classical theory of tachyons (special relativity extended ...dinamico2.unibg.it/recami/erasmo...

RIVISTA ])EL NUOVO CIMENTO ¥OL. 4, ~. 2 Aprile-Giugno 1974

Classical Theory of Tachyons (Special Relativity Extended to Superluminal Frames and Objects).

]~. I~ECAMI

I s t i t u t o di F i s i e a Teor ica dell' Un iver s i t h - C a t a n i a

I s t i t u t o N a z i o n a l e d i .Fisica N u e l e a r e - S e z i o n e di C a t a n i a

Centro S i e i l i a n o di l~is ica N u c l e a t e e di S t r u t t u r a della M a t e r i a - C a t a n i a

l~. I'~IIGNANI

I s t i t u t o di .Fisica del l ' U~d~:ersit~ - R o m a

(ricevuto il 2 0t tobre 1973)

210 1. Foreword. 211 2. Historical remarks. 212 3. Special relativity revisited: logical introduction to Superluminal reference

frames. 212 3"1. Postulates. 213 3"2. Duali ty principle. 214 3"3. Four-vector properties. 215 3"4. Superluminal inertial frames. 215 3"5, Generalized relativistic transformations. 217 3"6. Four-momentum. 218 3"7. Conservation laws. 218 4. Group G of g~'neralized Lorentz transformations (GLT). 218 4"1. The group elements. 219 4"2. Tensors. 220 4"3. Transcendent transformations. 222 4"4. GLT's matricial form. 223 4"5. Physical meaning (if the four subsets of G.

227 4"6. Par~meU'ization of the elements of G. 229 5. Generalized velocity composition law. 231 6. Geometrical interpretat ion of GLT's. Comparison of space and time

intervals. 235 7. Ant imat ter and tachyons. A (~third postulate,): the Dirac-Stiickelberg-

Feynmau-Sudarshan <~reinterpretation principle )) (RIP). 235 7"1. Bradyons, luxons, tachyons. 236 7"2. Four-momentum space. 238 7"3. The ((reinterpretation principle)): the thi rd postulate.

209

2 1 0 E. R~CAMI and ~. MIGNANI

241 7"4. Ant imat ter and matter.

242 8. C/5T and <( RIP >~. 242 8"1. Case of bradyons. 244 8"2. Case of tachyons. 245 8"3. Case of luxons. 245 8'4. About /5, ~ and C/ST nonconservation. 247 8"5. Again about GLT's plus RIP. 248 9. The (( taehyonization rule ~. 248 10. Description of Nature, physical laws and GLT's. 248 10"1. Interactions and objects. 249 10"2. Description and laws. 251 10"3. An example. 252 11. Crossing relations, CT, C/5{/' and all that.

11"t. G-invariant amplitudes in special relativity. 11"2. Effects of GLT's on reaction process descriptions. 11"3. Case of interactions among taehyons (with I V~I> c). l l '4. Case of interactions among bradyons.

252 254 )55 256 257 260 262 262 263

12. Causality and taehyons. 13. Digression. 14. Taehyon mechanics.

14"1. Rest mass and proper quantities. 14"2. Generalized relativistic Newton's law.

264 265 266

14"3. An application. 14"4. (~ Virtual particles ~> and taehyons.

15. Tachyons in the gravitational field. 266 266 268 270 270 271 272 275 276

15" 1. In t r o duetion. 15'2. Tachyons and gravitational field in special relativity.

15"3. On the so-called <(gravitational Cerenkov radiation >> and taehyons. 16. About electromagnetic Cerenkov radiation (ECR) from taehyons.

16"1. Taehyons do not emit ECR in vacuum. 16"2. The general problem.

17. Doppler effect for Superluminal astrophysical sources. 18. Generalization of Maxwell equations.

18" 1. AutoduM electromagnetic tensor. 277 18"2. ~[agnctic monopoles and taehyons. 281 19. Note on taehyon quantum field theory. 282 20. Miscellaneous remarks. 283 21. Brief discussion of the experiments. 284 22. Conclusions.

1. - F o r e w o r d .

Pub l i shed m a t e r i a l concern ing fas ter -~han- l ight objects ( tachyons) is r a the r

wide-spread a n d a l ready conspicuous, as well as contradictory~ so t h a t a rev iew

art icle seems in order. I n so doing~ we shall confine ourselves to the classical

t heo ry of S u p e r l u m i n a l objects , where classical me a ns <(relativistic * b u t no t

, q u a n t u m mechan i ca l >>.

C L A S S I C A L T H E O R Y OF T A C N Y O N S 211

For l ineari ty purposes, we are essentially exposing the theory elements

tha t we regard as correct, with large reference to our own contributions. There-

fore, m any papers are only briefly discussed. Some other works, tha t appear ~o us as inessential or incorrect, are not even mentioned, for compactness' sake.

