists in the creation of the pair ~ and 4 b y the object D; tachyon 4 is then absorbed at AI~ while g is scattered at C (transforming into tachyon ~); the event As is the emission, b y A itself, of tachyon 1 which annihilates at B with taehyon ~. Therefore, according to A, one has an initial pair creation at D, and a final pair annihilation at B, and tachyons 1, 4 (as well as events AI~ As) do not appear causally

68

E. RECA3~I

correlated at all. I n other words, according to A, the emission of I does not initiate a n y chain of events t h a t brings to the absorption of 4, and we are not in the presence of a n y , effect ~ preceding its own ~ cause ~>. Analogous, o r t h o d o x descriptions would be forwarded b y the other observers. F o r i~stance, the tachyons' and observers' velocities chosen b y Pn~h.'~ (3970) are such t h a t all tachyons will actually appear to observer so as moving in directions opposite to the ones shown in fig. 23. 9"2.3. C o m m e n t s . The comme~tts are the same. as in the previous subsect. 9"1. Notice t h a t the * ingredients )) t h a t allow us to give the p a r a d o x a solution are always the (( switching principle ~ (subsect. 5"12 ; see also ScrrwA_~Tz, 1982) and the t a c h y o n relativistic kinematics (sect. 6).

9"2.4. ( ( S t r o n g v e r s i o n s ) a n d i t s s o l u t i o n . L e t us formulate Pirani's p a r a d o x in its strong version. L e t us suppose t h a t t a c h y o n 4, when absorbed b y A at A~, blows up the whole l~boratory of A, eliminating even the physical possibility t h a t t a e h y o n i (believed to be the sequence starter) is subsequently emitted (~t A~). Following R o o t and TREFIL (1970; see also TnEFIL, 1978), we can see on the c o n t r a r y how, e.g., observers so and A will really describe the phenomenon. Observer so will see the l~boratory of A blow up after emission (at A~) of the antitachyon ~ towards D. According to so, therefore, the a n t i t a c h y o n I emitted b y B will proceed b e y o n d A (since it is not absorbed at A~) and will eventually be absorbed a t some remote sink-point U of the universe. B y means of a LT, starting from the description b y So, we can obtain (CALDrROLA and I~ECA~r, 1980) the description given b y A. Observer A, after h a v i n g absorbed at A~ the t a c h y o n 4 (emitted at D together with .3), will record the explosion of his own laboratory. At A2, however, A will cross the flight of a. tachyonic (( cosmic r a y ~ 1 (coming from the r e m o t e source g), which will annihilate at B with t h e a n t i t a c h y o n 3 scattered at C, i.e. with the a n t i t a c h y o n ~. 9"3. Solution o/ the Edmonds paradox. - The seeming paradoxes arising from the r e l a t i v i t y of the j u d g m e n t about (~cause )> and (~effect ~>h a v e been evidenced b y EDMONDS (1977b) in a clear (and amusing) way, with referellce to the simplest t a e h y o n process: the exchange of tachyons between two ordinary objects, at rest one w.r.t, the other. 9"3.1. T h e p a r a d o x . W e build a long rocket sled with a c 2 (and to subsects. 5"12 and sect. 6), however, the witness C - - w h e n questioned b y the police--will have to declare t h a t actually he only saw antitachyons come out of B's head and be finally absorbed b y A's pistol. The same would be confirmed by ]3 himself, were he still able to give testimony. 9"5.2. T h e s o l u t i o n ; and comments. L e t us preliminarily notice t h a t B and C (when knowing t a c h y o n mechanics) could at least revenge themselves on A b y making A surely liable to prosecution: t h e y should simply run towards A ! (cf. subsect. 3"12 and sect. 6). B u t let us analyse our paradox, as above expounded. Its main object is emphasizing t h a t , when A and B are moving one w.r.t, the other, both A and B can observe (( intrinsic emissions ~ in their respective rest frames (subsect. 6"10). I t follows t h a t it seems impossible in such cases to decide who is actually the beginner of t h e process, i.e. who is the cause of the t a c h y o n exchange. There are no grounds, in fact, for privileging A or B. I n a picturesque w a y - - a s B~LL p u t i t - - i t seems t h a t , when A aims his pistol at B (which is running away) and decides to fire suitable tachyons T, t h e n B is ~ obliged ~) to e m i t antitachyons T from his head and die.

72

~. ~eA~

To approach the solution, let us fu'st rephrase the p a r a d o x (follo~dng the last lines of subsect. 9"3) b y substituting two spherical objects for A's pistol and B's head. About the propecties of the emitters/absorbers of tachyons we know a priori only the results got in sect. 6; but, since this paradox simply exploits a particular aspect of the two-body interactions via t a c h y o n exchange, we have just to refer to those results. Their teaching m a y be interpreted as follows, if we recall t h a t we are assuming t a c h y o n production to be controllable (otherwise the paradox vanishes). The ta.chyon exchange takes place only when A, B possess suitable ~ tachyonie aptitudes ~), so as an electric discha.rgc takes place between A and B only if A, B possess electrical charges (or, rather, ~re at different potential levels). I n a sense, the couph~ of ~ spherical obj~ct~ ~) A, B can be regarded as resembling a Van de Graaff generator. The t a c h y o n spark is exchanged between A and B, therefore, only when observer A gives his sphere (the (~pistol ))) a suitable (( taehyonic charge ,>, or raises it to a suitable (( tachyonic potentiM ~). The person responsible for the t u c h y o n discharge between A and B (which may cause B to die) is, therefore, the one who iutentionMly prepares or modifies the ~(tachyonic properties ~> of his sphere: i.e., in the case above, it is A. I n the same way, if one raises a conducting sphere A to a positive (electrostatic) potential high enough w.r.t, the earth to provoke a t h u n d e r b o l t between A and a pedestrian B, he shah be the guilty murderer, even if the t h u n d e r b o l t electrons actually s t a r t from B and end at A. We have still to stress t h a t the original version of the Bell p a r a d o x was, in reMity, a tittle different, zkfter the discl~sion of the previous version, we are r e a d y to appreciate it immediately. I t is worthwhile to learn abo~tt it from the words used b y BEL~ (3985) himself in his c o m m e n t : (~In m y original version victim and witnesses remain at rest, and only the m n r d e r e r changes speed. Then he can profit also from a n y prejudice in f a v o u r of inertial observers. The version in which victim and witness run from the m u r d e r e r makes a good story. B u t I t h i n k I would prefer to commit such a m u r d e r while myself on the r u n past the s t a t i o n a r y victim and witnesses. Then, I would come to rest to agree with stationary witnesses, policemen, judges, and juries--pointing out t h a t in their common frame the death of the victim came before the pulhng of the trigger ~). ~Notice t h a t we have been always considering t a c h y o n emissions and a.bsoJ3)tions, b u t never t a c h y o n s catterings, since--while we know the t a c h y o n mechanics for the former, simple processes--we do not know y e t how tachyons interact with (ordinary) m a t t e r .

9"6. Signals by modulated tachyon beams: discussion of a paradox. 9"6.1. T i l e p a r a d o x . Still ((in macrophysics~), let us tackle at last a more sophisticaI~d paradox, proposcd b y ourselves (()ALD]:I~0LA~and I~ECA]~4:I,

CLASSICAL TACHYO~S

73

1980), which can be used to illustrate the m o s t subtle hints contained in t h e (( causality >>l i t e r a t u r e (cL, e.g., F o x et al., :[969, 1970). L e t us consider two o r d i n a r y inertial f r a m e s s--~ (t, x) and s ' z (t'~ x') m o v i n g one w.r.t, t h e other along the x-direction with speed u ~ c, and let us suppose t h a t s s e n d s - - i n its own f r a m e - - ~ signal along t h e positive x-direction to s' b y m e a n s of a m o d u l a t e d t a c h y o n b e a m h a v i n g speed V ~ c~/u (fig. 25). Aceordi~)g to s' t h e t a c h y o n b e a m will actually a p p e a r as an anti-

t!

/

~ B A~ /

/// //

X'

,/l

ll/ ~

X

Fig. 25.

t a c h y o n b e a m e m i t t e d b y s' itself towards s. W e can imagine t h a t observer s, when m e e t i n g s' a t O, h a n d s h i m a sealed letter a n d tells h i m the following: (( B y m e a n s of m y " t a c h y o n radio" A and s t a r t i n g at t i m e t, I will t r a n s m i t to y o u r " t a c h y o n radio" B a mnltifigured n u m b e r . The n u m b e r is w r i t t e n inside the envelope, to be opened only a f t e r the transmission ~>. Notice t h a t t h e (( free will ~ of s' is not jeopardized nor u n d e r question, since s' can well decide to not :switch on his t a c h y o n radio B. I n such a case we would be b a c k to t h e situation in subsect. 9"3: I n fact~ s would see his t a c h y o n s T b y - p a s s s' w i t h o u t being a b s o r b e d and proceed f u r t h e r into the space; s', on t h e c o n t r a r y , would see a n t i t a c h y o n s T coming f r o m the space a n d reaching A. I f s' knows e x t e n d e d relativity, he can t r a n s f o r m his description of the p h e n o m e n o n into t h e (( intrinsic description ~>given b y s, and find out t h a t s is (( intrinsically ~) e m i t t i n g a signal b y t a c h y o n s T. H e can check t h a t t h e signal carried b y those r T corresponds just to the n u m b e r w r i t t e n in a d v a n c e b y s. The p a r a d o x is actually m e t w h e n s' does decide to switch on his t a c h y o n radio B. I n f a c t (if t' is t h e L o r e n t z - t r a n s f o r m e d value of t, and At' --~ A ' O / V ' ) the observer s' at t i m e t t - At r would see his radio not only b r o a d c a s t t h e foretold multifigured n u m b e r (exactly the one w r i t t e n in t h e sealed letter, as s' can check s t r a i g h t after), b u t also e m i t simultaneously a n t i t a c h y o n s T towards s : T h a t is to say, t r a n s m i t t h e s a m e n u m b e r to s b y m e a n s of a n t i t a c h y o n s . To m a k e the p a r a d o x m o r e evident, we can imagine s to t r a n s m i t b y t h e m o d u -

74

E.

RECA~I

luted tachyon beam one of Beethoven's symphonies (whose number is shut up in advance into the envelope) instead of a plain number. l?urther related paradoxes were discussed by PArdI6 and I~ECA_W~(1976). 9"6.2. D i s c u s s i o n . Let us stress that s' would see the antitachyons T emitted by his radio B travel forward in time, endowed with positive energy. The problematic situation above arises only when (the tachyon emission being supposed to be controllable) a well-defined pattern of correlated taehyons is used by s as a signal. In such a case, s' would observe his taehyon radio B behave very strangely and unexpectedly, i.e. to transmit (by antitachyons T) just the signal :specified in advance by s in the sealed letter. He should conclude the intentional design of the tachyon exchange to stay on the side of s; we should not be in the presence of a real causality violation, however, since s' would not conclude that s is sending signals backward in time to him. We would be, on the contrary, in a condition similar to the one studied in subsect. 9"5.2. The paradox has actually to do with the unconventional behaviour of the sources/detectors of tachyons, rather than with causality; namely s', observing his apparatus B, finds himself in a situation analogous to the one (fig. 26) in which we possessed a series of objects b and saw them slip out sucked and (~aspired ~>by A (or in which we possessed a series of metallic pellets and saw them slip out attracted by a variable, controllable electromagnet A). From the behaviour of tachyon radios in the above Gedankenexperiment it seems to follow that we are in need of a theory formalization similar to Wheeler and Feynman~s (1945, 1949; see also FLATO and G~YE~CIN,1977, and GOTT I I I , 1974a). In particular, no tachyons can be emitted if detectors do not yet exist in the universe that will be able sooner or later to absorb them. This philosophy, as we already saw many times, is a must in ER, since tachyon physics cannot be developed without taking always into account the proper sources and detectors (whose roles can be inverted by a LT); it is not without meaning t h a t the same philosophy was shown (Wm~wL~]~ and F ~ s 1945, 1949) to be adoptable in the limiting case of photons. Let us recall t h a t --according to suitable observers--the two devices A, B are just exchanging infinite-speed tachyons (or antitachyons: an infinite-speed tachyon T going from A to B is exactly equivalent to an infinite-speed antitachyon T travelling from B to A). Any couple of bodies which exchange tachyons are thus reMizing~aeeording to those suitable observers--an instantaneous, mutual, symmetrical interaction. Thus tachyons can play an essential role at least as ~ internal lines ~>in bradyonic particle interactions (and vice versa, passing to a SuperluminM frame, bradyons would have a role as internal fines ot tachyonic particle interactions). This suggests that A and B can exchange that Beethoven's symphony by means of tachyons only if the inner structure of both A, B is (~already ,~suited

CLASSICAL TACHYO~S

75

f ~

b~

,,in,~

~aP"

a~

~ I

aaD~

,wl~ o

[~

-,

i

41~,,imm'. !

II

n . :.... :.% . = ~ ' .

Fig. 26.

to such an exchange; this again is similar to what discussed in subsect. 9"5.2, even if the situation is here more sophisticated. I t seems in conclusion t h a t tachyons are not suited for the transmission of really (t new ~> information. Of course, all problems are automatically (and simply) solved if we adopt the conservative attitude of assuming the tachyon exchanges between two bradyonic bodies A, B to be spontaneous and uncontrollable, as it happens in microphysies (as far as we know). For simplicity's sake, such a restrictive attitude might be actually adopted, even if unnecessary. See also, e.g., ]~AVAS (1974) and ~OL~rCK (1974). What seen in the last three parag~'aphs of this subsection can be of importance in connection with the dream of explaining the quantum phenomenology via classical physics (with tachyons). Actually, the fact that tachyons are just the objects sltitablc for transmitting a mutual interaction (in which it is meaningless to say whether A is influencing B, or B is influencing A) meets quite well the apparent requirements of QM and of QFT, that within a quantum system there exist Superlnminal connections (which seem to not imply controllable, Superluminal signals). See, e.g., STAI'I" (1985) and HEGEI~FELD (1985); see %lso \u (1976), Vm:cmr (1980), D'EsPAO~'A'r (1981) and R~a'DIJK and SELLE~ (t985). 9"6.3. F u r t h e r c o m m e n t s . When the signal does not consist of a well-defined pattern of taehyons, but is constituted by a few tachyons only --typically by ~ single taehyon--, we saw t h a t no paradoxes stu'vive. If on the contrary claims as the one put forth by NEWT0~ (1.967) were true, then one could send signals into the past even by ordinary antiparticles (which is not true, of cottrse; of. I ~ E c ~ and ~r 1975; IgECA:~, 1970). 5ioreover, to clarify further the terms of the paradox in subsect. 9"6.1, 9"6.2 above, let us explicitly recall that i) the ehrolmlogical order of events can be reserved by an ordinary LT along a spacelike path only; therefore, the order of the events along the A, B world-lines cannot change; ii) also the proper energies (rest masses) of A, B are Lorentz invariaut, together with their ~(jumps )>; iii) while s sees the total energy of A decrease, s' may see it increase (description details...) ; iv) the paradox in subseet. 9"6.1,9"6.2 is connected with the question whether the entropy variatioms and information exchanges are to be associated with the changes in the proper energies: in this case, in fact, they would not necessarily behave as the (~total energies )) (see CAI,DIICOLA and l%ncam, 1980, and PAv~IO and IgEc~-~I, 1976, where the paradoxical situations arising when one deals with macrotaehyons are furthermore discussed). We mentioned in the previous discussion (subsect. 9"6.2) that the behaviour of tachyon sources/detectors might appear paradoxical to us for the mere fact that we are not accustomed to it. To shed some light on the possible nature of such difficulties, let us report at last the following anecdote (Cso~A, 1970), which does not involve contemporary prejudices: (~For ancient Egyptians,

CLASSICAL TACI-IYONS

77

who knew only the Nile and its tributaries, which all flow South to North, the meaning of the word " s o u t h " coincided with tile one of "uI)stream" , and the meaning of the word " n o r t h " coincided with the one of " d o w n s t r e a m " . W h e n Egyt)tians discovered the E u p h r a t e s , which u n f o r t u n a t e l y happens to ttow N o r t h 1o South, t h e y passed through such a crisis t h a t it is mentioned ill the stele of Tuthmosis I, which tells ItS about that inverted water that goes downstream (i.e. towards the North) in going upstream ~>. See also, e.g., IIILOECOOleD (1960). 9"7. On the advanced solutions. - I~elativistic equations (both classical and qualital) arc kliowll to a d m i t ill general advanced, besides retarded, solutions. F o r ilistalice, Maxwell equations predict both retarded aml adwmccd electromagnetic radiations. Naiw,~ly, advanced solutions have been sometimes regarded as actually representil)g motions backwards in time. On the contrary, we know from the (( switching principle ~>(subsect. 2"1) alid the v e r y structure of S1r (see I)art I, sect. 2) t h a t the ~(advanced ~) waves or objects are nothing b u t alitiobjects or antiwaves travelling in the oi)i)osite si)~ce direction. W i t h i n E R , actually, when an equatioli admits a solution corresi)onding to (outgoing) particles or photons, t.hen a class of suitable GLTs tralisform such a solution into another one corresponding to (incoming) antii)articles or (anti)photons. In other words, if an equaLioli is G-covariant, it must a d m i t also of solutions relative to incoming antiparticles or I)hotolis, whenever it admits of solutions relative to outgoing particles or photons. This means t h a t all G-covariant relativistic equations must a d m i t both retarded and advanced solutions. W h e n confining onrselves to sublumilial wqocities u 2, v ~ < 1, the ordinary relativistic equations already satisfy such a requirement, for the reasons discussed in p a r t I (see, ill partictflar, subsect. 2"3, I)oint d)). We could, however, ask ourselves why do we usua.lly observe only, e.g., the outgoing, r a t h e r t h a n the incoming radiation. The clue to the question is in taking into account the initial conditions: In ordinary macroi)hysics some initial conditions are b y f'u' more probable t h a n others. F o r insta.nce, the equations of fluid dynamics allow to have on the sea surface both outgoing circula.r concentric waves and incoming circular waves tending to a centre. I t is known, however, t h a t the initial conditi(ms y M d i n g the former are more likely to be m e t t h a n those yielding the latter case.

10. - Tachyon classical physics (results independent of the SLTs' explicit form). According to subsect. 5"1, the laws of classical physics for iachyolis arc to be derived jlist b y at)plying a SLT to thc ordinary, classical laws of bradyons (this statemelit has been sometimes referred to as the ((rule of e x t e n d e d

78

E. RECAMI

relativity )): cf. PA]~:ER, 1969, and I~ECAMT and 1VIIG~ANI, 1974a). To proceed with, we lmcd nothing b u t the assumption in subsect. 7"1 ; i.e. we need only assuming t h a t SLTs exist which carry timelike into spacelike tangent vectors, and vice versa. I t is noticeable t h a t tachyou classical physics can be obtained in terms of purely real quantities. Subsections 10"1 and 10"2 below do contain important improvements w.r.t, review I. 10"1. Tachyon mechanics. - For example, the f u n d a m e n t a l law of bradyon dynamics reads (106)

Fs~c~

moC d s ]

dye re~

(fl~introduced by C ~ B o and FEI~A~I (1.962) for magnetic monopoles (A]~mDI~ 1968; FE~I~AI~I, 1978; DE FA~IA-t=~OSA et al., 1985, 1986).

84

11.

E. ~ E C A M I

Some ordinary physics in the light of LR.

l l ' ] . Introduction. A g a i n about CPT. - Looking for the SLTs in the ordinary space-time will pose us a new problem: finding out the transcendent t r a n s f o r m a t i o n ~f~: 5f4 which generalizes eq. (32) of subsect. S'5 to the 4-dimensional ca.se. However, after what we saw in p a r t I (sect. 2), we are already prepared to accept (cf. subsect. S']6 and 5"6) t h a t (37')

G~ G ~ --G~ G,

VGE G,

even in four dimensions. Actually, from iig. 5c) and 6 (now understood to hold in four dimcnsioHs), we see t h a t i) an ordinary LT = Z can carry from Ts to Ts, ii) if a SLT = L s exists t h a t carries from Bs to Ts, t h e n the subluminal t r a n s t o r m a t i o n Ls~]~.Zs will carry from Bs to Bs. Our general results in sect. 2 (e.g., eq. (10)) imply, therefore, t h a t eq. (53) will be valid also in four dimensions (.M_IG~A~I and ICECA:M:[, 1974b): (53 ~)

--I ~-PT----CPT~G;

i~l connection with eq. (53') see all the remarks already expounded in subsect. 2"3. As a consequence, the generalized group G in Minkowski space-time is expected to be the extension (PArdI5 and I ~ E c ~ , 1.977) of the proper, orthochronous (4-dimensional) Lorentz group ~q~+ b y means of the two operations CPT _ ~ - 1 ~nd ~f:

(120)

G = ~(~*+, el'T, o~).

I n our formalism, the operation CPT is a linear (classical) operator in the pseudo-Euclidean space, and will be a u n i t a r y (quantum-mechanical) operator when acting on the states space: cf. eq. (53'), and see I~ECA~I (1979a), C0S~A D E BEAUI~E6A~D (1983). F r o m w h a t precedes, and from fig. 5 and 6, we m a y say t h a t even in the 4-dimensional e n e r g y - m o m e n t u m space we have two symmetries: i) the one w.r.t, the h y p e r p l a n e E = 0, corresponding to the transition particle ~ antiparticle, and ii) the one w.r.t, tile light-cone, expected to correspond to the transition b r a d y o n ~- tachyon. I n any case, the (( switching procedure ~) (sect. 2 and subscct. S'12) will surely h a v e to be applied for both bradyons and taehyons also in four dimensions. L e t us, therefore, reconsider it in a more formal way. 11"2. A g a i n about the (~switching procedure ~). - This and the following subsections do not depend on the existence of tachyons. They depend essentially on our part I.