The mMn point we shM1 call a t tent ion to is generalization of speciM-relativ-

i ty theory to Superluminal (inertial) reference frames and tachyons, since according to us this consti tutes the only sound basis allowing a self-consistent development of the classical theory of faster-than-light objects.

Another point we shall investigate and clarify refers to the so-called (( causal paradoxes ,~; in fact they, even if easily solw~ble, gave rise to some confusion in the literature.

Par t icular a t tent ion shall be paid as well to rend explicit the symmetries

required by (( extended re la t ivi ty >): for example, the (( crossing relations )> and the ~/5~ theorem will be derived even for usual particle scattering (provided

their interact ion is rel~tivistically covariant). Moreover, extended relat ivi ty will be shown to clarify other points in s tandard physics, such as the connection between m:~tter and ant imat ter .

In extending mechanics and electromagnetism to tachyons, an interesting point will arise, in connection to generalized (( Maxwell equations ~>, tha t suggests a close link between the so-called magnetic monopoles and tachyons.

The experiments so far performed looking for tachyons are criticized. In :~ !irst reading, Subsect. 8"4 (iii) and Sect. 13 may be skipped.

Most of the present work is originM, either in the substance or in the form.

2 . - H i s t o r i c a l r e m a r k s .

Even in pre-rela~ivistie t imes the possibility of faster-than-light particles (FTLP) fascinated physicists ' minds. Among the first scientists occupied with this topic, let us ment ion Ttio~IsoN [1]~ HE_tV[~IDE [2], DE: COUDnES [3], and in part icular SO)[3IEm'ELD [4, 5].

Together wi~h re la t ivi ty theory [6], however, the conviction spread unfor- tuna te ly tha t light speed in vacuum was the upper limit of any velocity,

the early century physicists being misled by the evidence tha~ usual particles

(bradyons (*)) cannot overcome tha t velocity. They behaved like Sudarshan's

imaginary demographer s tudying the popnlalion pat terns of the Indian

subcontinent [10]: <, Suppose s demographer calmly asserts tha t there are no people Nor th of the HimMayas since none could climb over the mountain

ranges! That would be an absurd conclusion. People of central Asia are born

(*) The word (~bradyons, (from the Grcck {~&~, slow) for usuM (v<e) particles has been independently introduced in ref. [7-9].

212 F. RECAMI and R. MIG~A~I

there and live there: They did not have to be born in India and cross the moun- ta in ranges. So with F T L P ' s ~.

Moreover, TOLMA~ [11] believed to have shown, in his old (( paradox ~), tha t the existence of Superluminal particles allowed information transmission into the past (antitelephon).

Therefore one had to wait unti l the sixties before seeing our problem re-examined, by WIGNER[12], SCIIMIDT[12], TA~AIt:A [13], TERLETSKY[14] and above all by SUDARSHAST and co-workers [15]. After ref. [15], a number of

people started s tudying the subject, among whom (in USA, e.g.) FEIN-

BEnG [16], who introduced the te rm (( tachyons )~ (from the Greek ~Z,~q, last)

for FTLP ' s . For (~ particles ~> travelling at the light speed, in ref. []5] the

name (~ luxons ~) ([) has been proposed. The first experimental quest for tachyons was performed by ALV_~GER and

co-workers [17].

3. - Special relativity revisited: logical introduction to Superluminal reference

frames (*).

The very special-relativity theory, when approached with unbiased apti-

tude, will be shown to be able [18, 19] to describe both subluminal (s) and Superluminal (S) reference frames, as well as bo th bradyons (B) and

tachyons (T). Cf. also Sect. 10.

3"1. Postulates. - Let us set the relevant postulates of special re la t ivi ty in the following form [19, 20] (**):

Postulate 1. Space-time is homogeneous and space is isotropic.

Postulate 2. Physical laws of mechanics and electromagnetism are required [6] to be covariant when passing from an inertial frame to another frame in rectilinear, uniform relative motion (principle o/ relativity). We complete

this postulate, by adding tile reciuirement tha t the vacuum is covariant when

changing inertial frame.

:Notice explicitly tha t no mention of a limiting velocity is made in the foregoing, namely in the (( principle of re la t ivi ty ~ (PR). 111 this context, therefore, given a certain inertial reference frame, the class {I} of the inertial frames

a priori consists of all the frames moving with constant relative velocity u,

where -- c o < /ul < + ~ .

(*) Cf. also Sect. 10. (**) A third postulate (the ~reinterprctation principle ~>) will be added in the following: see Subseet. 7"3.

CL&SSIC&L THEORY OF TACHYONS 213

F r o m only the Pos tu la tes 1 and 2--i.e. wi thou t in te rven t ion of a n y as-

sumpt ion abou t the inva r i an t cha rac te r of the l ight speed [21] (*)- - the remain ing

(( principles )> of special r e la t iv i ty m a y ac tua l ly be deduced, as the ((reciprocity

principle>>[20], the l inear i ty of t r ans fo rma t ions [20] be tween two f rames

J~, ]2~{I}, and the existence of an inva r i an t square speed. Namely , f rom the

above-wr i t t en pos tu la tes it f o l l o w s - - b y ex tend ing the procedure in ref. [ 2 0 ] - -

t h a t a real q u a n t i t y exists, hav ing the d imensions of a ve loc i ty squared and

showing inv~riant va lue for all f rames ] ~ { I}:

(1) c ~ - - i nva r i an t q u a n t i t y .