CLASSICAL TACHY0)~S

~5

We shall indicate by ). Let us also call strong conjugation C the discrete operation (*) (121)

0 - C~o,

where C is the conjugation of all additive charges and Mo the rest mass conjugation (i.e. the reversal of the rest mass sign). /~ECAm and ZI!No (1976) showed t h a t formally (cf. fig. 3b)) ~- P5 ;

in fact, when dealing as usual with states with definite parity, one m a y write C-~~ --P/~Ps. Notice that in our formalism the strong conjugation C is a unitary operator when acting on the state space (cf. also VILELA-1Vf/-ENDES, 1976). For details and further developments see, e.g., besides the above-quoted papers, EDMONDS (1974), LAKE and I~OEDER (1975), PAV~I~ and REOA~ (1977), REOA~I (1978a), REOAlViI and RODI~IGUES (1982). Here we want only to show that, when considering the fundamental partides of matter as extended objects, the (geometrical) operation which reflects the internal space-time of a particle is equivalent to the ordinary operation C which reverses the sign of all its additive charges (PAV~I~ and I~EOAYg, 1982). 11"3. Charge conjugation and internal space-time reJlevtion. - Following PAV~I~ and R E C ~ (1982), let us consider in the ordinary space-time i) the extended object (particle) a, such that the interior of its ((world-tube ~>is a finite portion of Md; ii) the two operators space reflection, ~ , and time reversal, 3", that act (w.r.t. the particle world-tube W) both on the external and on the internal space-time: (123)

/ ~ = ~ i = ~i~, t 3- = 3-~3-I = 3-~3-~,

(*) In this sect. 11, for ]ormal reasons, we shall indicate the space-time symmetry operations (till now represented by Roman letters) by italics; however, C--= C, P=--P, T=--T, P=--P, T=--T, etc.

86

~.

Xo

X~

x

x

RECAM~

o

~T

a)

~ix1

x1

[~

xt

0

Xo -'cT

5~

x0

E

b)

x~

5~r

1

Fig. 28.

where ~z(3rz) is the i~ternal and ~E(3r~) tile external space reflection (time reversal). The ordinary p a r i t y _P and t i m e reversal T act on the c o n t r a r y only on the external space-time: P----~,

T--~.

The effects of ~ , '~z and ~ on the world-tube W of a are shown in fig. 28, and t h e analogmls effects of 3r E , 3r~, 3r in fig. 29. L e t us now depict W as a sheaf of world-lines w representing---let us s a y - its constituents (fig. 30a)). I n fig. 30 we show, besides the c.m. world-line,

Xo

.go

xo

9U u ..qtJ

9s I

a) X1

~=T

.9"

cP

<

/a

b)

1

Fig. 29.

CLASStCAL T,tCI1Y().N S

]]7

~Z

a)

b)

c)

Fig. 30.

also w~ ~ A alld w2---- B. The operation ~ Y will transform W inio a second worht-tube ~u consisting of the t r a n s f o r m e d world-lines ~ (see fig. 30b)). :Notice t h a t each ~ poinls in the opposite tim(; direction and occupies (w.r.t. the c.m. world-line) the position symmetrical to the corresponding w. If we apply the Stfickelberg-Feynman (subsect. 2"1), each worldline ~ transforms into ,% new world-line W (cf. fig. 30c)) which points in the positive time dh'ection, b u t represents now an anticonstituent. L e t us now explicitly generalize the (( switching principle ~>for extended particlcs as follows: W e identify the sheaf W of the world-lines w with the antiparticle g, i.e. W with the world-tube of g. This corresponds to assuming t h a t the overall time direction of a particle as a whole coincides with the time direction of its (( constituents ~>. A preliminary conchlsion is thai, the antiparticle g of a can be regarded (from t h e chronotopical, geometrical point of view) us derived from the reflection of the internal space-time of a. L e t us repeat w h a t precedes in a more rigorous way, following our sect. 2, i.e. recalling t h a t t h e transformation L = -- 1 is an actual (even if antichronous) Lorentz transformation, corresponding to the 180 ~ apace-time ((rotations>: P T ~ - 1. Now, to apply P T from the active point of view to the worhtt u b e W of tit. 30a) means to r o t a t e it (by :180~ in four dimensions) into (fig. 30b)); such a r o t a t i o n effects also a reflection of the internal 3-space of particle a, transforming i t - - a m o n g t h e o t h e r s - - i n t o its mirror image. The same result would be got b y applying P T front the passive point of view to t h e space-time in fig. 30a). r s we generalize the (~switching principle )> to the c~se of e x t e n d e d objects b y applying it to the world-tube W of fig. 30b). The world-tube 1'~ docs represe, n~. an (interHally ) particle not only going backwards in tint(;, b u t also carrying negative elmrgy; lherefore, the (( switching ~) does rigorolLsly t r a n s f o r m ~" into W (fig. 30c)), the anti-world-tube W representing (i. I n conclusion, (124)

--I --PT

: ~ Y ~ g 2 1 Y ~ =: ~ J - =

PT~IYl,

88

wherefrom, since P T : (125)

~. ~ c ~

C P T (subseet. 2"3), one derives

C~ r

.

As already anticipated, we have, therefore, shown the operation C, which inverts thc sign of (all) the additive charges of a particle, to be equivalelit to the (geometrical) operatioll of reflecting its internal space-time. Also the results reported in this subsection support the opinion that in theoretical physics we should advantageously substitute the new operations P - and T----3" for the ordinary oper~tions P and T, which are merely external reflections (for instance, onlly the former belong to the fidl Lorentz group). Besides our sect. 2, cf., e.g., review I, ld.ECA_M:I(1978e) and also COSTA DE BEAUICEGAI~D (1984). 11"4. Crossing relations. - Besides the C P T theorem, derived from the mere SR, from E:R only it is possible to get also the so-called (~crossing relations ~. Let us first recall that cross-sections and invariant sc,~ttering amphtudes can be defined (RECA_~_Iand M~G.~A_~,1974a) even at a classical, purely relativistic level. We are going to show (MIG~'ANI and REC~M:I, 1974a, 1975a) that,--within El~--the same fr is expected to yield the scattering amplitudes of different processes like (126a)

a ~- b --> c -[- d,

(126b)

a -!- ~ ->b -}- d

in correspondence, of course, to the respective, different domains of the kinematical variables. Let a, b, c, d be bradyonic objects w.r.t, a frame so. The two reactions (126a), (126b) amoug Bs are two different processes Pl, P, as seen by us; but they can be described as the same interaction ds -- d t ~ d~ among Ts by two suitable, different Super]uminal observers S~, S~ (review I; : R E C k , 1979a; C~])n~0LX and/~):cA_~r[, 1980). We can get the scattering amplitude A(pl) of Pl by applying the SLT(S~ -~ s0) -- Z] to the amphtude Al(dl) found by $1 when obser~ng the scattering Pl, i.e. A ( p l ) = LI[AI(dl)]. Conversely, we may get the scattering amplitude A(p2) of p, by applying the SLT(S~-~ so)~ L~ to the amphtude A2(d~) foun4 by S~ whelt observing the scattering p~, i.e. A(p~) = L~[A2(d2)]. But, by hypothesis, Al(dl) =A2(d2)--=A(ds). Then, it follows--roughly speaking--that (127)

A(p~) = A(p~)

for all reactions among bradyons of the kind (126a) and (J26b). Actually, in ordinary QFT the requirement (]27) is satisfied by assuming

CLASSICAL TACHYONS

~9

the amplitude A to be an analytic function t h a t can be continued from the domain of the invariant variables relative to (126a) to t h e domain relative to (126b). However, our requirement (127), imposed b y EI~ on the processes (126), has a more general nature, besides being purely relativistic in character. F o r f u r t h e r details see review I. At last, new were derived from EI~ ; t h e y might serve to check the relativistic covariance of weak and strong interactions (which a priori do not have to be relativistically covariant): of. 3/[Ia~A~I and RECAM_I (1974a, 1975a). 11"5. Further results and remarks. - Some results already appeared above; see, e.g., subseet. 9"7 on the i n t e r p r e t a t i o n of the (~advanced solutions ~>. Many f u r t h e r results will appear in p a r t IV (sect. 13), in connection with QM and elementary-particle physics: let us mention the ones related with the v a c u u m decays, v i r t u a l particles, a Lorentz-invariant bootstrap for hadrons, the wave-particle dualism, etc. Here, let ns only add the following preliminary observations. L e t ns consider (fig. 31) two bodies A and B which exchange (w.r.t. a frame c

_

Q

_

r

Fig. 31. so) a transcendent t a e h y o n T~ moving along the x-axis. F r o m fig. 3 and sect. 6 we have seen t h a t for t r a n s c e n d e n t particles t h e motion direction along AB is not defined. I n such a limiting case, we can consider Too either as a t a e h y o n T(v = q- ~ ) going from A to B, or equivalently as an a n t i t a c h y o n T(v ~ -- c~) going from B to A (el. also fig. 3). I n QM language, we could write (Pav~I5 and I~ECAMI, 1976) (128)

ITs) -~ alT(v

=

-~ ~ ) ) q- btT(v

=

--

c~)) ,

a s +

b 2 =

1.

Alternatively, it will be immediately realized t h a t so can interpret his observations also as due to a pair creation of infinite-speed tachyons T and (travelling along x) at any point C of the x-axis between A and B (MIG~A~I and I~CA3~, 19760; EDMONDS, 1976; CA];DrROLA and I~ECAHI, 1980): for instance, as the creation of a transcendent t a e h y o n T travelling towards (and absorbed by) B and of a transcendent a n t i t a c h y o n T travelling towards (and absorbed by) A. Actually, for each observer the v a c u u m can become elassitally unstable only b y emitting two (or more) infinite-speed tachyons, in such a way t h a t the t o t a l 3 - m o m e n t u m of the e m i t t e d set is zero (the total energy e m i t t e d would be automatically zero: see fig. 4, 5 and 6).

90

E. ~ECMAI

I t is interesting to eheck--cf, subsect. 5"6 and eq. (52) of subsect. S ' 1 2 - - t h a t ~ny (subluminal) observer s~, moving along x w.r.t, so in the direction A to B~ will just see a single (finite-speed) antitachyou T~ emitted b y B, passing through point C without a n y intcraction, and finally ~bsorbed b y A. On the contrary, any observer s2, moving along x w.r.t, so in the direction B to A~ will just see a single (finite-speed) t a c h y o n T~ e m i t t e d b y A, freely travelling from A to B (without a n y interaction at C), and finMly absorved b y B. In what precedes we m a y consider the masses of A and B so large t h a t the kinematical constraints, m e t in sect. 6, get simplified. In such a case, so, s~ and s2 will '~]1 see an elastic scattering of A and B. As we h a v e seen above, any observer So can describe the particular process ph u n d e r examination in terms either of '~ v a c u u m decay or of a suitabh~ t a c h y o n emission b y one of the two n e a r b y bodies A, B. One can alternatively adopt one of those two languages. ~ o r e generally, the probability of such v a c m t m decays must be related to the t r a n s c e n d e n t - t a c h y o n emission power (or absorption power) of matter. F u r t h e r m o r c , if A and B can exchange tachyons even when t h e y are v e r y far from each other, any observer s' (like s~ and s~) moving w.r.t, so will describe ph in terms either of an incoming, suitable tachyonic cosmic r a y or of the emission of a slfitable, finite-speed t'~chyon b y a material object. One of the conseql~ences, in brief, is t h a t the t a c h y o n cosmic flux is expected to h a v e for consistency a Lorentz-invariant 4 - m o m e n t u m distribution, just as depicted in fig. 10 und 5c). The large m a j o r i t y of (( cosmic ~ tachyons ought t h e n to appear to a n y observer as endowed with speed v e r y near to the light speed c (see also V~G~E~, 1979; KA_~OI and KA.~:EFUCHI, 1977). I n this respect~ it m a y be iateresting to recall t h a t an evaluation of the possibh~ cosmic flllx of tachyons yielded even if v e r y r o u g h - - a flux close to the neutrinos' one (MIG~A~I and I4EOA~I, ] 976a). As an e l e m e n t a r y illustration of other possible considerations, let us at last add the following. I f so observes the process (129a)

a --> b '-- t ,

where t is an antitachyon, t h e n - - a f t e r a suitable L T - - t h e new observer s' can describe the s%me process as (129b)

a ~--t-~b.

If, in eq. (129a), the e m i t t e d t had travelled till absorbcd b y a (near or jar) detector U, t h e n in eq. (129b) t must, of course, be regarded as emitMd b y a (near or jar) source U. If A~ is the mean life of particle a for the decay (129a), measured b y so, it will bc the Lorentz transform of the averagc time A~' t h a t particle a must spend according to s' before absorbing a (( cosmic ~ t a c h y o n t and trasforming into b.

CLASSICAL TACI~YONS

91

I ) A ~ III General Relativity and Tachyons. 12. - About tachyons in general relativity (GI~). 12"1. Eoreword, and some bibliography. - Spacehke geodesics are (( at home >>in general relativity (GI~), so that tachyons have often been implicit ingredients of this theory. Some papers dealing with tachyons in GR have been already quoted in subsect. 10"2; other papers are ~ISLLEI~ and Wm~ELE~ (1962), FosTE~ and I~Au (1972), I~Auand F0STE~ (1973), LEI]~0WITZand I~0SE~ (1973), BA~EgJEE (1973), GOTTIII (1974a, b), ARCIDIACONO (1974), GOLD01~I (1975a, b, e, 1978), DAVIES (1975), LAKEand I~OED]~ (1975), I~Au and ZI~IWE~AN (1976, 1977), PArdI6 and l~EcA~g (1977), DE SA]~BATAet al. (1977), BANE~JEE and C~OUDH~I (1977), S~IVASTAVA and PAT}tAK (1977), S~IVASTAVA (1977), G~wVICH and TA~ASEVIC~ (1978), K0WALCZu (1978), I~ECA~ (1978a), CA:~_E~ZIND (1978), MILEWSKI (1978), Jom~I and SRIVASTAVA(1978), DHUI~ANDgA:a (1978), DtIUlCANI)HAR and NA~LI~A~ (1978), CAST0~INA and I~ECA~ (1978), ~AI~LIKAR and D]t~I~ANDHAtr (1978), I~EOAlV~ and S]~A~[ (1979), DADmCH (1979), MA:LLE~ (1979), LXUmCIC et al. (1979), P~ASAD and SI~ItA (1979), RAY (1980), S~ANKS (1980), TALUKDAIr et al. (1981), BA~ERJI and !VIANDAL (1982), MA~N and MOF~AT (1982), S~IVASTAVA (1982, 1984), ISmKAWA and M/YASBITA (1983), ]~ISl~O~ (1983), GU~IN (1983, 1984, 1985). For instance, S ~ (1974) calculated--see subsect. 10"2--the deflection of n neutral tachyon (coming, e.g., from infinity) in the field of a gravitating body like the Sun. He found the deflection towards the San to decrease monotonically for increasing tachyon speeds, and at infinite speed to be half as much as that for photons. Later on C o ~ and LATI~OP (1978) noticed that the ordinary principle-of-equivMence calculation for the deflection of light b y the Sun yields, by construction, only the deflection relative to the trajectories of infinitely fast particles (purely spatial geodesics); the total deflection will thus be the sum of the deflection given by the principle of equivalence and the deflection of the infinite-speed tachyons. This does solve and eliminate the puzzling discrepancy between the deflection of light evaluated by EINSTEIN in 1911, using the principle of equivalence only, and the one calculated four years later using the full theory of GI~. In the first calculation EINSTEIN (1911) found a deflection of one-half the correct value, since the remaining one-half is exactly forwarded by the deflection of the transcendent tachyons. We shall here confine ourselves only to two topics: i) tachyons and blackholes, if) the apparent Superluminal expansions in astrophysics.

92

~. ~ECAMI

Let us recMl tha.t the space-times of SI~ and of G/r are pseudo-l-Ciemannian (subsect. 4"3.5); a priori, one may thus complete the ordinary G~ transformation group (_'~[OLLEIr 1962; SACHS anti Wu, 1980) by adding to it co-ordinate tra.nsformations which invert the geodesic type. 12"2. Black-holes and tachyons. 12"2.1. F o r e w o r d . Black-holes (see, e.g., HAWr~G a.nd ELLIS, 1973) are naturally linked to tachyons, since they are a priori allowed in classical physics to emit only t~chyons. Black-holes (BEI) offer themselves, therefore, as suitable sources and detectors (see subsect. 5"12-5"14) of tachyons; and t~chyonic matter could be either emitted and reabsorbed by a BH, or exchanged between BI~s (see PAu and t~.ECA'~[, 1977; DE SABBATA et al., 1977; KAr~LIK~ and DHUlCA:NDIIAIr ]978; CASTOtCINA and ]~ECA_MI~1978; I~.ECA~, 1979a; ~.ECA:~IX and StfArI, 1979; BARUT et al., 1982). This should hold also for hadrons (subsect. 6"13), if they can actually be regarded as t( strong BUs ~ (A_~mhxv et al., 1983; I~EOA~_t,1982a; CAS!I'OYCINAand ICEOA3~,1978 ; SALAM~] 978 ; SALA3~ and STRATIIDEE~ 1978; C&LDLrCOLA et al., 1978). 12"2.2. C o n n e c t i o n s b e t w e e n B~ts a n d Ts. But the connection between BHs and taehyons is deeper, since the problem of the transition outside/inside the Laplace-Schwarzschild horizon in GR in muthematicMly analogous to the problem of the transition bradyon/tachyon in S/~ (REO~t, 1978a, ]979a). Let Its start by recalling some results in the appendix B of IIawking and Ellis (1973). The vacuum metric in the spherically symmetric case reads (130)

ds -~--- --/~-'(t, r) dt ~ + X2(t, r) dr 2 4- Y2(t, r) dO

with d Q ~ d 0 2 + S l n" - ~0 d ~ .~ When

(131.)

as-"

]

Y,"

Y , , < 0 , cq. (130) becomes ( G - - - - c :

1)

2 _dt + l - r/

which is the known, unique (static) Schwarzschild metric for r > 2m. When Y;~ :Y..~> 0, eq. (130) yields on the contrary the (spatially homogeneoits) solution (131b)

ds2 - - - ( 1 - - 2~. )dr~

(1-- 2-~)-~dt~ -'- t2dD ,

which is (part of) the SchwarzschihI solution for r < 2m, since the transform~tion t ~-~r carries eq. (131 b) into the form (131a) with r < 2m (see also GOLDO~I, 1975c). In other words, the, solution (131a) holds apriori for r ~ 2m; inside the horizon,

CLASSICAL TAC}IYOI~"S

93

however, it is ((reiltterl)reted ~> into the f o r m (131b), b y inverting the roles of t a n d r. I n such a w a y one obtains t h a t the metric does not ch~nge signature. I n the two-dimensional case, however, we haw; seell (subsect. 5"6) t h a t the transf o r m a t i o n t ~ - x is just the effect of eqs. (39") w h e n U --+ ~ , i.e. is just the t.ranscendent (Superluminal) L o r e n t z t r a n s f o r m a t i o n (cf. also eq. (39')). A n d il) four dimensions the operation t ~-~ r would h~ve the s a m e efteet exl)ected f r o m a (4-dimensional) t r a n s c e n d e n t ~ . W e reached the point where it becomes agaiu essential the fact t h a t t,he space-time of Gt~ is p s e u d o - t ~ i e m a n n i a n (SACIIS and WI:, 1980), a n d noli l~iemamiian. ~ a m e l y , if one wishes to m a k e use of the t h e o r e m s of ]~iemannian

94

~. ~ECAMI

geometry, one has to limit the grollp of the admissible co-ordinate transformations: see M o d e m (1962), p. 234, CA~E~ZI~]) (1970), HALPEZ~ and MALL'~ (1969). This was overlooked, e.g., b y KOWALCZY~SK~ (1984). I n a pseudo-I~iemanuian---or L o r e n t z i a n - - s p a c e - t i m e we m a y have coordinate transformations even changing the ds 2 sign. Therefore, in order to be able to realize whcther we are deMing with a ~ b r a d y o n ~>or a ~ t a c h y o n ~ we m u s t - - g i v e n an initial set of co-ordinates (~, fl, 7, ~) and a space point P - confine onrselves to the general co-ordinate transformations which c o m p l y with the following requirement, i f co-ordinates (~, fl, 7, ~) define at P a local observer 0, t h e n a new set of co-ordinates (s fl', ~', 5') is acceptable only if it detines at the same lP a second local observer 0 ' which (locally) moves slower than light w . r . t . O . To usc Moller's (]962) words, a n y ((refl~rence frame~) in GI~ can bc regarded as a movirlg fluid; and we must limit ourselves only to the general co-ordinate transformations leading to a. ((frame ~ (a', fl', 7', (Y) t h a t can be pictured as a (~real ~) fluid: This means t h a t the velocities of the (( points of rcference ~ - t h e fluid p a r t i c l e s - - m u s t always be smaller t h a n e relative to the local inertial observer. This has to hold, of course, also for the initial ((frame ~ (a, fl, 7, 8). F o r instance, once we introduce everywhere the co-ordinates (132), we cannot pass (inside the horizon) to co-ordinates (]33b). I n terms of the co-ordinates (]31a), or r a t h e r of the co-ordinates (]32), defined everywhere (for r 2m), a falling b o d y which is a b r a d y o n B in the externM region would seemingly be a t a c h y o n T in the internal regioll (see also GOLDO.NI, 1975c). This agrees with the fact that, when adopting suitable co-ordinates bearing a particularly direct physical meaning, m a n y authors verified t h a t any fMling b o d y does reach the light speed c--in those co-ordinates-on any Schwarzsebild surfaces (see, e.g., ZELtDOVIClt and :[~OVI~OV, 1971; MAI~KLEY, 1973; JAFFE mild SIIABIRO, 1974; CAV_A2,LEI~I-~nd SPI-NELLI, 1973, 1977, 1978; MILEWSKI, 1978). Ill particular, the co-ordin'~tes r, t of the distant observer have no direct significance when looking at the speed of a falling body. F o r instance, DE SA]~BA~rA et al. (1977), following SALTZlW~ and SAI,TZ]~A~ (1969), choose at each space point ]) (r, 0, ? constant) outside the horizon the local i r a m e 2:(X, T) at rest with respect to the horizon and to the Schwarzschild metric (~g~/~T ---- 0). Of course, frames 27 are not inertial. Then one immediately gets (see, e.g., the book b y LI~HT~A~ et al., 1975) t h a t the s t a t i o n a r y observer 27 measures the velocity d R / d T = ( 1 - - 2 m / r ) - ~ d r / d t so t h a t , independently of the initial velocity, this locally measured speed approaches t h a t of light as r approaches 2m. I t should not look strange t h a t a fMling b o d y would reach the light spced for r ~ 2m w.r.t, the local s t a t i o n a r y frame X~, since the local inertial frame would also move with the spced of light w.r.t. Z ~ . L e t its recall within SR t h a t , given a frame so, if we are in the presence of a b o d y B with speed v = e - - e l - d c and of a second frame s' with speed u = e - - e ~ - - > e wliere s~ = qsx (for simplicity we refer to t.he case of collinear motions), the

CLASSICAL TACYIYONS

~

speed v' of B w.r.t, s' will be (134)

v'= c -

which can yield any real values. If @~ 0, then v'-+ c; but if @= 1, then v' -+ 0. And, when v -~ c, the energy of the falling body B does not diverge in X~ ; aetually~ the total energy E of a test particle B is invariant in the local frames ~. For instance~ in the frames ~ where dT is orthogonal to the space hyperplane, it is E = m o ~/-~oo/V/~----v ~. 12"2A. A r e f o r m u l a t i o n . Obviously, part of what precedes does not agree with the conventional formulation of G~ based on l~iemannian geometry, where space-time is supposed to be a smooth, paracompact, simply connected manifold with metric. ]~ECKMIand SI~_~H (1979) proposed a new formulation, where ((( metric-induced ~>) changes o] topology are allowed when passing from a space-time patch to another (see also Scmv~vTz]~, 1968; IVAN]~NKO, 1979; I~OSE~ 1970; WHEEL]~]r 1968; GSBEL, 1976). Whitin :such a formnlation~ they concluded that an (( external observer ~) will deem a fulling body to be a bradyon for r > 2m and ~ tachyon for r < 2m. Conversely, a body which is a tachyon for r > 2m will be deemed a bradyon for r < 2m; bat such will, lof course, be able to come out from a BIt, transforming again into tachyons (cf. also CU~INGHA~, 1975). Notice that, a priori, the 2c), b) the fact t h a t only Superluminal expansions (and not approaches) are observed, c) the fact t h a t W is always Superluminal and practically does not depend on ~, d) the relative-motion colline~rity, e) the fact t h a t the flux density ratio does depend on v and t, since the observed flux differential intensities for the two images as a function of time are given by the formulae (CAsTELLX~'0, 1984) ~]mdt)

dI,dv -- 4~d: V2 --]VL f (139)

f,.-

[(+

X(Vo)~0 F,dv~

~/~t,(t)

1 R(r,:)

T 1

(i = 1, 2),

2 )]' i

,

the integration extrema being (140a)

m~(t) ~ K {VG [VTG']-t :J=1}, 2

(1.40b)

M~(t)--K,~

VT(G'--2L i -i:~L~-/~-- 2%/V2_ :~)]89• 1 ,

where d is the (~proper distance }>OH at the reception epocb (fig. ]5a)) ; L ~ l/d; T ~--ct/d; K ~V/~--lR(t*)/R(t); G --%/~----1 ~- VT and G' --2G-- VT. All equations (139)-(140) become dimensionally correct provided t h a t V/e is substituted for V. B u t the present model remains disfavoured since i) the Superluminal expansion seems to regard not the whole quasar or galaxy, but only a > of it; it) at least in one case (3C273) a.l~ object was visib]e there, even before the expansion started; iii) it is incompatible with the acceleration seemingly observed at least in another case (3C345). ~evertheless we exploited somewhat t h a t case, since, A) in general, the above discussion tells us how it wolfld appear a single Superluminal cosmic source; B) it might still regard part of the present-type phenomenology(*); C) and, chiefly, it must be taken into account for each one of the Superluminal objects considered in the following models. (*) In 1986, e.g., two quasars very similar to each other but with a large angular separation have been observed. It would be interesting to measure their relative recession speed, and so on.

CL&S SICAL TACH~YONS

101

12"5. The models with more than one radio source. - We recalled in subsect. 12"2 that black-holes can classically emit (only) tachyonic matter, so t h a t they are expected to be suitable classical sources--and detectors--of tachyons (PAv~I6 and I~ECAMI,1977; DE SABBATAet al., 1977; N ~ L ~ and D n ~ A ~ D~A~, 1978; I~ECA~, 1979a; I~EtA~ and S~_&H, 1979; BA~UT et al., 1982). Notice that, vice versa, tachyons can not only enter the horizon of a blackhole, but also come out from a horizon. As we already said, the motion of a spacelike object penetrating the horizon has been already investigated, within Gl%, in the existing literature (see the end of subsect. 12"2.4). We Mso saw in subsoet. 5"18 (fig. 14) and in subscct. 10"2 that, in a (( subluminal }>frame, two tachyons may s e e m - - a s all the precedent authors claimed-to repel each other from the kinematical point of view, due to the novel features of tachyon mechanics (subsoet. 10"1, cqs. (109b), (109c)). In reality, they will gravitationally attract ouch other, from the energeticM and dynamicM points of view (subseet. 10"2; see also fig. 4a)). From subsect. 10"2 a tachyon is expected to behave the same w a y also in the gravitational field of a bradyonic source. If a central source B (e.g., a black-hole) emits, e.g., a SuperluminM body T, the object T under the effect of gravity will lose energy and, therefore, accelerate away (subsect. 5"3). If the total energy E = moc~/V/V 2 - 1 of T is larger than the gravitational binding energy E, it will escape to infinity with finite (asymptotically constant) speed. (Since at infinite speed a taehyon possesses zero total energy--see fig. 5e) and subscct. 6"14--, we m a y regard its total energy as all kinetic). If on the contrary E < E, then T will reach infinite speed (i.e. the zerototal-energy state) at a finite distance; afterwards the gravitational field will not be able to subtract any more energy to T~ and T will start going back towards the source B, appearing now--possibly--as an antitachyon T (subsect. 5"12 and 11"2). I t should be remembered (subsect. 11"5 and eq. (128)) that at infinite speed the motion direction is undefined, in the sense that the transcendent tachyon can be described either as a tachyon T going back or as an untitachyon going forth, or vice versa. We shall see, on another occasion (subsect. 13"2), that a tachyon subjected, e.g., to a central attractive elastic force F = -- kx can move periodically back and forth with a motion analogous to the harmonic one, reversing its direction at the points where it has transcendent speed (and maybe alternately appearing--every half an oscillation--now as a tachyon and now as ~n antitachyon). Let us consider, in general, a tachyon T moving in space-time (fig. 32) along the spacelike curved path AP~ so to reach at P the zero-energy state. According to the nature of the force fields acting on T, after P it can proceed along PB (just as expected in the above two cases, with attractive central forces), or along PC, or along PD. In the last case, T wolfld appear to annihilate at t ) with an antitaehyon emitted by D and travelling along the curved worldline DP (subsect. 5"12 and 11"2; see also DAWES, 1975, p. 577).

102

~. RECAMI

-A

o

Fig. 32. I t is clear t h a t the observed (( Superluminal ~>expansions can be explained i) either b y t h e splitting of a central b o d y into two (oppositely moving) collinear t a c h y o n s T1 and T2, or b y the emission f r o m a central source B of ii) a t a c h y o n T, or iii) of a couple of t a c h y o n s T1 a n d T, (ill the l a t t e r case, T~ and T~ can for simplicity's sake be considered as e m i t t e d in opposite directions with t h e s a m e speed). I n thi~ respect, it is interesting t h a t NE'E.~AN (1974) regarded quasars ---or a t least their dense c o r e s - - a s possible white holes, i.e. as possible (( lagging cores ~ of the original expansion. F o r simplicity, let us confine ourselves to a flat s t a t i o n a r y universe, 12"5.1. T h e c a s e ii). I n t h e case ii), be O the observer and a the angle b e t w e e n B O a n d t h e m o t i o n direction of T. :Neglecting for the m o m e n t the

Fig. 33. g r a v i t a t i o n a l interactions, the observed apparent relative speed b e t w e e n T a n d B will, of course, be (see fig. 33)

(141)

W=]_9

V sin a

Vcosa

(V>

1).

L e t us assume V > 0; then, W > 0 will m e a n recession of T f r o m B, b u t W < 0 will m e a n approach. Owing to the cylindrical s y m m e t r y of our p r o b l e m w.r.t. BO, let us confine ourselves to values 0 < ~ < 380 ~ L e t us m e n t i o n once m o r e t h a t W -+ oo w h e n cos a -> 1/V ((( optic-boom )>situation). I f the emission angle a of T f r o m B w.r.t. BO has t h e value ~ = ~ , with cos ab ~ 1IV (0 <

103

CLASSICAL TACI-IYONS

< a~ < 90 ~ b ~ (~boom))), t a c h y o n T ~ppears in t h e optic-boom phase; b u t the recession speed of T f r o m B would be too ldgh i~ this ease, as we saw in the, previous subsection. IncidenLuny, to ~pply the results got iu subsect. 12"4 to the Superluminal object T (or T~ and T~ in the other cases i), iii)), one h~s to t a k e account of t h e fact t h a t the present t a e h y o n s are born at a finite time, i.e. do not exist before their emission from B. I t is t h e u i m m e d i a t e to deduce t h a t we shall observe a) for a > a~, i.e. for a~ < a < 180 ~, the object T recede f r o m B; b u t b) for 0 < ~ < ~ , t h e object T approach B. More precisely, we shall see T receding from B with speed W > 2 when

V t- V S - V ' - - 4 2(V I 1)--- < t g , ~ <

(142)

2(V ,~- 1)

'

1 arccos ~ < a < 180 ~ .

I t should be noticed t h a t eq. (141) can yield values W > 2 whenever V > 2/v/5: in particular, therefore; ]or all possible values V > 1 of V. Due to eqs. (142), the ~(emission direction ~) c~ of T m u s t be, however, contained inside a certain suitable solid angle: a~ < a < a~; such a solid angle always i~cluding, of course, the optic-boom dh'ection ~ . ) ' o r instance, for V - > I we get 0 < t g (o~/2)< ~-, a>c% = 0, w h e r e f r o m ([43)

0 < ~ < 53.13 ~

in such ,~ case, we shall never observe T a p p r o a c h i n g B. On the V-~oo we get 89 -- %/5) < ~g (a/2) < 89 -]- ~/5), ~ = 9 0 ~ < f r o m -- 63.44 ~ < a < 116.57 ~ a > 9 0 ~ t h a t is to say 9 0 ~ we a d d the r e q u i r e m e n t , e.g., W < 50, in order t h a t 2 < W < to exclude in eq. (143)--for V - ~ l - - o n l y the t i n y angle 0 < Lh~t in conclusion 2.29~

~ < 53.13 ~

(V --~ 1); c o n t r a r y , for 180 ~ where116.57 ~. I f 50, we h a v e ~ < 2.29 ~ so

(V- ~ ~).

Tim same r e q u i r e m e n t 2 < W < 50 will not a f f e c t - - o n the c o n t r a r y - - t h e a b o v e result 90~ < 116.57 ~ for the case V - 7 c~. Similar calcldations were p e r f o r m e d also b y F ~ . ~ m L S ~ , ~ et al. (1983). The present case ii) suffers f r o m some difficulties. First, for ~ > ~2 (for instance, for 53~162 < 180 ~ in the, case V-> 3) we should observe recession speeds with 1 < W < 2, which is not s u p p o r t e d b y the d a t a ; b u t this can be understood in terms of the ])oppler-shift selection effects (see subsect. 10"4 and BLAR'DlrOaD et al., 1977). Second, for a < ~ one should observe also Superluminal a p p r o a c h e s ; only for V ~ 1 (V ~ 3) it is at. _~ 0 a.nd, therefore, such Superluminal approaches are not predicted.

104

~. 2ECAMI

]n cotmhlsiol b this model ii) :~,ppears acceptable only if tile emission mechanism of T from 13 is such t h a t T has v e r y large kinetic energ~~, i.e. speed V ~ 1. 12"5.2. T h e c a s e s i) a n d iii). Let us pass now to analyse the cases i) a:nd iii), still assuming for simphcity T 1 and T., to be emitted with the same speed V in opposite directions. Be a again in the range [0, 180~ In these cases, one would observe faster-than-light recessions for c~> %. W h e n c~< %, on the contrary, we would observe a single t a c h y o n T ~- T~ reaching the position B, passing it, and continuing its motion (as T~= T~) b e y o n d B with the same velocity b u t with a new, different Doppler shift. One can perform calculations analogous to the ones in subsect. 12"5.1; see also I~.~KELSTEIN et al. (1983). I n case i), in conclusion, we would never observe Superluminal approaches. F o r a < ~b we would always see only one body at a time (even if T --:- T~ might result as a feeble radiosource, owing to the red-shift effect): the motion of T wouhl produce a variability in the quasar. For ~ > ab, as already mentioned, wc would see a Superluminal expansion; again, let us recall t h a t the cases with 1 < W < 2 (expected for large angles e only) could be hidden b y the Doppler effect. Case iii) is not v e r y different from case it). I t becomes ((statistically ~) acceptable only if, for some astrophysical reasons~ the e m i t t e d tachyonic bodies T~ and T~ carry v e r y high kinetic energy ( V ) 1 ) . 12"6. Are (~superluminal~) expansions Superluminal? - I f the e m i t ~ d tachyonic bodies T (or T1 and T~) carry away a lot of kinetic energy ( V ~ 1), all the models i), it), iii) m a y be acceptable from the probabilistic point of view. Contrariwise, only the model i ) - - a n d the model iii), if ]3 becomes a weak radiosource aft(;r the emission of T1, T~--remains statistically viable, provided t h a t one considers t h a t the Doppler (;fleet can hide the objects e m i t t e d at large angles (say, e.g., between 60 ~ and 180~ On this point, therefore, we do not agree with the conclusions in FIS-KEI, STEIN et al. (1983). I n conclusion, the models implying real Superluminal motions investigated in subsect. 12"5 seem to be the most viable for explaining the a p p a r e n t (( S u p e r h m i n a l e• ~); (;specially when taking account of the gravitational interactions between B and T, or T1 aud T~ (or among T~, T2, B). Actually, if we take the gravitational a t t r a c t i o n between B and T (subsect. 10"2) into acc(umV--for simplicity, let us confine ourselves to the case it)--, we can easily explain the accelerations, probably observed at least for 3C3~5 and m a y b e for 3C273 (Sm~NaLIN and Yo~-aZlI~N, 1983)(*). (*) Notice that, i] very high gravitational attractions are opera~ing, then our models i)-iii) can even prcdic~ a Superluminal approach (following the Supcrlulninal expansion).

CLASSICAL TACItYONS

105

Some calculations in this direction have been recently performed also by SHE~'GHN et al. (1984) and (L~o (1984). But those authors (lid not compare corrcctly their evaluations with the, data, since they ovcrlooked that--because of the finite value of the light speed--the images' apparent velocities do not coincide with the sources' actual velocities. The values W0 calculated by those attthors, therefore, have to be corrected by passing to the values W : : ~-Wo since/(1--cosec); only the values of W arc to be compared with the observation d~ta.. A l l t.he calcul'~tions, moreover, ought to be corrected for the universe cxp~msion, ttowcver, let us re.call (subsect. 12"4) t hat in the homogeneous isotropic cosmologies--- expansions--the angular expansion rates are not expected to be modified by the expansion, at least in t.hc ordinary observational conditions. While the corrections due to the universe, curvature would be appreciable only for very distant objects.

Tachyons in Quantum Mechanics and Elementary-Particle Physics. 13. - The possible role of tachyons in elementary-particle physics and quantum mechanics. I n this review we purported (subsect. 1"1) to confine ourselves to the classical theory of tachyons, leaving aside their possible quantnm field theories (el., e.g., BROID0 and TAYLOI~,1968). We have already met, however, many instances of the possible rolc of tachyons in elementary-particle physics. And we want to develop some more such a.n aspect of tachy()ns in the present sectiom In subseet. 1"1 we mentioned, moreover, the dream of reproducing the quantu m behaviour at a classical level, i.e. wit.hin a classical physics including tachyons (and suitable extended-type models of elementary particles). In the present section we shaft put forth also some hints pointing in such a direction. l~ight now, let us only recall from sect. 9 and 6 t hat tachyons are the suitable carriers of mutual, symmetrical interactions between A and ]3, rather than of ~ signals ~>, i.e. of controllable, interpretable messages sent either by A to ]3, or by ]3 to A. If one accepts t hat fa.ct, then--~s we have seen--one is left within SI~ with no problem at all. But such tachyonic influel:ces, which already allow ~s to abandon the (~strong locMity ~> prejudice, would be enouglJ to interpret quantum mechanics in agre in subnuch~ar interactions, i.e. that tachyollS can be the ~ realistic )~classical carriers of elastic and inelastic interactions between elementary particles (SUDA_~SIIAN, 1968; ICECAMI, 1968; CI~AVELI~ et al., 1973; see also all the ref. (s) and (9) in MACeAI~O~E and I~ECAm, 1980b). Actually, it is known that the ~(virtual particles ~}exchanged between two elementary particles (and, therefore, realizing the interaction)must carry a negative four-momentum square, for simple kinematical reasons (review I):

(144)

t :---: p . p "

-- B ~ -- p~ < 0,

just as happens for taehyons (cf., e.g., subsect. 6"1, eq. (29e)). Long ago it was checked (ICECAMI, 1969; OLX~OVSI~u and Ir 1969) whether virtual objects could really be regarded as faster than light, at least within the so-called peripheral models (( with absorption ~) (see, e.g., DAa~, 1964). To evaluate the effect of the absorptive channels in the one-particle exchange models, one has to cut out the low partial waves from the Born amplitude. Namely, an impact parameter (Fol~rier-Bessel) expansion of the Born amplitudes is used, and a cut-off is imposed at a m i n i m a l radius R which is varied to fit the experimental data. While considering--for example---different cases of pp interactions via K-meson exchange, values of R were found ranging from 0.9 to 1.1 fm, i.e. much larger than the K-meson Compton wave-length. The same kind of model (at a few GeV/c, with form factors) was also applied to pion-nucleon reactions via o-meson exchange; and also for the p a value (R ~ 0.8 fro) much grea~er tha.n the p-meson Compton wave-length was found.

C~ASSICA~ ~ e H ~ o x s

107

E v e n if such rough tests are meaningful only within those models, one deduced the v i r t u a l K and p mesons of the mtcleon cloud to t r a v e l faster t h a n light: 2 for instance, in the first case, for t ~ -- rex, one finds > 1.75 c. According to WIG~ER (1976), (( there is no reason to believe that interaction cannot be transmitted ]aster than light travels >>. This possibility was considered in detail b y VA~ DA~ and W I ~ E ~ (1965, 1966) already in the Sixties. See also A~uDx~ (1971), CosTA DE BEA~-~E~A~D (1972), MATEEWS and SEETHAI~A~A~ (1973), FLATO and GuE~I~ (1977) and Sm:RoKov (1981). And a n y action-at-a-distance t h e o r y (see, e.g., SUD~SEX~, 1970d; VoLKov, 1971; LEITER, 1971b; ]=[0YLE and NAI~LXKA_R,1974) implies the existence of spacelike objects, since the infinite speed is not i n v a r i a n t (subscct. 4"1). Moreover, i] hadrons can really be considered as ((strong black-holes >) (subsect. 12"2.1), t h e n :strong interactions cau classically be mediated only b y a t a c h y o n exchange, i.e. the strong field (( quanta >> should be Superluminal (*). I n a n y case, we van describe at a classical level the v i r t u a l cloud of the hadrons as m a d e of t a c h y o n s (see also S U D ~ S t L ~ , 1970b), provided t h a t such tachyous, once emitted, are--(( strongly >>--attached b y the source hadron, in analogy with what we discussed for the ordinary gravitational case (subsect. 12"5). F o r the description in terms of a ((strong gravity >) field, see, e.g., SALA~ (1978), SIVA~A~ and Sz~]tA (1979), I~EOA_M~ (1982a, b) and references therein, and AZvrM~ZAZUet al. (1983). I n fact, if the a t t r a c t i o n is strong enough, the emitted tachyous will soon reach the zero-energy (infinite-speed) state; and afterwards (cf. fig. 32) t h e y will go back, till reabsorbed b y the source hadron. Notice t h a t transcendent tachyons can only take energy from the field. Notice, moreover, t h a t classical tachyons subjected to an a t t r a c t i v e central field can move back and f o r t h in a kind of tachyonic harmonic motion (see fig. 34), where the inversion points just correspond to the infinite speed (ef. subsect. 12"5; see also A~rt~o~ov et al., 1969). Finally, let us consider a h a d r o n emitting and reabsorbing (classical) tachyons. I t will be surrounded b y a cloud of outgoing and incoming

T

T

V=o~ [

T aC6"6"6"r T

Y

7

: V=oo

r

Fig. 34.

(*) Incidentally, in such an approach hadrons should be regarded as pulsating objects (cf. R~cA~, 1982b), with a period of about 10-~3 s, that pulsation being possibly related with the zitterbewegung (see, e.g., H~sT~]~s, 1985) or, more generally, with the existence of a .

108

~.:. m,:CAm

taci~yons (*). I n the (, contimlous ~ approximation (and spherically symmetric case), t h a t clond can be described b y the spherical waves:

I o:: cxp [ +

(1,15)

imor]p

r

We are, of course, confi~fing ourselves to subluminal frames o~fly. We can find out, however, the results forwarded b y EI4 formally b y putting for tachyons mo ~ :~ i# (# reM). I t is noticeable t h a t from eq. (145) we get, then, the Yuk,~wa potential b y setting m := ~= i# fl)r tile outgoing and m --= -- i# for the incoming waves

(145')

:

Ioc

exp [--/~r] I~ ; r

I

in other words, at the static limit, the Yukawa. potential can be regarded am the (~contirmous ~) (classicM) description of a flux of outgoing tachyons and incoming a n t i t a e h y o n s : see ()ASTOI~INAand I{ECA~ (:1978). Se also I=[ADJIOA:N_~OU (1966), ~ElC~E~TI and VEI~DE (1966), YAM_AMOTO (1976), EI~ICSE~" 'rod VOYE~LI (1976), FLA~:0 and G L r E ] ~ (1977) and :FEDERIGHI (1983). W h e n two h~drons come close to ea.ch other, one of the cloud t a c h y o n s - - i n s t e a d of being reabsorbed b y the m o t h e r h a d r o n - - c a n be absorbed b y the second h a d r o n , or vice versa (thin s t a t e m e n t in frame dependent). T h a t would be the simplest hadron-hadron interaction. The actual presence of a t a c h y o n exchange wouhl produce a resonance peak in the scattering amplitude as a fm~ction of the momentl~m transfcr t ~ (p~ -- p2) ~ ( S U D ~ S m ~ , 1969a, b, 1970c). Precisely, it would produce a (( negative t e n h a n c e m e n t )), fixed when s -~ (p~ + p2) 2 varies, and possibly to be found also in other similar processes (D~A~ and SUDARSIIA~, 3968; GLOCK, 1969; BALDO et al., 1970), mlless the tachyons appear to possess a v e r y large width (BuGI~ff et al., 1972; nee also KI~0LIKOWSKI, 1969). A positive theoretical evide~lce wan p u t forth b y GLEESON et al. (1972a). See also VA~ DE~ S P ~ (t973), g u n (1973), AK~UA (1976), E~ATSU et al. (1978), review I, p. 266, and ]~ALDO et aI. (1970). Before closing this section, let us reca.ll tha.t long ago (l~nch:v~, 1968, 1969) it was suggested t h a t the unst'~blc particles ((( re. onanccs )~), bearing masses M*: M + i F formally complex, might be compounds of bradyons and tachyons. W e shall come back to this point in subsect. 13"5 (see also, e.g, SU])A~StIAN, 1970d; :EDMONDS,1974; KESZTH]~]LHYI and ~A6Y, 1974).

(*) Let us remark, by tlle way, that such continuous processes of tachyon cnfission and absorption could be one cause for l,he stochastic motion that in classical physics is to be added to the detcrministic global motion in order to get thc Schr6dinger equation (cf., e.g., CAv~LxRI, 1985).

109

CLASSICAL TACIIYO_N'S

The considerations a b o v e can be interesting also with regard to ttegerfeld's (t985) result t h a t , in relativistic q u a n t u m theory, a relativistic particle or s y s t e m t h a t at t - - 0 is locMizcd with exponentiMly b o u n d e d ta, il at later times (( violates causality ~), in the sense t h a t the assumption ol finite propagation speeds leads to a contradiction. More in general, for t h e possible connections b e t w e e n Superhnninal motions a n d the (( q u a n t u m p o t e n t i a l )> ( B m r ~ a n d VIGIE~, 1954, 1958), see, for instance, Vmi~:Ir (1979, 1980). See a.lso STAPP (1977, 1985) and D'EsPA(~.N'AT (1981). 13"3. P r e l i m i n a r y applications. Neutrinos and tachyons. - I f subnuclear interactions are considered as m e d i a t e d b y quanta, no ordinary (bradyonic) particles can be the carriers of the transferred e n e r g y - m o m e n t u m . We h a v e seen, on t h e contrary, t h a t classical t a e h y o n s can a priori act as the carriers of those interactions. As p r e l i m i n a r y e x a m p l e s or applications, ]el us consider t h e v e r t e x An3-+ - + p + % of a suitable one-particle exchange diagram, a n d suppose the exchanged particle (internal line) % to be a t a c h y o n i c pion, instead of a (( virtl]al ~ pion. Then, f r o m subsect. 6"3 and 6"8 we should get 1232 ~ - 938 ~ = 1402 -IL 2-1232.~/v~,p[ 2 - 1 4 0 ~, and, therefore (MAccA]C~0~E and ICEc)=~g, 1980b), (146)

[p[~

- 287 M e V / e ,

E ~ - - 251 M e V ,

so t h a t , in t h e c.m. of the As3(1232), t h e t o t a l energy of the t a e h y o n pion is predicted to be centred a r o n n d 251 MeV. Again, let us consider the decay ~ -+ ~ -~- ~T raider the hypothesis, now, t h a t ~T be a t a c h y o n - n e u t r i n o , with m , ~ 0, v ~ c. I t has been shown b y CAWr~EY (1972) t h a t such a hypothesis is not inconsistent with the e x p e r i m e n t a l data., and implied for the m u o n - n e u t r i n o a mass m ~ < 1.7 MeV. I n the two limi]~ing cases, from subsecl.. 6"3 a n d 6"8 in the c.m. of t h e pion we shmfld get (MACCARR o ~ and t%ECA~-UI, 1980b) (147a)

m, =- 0

~ ]p], ~_ 29.79 M e V / c ,

(147b)

my= ].7~

IPlv ~ - 2 9 . 8 3 N e V / c ,

vv ~ c,

v,_~l.0016c.

I t is interesting to notice t h a t , in t h e case of t a c h y o n - n e u t r i n o s , the m a g n i t u d e of t h e neutrino impulse increases w.r.t, t h e light speed case. L e t us recall once m o r e f r o m subseet. 6"13 t h a t / o r instance a n y elastic scattering can be (, realistically ~>m e d i a t e d b y a suitable t a c h y o n exchange during the approaching p h a s e of l~he two bodies. ]in tlle c.m.f. (]Pa[ = [P~[ ---- IPI) we would obtain eq. (82): m oo

(82)

cos0o.m.

-

1 - - .;i, p, ~- ~ - 2

110

~. ~ c A ~

so that, for each discrete value of the tachyon rest mass ma (subsect. 5'1), the quantity 0 too assumes a discrete value, which is merely a function of ]P[. (Let us recall, with regard to this result, also Landd's (1965, 1975) ideas.) We have always neglected, however, the mass width of the tachyons. For further considerations about tachyons and virtual fields see, e.g., YA~ DES SPUY (1978) and SOUSEK et al. (1981). Tachyons can also be the exchanged particles capable of solving the classical-physics paradoxes connected with pair creation in a constant electric field (ZEL'DOVIC~, 1974a, p. 3r and 1972). For joint probability distributions in phase-space and tachyons see, e.g., K~tiGE~ (1978, and references therein), where the ordinary formalism w~s generalized to the relativistic case and shown to yield a unified description of bradyons and tachyons. 13"4. Classical v a c u u m instabilities. - We saw in subsect. 11"5 that the vacuum can become unstable, at the classical level, by emitting couples of zeroenergy (infinite-speed) tachyons T and T. For a discussion of this point (and of the possible connection between the cosmic tachyoa flux and the tachyon emittance of ordinary matter) we refer the reader to subsect. 11"5 (and fig. 31). See also M~G~A~I and I~ECA~ (1976a), as well as fig. 32 in our subsect. 12"5. Here let us observe t h a t the probability of such a decay of a vacuum ~ bubble ~ into two collinear transcendent tachyons (T and T) is expressible, according to Fermi's golden rule (FE~lv~, 1951), as ~-1oc moc/87ch, where mo is the tachyon rest mass (both tachyons T and T must have the same rest mass, due to the impulse conservation; remember that for transcendent tachyons IPl ---- race); bat we are unable to evaluate the proportionality constant. More interesting appears considering, in two dimensions (sect. 5), an ordinary particle P ~- P~ harmonically oscillating in a frame ]' around the space origin O'. If the frame ]' moves SuperluminMly w.r.t, another frame ]o(t, x), in the second frame the world-Hue of point O' is a spacelike straight line S; and the world-line of the harmonic oscillator P --= PT (now a tachyon with variable velocity) is depicted in fig. 35. Due to what we saw in subseet. 5"125"14,--as well as in sect. 11, subsect. 12"5 and 13"2~, the ~ sabluminM ~ observer fo will see a vacuum fluctuation propagating in space, with vacuum decays (pair creations of transcendent taehyons) in correspondence with the events 01, 02, Ca..., and with analogous pair annihilations (of transcendent tachyons) in correspondence with the events A1, A~, As... (fig. 35). Cf. also WI~vr~L (1971b) and CATA~A et al. (1982). :Notice that each vacuum instability C is just a vacuum decay into a taehyon T and an antit~chyon T, having the same rest mass and oppositely moving with infinite speed; such a process is perfectly allowed by classicM mechanics (see, e.g., subsect. 11"5). Analogously, each event A is nothing but the annihilation (into a ~ vacuum bubble ~>) of transcendent T - T pair.

e~ASSZCAZTae.YO~S

111

I

t A3

_ _~2_ _

r

....

_ ....

X

Fig. 35.

This is another example of classical description of a typically (( quantM }> phenomenon, i.e. of ~ phenomenon usually regarded ~s belonging to the realm of quantum field theory (QFT). See also, e.g., NAmer (1950), 3lAngm~i~ (1977), FVKUDA (1978), S~Au and MILLEI~ (1978) and SOU6EK (1981). Let us remark, at this point, that in ordinary theories the possible presence of tachyons is not taken into explicit account. I t follows that the ordinary vacuum is not relativistically invariant, if tachyons on the contrary exist (and, let us repeeot, if account of them is not explicitly taken): cf., e.g., subsect. 3"17 and fig. 13. The fact that in the usual theories the ordinary concept of empty space may not be Lorentz invariant was particularly stressed by NIELSEn (1979), who noticed that, if some large region in space is empty of tachyons as observed from one frame, there is no guarantee that it will be so seen from another frame of reference. NIELSEn et al. (see, e.g., :NIELSEn and NINO~frYA, 1978; ~_~-IELSEN, 1977) als0 developed noninvariant theories, even if independently of the above observations. 13"5. A L o r e n t z - i n v a r i a n t bootstrap. - The idea that tachyons may have a role in elementary-particle structure has been taken over by many authors (see, e.g., I~ECAMI, 1968, 1969a; HANA31OTO, 1974; AIC~BA, 1976; I~AFA~ELLI, 1976, 1978; VAn ])E~ SPRY, 1978; CAS~O~InA and I~ECA~, 1978; SzA~0SI and T~EVlSAn, 1978; see alsot~0SEN and SZA3~0SI, 1980, and ref. (s,9) in MACCA~OnE and I~ECAMI, 1980b). One of the most interesting results is probably the one by CO~BE~, who succeeded in building up a Lorentz-invariant of hadrons or of hadronic ((resonances)) (Co,BEn, 1977a, b, 1978a, b). Let us describe his approach by following initially CASTORIn~ and I~ECAm (1978). COI~BEN started from the known fact that a free bradyon with rest mass M and a free tachyon T with rest mass m caD_trap each other in a relativisticMly invariant way; if M > m, the compound particle is always a bradyon B*. If the two particles have infinite relative speed, and P, p are their four-momenta,

112

~. ~ECA~*

t h e n (subsect. 11"5) (148a)

p~P,----0~p

•

In such a case the mass M* of the compound bradyon B* is (subsect. 6"3 and 6"5) (148b)

M* : % / ~ - -

m ~,

as easily follows from cqs. (58), (59), or from eqs. (64), (65). L e t us now assumc that, inside the (~composite hadi'on ~, the t a c h y o n T feels a strong field similar to the gravitational one (see, e.g., l ~ n c ~ , 1982a, and references therein); let us assume moreover t h a t the trapped t a c h y o n has already reached an equilibrium state and is revolving along a circumference around the bradyon B (see also S~P~L~S, 1983). F r o m subscct. 6"1~ and 10"], we t h e n derive t h a t a n y bradyon-tachyon compound--in its lowest-energy state (~ ground state ~))--is expected to be constituted by a t a c h y o a T having divergent speed w.r.t, the bradyon B, so t h a t condition (148a) is satisfied. T reaches in fact its minimal energy when its speed diverges, i.e. the fundamental state of the system should correspond to a transcendent periodic motion of T. One also derives t h a t the trapping force, which holds T on a circular orbit, tends to zero when T tends to have infinite speed. I n such u case the interaction is negligible, even i / t h e ~ sel/-trapping ~) keeps itsel]. Under condition (148a), therefore, one m a y consider the B-T compound us a couple of two ]ree particles! Actually C o ~ E ~ (1978a), by using the q u a n t u m language, considered two particles satisfying the equations (M > m)

(149a)

Oy~ -~ K2~B

(1~9b)

[~YJT ---- -- k~ ~PT

(K ~ Melt), (k ~ melt),

and such t h a t , if v/--~ V2B~PT, (149c)

O ~ --~ (K ~ - k 2)

k

=

Equation (149e) comes from postulating the invariant interaction ~, ~B ~ ~T = 0, which is nothing b u t the quantum-field version of condition (148a); in fact, applied to the eigenstates of energy and m o m e n t u m , it just implies eq. (148a). (Cf. also eqs. (149c) and (148b).) Plane timelike and spaeelike waves can, therefore, (( lock )) to form a plane wave, t h a t is timelike when M ~ m. ~otice t h a t everything still holds when we substitute D(~)=-~, - (ie~/~c)A~ for ~ , I t would not be possible to combine two timelikc, states in this way, because applying the condition ~ B ~ - - - - 0 (or D(~)y~nD(f~ : 0) to such states leads to imaginary m o m e n t a and exponentially increasing (not normalizable)

CLASSICAI~

113

TACtIYONS

wave functions. This corresponds: of course, to the classical fact t h a t condition (148a) cannot be satisfied by two bradyons. On the contrary, a bra(ty(m B can combine in a rcl'~tivistically invariant way with more t h a ~ one tachyon to yield another b r a d y o n B*. Actually, due to conditions of the t y p e of eq. (]48a), it can trap no more than three taehyons, getting eventually the mass (t48c)

M*

=

2 ~/'M~--m~2 m~--m~,

provided t h a t it is real. In such a situation, the three transcendent tachyons T1, T.~, T3 c~m be im~gined ~s moving circularly a r o u n d the axes x, y, z, respectively (tim circle centres always coinci(ling with B). Going back to the q u a n t u m field language (Co~BE~, t977a, 1978b), the extra, conditions ~ u % ~ , ~ J = 0 (i, j : 1, 2, 3, i r require l~he functions ~u~0~ to be orthogonal to each other in space. 15Iore generally, setting M : me, the conditious ~ u V ~ u ~ : : 0 (a, fl = 0, l, 2, 3, ~ fl) imply theft no more than three spaeelike states can be s u p e r i m p o s e d on one timelike state to yield another pa.rticle. (Cf. also 1-)RE:PA ICATA, 1976; IIotl, 1976; ~PAGELS~]976.) IU QFT a b r a d y o n ~t rest is describe(I, as usual, b y a wave function periodic in time and independent of position. A transcendent tachyon, on the contrary, corres])onds to a w~ve functio:~ static in time and periodic in space: t~ lattice (el. a~lso sect. 8). In(;ide.qt~tly, the interaction between a br~dyon ~n(i ~ transcendent t a c b y o n is, therefore, almlogous to the scattering of a wave b y a diffraction grating (CerumEN, 1978a). The three values of the lattice spacings in the three directions of space m a y be regarded as corresponding to the masses of lhe three spacelike states t h a t can combiue in the above way with one timelike state ((~(mnE~~, 1978b). B y resorting to eqs. (148b), (148c) and to suitable q u a n t u m n u m b e r considerations, COICBE.~ (1977a, b, 1978a, b) found masses and q u a n t u m numbers of a host of hadrons as composed of other (sub- aud Super-lumina.1) hadrons, thus realizing a rclativisl,ic~.lly invariant bootstrap (CHEw, 11968). There are a. n u m b e r of ex~mples which appear to verify this, especially in the spectrum of the K-particles ~md thc s ~ 0 mesons l h~t preferentially decay into K K : we refer the reader to 1:he interesting tables published b y CORBE~ in his above-mentioned pqpers, which also contain f u r t h e r details ~nd comments. CO]r " found also the ma~'.s differences among the members of various isospin multiplets by binding Superluminal leptons to suitable subluminal hadrons. I t would be interesting t r y i n g to generalize snch an approach even to the qua.rk level. Actually, m a n y a~tthors suggested t h a t q u a r k s - - m o r e generally, the elementary-pa.rticlc c o n s t i t u e u t s ~ m i g h t be tachyons (see, e.g., HA.~A~OTO, 1972; ?r and ~],~]CAMI, 1975b, p. 539; GuE~'~, 1976; SOU5EK, 1979a, b; see 'also BROW~- and I~go, ]983). RAFA~ELLI (i974~ 1976, 1978) showed t h a t

114

~:. n ~ c ~

in classical relativistic physics there exists the possibility for a description of an elementary particle which has constituents, if those constituents are tachyons. Free spinning tachyons are then the candidates for elementaryparticle constituents. And in the range of Superluminal velocities the theory of a free spinning point particle admits uniquely of a linearly rising trajectory, naturally yielding the constituent confinement (see also subsect. 12"2 and I~ECA~K, 1982a). ~oreover, we shall see--subsect. 15"2--that the duality between electric and magnetic charges is possibly a particular aspect of the bradyons/tachyons duality; and authors as TzE (1974) and BA~UT (1978C) Cmderlined the connections between electromagnetic and dual strings; possibly, a link can thus be found between tachyons and hadron .structure (MI(~-A~I and t ~ : c ~ K , 1975b). Let us add that~ more generaily~ quarks have been identified (P~IsI~ 1978) with nonconventional (( monopoles ~)~ i.e. with the monopoles of the field which mediates strong interactions inside hadrons. Asid% it stands the ((electromagnetic ~) approach by JEtILE (1971, 1972), who noticed that---while the introduction of a gauge variable single-valued in space implies charge conservation but does not change the physical situation-a (~pseudogauge )) transformation (with a variable 0 which is single-valued only modulo 2~) is equivalent on the contrary to the introduction of a quantized flux hc/e (see also BERNARDINI~1982). I t is~ namely~ a transformation from A ' -- O, 3~0' 0 to A ~ Ok, where Ak -- (?ic/e) ~kO 0 is invariant; and, because of (e/~v)~ ~ A ~ d r ~ = • one may assume the flux line (singularity of 1~3

3k0) to be a closed loop. A more detailed analysis shows that such singularity loops spinning with velocities larger than c permit a consistent formulation of leptons, as well as of quarks and hadrons, in terms of electromagnetic fields and their probability amplitude distributions. The topological structure of those fields (toroidal field lines) represents the internal quantum numbers in particle physics. 13"6. May classical microtachyons appear as slower-than-light quantum particles? - We have seen in subsect. 8"4 that, if a taehyon P~ has a very large intrinsic (i.e. measured in its own rest frame) li/etime At(P~), just as it is for macroscopic and even more cosmic objects~ then PT can actually be associated with Superluminal motion (subsect. 8"1). On the contrary, we saw that, if the intrinsic lifetime A--t(P~) of PT is small w.r.t, the observation time duration of tachyon PT (as it often happens in microphysics), then PT will actually appear endowed with a slower-than-light (( front velocity )>~or (~group velocity )), v . Only its (( phase velocity ~>Vv will be Superluminal: v V ~ = 1; cf. eq. (100). In subsect. 8"5 we noticed some formal analogies between these classical tachyons and the (( de Broglie particles )> met in QFT. The analogies become more strict when we analyse the appearance of a tachyonic particle endowed with an additional oscillatory movement~ for example (and for simplicity)

CLASSICAL TACIIYON8

ll5

along the motion line (GAt~IJOClO, 198~t). Let us recall t h a t the ~(see eq. (99), subsect. 8"4) : the whole wave may be possibly regarded, in a sense, as a 1) (150a) 150b)

SLT(U) = • 5 P . L T ( u ) [ 5P~3~4--il, SPEGJ

(ul[U , u = l / U ) ,

i.e. a generic SLT, corresponding to a velocity U, will be formally expressed by the product of the dual (subluminul) LT, corresponding to the velocity u]l U, with u : e~/U, by the matrix il. Due to the imaginary unit i, the SLTs satisfy eq. (15) with the minus sign. The operation Sp ~_ Sf4 -- il plays the role of (( transcendent SLT >>, since when u--~0 we get

(151)

SLT(U = ~)---- = k i l .

Incidentally, the transcendent (~transformation ~>~ is simply given b y eq. (150b), and does not affect the speed u (nsmely, does not operate a n y change fl ~ 1/fl, differently from what stated in some previous p~pers). I t is i m p o r t a n t to stress t h a t the group properties and space isotropy can be preserved only by an operator ~ which is a 4 • 4 matrix symmetrical w.r.t. all the possible axis permutations (MACCA~o~v. and l~v.cAlv~, 1982a, 1984). The expression of 5P t h a t appeared in review I w~s suited only for the simple case of eollinear boosts (and the GLTs as written in review I formed a group only for eollinear boosts). Misunderstanding this fact ~nd overlooking some recent literature (e.g, MACCAI~tCONE and I~ECAlVII~1982b), 1VIARCHILDON et al.

123

CLASSICAL T A C H Y O N S

(1983) a d o p t e d t h e expression for 5 # g i v e n in r e v i e w I also for t h e case of generic (nonco]linear) SLTs. T h e y were led, of course, to i n c o r r e c t conclusions. T h e g r o a p G of t h e generalized L o r e n t z ( ( t r a n s f o r m a t i o n s ~ (GLT), bo~h sub- a n d S u p e r - l u m i n a l , will be

(152)

{

6 :

s174

z(4),

z ( ~ ) =- { f f ~

1) ~ + z , - ~, + i, - i;

thi:s is analogo~ls to w h a t seen in subsect. 5"6, b u t n o w o~fr is t h e ]our-dimensional p r o p e r o r t h o c l i r o n o a s L o r e n t z group. A g a i n we h a v e t h a t , if G ~ 6 , t h e n (VG ~ G) also -- G E G a n d SfG ~ G; ci. eqs. (37). I n p a r t i c u l a r , given u c e r t a i n L T ~ JL(u) a n d t h e S L T : + i Z ( u ) , one h a s [iL(u)] [iL-~(u)] ~ [iL(u)]9ILL(-- u)] ~ -- 1, while, on t h e c o n t r a r y , it is [iL(u)] [-- iZ-~(u)] ~ [iL(u)]. 9[-- i Z ( ~ u)] ~ + 1 ; this shows t h a t (153)

[iL(u)] -~ = -- iL-~(u) = -- i L ( - - u ) .

T h e g r o u p 6 is n o n c o m p a c t , n o n c o n n e c t e d a n d w i t h discontinuities on t h e l i g h t - c o n e ; its c e n t r a l elements, m o r e o v e r , are C = ( + 1, -- 1, + i l , - il). L e t us recall f r o m subsect. 11"1 t h a t -- 1 ~ P T = C P T ~ G, a n d t h a t G = #(~r CPT, 5P). See also subsect. 11"3. Of course, also d e t ~9~ = + 1, 5P~5P = -- 1 a n d ~= ~ E 6 (cf. eq. (150b)). I n t h e p a r t i c u l a r case of a b o o s t along x, our S L T s , eqs. (150), c a n be ]ormally w r i t t e n ( U = 1/u) (see MACCAI~O~]~ a n d l~gCh~I, 1984; MACOAm~O~ et al., 1983, a n d references therein) dt -- u dx

(154)

dx' =

dx -- u dt :~ i V- Y -- - u ~

dy' =

:~: i d y ,

dz ~ ~

dx -- U dt

dt -- U dx

: F iV ~- - ~-

'

~: i dz

( S u p e r l u m i n a l case, u ~ < 1, U s > 1, u ~ 1 / U ) , w h e r e we t o o k a d v a n t a g e of t h e i m p o r t a n t identities (41): see sub sect. 5"6. N o t i c e t h a t u n d e r (( t r a n s f o r m a t i o n s ~>(154) for t h e r o a r - v e l o c i t y (snbsect. 7"2) it h a p p e n s ! ! that u,u, =u , u , ; eqs. (151) are, t h e r e f o r e , associated w i t h S u p e r l u m i n a l m o t i o n s , a:s we shall see b e t t e r below. One s h o u l d n o t confuse in t h e following t h e b o o s t speeds u, U w i t h t h e f o u r - v e l o c i t y c o m p o n e n t s u , of t h e considered object. L e t Its u n d e r l i n e t h a t ol~r (~f o r m a l ~ S L T s , eqs. (154), do f o r m a g r o u p , G, t o g e t h e r w i t h t h e o r d i n a r y ( o r t h o c h r o n o n s a n d a n t i e h r o n o u s ) LTs. I t s h o u l d be n o t i c e d t h a t t h e generalized L o r e n t z > i n t r o d u c e o n l y

124

~. n~cxm

real or purely imaginary quantities, with exclusion of (generic) complex quantities. L e t us moreover stress t h a t the transcendent transformation 5 ~ does not depeud at this stage on a n y spatial direction, analogously to the transformation LT(u = 0) = 1. This accords with the known fact (subsect. 3"2) t h a t the infinite speed plays for Ts a role similar to the one of the null speed for Bs; more generally, the dual correspondence (subsect. 5"11)

holds also in four dimensions. (See also the beginning of sect. 14.) 14"3. Preliminary expression o] GLTs in ]our dimensions. - Subsections 5"8 and 5"9 can be e x t e n d e d to four dimensions (see MACCA~RO~n et al., 1983). F i r s t of all (156)

6 = ~|

~*+, !

where ~

is the discrete group of the dilations D: x, = 0x, with 0 = • 1, • i. Then, b y using the formalism of subsect. 5"8, we can end up with eqs. (45)~ valid now also in /our dimensions. I n terms of the light-cone co-ordinates (46) and of the discrete scale p a r a m eter 9, the GLTs in the case of generalized boosts along x can be written

(~57)

{

d~':

9~

9adS,

•177

d $ ' = 9a-~d$, 0 and worlds, and the tachyon rest mass is real; he succeeded, e.g., in producing the (~tadpoles ,> dynamically (without supposing a nonzero vacuum expectation value of the fields). Passing from the slow to the fast worlds, however, means interchanging time with space. And in four dimensions, which space axis has the time axis to be intercharged with? The approach mainly followed by us is equivalent to answer: (~with all the three space axes )~, so to get transformations preserving the quadratic form, except for its sign (see eq. (15), subsect. 14"2) ; afterwards, one has to tackle the appearance of imaginary transverse components. In order to avoid such difficulty, GOLDONIintroduced a different metric signature for each observed tachyon, ending up with the four independent space-time

128

r. n~cA~

metric signatures ( - - - - - - + ) , ( A - - - - - - - ) , (---[-----), (----~---). It follows t h a t tachyons are not observable in Goldoni's approach, except for the fact t h a t t h e y exchange with bradyous (only) internal q u a n t u m numbers. Some consequences for Q F T m a y be appealing; b u t we deem t h a t this approach t o o - - a t the relativistic level avoids facing the real problem b y a (( trick )). ~evertheless, r a t h e r valuable seem the considerations developed b y GOLD0.'~I (1975a, b, c) in general relativity. 14"5. A simple application. - L e t us go back to subscct. 14"4 and apply it to find out, e.g., how a four-dimensional (space-time) (( sphere ~)t 2 + x ~ ~- y2 ~_ z ~ : A ~, t h a t is to say (161)

89

-t- 89 ~_ y~ + z2 = A 2 '

deforms u n d e r Lorentz transformations. I n the ordinary subluminal case (eqs. (157) with @= + 1), eq. (161) in terms of the new (primed) co-ordinates can be r e w r i t t e n as (0 < a < ~- cr (162a)

89

'2 + 89

'2 -4- y,2 _~ z,2 = A ~

(subluminal case),

which in the new frame is a four-dimensional ellipsoid. I n the ease of a Superluminal boost (eqs. (158 bis)), eq. (161) becomes, in terms of the new primed co-ordinates (0 < a < + oo), (162b)

89

~_ 89 2~,~_ y , 2 _ z,2._ A 2

(Superluminal case),

which in t h e new frame is now a four-dimensional hyperboloid. Notice explicitly, however, t h a t the present operation of transforming u n d e r GLTs a four-dimensional set of events has nothing to do with what one ordinarily performs (in fact, one usually considers a world-tube and t h e n cuts it with different three-dimensional hyperplanes). 14"6. A n s w e r to the (( Einstein problem ~ o] subsect. 3"2. - W e h a v e still the task of i n t e r p r e t i n g physically the SLTs as given b y eqs. (]50), (154). Before going on, however, we wish to answer preliminarily the ((Einstein problem ~)mentioned in subsect. 3"2 (cf. eq. (12)). W e have seen in subsect. 5"6, and later on in connection with cqs. (154 bis) (subscct. 14'3), t h a t eq. (12) is not correct, coming from an uncritical extension of LTs to the Superluminal case. L e t us consider an object with its centre at the space origin 0 of its rest frame; be it intrinsically spherical or, moro generally, let it have the intrinsic sizes Axo----2xo ~ 2r, Ayo ~- 2yo and AZo -~ 2zo along the three space axes, respectively. Instead of eq. (12), for the size along the boost motion line x the t h e o r y of E R yields the

129

CLASSICAL TACII YO_N'S

real expression (Xo - - r ) (163a)

A x ' ~ : Axo % / ~ 2

1

(U-~ 1).

~ o problems arise,, therefore, for the object size along the x-axis. We m e e t problems, however, for the transverse sizes, which become imagillaries ~ccording to eqs. (154 his): (163b)

A y ' --- l A y o ,

Az' = iAzo .

B u t let us go back to subsect. 8"2 and fig. 19. If the considere4 object P -- P~3 is ellipsoidal in its rest frame, then, when SuperluminM, I~ ~_ 1)v will appear to be spread over the whole space confined bctwc(m the dollble indefinite corm ~ : y2/y2o + z2/z~ ~ ( U t - - xp/x~(U'-'-- 1), and the two-sheeted hyperboloid ~ f : yO./y~ -~ z2/z~ ( U t - x ) 2 / x ~ ( U " - - 1 ) - ~! (cf. lCECAYJI and MACCAI~RONE, 1980). See fig. 17. The distance 2x'o between the two vertices Vt and V2 of d/f, which yields the linear size of PT along x, is 2x'o ~ 2xo X / ~ / 2 - 1. F o r instance, for t ~ 0, the position of the two vertices of J f is given b y V~.~ -= ~ xo v / U 2 - 1. This, i~lcidcnt~lly, clarifies the meaning of eq. (163a). L e t us now tm'n our a t t e n t i o n to the transverse sizes. Tile quantities Y0 and zo correspond to the intersections of the initial ellipsoid with the iuitiM axes y and z, respectively (for t ~ 0). We have t h e n to look in the tachyonic case for the intersections of . ~ with the transverse axes y and z. Since these intersections are not real, we shall formally get, still for t = 0, Y'o z

iyo ,

zo --- izo ,

which do explain the meaning of eqs. (163b). I n fact (see fig. 38), the real quantities y~/i = Yo and z~/i = z o have still the clear, simple meaning of semiaxes of J r . In other words, the quantities lY:[---- Y'o/i and [z:l-~ z:/i jast tell

Iy:/i

~) Fig. 38.

Ct=0)

b)

130

~. ~ C A ~

us the shape of the t a c h y o n relevant surface (they express the transverse size of the (~f u n d a m e n t a l rectangles )>, i.e. allow us to find out the f u n d a m e n t a l a s y m p t o t e s of PT). See I~ECA_M~ and MACCAI~0~E (1980); see also Com~E~(1975), GZADKICK (1978a, b), TE~LETSKu (1978), GOTT I I I (1974a, b) and FLE~-~Y et al. (1973). E v e n if in a particular case only, we have practically shown how to interpret also the last two equations in (3.54 bis). We shall come back to this point; b u t let us add here the following. E q u a t i o n s (154 bis) seem to t r a n s f o r m each ellipsoidal (or spherical) surface 5~ into a two-sheeted hyperboloid ~ . L e t us now consider the intersections of a n y surface 5 ~ (see fig. 39a)) and of the corresponding 5f~ (fig. 39b)) with all the possible transverse planes x -- 5. I n fig. 39, Y I I I

I r I

I I T

I

I 1

I I

I I

i

i

i

2z R3~' R~R~ i I I I I I

Z

' I I

I I I I I I I I I

a)

,y

(t = O)

/ I

J I I

I

x

R31 V1 [

/

I

J

Z

I I I I

b)

Fig. 39. for simplicity, the case of a Superluminal boost along x with speed V -~ c V/{ and t ---- 0 is considered; so t h a t OV 1 ---- O1~ = xo -- r and all quantities ()1~ have the same value b o t h in a) and in b). I t is immediate to realize t h a t , when the intersections of ~1 with the plane x -- 5 are real, t h e n the corresponding intersections of 5z~ are imaginary (with the same magnitude), and vice versa. l~amely, in the particular case considered, the intersections of 5f~ are real for 141 < r and imaginary for 151 > r, while the intersections of 5f~ are on the contrary imaginary for ]51 < r and real for ]xl > r. I t is easy to u n d e r s t a n d t h a t eqs. (].54 bis) operate in the planes (x, y) and (x, z) a mapping of ellipses # into hyperboles h, in such a way t h a t the real part of # goes into the imaginary part of h, and vice versa (see CALDn~OLA et al., 1980). Cf. also fig. 37. 14"7. A n auxiliary six-dimensional space-time ]5(3, 3). - Equations (150), as well as (154), call imaginary quantities into play and, therefore, seem to require

CLASSICAL

TACI~YONS

13i

an 8-dimensional space C A (i.e. a 4-dimensional complex space-time) us the kinematical background. However, an essential teaching of SI~ appears to be that the four-position is given by one real and three imaginary co-ordinates--or vice versa---so that formally (with c--= 1) time----ixspace. As noticed b y MINKowsxI (1908) himself, one might formally write ls ---- i • (3 X 10 s) m. As a consequence, to interpret the SLTs it can be enough to assume (temporarily, at least) a 6-dimensional space-time M(3, 3) as background; this was first suggested in MIGNA~I and t~ECAMI (1976b). Ever since, much work has been done on such spaces, with or without direct connection with the SLTs: see, e.g., DATTOLI and XV[IG~ANI (1978), VY~N (1978), PA}'PAS (1978, 1979, 1982), ZK~o (1979, 1983), STlC~AD (1978, 1979a, b, 1980), PAV~I~ (1981a, b), JOH~S0N (1981), FRO~I~G (1981), LEWIS (1981), PArrY (1982), CON]~0~TO (1984) and particularly CoLE (1978, 1979, 1980a, b, c, d, e); see also TO~TI (1976), JA~CEWlCZ (1980) and MACCAI~0NE and RECA~ (1982b). The idea of a possible multidimensional time, of course, was older (see, e.g., Bu~G~, 1959; Do~LI~G, 1970; KALITZ~, 1975; DEME~S, 1975). Alternative approaches, that can be promising also w.r.t, tachyon theory, m a y be the ones which start from a complexification of space-time, via the introduction ab initio either of complex numbers (GI~EGOtr1651961, 1962 ; SUDA~SHAN, 1963; review I; YACC~INT, 1974; 1V~GNA~r and I~ECA~, 1974v; COLE, 1977; K.~LNAY,1978; MOSKALENKO ~nd MOSXALENKO, 1978 i see ~lso ~OSEN, 1962; DAS, 1966; S~N, 1966; KXL~AY and TOLEDO,1967; ~BALDOand ~ECAMI, 1969; t~ECAM~, 1970; OLXHOVSKY and t~ECAY~, 1970c; HA~SEN and NEW~AN, 1975; HEsrENES, 1975; PLEBA~SKIand SCH/_LD,1976; CHARON,1977; IMAEDA, 1979, and SAcgs, 1982), or of octonions (see, e.g., CASALBUO~-s1978), or of twistors (see, e.g., PEN~OSE and MCCALLVXVf,1973; HA~SEN and IqEW~AN, 1975) and quaternions (see, e.g., EDD/[0NDS,1972, 1977a, 1978; WEI~GAtCTEN,1973; MXGXNANI, 1975, 1978; IMAEDA,1979). The most promising approaches are probably the last one and those making recourse to more general Clifford algebras: see the end of subsect. 13"8 and the beginning of this sect. 14 (and Sov~E:~, 1981). Let us mention, incidentally, that transformations in Ca-space ~re related to the group SU~ of (unitary) intrinsic symmetries of elementary particles. It is not without meaning, possibly, that the M(3, 3) formalism has been used to express the law of (( trichromatism ~) (DE~E~S, 1975). Let us coufine ourselves to boosts along x. W e are left with the problem of discussing the formal equations (154). Let us consider ( ~ A c C ~ 0 ~ E and I~ECA~, 1984) the GLTs, eqs. (152), as defined in M~ ~ M(3, 3) -- (x, y, z, t~, t~, t~); any observer s in M~ is free to rotate the triad (t} = (t~, G, t~) provided that {t} 2 {x} - - (x, y, z). In particular, the initial observer so can always choose the axes t~, t~, t~ in such a w a y that, under a transcendent Zorentz trans/ormation (without rotations: "~DLLEtr 1962, p. 18-22, 45-46) 5~ ~ S f ~ , it is x--->t~, y-->t~, z--->t~, t~-->x, t~--->y,

132

~. ~ c x ~ z

L -+ z, in agreement with the now 1 is the six-dimensional Moreover, if observer so, along x~, rotates {t} so t h a t sect. 4 and the following), t h e n to operate as in fig. 40b).

fact t h a t the ]ormal expression of ~ = il (where identity) is fildependeut of a n y space direction. when aiming to perform a Superluminal boost tj -- t (axis t being his ordinary time axis: see any transcendent boosts can be formally described

f

J

+ ~tz=;z

J E(3)

E(3)

/

a)

b)

Fig. 40. W h a t above means t h a t the imaginary unit i can be regarded as a 90 ~ r o t a t i o n operator also in Me; from the active point of view, e.g., it carries x -- (x, y, z) ~- (t~, t~, tz) -- t. H e r e t h e meaning of i, for one and the same observer, is analogous to its meaning in SlY, where it is used to distinguish t h e time from the space co-ordinates, which are orthogonal to time. Therefore, (164)

i ----exp [i~/2]

~ tt ~-~- xx

(two-dimensional case), (six-dimensional case).

~ o t i c e t h a t in ms the GLTs are actually (linear) $rans]ormations, and not only mappings. W h a t precedes (see r e.g., eq. (164)) implies t h a t (165a)

ds~~ ---- + ds~

for L T , ,

(165b)

ds.'2 . . . .

for SLT6,

ds~

with obvious meaning of the symbols. The GLTs, as always, can be considered either from the active or from the passive point of view (in the latter case,

CLASSICAL TACHYO~S

133

t h e y will keep the 6-vector fixed and (( r o t a t e ~ on the c o n t r a r y the six axes without c h a n g i n g - - n o t i c e - - t h e i r names during the (~rotation O. The subluminal LTs in Yle, to be reducible in four dimensions to the ordin a r y ones in agreement with SR (ds: '~ = ~- ds~), must be confined to those t h a t call into p l a y one t i m e axis, let it be t ~ t~, while t~ and t3 remain unchanged (or change in 1~6 only in such a way t h a t at'22 ~- dt'a2 = dt~ + d 0 . As a consequence, because of eqs. (150), also the SLTs in M8 must comply with some constraints (gee MXCCAI~I~ONE and I~EcAsr[, 1984). F o r instanc% when the boost speed U tends to infinity, the axis t' ---t'1 tends to coincide with the ! boost axis x~, and the axis x~ with the axis t, -- t. As to the signature in ~ e two alternative conventions are available. The first one is this : we can paint in blue (red) the axes called tj (xj) b y the initial observer so, and state t h a t the blue (red) co-ordinate squares m u s t always be t a k e n as positive (negative) for all observers, even when they are (~r o t a t e d )) so as to span the region initially spanned b y the opposite-colour axes. Under such a convention, a t r a n s c e n d e n t SLT acts as follows:

I+! 4+

dt~ --> dr,' :-= dz 7 dt~ -+ dt~ = dy I dt~ --> dt~' = dx

(166)

dx --> d x ' ~ dt~

J

(under ~ ) .

dy -+ d y ' = d 6

~-I

dz -~ d z ' -

dt~

t-

Notice t h a t no imaginary units enter eqs. (166). The previous discussion on the action of o~' in Me was p e r f o r m e d with such a metric choice. The second possible convention (still w i t h o u t changing t h e n a m e s - - l e t us repeat----of the axes tj, x~- during theh" ~,rotation ,~) would consist in adopting the opposite six-dimensional metric in the r.h.s, of eqs. (166); it corresponds to changing the (, axis signatures ~ during their rotation:

/§

d6 -> i dt.' --- i dz 7 dry -~ i dt~ = i dy

-i(167)

[

dt~ ~ i dt~' ---- i dx

_ (under 3~).

dx --> i dx' ---- i dt~ dy - > i dy' = i dt~

dz --~ i dz' ---- i dtz

§ §

Such a second convention implies the appearance of imaginary units (merely due, however, to the change of metric w.r.t, cqs. (166)).

134

~.

~XCAMI

I n a n y case, the axes called tj b y the subluminal observer So, and considered b y So as subtending a three-temporal space (t~, t~, tz) L (x, y, z), are regarded b y the Superluminal observer S'~, and b y a n y other S', as spatiM axes subtending a three-spatial space; and vice versa. According to our second postulate (sect. 4) we have now to assume t h a t so has access only to a 4-dimcusionM slice M~ of Ms, W h e n so describes bradyous B, we h a v e to assume M4 --- (tl ---- t; x, y, z), so t h a t the co-ordinates t~, t3 of a n y B are not observable for so. W i t h regard r SLTs we must, e.g., specify, from the passive point of view, which is the (( observability slice )) MI of M~ accessible to S r when he describes his own bradyons. B y checking, e.g., eqs. (166) we reMize t h a t only two choices are possible: either i) M~4----(t~; x', y', z'),

or ii)

(t:,

t:, x').

The first choice means

ass

ing

that each

while r o t a t i n g carries with itself the p r o p e r t y of being observable or unobservable, so t h a t the axes observable for S' are the t.ransforms of the axes observable for So. The second choice, on the contrary, means assuming the observability (or unobservabilit.y) of each axis to be established b y its position ill M6 (as judged b y one and the same observer), so t h a t two of the axes (i.e. t~, t:) observable for S' are the transforms of two axes (i.e. t~, t/) unobservable for So. I n other words, the first choice is M'4 • Ms, while the second choice is M~ ~- ~ a (in 518, when it is referred to one and the same observer). ~Totice t h a t , roughly speaking, the above properties of the two choices get reversed when passing to the active point of view. The ]irst choice does not lead automaticMly, from eqs. (165) in six dimensions, to the ds~ invarianee (except for the sign) in four dimensions. I t moreover calls all six co-ordinates into play~ even in the case of sublnminal LTs obtained through suitable chains of SLTs and LTs. This choice, therefore, could be adopted only when whishing to build np a ~ruly six-dimensional theory. The resulting t h e o r y would predict the existence in M~ of a (( t a c h y o n corridor ~) ~nd would violate the light speed invariance in M:: in such a seuse it would be similar to Antippa's (1975). The second choice, once assumed in M6 t h a t ds~~ ---- -- ds~ for SLTs, does lead alltomatically also to ds~' ---- -- ds~ in four dimensions (M_acc~a~o:vE and ICEc~[r, 1984). Moreover, it calls actually into play four co-ordinates only, in the sense t h a t (of., e.g., eqs. (166)) it is enough to kimw initially the co-ordinates (t; x, y, z) in Ma in order to know finally the co-ordinates (t:, t', t',; x') in ~-~I~. We a d o p t the second choice since we w a n t to t r y to go back from six to four dimensious, and since we like to have the light spced invariance preserved in four dimensions even u n d e r SLTs. The (( square brackets )~ appearing in eqs. (166), (167) just refer to such a choice. To go on, let us s t a r t b y adopting also the signature--first c o n v e n t i o n ~ associated with eqs. (166). I f we consider in M~ a (tangent) 6-vector dv lying on the slice Md(t, ~ t; x, y, z), t h e n a S L T - - r e g a r d e d from the active point of view--will ((rotate ~ dv into a vector dr' lying on the slice M'4(t,, t,, t~; x):

CLASSICALTAC~YO.~'S

135

see fig. 41. In other words, any S L T - - a s given by eqs. (150), (154)--leads from a bradyon B with observable co-ordinates in M(1, 3) -- (t; x, y, z) to a final t a c h y o n T with ~(observable )~co-ordinates in M'(3, 1) -- (t~, t2, t3; w), where the w-axis belongs to E(3) -- (x, y, z) and the t-axis belongs to E'(3) -_---(t~, t2, t3): see fig. 40a). F o r m a l l y : (1, 3) SLT (3, ] ). F r o m t,he p a s s i v e point of view, the initial

/

l;

f

!

t~zmZt

/

Fig. 41.

observer so has access, e.g., only *,o the slice (t= ~_ t; x, y, z), while the final observer S' (e.g., S'~) has access only to the slice (t~, t~, t =, " x'), so t h a t the coordinates t~, t, (and y', z') are not observable (see also POOLE et a l , 1980, and SoBczYK, ]983). Notice t h a t x' comes from the (~rotation~> of the boost axis. At this point two observations are in order: 1) Our second postulate (sect. 4 and subsect. 14"2) requires observer S' to regard his space-time (t', t~, t~'; x') as related to three space axes and one time axis; actually renaming them, e.g., ~i, ~'2, ~ and z', respectively. This consideration is the core of our interpretation, i.e. the basis ~or understanding how S' sees the tachyons T in his M't. 2) The principle of relativity (sect. 4) requires t h a t also so describe his tachyons (in M4) just as S' describes his tachyons (in M~); and vice versa. If we u n d e r s t a n d how S' sees his tachyo~m in M:, we can immediately go back to the initial M(1, 3) and forget about six dimensions. In connection with M:, the effect of a Snperluminal boosts along x will be

136

~,:. ~ c A m

the following:

dx -~ d x ' =

dx -- U dt ~: ~ / U 2 _ 1

dt - - U d x dt~ -.~- dt" = T - - -

x/U~- ]

(]68)

(U~l/u,

u2). It will seem to contain complex quantities only in its (partially) reinterpreted ]orm. B u t this is a (( local ~>f a c t relative to t h e ]inal frame, and due to a trivial effect of t h e r e l e v a n t space rotations: its i n t e r p r e t a t i o n is p a r t l y related to fig. 42 (in t h e following). L e t us also recall t h a t in the case of a chain of GLTs the i n t e r p r e t a t i o n procedure is to be applied only at the end o] the chain (the r e i n t e r p r e t a t i o n , being f r a m e dependent, breaks the Lorentz invarian~.e). W e h a v e just to c o m p a r e t h e m a t r i x in eq. (172) with t h e m a t r i x in eq. ([69), including in it its i m a g i n a r y coeffmient, in order to get t h e i n t e r p r e t a t i o n of eqs. (169). Such a r e i n t e r p r e t a t i o n will proceed, as usual, in two steps; t h e fix'st consisting now in the i n t e r p r e t a t i o n of t h e t i m e co-ordinate a n d of the space co-ordinate along 5; the second one consisting in the i n t e r p r e t a t i o n of the imagina r y space co-ordinates transverse to 1. F o r instance, let us c o m p a r e eq. (169) with eq. (172b), a p a r t f r o m their double signs:

[ dt .... i y d t 4- iyunsdx*,

(169)

l

dx '~ := -- iuyn" dt 4- i5~ dx" -- i(y -- 1) n ~n, dx*; dt' = - - u ? d t - -

?n, dx*,

d x ' ~ - - $n'dt 4- i~:dx'4- (u 7 4- i)n'n~dx'.

(172b)

J~irst step. Recipe: You can eliminate the imaginary unit in all the addenda containing ~ as a multiplier, provided that you substitute t ]or r, and r, /or t (notice t h a t rj, = =

/''n

---- --

nsXS).

gecond step. I n the second equations in (169) and (172b) if we p u t r = x (xyz) a n d r' -- x' -- (x'y'z'), we can write r = r, 4- r• where r, -- (r,)n and r• = r - r~, n---- r - ( r . n ) n . Then eq. (172b) can be w r i t t e n in integral f o r m as r ! = r,! , r•/ 9r d- ir• a n d - - a f t e r h a v i n g applied t h e ! (~first step ~>r e c i p e - - w e are left only with r x = i r . , i.e. only with the i m a g i n a r y t e r m s (not containing ? as a multiplier) (1.73)

d(r~)' = id(r•

----i(5~ 4- n~n~)dx s ,

which e n t e r only t h e expression dx 'r. (Of course, rz is a space vector orthogonal to I and, therefore, corresponds to t w o f u r t h e r co-ordinates only). Since eqs. (173) refer to the space co-ordinates orthogonal to the boost direction, t h e i r i m a g i n a r y (( signs ~)h a v e to be i n t e r p r e t e d so as wc did (fig. 19) in subscct. 14"6 (and 14"9) for t h e t r a n s v e r s e co-ordinates y' and z' in the case of Superluminal x-boosts: see fig. 42.

CLASSICAL

TACIIYONS

Y

141

~')'/ x

Fig. 42.

This means t h a t , if the considered SLT is applied to a body P~ initially at rest (e.g., spherical in it,s rest frame), we shall finally obtMn a b o d y P'r moving along the mot,ion line 1 with Superluminal speed V : U, such a b o d y P'r being no longer spherical or ellipsoidal in shape, b u t appearing on the c o n t r a r y as confined between a t'wo-sheeted hyperboloid and a double cone, b o t h having as s y m m e t r y axis the boost motion line 1. Figure 42 refers to the case in which P~ is intrfllsicMly sphericM; and the double-cone semi-~ngle ~ is given b y tg ~-----= (V 2 - 1) -89 More in generM, the axis of the t a c h y o n shape will not coincide with 1 (but will depend on the t a c h y o n speed V = U). / More precisely, the vector r• apart from its imaginary (~sign )),--i.e. the vector r • be described iby the two co-ordinates ,~-(')~ Yo, r~(~),= Z o just as in subsect. 14"6 and 14"9: see fig. 38 and 42. We see once more t h a t this reinterpretation ((second step ~) works only in particular special cases. To clarify a bit more the present., situation, MACO~RO~E et al. (1983) emphasized the following points: i) one is not supposed to consider (and reinterpret) the GLTs when t h e y are applied just to a ((vacuum point ~); actually, we know f~'om St~ t h a t each observer has a right to consider the v a c u u m as at rest w.r.t, himself; ii) one should t h e n a p p l y - - a n d eventually reinterprel~--the GLTs, in part,icular the SLTs, only to transform the spacetime regions associated with physical objects; these are considered as extended objects (Ks 1978), the pointlike situation being regarded only as a limiting case; iii) the e x t e n d e d - t y p e object is referred to a frame with the space origin in its centre of s y m m e t r y . Many problems remMn still open, therefore, in connection with such a (~ second step ~> of the interpretation (cf. subsect. 14"]4-14"16). 14"11. Preliminaries on the velocity composition problem. Let us apply a SLT in the form (172a) along the generic motion line 1 with SuperluminM -

142

~. s ~ e A ~ x

speed U --= 1 / u (U ~ > 1, u ~ < 1) to a b r a d y o n P~ having initial fom'-velocity u , and velocity v. Again, one should p a y a t t e n t i o n to not confuse t h e boost speeds u, U with the four-velocity components u*, of PB- F o r the purpose of generality, v and U should not be parallel. W e get { u '~ ~

(174)

l

-- f(u ~

u" ~- -

U%),

p(u~ -

Uu~

~§

iu~ ,

where u R - ~ - - u ~ n , , u .r ~ U r + u ~ n ~ u ~ - ~ U r - - uj~n ~, and n is still the unit / vector along l, while y ---- (U ~ - 1) - i so as in eq. (172a). ~ o t i e e t h a t u o is real, l I$ while the second equation in (174) can be rewritten as (u N= u n~) J u~ = - f ( u ,

(175)

,

t

- tTuO).- - ( u , , -

I

~uO)/Vu ~ - 1

I

( U S > 1), I

/

where u u is real too and only u• is purely imaginary; u,, u,, (u j , u• are the longitudinal (transverse) components w.r.t, the boost direction. If we define the 3-velocity V' for tachyons in terms of the 4-velocity u~ (j----1,2,3): V 'J (176)

1

u'J--V=--;=---"~ :l

u'~

V'-V . . . '~. ~. 1

'

eqs. {175) yield ,

U - - v,

1 - - uv~

(U~

(177)

1, u 2 ~ 1, U - ~ I / u ) .

v', = i v~ V u ~ - 1 _ iv• V i - u~ Uvn - - I v H- - u

I t m a y be noticed t h a t V,' = 1/~ u, V•' = i~l[~,~, where ~ is tile t r a n s f o r m of v u n d e r the d u a l (subluminal) Lorentz transformation L ( u ) with u = 1 ] U , / I u I]U. Again V~ is real and V~ pure imaginary. H o w e v e r , V '~ is a l w a y s positive so t h a t IV'l is real and even more Superluminal; in fact,

(17s)

v '~ z

v',~+ v ? = v,~/ 2 - I v l t ~ >

1.

More in general, eqs. (177) yield for the magnitudes (179)

1-

V '~ - - - ( 1 - - v ~

U 2)

(1-- U.v) ~

(v2~l,

U2,V'2~1),

which, incidentally, is a G-eovariant relation. L e t us recall t h a t eqs. (174), (175) and (177) have been derived from the (partially) reinterpreted form of

CLaSSiCaL TAC~u

143

the SLTs ; therefore, t h e y do not possess group-theoretical properties a n y longer. For instance, eqs. (177) cannot be applied when transforming (under a SLT) a speed initially Superlnminal. Eqnation (179) shows t h a t under a SLT a bradyonic speed v goes into a tachyonic speed V'. B u t we have still to discuss the fact t h a t the t a c h y o n 3-velocity components transverse to the SLT motion line are imaginary (see the second equation in (177)). We shall proceed in analogy with subsect. 14"6 and 14"10. L e t us initially consider, in its c.m. frame, a spherical object with centre at O, whose external surface expands in time for t > 0 (0),

where R and v are fixed quantities. In Lorentz-invariant form (ior the subluminal observers), the equation of the (~bomb ~) world-cone is (MACC~I~I~O~E et al., 1983) (180')

! (x~ ~- bz)(x" -~ b') < [(x~ -~u~ub')~uz] 2< ( 1 - - v 2 ) -~ (x~ ~- b~)(x" -~ b~),

I x~>0, where x, ~ (t, x, y, z) is the generic event inside the (truncated) world-cone, vector ug is the ((bomb)~ centre-of-mass four-velocity and b~ ~ u , R / v . One can pass to Superlnminal observers S' just recalling t h a t (subsect. 8"2) the SLTs invert the quadratic-form sign (eL, however, also subsect. 8"3). I f S' just moves ~long the x-axis with Snperlnminal speed -- U, the first limiting equality in eq. (180') transforms, as usual, i'nto the equation of a double cone symmetrical w.r.t, the x-sxis and travelling with speed V ~- U along the axis x ~ x'. The second inequality in eq. (180') transforms on the contrary into the equation (181)

(1-- v ~ V~)x ' 2 - ( V ~ - l ) ( y '~ ~-z ' 2 ) - 2 t ' V ( 1 1~ v : < 1). V 2') 3 =

' --:-

ivy,3

•

i

%'~ V1 -- v -~

14"14. I s linearity strictly necessary ? - We might have expected t h a t tra.ns/ formations Y : x, -> x~ mapping points of M 4 into points of M4 (in such a way t h a t ds ~ - + - ds ~) did not exist. Otherwise real linear S~Ts: dx, -+ dx~ of the t a n g e n t vector space associated with the original ma.nifold map f should h.~ve existed (I~Im)~,m, 1966; S_m~z, 1984). B u t we saw, already at the end os subseet. 3"2, t h a t real linear SLTs (meeting the requirements ii)-iv) of subsect. 4"2) do not exist in four dimensions. Oil the contrary, the results in subsect. 8"2, as well as in subsect. 14"6 and 14"11, seem to show t h a t in the Supcrluminal case in M 4 we have to deal with

CLASSmAL TAC~YONS

149

mappings t h a t transform manifolds into manifolds (e.g., points into cones). In subscct. 8"3 we inferred the SLTs: dx, --, d x, ' to be linear b~tt not real; just as we found in the present sect. 14. We may, h o w e v e r ~ a n d perha.ps more soundly--, ma.ke recourse to nonlinear (but real) SLTs. ' real but not linear~ then S u p c r h m i n a l If we consider SLTs: d x - ~ d x~, maps 3-': M~ --~ .'~I~ (carrying points into points) do not exists. We '~h'eady realized this. The i m p o r t a n t point, in this case, is t h a t the (( Supcrluminal mappings >)J- (transforming then manifolds idle manifolds) be compatible with the postulates of SI~ ; in particular (subsect. 4"2), i) trans]orm inertial motion into inertial motion, ii) preserve space isotropy (and homogeneity), iii) preserve the light speed invariance. To meet the group-theoretical requirements, we have to stick to eqs. (154) and to their integral form. But their reinterpretation--accomplished in this sect. 14 and anticip~tcd in sect. 8---does comply with conditions i)-iii) above. For example., it leads from a pointlike b r a d y o n moving with constant velocity to a t a c h y o n spatially extended, b u t still travelling with constant velocity. The problem is now to look for real, nonlinear SLTs (i.e. mappings of the tangent vevtor space), and substitute them for the linear nonreal eqs. (154 bis); with the hope t h a t the new (nonlinear) SLTs can yield more rigorously the same results m e t before, thus solving the problems lef~ open by the previously in interaction processes-any experimental project going beyond simple kinematical evaluations ought to take account (Co,BEN, 1975) also Of the drastic ~ deformations> caused by the huge velocity of the observed objects w.r.t, us: see, e.g., the results on the tachyon shape presented in subseet. 8"2 and 14"6. As noticed by BA~um (1978), one may wonder if we have really correctly looked for taehyons so far. Within the classical theory of taehyons, it would be important to evaluate how charged taehyons would eleetromagnetieally interact with ordinary matter: for instance, with an electron. The ealcul~tions can be made on the basis of the generalized Maxwell equations, either in Corben's form or in Mignani and l~ecami's (subsec~. 15"1). I t we take seriously, however, sect. 8 on the shape of taehyons, we have to remember t hat a pointlike charge will appear--when Superluminal--to be spread over a double cone ~; it would be nice (see subsect. 10"3) first to know the L-function of the space-time co-ordinates yielding the distribution of the taehyon charge density over c~. Let us recall, moreover, that y-y interactions should a priori produce both couples of bradyons and couples of taehyons. At last, for the possible role of taehyons in astrophysics, see sect. 12.

16.

-

Conclusions.

In this paper, after having shown that BE can be adjusted to describe both particles and antiparticles, we have presented a review of taehyons with particular attention to their classical theory. We first exploited the extension of Sl~ to taehyons in two dimensions:

160

~. R~CA~I

an elegant model theory which allows a better lmderst~nding also of ordinary physics. Then we passed to the four-dimensional results (particularly on tachyon mechanics) which can be derived without assuming the existence of Superluminal reference frames. We discussed, moreover, the localizability and unexpected apparent shape of the t~chyonic objects, and carefully showed---on the basis just of taehyon mechanics--how to solve the common causal paradoxes. In connection with GE, particularly the problem of the apparent Superluminal expansions observed in astrophysics has been reviewed. Later on we examined the important issue of the possible role of tachyons in elementaryparticle physics and in quantum mechanics. At least we tackled the (still open) problem of the extension of the relativistic theories to tachyons in four dimensions, and reviewed the electromagnetic theory of tachyons: a topic that can be relevant also to the experimental side. A few conclusions may be the following. Most tachyon classical physics can be obtained without resorting to Superluminal observers; and in such a classical physics extended to tachyons the ordinary causal problems can be solved. The elegant reslflts of EIr in two dimensions, however, prompt us to look for its multidimensional extensions (i.e. to try understanding the meaning and the possible physical relevance of all the related problems: sect. 14). Tachyons m~y, for instance, have a role as objects exchanged between elementary particles, or betwecn black-holes (if the latter exist). They can also be classically emitted by a black-hole and have, therefore, a possible role in astrophysics. For future research, it looks~ however, even more interesting to exploit the possibility of reproducing quantum mcehalfics at the classical level by means of tachyons. In this respect even the appearance of imaginary quantities in the theories of tachyons can be a releva.nt fact, to be furthcr studied.

*** The author thanks, for encouragement, A. BA~UT, P. CALDIROLA,A. GIGLI~ M. JAI~KEE, P.-O. L~JWDIN~N. I~0SEiN', D. SCIA_MA~G. SUDARSIIA~N,A. VAN DEE MEICWE, C. VILLI, and particularly 1~. A. ICiccI and D. WmKr~'S0~. He thanks moreover, for discussions~ A. AGODI~ M. ANILE, J. S. BELL, l~. BROWN~ A. CASTELLLN0, E. GIAN]NETT0~ _A_. IiNSOLIA, A. ITALIAI~0, A . J . ]~[~LNAY, G.D. ]~[ACCARIt0NE~ A. MAIA~ Ir M/GNAI~I~ M. PAV~I~, A. I~IGAS, M. ROD0~), W.A. I~ODRIGUES, M.A. :F. I~0SA, F. SELLERI, M.V. TENOI~IO, and particularly P. S~mz. He is grateful to P. PAPM~Iand the Redazione of this journal for the generous co-operation. At last, the author exprcsses his thanks to Mr. F. AgEIVCr for his help in the numerous drawings, and to Dr. L. 1-r BALDI_NIfor the kind collaboration over the years.

CLASSICAL TACI-IYONS

161

T h e first s t i m u l i to w r i t e t h i s r e v i e w c ~ m e f r o m t w o kind~ p r e l i m i n a r y i n v i t a t i o n s : one i n 1976 b y K . PAULUS (on b e h a l f o n t h e I n s t i t u t e of P h y s i c s ) , a n d o n e i n 1980 b y Sir D. W I L K ~ S O ~ (editor of t w o series of b o o k s for O x f o r d U n i v e r s i t y Press). T h e p r e s e n t w o r k a p p e a r e d I N F N / A E - 8 ~ / 8 ( F r ~ s c ~ t i , 21 A u g u s t 1984).

in preprint form

as r e p o r t

Note added in p~oo]s. Two recent papers (~) called a t t e n t i o n to a series of experimental works (2)--dated 1971 to 1984--which suggest t h a t the muon-neutrino coming from pion decay (7:+-+ -~ ~++v) might be tachyonic, in the sense t h a t its four-momentum square results to be negative (even if by two standard deviations only: (~world average)> based on ref. (~)). More recent data (s), adopting a new precision measurement of the r~ mass, yielded the value (N1)

p2

p~p~ = ( _ 0.097 ~ 0.072) (MeV)2/c 2 .

Actually, the idea t h a t neutrinos could be tachyons, at least in some cases, has a long story: see subsect. 11"5 a n d 13"3 of this review a n d references therein. For instance, it is immediate to show (4) t h a t in the pion centre of mass (in which Ipl~ = Ipl~) it is (N2a)

lp]v~29.7901 ~ e V / c ~ llalo,

if m v = 0,

vv=c ;

(N2b)

[Ply< IPlo,

if ~

0,

v~ lP[o,

if ~n. # 0;

vv > c.

The latest experimental data (N1) are compatible with a tachyon-neutrino with mv_~0.31 MeV/e; from eqs. (55'), (55") it follows that in the pion e.m. V / e ~ v ~ / e ~_ 0.000055. Let us mention some consequences, t h a t could be experimentally tested. Due to the (( switching procedure )~, when the pion speed in the laboratory is v= < < e~/V ~ 0.999 945c, we shall start seeing some pion ((decays ~ as processes : : - ~ - + -+ ~, the condition being t h a t v=- V < - - c2 (where v= is measured in the laboratory a n d V in the pion c.m.). E.g., for pions with a laboratory energy E ~ 31.2 GeV, one event every ~ 4 5 0 0 0 decays will appear in the laboratory as a tachyon absorption in flight, ~ + ~ ~ ~, corresponding to a mean life in flight of A~' 0.26083 s. Since the electron-neutrino seems to be bradyonic, if the muon-neutrino is on the contrary faster t h a n light, interesting consequences then follow w.r.t, neutrino oscillations. For simplicity's sake, let us consider Majorana neutrinos with a finite mass. (1) A. C~ODOS, A. I. HAUSEI~ and V. A. KOST:ELECKY: Phys. Lett. B, 150, 431 (1985); H. VAN DA~, u J. NG a n d L. C. BI~D~N~IA~: Phys. Lett. B, 158, 227 (1985). (2) E . V . SI~RV~ a n d K. 0. H. Zloe]~: Phys. Lett. J~, 37, 114 (1971); G. BACKE~STOSS et al.: Phys. Lett. B, 43, 539 (1973); D. C. L v et at.: Phys. Rev. Lett., 45, 1066 (1980); H. B. AND~n~IZB et al.: Phys. Lett. B, 114, 76 (1982); R. AreOLA et al.: Phys. Lett. B, 146, 431 (1984). (3) B. J]~CKI~L~ANN et al.: Phys. t~ev. Lett., 56, 1444 (1986). (4) G.D. MACCAI~RONEa n d E. RECAMI: Nuovo Cimento A, 57, 85 (1980), p. 99.

162

~. ~nc~

I u the s t a n d a r d formulae (~), the relevant quantity ~ve can ordinarily reach only valucs of the order of ~ 10 ~ (eV) ~, whilst under the present hypothesis it could be of the order of 2.5- 10~ (eV) 'z. This means t h a t , if the muon-neutrino is Supcrluminal, the position of the first oscillation m a x i m u m corresponds to values L/I~ various orders of magnitude smaller than ordinarily expected (whcl'e E, L are the neutrino energy and the distance from the source, respccti,cely). Even more, the coherence between the two ~mass eigenstates ~> of the muon-neutrino (a condition necessary to interference) would be lost b o t h in the solar neutrinos and in the reactor experiments. I t would be expected to r e m a i n still satisfied only for the cosmic radiation. Notice t h a t for a tachyon-neutrino the mass upper limit is about (0.44-: 0.49) MeV/c ~ (95% confidence level). The ordinary astrophysical arguments, setting stringent limits on the masses of bradyonic neutrinos, do not hold good in our case. F u r t h e r details can be found in ref. (s). Here let us only recall t h a t : i) if the existence of Supcrluminal neutrinos will be confirmed, then there should exist both sub- and Superl u m i n a l neutrinos, even if behaving differently; ii) also tachyons can be associated with semi-integer or integer spins, since in the spacelike case we are allowed to make recourse to n o n u n i t a r y representations of the Poincard group; iii) faster-than-light electric charges (dipoles) ought to behave as (Superluminal) magnetic poles (dipoles); iv) if neutrinos possess a finite mass but arc tachyonic, then the rclativistically i n v a ~ a n t distinction between (left-handed) neutrinos and (right-handed) antineutrinos still holds, since in E R the helicity gets reversed together with the particle/antiparticle character.

(s) See, e.g., V. F~,~a~_~-~o and B. SXITTA: Neutrino oscillation experiments, report I N F N / A E - 8 5 / 6 (Frascati, 1985). Cf. also D. H. P~]~];~Ns: Introduction to High Energy _Physics (Addison-Wesley, London, 1982). (s) E. GIANN~TTO, G . D . MACC),RI',ON:E, R. M~A_~-~ a n d E. I { ~ c 2 ~ : -possible consequences /or neutrino oscillations o] a tachyonic muon-neutrino, preprint I)P/777 (Physics Department, Catania University, 1986), to bc published. Sec also E. RI~CAMI, R. M~G_~-A~I a n d G . D . M~CC~R~ON~: Are ranon neutrinos ]aster-than-light particles? (Comments on a recent paper by Chodos et al.), report INFN/AE-86/1 (Fraseati, 1986).

REFERENCES

~BERS E., I . T . GJ~oDsKY and R . E . iNolvro~: Phys..Rev., 159, 1222 (1967). ~GOl)X A.: Lezioni di fisiea te~ica (Catania University, unpublished) (1972). AGvI)I~ J. L.: .Letl. )~uovo Cimento, 2, 353 (1971). A(~uDI.~- J . L . and A. M. P~,ATZ],:CK: _Phys. Lett. A, 90, 173 (1982a). AG~DIN J . L . and A. M. PS,ATZ~CK: _Phys. Rev. D, 26, 1923 (1982b). AIIARONOV Y., h_. KOI~AIr and L. SVSSKI~I): Phys. Rev., 182, 1400 (1969). AK~BA T.: _Frog. Theor. Phys., 56, 1278 (1976). A~A~AIr RA)~*~;JAM G. and N. ~NAMAS~VAYAM:f~ett..Nuovo Cimento, 6, 245 (1973). A_LAOAR RAMA-~VZA~ G., G. A, SAV~II~AZ a n d T . S . S I I A ~ n A : Pram~na, 21, 393 (1983). ALV)iG~ T., J. BLO.'~QvIS~ and P. E ~ A h - N : Annual Iteport o] .Nobel l~,esearch Institute, Stockholm (unpublished) (1963). A L V ) I ~ T., P. ER~A_~~ and A. K ~ I ~ : Annual l~epo~'t o] Nobel Itesearch Institute, Stockholm (unpublished) (1965).

CLASSICAL TACHYO~S

163

ALV~GER T., P. E ~ A N ~ and A. KEREK: preprint (Stockholm, Nobel Institute) (1966). ALVldGER T. and M . N . K~EISLER: Phys. Rev., 171, 1357 (1968). A~A~I)I E. : in Old and 2/ew Problems in Elementary Particles, edited b y G. P v e e I (New York, N . Y . , 1968), p. 1. A~xLDI E. and N. CABIBBO: in Aspects o] Q~ant~m Theory, edited b y A. SALA~ and E . P . W I ~ X R (Cambridge, 1972). A ~ I R A Z v P., E. RECA~I and W. RODRIGVES: 2/~OVO Cimento A, 78, 192 (1983). A~TIPPA A . F . : 2/uovo Cimento A, 10, 389 (1972). ANTIP~A A . F . : Phys. Bey. D, 11, 724 (1975), A~TleeA A . F . and A. E. EVERETT: Phys. Rev. D, 4, 2198 (1971). A~TIP~A A . F . and E . A . EVERETT: Phys. Roy. D, 8, 2352 (1973). ARCI~)IACO~VO G.: Collect. Math. (Barcelona), 25, 295 (1974). ARo~s M . E . and E. C. G. SVDAI~SHA~: Phys. Bey., 173, 1622 (1968). ARZELI~S H. : La cindmatique relativiste (Gauthier-Villars, Paris, 1955), p. 217. ARz~LII~S H.: C. R. Acad. Sci., 245, 2698 (1957). AnzELIl~S H.: Dynamique rclativiste, Vol. 2 (Gauthier-Villars, Paris, 1958), p. 101. ARZELIi~S H.: C. R. Acad. Sci., Set. A, 279, 535 (1974). H.: Phys. Today, 25(11), 15 (1972). H., PH. CO~BE a n d P. SORBA: Rep. Math. Phys., 5, 145 (1974). M. and E. RECAMI: Lett. 2/uovo Cimento, 2, 643 (1969). M., G. FONTE a n d E. RECA~I: Lett. 2/uovo Cime~to, 4, 241 (1970). BALTAY C,, G, FEII~BEI%G, ~ . YEH and R. LINSKER: Phys. Rev. D, l , 759 (1970). BASERZEE A.: Curt. Sei. (India), 42, 493 (1973). BAWEI~X~E A. and S . B . DTJTTA CItOUDH~JRY: Aust. J. Phys., 30, 251 (1977). BANERZI S. and D. R. MA~I)~L: J. Phys. A, 15, 3181 (1982). BARAS~V.NKOV V. S. : Soy. Phys. Usp., 17, 774 (1975) (English translation of Usp. ffiz. 2~auk, 114, 133 (1974)). BAR~ARD A. C. L. and E . A . SALLIN: Phys. Today, 22(10), 9 (1969). BARRET T . W . : 2/novo Cimento B, 45, 297 (1978). BARTLETT I). F. and M. LAtIANA: Phys. Bey. D, 6, 1817 (1972). B~a~TLETT D . F . , D. Soo and M. G. W~ITE: Phys. Rev. D, 18, 2253 (1978). BARVT A. 0. : in Tachyons, Monopolcs, and Related Topics, edited b y E. RECAMI (NorthHolland, Amsterdam, 1978a), p. 143. BARVT A. 0 . : Phys. Lett. A, 67, 257 (1978b). B ~ V T A. O. : in Taehyons, Monopoles, and Related Topics, edited b y ]~. RECAMI (NorthHolland, Amsterdam, 1978c), p. 227. BARVT A. 0 . : JSett. Math. Phys., 1D, 195 (1985). BAR~T A. O. and I . H . DVRv: Proe. R. Soc. London, Ser. A, 333, 217 (1973). BA]aI~T A. 0. and J. N&G]~L: J. Phys. A, 10, 1223 (1977). B ~ V T A. 0., G . D . MACC~mROXE and E. R]~CAMI: 2/uovo Cimento A, 71, 509 (1982). BARVT A. 0., C. K. E. SC~NEI])ER and R. WILson: J. Math. Phys. (2/. Y.), 20, 2244 (1979). BELL J. S.: private communication (1979). B~LL J. S.: private communication (1985). B~NFORD G . A . , D . L . BOOK and W . A . NEWCOMB: Phys. Rev. D, 2, 263 (1970). B]~RLEY D. et al. : in Berkeley 1974 Proceedings P E P Sum~ner Study (Berkeley, Cal., 1975), p. 450. B]~I~NA~DI~I C.: 2/uovo Cimento A, 67, 298 (1982). BE~zI V. and V. Go~INI: J. Math. Phys. (2/. Y.), 10, 1518 (1969). BHAT P. N., N . V . GOPALAK~IS~NA~, S. K. GVl~TA and S. C. TONWAR: J. Phys. G, 5, L13 (1979). BACRY BACRY BALDO BALDO

164

~. R~CA~

BILANIVK O.M. and E. C. G. SVDABSHA~: Phys. Today, 22(5), 43 (1969a). BILA~IVK O.M. and E. C. G. SVDARSHA~: ~ature (London), 223, 386 (1969b). BILA~IVK O.M., V. K. D]~SHPA~DB and E. C. G. SVDABSHA~: Am. J, Phys., 30, 718 (1962). BILANII~K 0. M., S.L. BROWN, B. Dx W I ~ , V.A. Na~wco~, M. SACHS, E. C. G. SVDARSHAN and S. YOSHIKAWA: Phys. Today, 22(12), 47 (1969). BILANIVK 0. M., P . L . CSONKA, E . H . K]~RNXR, R . G . N]~WTON, E. C. G. SVD~RSI~A~r and G.N. TSANDOULAS: Phys. Today, 23(5), 13 (1970); 23(11), 79 (1970). B I I ~ A J. A., M. H. COH]~N, S, C. UNWlN and I. I. K. PAvLINY-To~lz: Nature (London), 306, 42 (1983). BJORKE~r J . D . and S.D. Dm,3LL: Relativistic Quantum Mechanics, Vol. I (McGrawHill, New York, N . Y . , 1964), p. 86. B;IORKEN J . D . , J . B . KOGV~ and D . E . Solemn: Phys. Rev. D, 3, 1382 (1971). BLANDFORD R . D . , C.F. M c K ~ and M. J. R ~ s : Nature (London), 267, 211 (1977). BLIID~AN S.A. a~nd M.A. RVD]~R~A~r Phys. ~ev. D, 1, 3243 (1970). BOHH D.: The Special Theory o] Relativity (New York, N . Y . , 1965). BOH~ D. and J . P . VIGI~R: Phys. Rev., 96, 208 (1954). BoHI~ D. and J . P . VIGI]~R: Phys. Rev., 109, 882 (1958). BOLO~OVSKY B.M. and V . L . GINZBVRG: Usp. ffiz. _~auk, 1{}6, 577 (1972). BO~DI H.: Relativity and Common Sense (Doubleday, New York, N . Y . , 1964). Bo~A~Av M. : Ronda 1980 Proceedings. Fundamental Physics (1980), p. 1. B!aOIDO IV[.~r and J. O. TAYLOR: Phys. Rev., 17, 1606 (1968). BRow~ G. E. and H. RHO: Phys. Today, 36(2), 24 (1983). B R o w ~ I. W. A., R . R . CLARK, P. K. MOOR~, T. W. B. MIIXLOW, P. N. ~u M . H . COI~]~N and R . W . POI~CAS: s (London), 299, 788 (1982). BI~Ol~I~ A. I., L . L . JXXKOVSKY and N . A . KOBYL~NSKY: Lett..Yuovo Cimento, 5, 389 (1972). BIILm~cK A. R. and C.A. Hlms~ : Answer to Agudiu and Platzeck, preprint (Adelaide University, Adelaide, 1984). Bll~ov, 3s Br. J. Philos. Sol., 9, 39 (1959). CAnI~O N. and E. FP.~R)mI: Nuovo Cimento, 23, 1147 (1962). CALDIROLA P. and E. RECA~I: J~pistemologia (Genova), 1, 263 (1978). CALDIROLA P. and E. RXCA~I: in Italian Studies in the Philosophy o] Science, edited b y M.L. DALLA C ~ I ~ A (Reidel, Boston, Mass., 1980), p. 249. CALDIROLAP., G. D. MACCARRO~ and E. R~CA~I: Lett. Nuovo Cimento, 29, 241 (1980). CALDInOLA P., M. PAV~I~ and E. R~CA~I: Nuovo Cimento B, 487 205 (1978). CAM~ZI~) M.: Gen. Eel. Gray., 1, 41 (1970). C A ~ Z l N D M.: in Taehyons, Monopoles, and Related Topics, edited by E. RxcA~I (North-Holland, Amsterdam, 1978), p. 89. CAO SH.-L.: preprint (Beijing Normal University, Bejjing, 1984). C~_~]~Y A . L . , C.M. EY and C.A. HvI~sT: Hadronic J., 2, 1021 (1979). CAI~OL A. et al.: in Berkeley 1975 Proceedings P E P Summer Study (Berkeley, Cal., 1975), p. 176. CASALBITONI ]~. : in Taehyons, Monopoles, and Related Topics, edited by E. R~cA~I (North-Holland, Amsterdam, 1978), p. 247. CAS~LLI~O A. : Un approccio teorico allo studio di alcune apparenti espansioni superl~minal,i in astro]isiea, MS. Thesis, supervisor E. R]~cA~I (Catania University, Physics Department, 1984). CAS~OmXA P. and E. R~CAI~t: Lett. Nuovo Cimento, 22, 195 (1978). CATARA F., M. CONSOLI and E. EB~I~L~: Nuovo Cimento B, 70, 45 (1982). CAVALI~I~ A., P. MOl~ISO~ and L, SAI~Ol~I: Science, 173, 525 (1971).

CLASSICAL TACttYONS

165

CAVALL~nI G.: JSett. 2~eovo Cimento, 43, 285 (1985). CAVALLEnI G. and G. SPIN'E.LLI: Lett..37uovo Cimento, 6, 5 (1973). CAVALL~nI G. and G. SPI~ELLI: Phys. t~ev. D, 15, 3065 (1977). CAVALL~nI G. and G. SI'I~ELLI: Lett. Nuovo Cimento, 22, 113 (1978). CAWLEu R . G . : Ann. Phys. (N. Y.), 54, 132 (1969). CAWLEY R . G . : rnt. J. Theor. Phys., 3, 483 (1970), p. 491. CAWLE:~ R . G . : Lett. Nuovo Cimento, 3, 523 (1972). CI~AnO~r J. E. : Thdorie de la relativitg complexe (A. Michel, Paris, 1977). CI~EW G . F . : Science, 161, 762 (1968). CiBonowsKi J.: preprint (Institute of Experimental Physics, Warsaw, 1982). CLAVELLI L., S. FEIISTER and J . L . UI~E~SKu _/Yuel. Phys. B, 65, 373 (1973). COI~E~ M. H. and S.C. U~lwI~: in Proceedings o] the IAU Symposium 1~o. 97 (1982), p. 345. COlZEN M.H., W. CAN~ON, G . H . PlmC]~LL, D . E . SI~A~F]~, J . J . Bt~oD]~I~IC]L K . I . K E L L E I ~ A ~ and D . L . JAIS)ICEY: Astrophys. J., 170, 207 (1971). COI~EN M.H., K . I . K~LLERMAI'~N, D. ]3. SHAFFER, l~. 1~. LII'~t0I:ELD,A . T . MOFFET, J . D . ROlVINEY, G.A. SEIELSTAD, I. I. K. PAVLINv-ToTH, E. PnEI~SS, A. WITZ~L, R . T . SCHILLIZZI and B . J . GELI)ZA~IL~n: Nature (London), 268, 405 (1977). COLE E . A . : 1Vq,ovo Cimento A, 40, 171 (1977). COLE E . A . : s Cimento B, 44, 157 (1978). COL~ E . A . : Phys. I~ett. A, 75, 29 (1979). COL]~ E . A . : J. Phys. A, 13, 109 (1980a). COLE E . A . : ~Vq~ovo Cimento B, 55, 269 (1980b). COL~ E . A . : Phys. I~ett. A, 76, 371 (1980c). COL~ E . A . : JUett. ~Tq,ovo Cimento, 28, 171 (1980d). COLE E . A . : JVuovo Cimento A, 60, 1 (1980e). COL]S E. A.: ~Vnovo Cimento B, 85, 105 (1985). See also COLE E. A. and I. M. STAZZI: Lett. Nuovo Cimento, 43, 388 (1985). COLE~A~ S. : in Aeausality in Snbnuclear Phenomena, edited by A. ZlCmC~II, Pa~t A (Academic Press, New York, N . Y . , 1970), p. 283. C o H ~ R . P . and J. D. LATm~OI~: Am. J. Phys., 46, 801 (1978). CONrOI~TO G. : preprint (Cosenza University, Department of Mathematics, 1984). COn~E~ H. C.: Lett. Nuovo C~mento, 11, 533 (1974). COI~]~E~ H . C . : s Cimento A, 29, 415 (1975). CoRm~x H . C . : Int. J. Theor. Phys., 15, 703 (1976). COl~B]~ H . C . : I~ett. ~ o v o Cimento, 20, 6~5 (1977a). Col~ H . C . : three preprints (West Hill, Ont. : Scarborough College, August, September and November, 1977b). COI~B~X H. C. : in Tachyons, Monopoles, and Re~ated Topics, edited by E. R~cAmI (North-Holland, Amsterdam, 1978a), p. 31. CORBE~ H . C . : I~ett. Nuovo Cimento, 22, 116 (1978b). COSTA I)E B~AIm~AX~) 0.: 2'ound. Phys., 2, 111 (1972). COSTA a)E B~A~R~AI~I) 0.: Int. J. Theol. Phys., 7, 129 (1973). COSTA I)~ B~AVl~EO~]) 0. : in Old and ~Vew Qq~estions in Physics, Coscaology, Philosophy: Essays in Honor of W. Youvg~aq~, edited by A. VA~ I)~1~ M~nw~ (Plenum Press, New York, N . Y . , 1983), p. 87. C o s t a DE BEAtmE~AI~I) 0. : in The Wave-Particle Dq*alism, edited by S. D i ~ n et al. (Reidel, Dordreeh*, 1984), p. 485. COSTA a)]~ BEAVR]~,CJ~) 0., CH. I~BEI~ and J. RICAnI): Int. J. Theor. Phys., 4, 125 (1971). CSO~CKA P . L . : Nucl. Phys. B, 21, 436 (1970). CII~NI~GI~AM C. T.: preprint DAP-395 (Calteeh, Pasadena, Cal., 1975).

166

E. ~EC~M~

DADttICH N.: Phys. T~ett. A, 70, 3 (1979). DANB~G J . S . , G. R. KALBFLEISCH, S . R . ]~ORENSTI'~IN, ]~. C. STI~AND, V. VANDE]RBV~G, J . N . CIL~VMAN a n d J. LYS: Phys. Rev. D, 4, 53 (1971). DAR A.: Phys. _I~ev. Left., 13, 91 (1964). DAs A.: J. Math. Phys. (PV. Iz.), 7, 45, 52, 61 (1966). DATTOLI G. and R. MIGNANI: Lett..Nuovo Cimento, 22, 65 (1978). DAVIES P. C. W . : Nuovo Cimento B, 2S, 571 (1975), p. 577. I)E FARIA-RosA M. A., E. RECA-MI and W. A. ROD~IG~ES: submitted for publication (1985). DE F A ~ - R O S A M. A., E. RECP~MIand N. A. ROD~IGt~S : Phys. Lett. B, 177, 233 (1986). D].~LL'ANzoNIo G . F . : J. Math. Phys. (7(. IZ.), 2, 572 (1961). DEM~.RS P.: Ca~. J. Phys., ~3, 1687 (1975). DENT W . A . : Seience, 175, 1105 (1972). DE SAsB~'rA V.: in Taehyons, Monopoles, and Berated Topics, edited b y E. RECAMI (North-Holland, Amsterdam, 1978), p. 99. DE ShBB~TA V., M. PAVg~ and E. RECA~a~: Left..~uovo Cimento, 19, 441 (1977). DES Cou])]~I.:S T~I.: A~'ch. ~derl. Sei. (II), S, 652 (1900). D'EsI'AGNAT B.: .Fo~end. Phys., 11, 205 (1981). DH-~R J. and E. C. G. S~DAICSI~N: Phys. Rev., 174, 1808 (1968). DHURANDH~ S. V.: J. Math. Phys. (N. Y.), 19, 561 (1978). D~nAND~KA~ S.V. and J. V. N ~ L I X A P : Gen. l~el. Gray., 9, 1089 (1978). ])I J o 2 I o M.: Nuovo Cimento B, 22, 70 (1974). D~RAC F. A. M.: Proe. R. Soe. London, Ser. A, 133, 60 (1931). DO~LING J . : Am. J. Phys., 38, 539 (1970). DUFFEY G. H.: .Found. Phys., S, 349 (1975). DVF~EY G. H.: .Found. Phys., 10, 959 (1980). E C K ~ G.: Ann. Phys. (PV. Y.), 58, 303 (1970). EDMONDS J . D . : ~ett. Nuovo Cimento, S, 572 (1972). EDMONDS J . D . : .Found. Phys., 4, 473 (1974). EDMONDS J . D . : .Found. Phys., 6, 33 (1976). EDMONDS J . D . : Found. Phys., 7, 835 (1977a). EDMONDS J . D . : Lett. Nuovo Cimento, 18, 498 (1977b). EDMONDS J. D. : in Tachyons, Monopoles, and Related Topivs, edited b y E. RLCAMI (North-Holland, Amsterdam, 1978), p. 79. EEG J. 0 . : Phys. ~Yorv., 7, 21 (1973). EINST],:I~- A.: Ann. Phys. (Leipzig), 17, 891 (1905). EINSTEIn" A.: An~. Phys. (Leipzig), 3S, 898 (1911). EINSTEIN A. and P. BERG-MANN: An~. Math. (N. Y.), 39, 683 (1938). ELDE~ J . D . : Phys. Today, 23(10), 15, 79 (1970). EPSTEIh" R . L . and M . J . GELLEd: ZTatuq'e (London), 265, 219 (1977). EY C.M. and C.A. HUI~sT: .Nuovo Cimento B, 39, 76 (1977). ENATS~: II., A. TAKENAKA and M. OKAZA~I: Nuovo Cimento A, 43, 575 (1978). ERIXSEN E. and K. VOY~.NLI: .Found. Phys., 6, 115 (1976). EVerETT A. E.: Phys. I~ev. D, 13, 785, 795 (1976). FEDERIGHI I.: Boll. Soc. Itat..Fis., 130, 92 (1983). F~:INBE~G G.: Phys. 17ev., 1Sg, 1089 (1967). FEINBEIr G.: Sci. Am., 222(2), 68 (1970). )~INSEBG G.: Phys. Rev. D, 17, 1651 (1978). FEINBF~I~G G.: Phys. Rev. D, 19, 5812 (1979). FELI)MAN L . M . : Am. J. Phys., 42, 179 (1974).

CLASSICAL ~C~YO~S

167

F]~R~ E.: Elementary Particles (Yale University Press, New Haven, Conn., 1951). F ~ n ~ n I E. : in Taehyons, Monopoles, and Related Topics, edited by E. RECAMI (NorthHolland, Amsterdam, 1978), p. 203. FEnR:~TI I. and M. V]~!aD]~: Atti Accad. Sci. Torino, C1. Sci. .Fis. Mat. )[at., p. 318 (1966). FEYNlVlA~ R . P . : Phys. Rev., 76, 749, 769 (1949). FIGV~InXDO V . L . : Ph.D. Thesis, work in progress (IMECC, Campinas State University, 1985). FII~!KiELST]~IlqA. M., V. ,]'A. KI%EINOVICHand S. N. PAN])~v : report (Space Astrophysics Observatory, Pulkovo, 1983). Fi~z~ B. and M. P~s~onI: Calcolo tensoriale e applicazioni (Zanichelli, Bologna, 1961). F L ~ O M. and M. G v ] ~ I ~ : Helv. Phys. Aeta, $~, 117 (1977). F L ~ n r N., J. L ~ I T ~ - L o ~ s and G. O ~ , ~ c H ~ n : Aeta Phys. Austriaca, 38, 113 (1973). FOSTER J. C. and J. R. R~Y: J. Math. Phys. (N. Y.), 13, 979 (1972). F o x R., C. G. KVe~R and S. G. LI~'SON: 2Vature (London),223, 597 (1969). F o x R., C. G. Ku~ER a n d S. G. LIPSO~: Proe. 1~. Soe. London, Ser. A, 316, 515 (1970). F n ~ x P. and It. Ro~m~: Ann. Phys. (Leipzig), 34, 825 (1911). FRE]~D K.: J. Chem. Phys., $6, 692 (1972). FRO~I~G H . D . : Spee: Sei. Technol., 4, 515 (1981). FRO~SDAL C.: Phys. Rev., 171, 1811 (1968). FRO~S])AL C.: Phys. Bey., 182, 1564 (1969a). FRO~S])AL C.: Phys. Rev., 18~, 1768 (1969b). FVKVD~ R.: Phys. Lett. B, 73, 33 (1978). FVLL]~R R . W . and J . A . WH]~L~R: Phys. l~ev., 128, 919 (1962). CxALILEI G . : Dialogo ... sopra i due massimi sistemi del mondo: Tolemaieo e Copernieano (G. B. Landini, Firenze, 1632). CxALILEI G , : Dialogue on the Great World Systems: Salusbury Translation, edited by G. D]~ SA~TILLA~A (Chicago University Press, Chicago, Ill., 1953), p. 199. CxAICUCCIO A.: private communication (1984). G A ~ C c i o A., G. D. lV[ACC~RO~]~, E. R~eAMI and J . P . u Lett. Nuovo Cimento, 27, 6O (1980). GAw~I~ L . L . : Int. J. Theor. Phys., 19, 25 (1980). GIACOMELLI G.: in Evolution o] Particle Physics, edited b y M. C o ~ v ~ s I (New York, N . Y . , 1970), p. 148. GIA~N]~TTO E., G. I). MACCAI~RON:E,1~. MIGNA~!I a n d E. RECAMI: preprint PP/777 (Physics Department, University of Catania, 1986). GIrArD R. and L. MARCmLDO~: _Found. Phys., 14, 535 (1984). GLADKIKI-I V . A . : Pizika (ls. Tomsk Univ.), 6, 69, 130 (1978a). GLA])KI~:~ V. A.: .Fizika (Is. Tomsk Univ.), 12, 52 (1978b). GLEXSON A . M . and E. C. G. SVDARSHA~: Phys. t~ev. D, 1, 474 (1970). GL~SON A. M., M. G. GV~DZlK, E. C. G. SUDAnSHAN and A. PAONAM]~TA : Phys. Rev. A, 6, 807 (1972a). GL~ESO~ A . M . , M . G . GuN])zI~:, E. C. G. S v D ~ n s n ~ and A. PAGI~AMEI~TA:~ields Quanta, 2, 175 (1972b). GLi)c~: M.: ~uovo Cimento A, 62, 791 (1969). G6BXL R.: Commun. Math. Phys., 46, 289 (1976). G6D~L K. : in A. ~instein: Philosopher-Scientist, edited b y P. A. S c ~ (Open Court, L a Salle, Ill., 1973), p. 558. GOLD~ A . S . and F. S~IT~: t~ep. Prog. Phys., 38, 731 (1975), p. 757. GoLDo~I R.: Lett. Nnovo Cimento, 5, 495 (1972).

168

~. ~c_~fi

GOLDOM R.: Nuovo Cimento A, 14, 501, 527 (1973). GOLDO.~[ R.: Aeta Phys. Austriaca, 41, 75 (1975a). GOLDO~ R.: Aeta Phys. Anst~'iaca, 41, 133 (1975b). GOLDOW~ R.: Gen. Rel. Grav., O, 103 (1975c). GO~DO~ R. : in Tachyons, Monopoles, and Related Topics, cditcd by E. R~c~_~i~ (NorthHolland, Amsterdam, 1978), p. 125. GO~DR~h~D J.-C.: Report CEA-B18-199 (CEN, Saclay, 1971), in French. GOR~N~ V.: Com/mun. Math. Phys., 21, 150 (1971). Go~I V. a n d A. Z~CCA: J. Math. Phys. (N. ~.), 1], 2226 (1970). GoT'r I I I J. R.: Nuovo Cimento B, 22, 49 (1974a). GOTT I I I J. R.: Astrophys. J., 187, 1 (1974b). G~n~.'~n].:n~ O . W . : J . Math. Phys. (N. :Y.), 3, 859 (1962). G~EGoRY C.: Nature (London), 192, 1320 (1961). G R ~ o ~ r C.: Phys. l~ev., 125, 2136 (1962). Ga~;GoI~r C.: Nature (London), 206, 702 (1965). G~EGOICV C.: Nature Phys. Sci., 239, 56 (1972). G~r L . T . and R . F . S ~ r Phys. lgev. Lett., 20, 695 (1968). Gao-~ 0.: Lett. Nuovo Cimento, 23, 97 (1978). G~.'~ ~.: private communication (1979). G~:As~ ~ . : Sobre la importancia del movimento en el concepto de la realidad fisica (Ed. Valencia 2000, Valencia, 1983). G ~ N ~ - M.: Phys. Letl. B, 62, 81 (1976). G~R~:wc~ L. E. and S.V. TARAS~WC~: Soy. Astron. Lett., 4, 183 (1978). Gu]~_-u V. S.: private communication (1983). GI;RI~,- V. S.: Fizika (Alma-Ata), 16, 87 (1984). Gt;RIN V.S.: Pram~na, 24, 817 (1985). HAD,)'IOAKNOU F . T . : Nq~ovo Cimento, 44, 185 (1966). HAGSTON W. E. and I . D . Cox: ~'oq~nd. Phys., ]5, 773 (1985). Hhlr~ E.: Arch. Math. Phys., 21, I (1913). ][ALPb]R); ~ . and S. 51A~J~.'~: Co-ordinate conditions in general relativity, report (Colgate University, Hamilton, N. Y., 1969). IIA~A~OTO S.: Prog. Theor. Phys., 48, 1037 (1972). ][X~AMOTO S.: Prog. Theor. Phys., 51, 1977 (1974). II):~-~-i H. and E. III:G~TOBLEr~: in Tachyons, Monopoles, and Related Topics, edited b y E. R],;cA~I (North-Holland, Amsterdam, 1978), p. 61. I IANs],:N R. 0. and E . T . N~w:~A~: Gen. ~el. Grav., 6, 361 (1975). IIAvAS P. : in Causality and Physical Theories, edited b y W. B. RoI,~c~; (Ncw York, N . Y . , 1974). HAWKL'~G S. W. and G. F. R. ELLIS : The Large-Scale Structure o/Space-Time (Cambridge University Prcss, Cambridge, 1973). H~AVIS~D].; 0 . : Electrical Papers, Vol. 2 (London, 1892), p. 497. II~G~:m~']~LD G . C . : Phys. Rev. D, 10, 3320 (1974). Itl~G~m~'].;~I) G . C . : Phys. Rev. Lett., 54, 2395 (1985). H~s~xB~:~c, W. : in Aspects o] Quantum Theory, edited b y A. SA~.~)~ and E. P. WiGN:E:R (Cambridge University Press, Cambridge, 1972). H~ST],:~-].:S D.: Space-Time Algebra (Gordon a n d Breach, New York, zN. Y., 1966). IIES'rE~:S D.: J. Math. Phys. (N. Y.), 16, 556 (1975). [IESTENES D.: ~'ound. Phys., 15, 63 (1985). ]~[:ETT:EL R.O. and T. hi. H].;LHW~LL: _~YUOVOCimento B, 13, 82 (1973). H~LGEVOO~D J. : Dispersion Relations and Causal Description (North-Ilolland, Amsterdam, 1960), p. 4.

CLASSICAL TACtIYOI~S

1~9

HOH F. C.: Phys. Rev. D, 1~, 2790 (1976). HONIG E., l~. LAKE and R . C . RO]~])E~: Phys. l~ev. D, 10, 3155 (1974). HOYLE F. and J. V. NAI~LIKAtr: Action-at-a-distance (Freeman, San Francisco, Cal., 1974). HUARD S. and C. IMBEI~T: Opt. Commnn., 2~, 185 (1978). I~A~OWSKI W. V.: Phys. Z., 21, 972 (1910). I~A]~DA K.: Nuovo Cimento B, ~0, 271 (1979). ISHIKAWA K . I . and T. M~u Gen. l~el. Gray., 15, 1009 (1983). ISRA]~L W . : Phys. Rev., 16~, 1775 (1967). IVANENKO D. D.: in Relativity, Quanta and Cosmology, edited b y F. D]~ FIN~S and M. PANTALEO, Vol. 1 (Johnson Reprint Co., I~ew York, N . Y . , 1979), p. 295. JACKIW R. and C. R]~BBI: Phys. Rev. Lett., 37, 172 (1976). JADCZYK A. Z. : preprint No. 213 (Wroclaw University, I n s t i t u t e of Theoretical Physics, 1970). JAFF]~ J. a n d I. SI~A~IRO: Phys. t~ev. D, 6, 405 (1974). JAMmeR M. : in Problems in the ~'oundations of Physics, Proc. S. 1.1~., Course L X X I I , edited b y G. TO~ALDO DI F~ANClA (North-Holland, Amsterdam, 1979), p. 202. JANCEWICZ B.: Electromagnetism with use o/ bivectors, preprint (Wroclaw University, D e p a r t m e n t of Theoretical Physics, 1980); see also Eur. J. Phys., l , 179 (1980). JANIS A . I . , E . T . N]~WMAN and J. WINICO~R: Phys. l~ev..Lett., 20, 878 (1968). JEHLE M.: Phys. Rev. D, 3, 306 (1971). JEHL~ M.: Phys. l~ev. D, 6, 441 (1972). JOHNSON L. E. : External tachyons, Internal bradyons, unpublished report (New Concord, 0., 1981). JoHnI V. B. and S. K. SnlVASTAV~: preprint (Gorakhpur University, Physics Department, 1978). JoN]~s L . W . : Rev. Mod. Phys., ~9, 717 (1977). Jo~v]~s R. T. : J. ~ranklin Inst., 27~, 1 (1963). JORDAN T . F . : J. Math. Phys. (~. Y.), 19, 247 (1978). Jv]~ C.: Phys. Rev. D, 8, 1757 (1973). KALITZlN N. : Multitemporat Theory o] Relativity (Bulgarian A c a d e m y of Sciences, Sofia, 1975). Ks A. J. : in Tachyons, Monopoles, and Related Topics, edited b y E. RECAMI (North-Holland, Amsterdam, 1978), p. 53. KiLNAY A . J . : Lett. 2~uovo Cimento, 27, 437 (1980). KJ~LlVAY A . J . and B . P . TOLEDO: s Cimento, 48, 997 (1967). KA~OI K. and S. ]~AMEFIJCttI: ~Lett. ~Vuovo Cimento, 19, 413 (1977). KASTnVP H . A . : Aqva. Phys. (Leipzig), 7, 388 (1962). KELLEn J.: Int. J. Theor. Phys., 21, 829 (1982). KELLER J . : Int. J. Theor. Phys., 23, 817 (1984). KELLnnMA~N K . I . : Ann. N. Y. Acad. Sei., 336, 1 (1980). KESZTH~LRYI T. and K . L . NAGY: Aeta Phys. Acad. Sci. Hung., 37, 259 (1974). KmcI~ D.: Umsch. Wiss. Tech., 77, 758 (1977). KI~ZRNI~S D . A . and L. POLY~CH~K0: SoY. Phys. J E T P , 19, 514 (1964). KII~ZH~I~S D . A . and V . N . SAzo~ov: in Einsteinian Symposium, 1973, Academy o] Sciences, USSR (Nauka, Moscow, 1974), in Russian. KLEIN 0 . : Z. Phys., 53, 157 (1929). KNIGHT C.A., D . S . ROBERTSO~q, A~ E . E . ROGERS, I . I . SHAPIRO,t . R. W~IIT:N]~X, T. A. CLA~K, R. M. GOLI)ST~I~, G. E. MAnA~DI~O and N. R. VA~D~nBE~G: Science, 172, 52 (1971).

170

E. R ~ c ~ I

KoRr~ D. and Z. FRIED: ~UOVO Cimento A, 52, 173 (1967). KOWXLCZY~SKI J. K.: Phys. T~ett. A, 65, 269 (1978). KowALczv~sxi J. K . : Phys. Lett. A, 74, 157 (1979). KOWALCZV~SKI J. K.: Int. J. Theor. Phys., 23, 27 (1984). K~EISL~R M . N . : Phys. Teach., 7, 391 (1969). KREISLER M . N . : Am. Sci., 61, 201 (1973). KRSHKOWSKI W . : report (~P ~ No. 1060/VII/PtI (Institute for Nuclear Research, Warsaw, 1969). K~)G~R J. : in Taehyons, Monopoles, and Related Topics, edited b y ]~. R]~cAMI (No~'thI-Iolland, Amsterdam, 1978), 1o. 195. KYSXLKA~A.: Int. J. Theor. Phys., 20, 13 (1981). LAKE K. and R . C . ROEDER: Lett. Nq~ovo Cimento, 12, 641 (1975). LALA~ V.: B~ll. Soc. Math. l~r., 65, 83 (1937). LANI)AV L. and E. LIFSHITZ: Mdcanique (Mir, Moscow, 1966a). LAlqDAV L. a n d E. LIFSHI~Z: Thdorie du chaq~2 (Mir, Moscow, 1966b). LA~I)~ A.: Am. J. Phys., 33, 123 (1965). LA~])~ A.: Aq~. J. Phys., 43, 701 (1975). LAPLACE P. S. : Mdcanique celeste, in Ouvres (Imprimerie Royal, Paris, 1845), Tome IV, Book X, Chapt. V I I , p. 364. L ] ~ T . D . : Are blael~-hotes black bodies?, preprint CU-TP-305 (Columbia University, l~ew York, N . Y . , 1985). LEIBOWITZ E. and ~ . ROSE~: Gen. t~el. Gray., 4, 449 (1973). L]~IT]~R D.: T~ett. Nr Cimento, 1, 395 (1971a). L]~ITER D.: ~novo Cimento A, 2, 679 (1971b). LE~K:E H.: ~uovo Cime/ato A, 32, 181 (1976). L ] ~ K ~ H.: Int. J. Theor. Phys., 16, 307 (1977a). LE~KE H.: Phys. Lett. A, 60, 271 (1977b). LEwiS B. L.: r e p o r t (Naval Research Laboratory, Washington, D.C., 1981). LIAO~V L. and X. CgO]~G~I~: in International Con]erence on l~etativity and Gravitation (GR10), edited b y B. B~R~O~TI, F. DE FELICV, and A. PASCOLI~I (C.~q.R., Rome, 1984), p. 749. LIOttTMAhr A. P., W. H. PRESS, R . H . PRICE ~nd S . A . T]~UKOLSKI: Problem Book in Relativity a~d Gravitation (Princeton University Press, Princeton, N. J., 1975), p. 405. LzvBIClC A., K. PISK and B . A . LOGAn: Phys. /~ev. D, 20, 1016 (1979). LVCRE~IVS CA~O T. (ca. 50 B.C.) : De t~erum J~atura, edited b y M. T. CICERO (Rome), Book 4, lines 201-203. LUGIATO L. and V. GORI~I: J. Math. Phys. (~. Y.), 13, 665 (1972). ~IACCARRO~E G . D . and E. RECAMI: ~ound. Phys., 10, 949 (1980a). MACCARRO~E G . D . a n d E. RECAMI: N'~OVO Ci~nento A, 57, 85 (1980b). MACCA~RO~E G. D. and E. RECAMI: report INFN/AE-82/12 (II~IFN, Frascati, 1982a), p. 1. MAOCA_aEoNE G . D . and E. R]~CAMI: Zett. Nuovo Cimento, 34, 251 (1982b). MACCA~RO~-E G . D . and E. RxCA~I: ffound. Phys., 14, 367 (1984). MACC~]COSV. G . D . , M. PAv~I~ and ]~. R~CAMI: s Ciqnento B, 73, 91 (1983). MA~KL]~ F . : Am. J. Phys., 41, 45 (1973). ~r E.: Nnovo Cimento, 9, 335 (1932). ~[ALTSV.V V. K.: Teor. Mat. •iz., 47, 177 (1981). M A ~ R . B . and J . W . MOFFAT: Phys. leer. D, 26, 1858 (1982). MAN~HErM P. D.: preprint SLtkC-PUB-1885 (SLAC, Stanford University, 1977).

CLASSICAL TACI-IYONS

171

MARCItILDOI~ L., A . F . ANTIt)I'A and A . E . Ev]~R]~T~: Phys. Rev. D, 27, 1740 (1983). M~CHILDON L., A. E. EVERE~T and A. Y. ANTI~'~'A: s Cimento B, 53, 253 (1979). MAnQV~S G.C. and J . A . SwI~CA: Nucl. Phys. B, ~3, 205 (1972). MARSCE~n A . P . and J. S. SCOTT: Publ. Astron. Soc. Pac., 92, 127 (1980). M A r x E.: Int. J. Theor. Phys., 3, 299 (1970). MATI~EWS P . M . and M. S]~THARA~A~: Phys. Rev. D, 8, 1815 (1973). McLAV~HI~ D.: J. Math. Phys. (~. Iz.), 13, 784, 1099 (1972). M ~ s K Y M . B . : Commun. Math. Phys., 47, 97 (1976). MI~NANI R.: Lett. Nuovo Cimento, 13, 134 (1975). MI~NANI R. : Tachyons, Monopotes, and Related Topics, edited b y E. R~cA~I (NorthHolland, Amsterdam, 1978), p. 67. MIGNA~I R. and E. R~CA~I: Nuovo Cimento A, 14, i169 (1973a); Erratum, 16, 208 (1973). MIG~ANI R. and E. R]~CA~I: Lett. 2Vuovo Cimento, 7, 388 (1973b). MI~A~I R. and E. R]~CAMI: 2iuovo Cimento A, 24, 438 (1974a). MIG~qA~I R. and E. R]~cA~I: Lett. Nuovo Cimento, 11, 421 (1974b). MIGNANI R. and E. R]~cA~I: Lett. ~uovo Cimento, 9, 357 (1974e). MIGNA~I R. and E. R]~cA~I: Lett..Nuovo Cizaento, 9, 367 (1974d). MIGNANI R. and E. R]~CAMI: Nuovo Cimento B, 21, 210 (1974e). MIGNANI R. and E. R]~cA~I: Gen. Rel. Gray., 5, 615 (1974]). MIGNANI R. and E. R]~CAMI: Int. J. Theor. Phys., 12, 299 (1975a). MIGNA~I R. and E. R]~CA~I: _~uovo Cimento A, 30, 533 (1975b). MI~NANI R. and E. R]~CA~I: Lett. :Yuovo Cimento, 13, 589 (1975e). MIGNANI g. and E. R]~CAMI: Phys. Lett. B, 65, 148 (1976a). MIGNANI R. and E. R]~CA~I: Zett. JVuovo Cimento, 16, 449 (1976b). MIGNAI~I R. and E. R]~CAMI: Lett. 2~novo Cimento, 18, 5 (1977). MIGNANI R., E. R]~CAMI and U. LOM]SAI~DO: Lett. Nuovo Cimento, 4, 624 (1972). MiL]~WSXi B.: private communication (1978). MILLIOn J. G.: Phys. Rev. D, 19, 442 (1979). MINI~OWSl~I H. : Space and Time: address delivered at the L X X X Asse/~bly o/ German Scientists and Physicians (Cologne, September 21, 1908). MOLL]~R C. : The Theory o/ t~elativity (Oxford University Press, Oxford, 1962), p. 234. MooR]~ R . L . , A. C. S. RXADH]~AD and L. BAATH: .Nature (London), 306, 44 (1983). MOS]~AL~NKO V . A . and T . V . MOSKAL~N]~O: IZV. Akad. Nauk Mold. SSR, Set..Fiz.Tekh. Mat. :Yank, p. 1 (1978). MVKV~])A N. : Completeness o/ the solutions o/ the Ma]orana equations, preprint (Tara Institute, Bombay, 1969). Mv~PHY J. E. : Taehyons /ields and causality, preprint (Louisiana State University, New Orleans, La., 1971). MYSAX L. and G. Sz~K~]~s: Can. J. Phys., 44, 617 (1966).

NAMBT: Y.: Prog. Theor. Phys., 5, 82 (1950). NARANAN S.: Lett. J~uovo Cimento, 3, 623 (1972). NA~LIKAt~ J . V . and S.V. DHVRA~DEA]~: Pram~na, 6, 388 (1976). XA~LIKA]~ J . V . and S.V. DHVnA~DHAn: Lett. Nuovo Cimento, 23, 513 (1978). N~LIXA]~ J. V. and E. C. G. SUI)A~SEA~: MOrt. Not. R. Astron. Soe., 175, 105 (1976). N]~%MAN Y. : in High-Energy Astrophysics and its Relation to Elementary Particle Physics, edited b y K. Bm~CH]~ and G. SETTI (MIT Press, Cambridge, Mass., 1974), p. 405. NEWTON ]~. G.: Phys. Rev., 162, 1274 (1967). N:~WTO~ R . G . : Science (Am. Ass. Adv. Sci.), 167, 1569 (1970). NIELS]~N H. B. : in ffundamentals o/the Quark Model, edited b y Y. BAI~BOI/Ir and A. T. DAVIES (Scottish Universities Summer School, 1977).

172

~. ~CAMI

NIeLSeN II. B.: in l'achyons, Monopoles, and Related Topics, edited b y E. R ~ c ~ r (North-tIolland, A m s t e r d a m , 1978) p. 169. NrEr~s~,;~- H. B. : private communications (1979). N~LS~,:N H. B. and M. MI.~-O~l~rA: preprint NBI-HE-78-10 (Niels Bohr Institute, Copenhagen, 1978). NIELSen" N. K. and P. 0La~SE_~: Nucl. Phys. _B, 14.4, 376 {1978). NISHIOKA M.: Hadronic J., 6, 794 (1983). OLXlrOVSI