3"2. Duality principle (**). - At this point , exper ience is known to have

u n d o u b t e d l y yie lded t h a t the inva r i an t speed c is real, and ac tua l ly is the

speed of l ight :

(2) c = speed of l i gh t .

I n the cases in which c m a y be a s sumed to be (practically) infinite, we would of course get in the fol lowing the Galilean physics.

Le t us now choose a par t icu lar inert ial f rame so. The light speed c - -because

of its i n v a r i a n t - q u a n t i t y cha rac te r - -Mlows an exhaus t ive par t i t ion (**) of f rames

J e{1} in two subclasses {s}, {S} of f rames h a v i n g speeds u < c and U > c relative to so, respect ively. Fo r s implici ty, in the fol lowing we shall consider

ourselves as (( t he observer so >). F r a m e s s c {s} will be called subluminal , and

f rames s ~ {S} Super luminal . The relat ive speed of two f rames s~, s~ (or S~, S~)

will a lways be smaller t h a n c; and the re la t ive speed of two f rames s, S will

a lways be larger them c. The i m p o r t a n t point is t h a t the above, exhaus t ive

pa r t i t i on is i nva r i an t when So is m a d e to v a r y inside {s} (or inside {S}); on the

con t ra ry , when we I)ass f rom So E {s} to a f r ame So ~ {S}, tile subclasses {s},

{S} are in te rchanged one wi th the o ther [19, 22, 23] (***). At the present t ime,

we neglect lumina l fraHleS (u U - c) as (( unphys ica l ~>, even if ma thema t i ca l

use of ( ( inf in i te-monlentum f rames >~ has spread ou t r ecen t ly (***).

(*) As early as 1911, it has been shown [21] (in the framework of standard relativity) that the assumption of the existence of an invariant velocity is not necessary in order to derive the ~, Lorentz transformations ,). (**) See also Sec~. 10. (***) The most original feature of paper [22]--i.e. introducing unidircctionality in time for bradyons and ill space for tachyons--docs not nleet our agreement, since irreversibility seems essentially to be a statistica7 output. The correct part of that article, confined to a bidimensional space-time, inainly coincides with the results in ref. [18]. (***) In any case, from such frames the space-time should appear as a bidimensional space, projection of a suitable 3-dimensional hypersurfaee onto a plane normal to the luminal ray direction. On such a plane, both objects and photons appear immobile. Of course, these frames are useful, e.g., for investigating the (at rest) (( partons )>, otherwise relativistic subpartieles.

214 ~ . R:ECAMI a n d 1~. M 1 G N A N I

We can immediately deduce a (( dual i ty principle)) [18, 23-25], which may be briefly put in the form (, the terms B, T, s, S do not have an absolute meaning,

bu t only a relative one ~>. Disregarding the fact tha t the (~ dual i ty principle )) follows from the previous (extended)<(principle of re la t ivi ty ~) led MARI- WALLA [26] tO assume tha t bradyonic and tachyonio characters are absolute,

and therefore to the impossibility of defining Superluminal frames. For a crit- icism of this paper see also ref. [19].

3"3. Four-vector propert ies . - Firs t of all, let us explicitly notice tha t we shall neglect, as in the following, space-time translations, i e. consider only

restricted Lorentz transformations. Al l frames are supposed to have the same

event as their origin. Also remember that , in Minkowsky space, bradyons are characterized by timelike world-lines, luxons by lightlike world-lines and tachyons by spacelike world-lines.

~Now, the linear t ransformations L, making transitions between two (inertial) frames ]1, ]2 E{I}, and satisfying Postulates 1 and 2, must preserve the four-

vector magnitudes, apar t f rom the s ign [23, 18, 22]. This point is proved, e.g., in ref. [27], us u consequence of light speed in-

variance. Therefore, t ransformations L between two inertial frames f~, f~

must be such tha t (*)

(3) 12 /2 Xo - - x = ± (Xo ~ - x ~1

for every four-vector x _-- (xo, x) , where x means either four-position, or four- momentum, or four-velocity, or four-current density, and so on. In the par- t icular case of chronotopical vectors, eq. (3) will read

(4) c°~t,~ _ x,~ = ~ (c~t ° ' - x ~) ,

or be t te r (by using G, = ~ and Einstein 's notations)

(5 ) c2 t '2+ (ix')2 = ~ [c2t2+ (ix)2].

In the following, we shall always avoid explicit use of a metric tensor (**) - - a s well as in eq. (5) - -by writing the generic chronotopical vector us x _=_ (Xo, Xl, x2, x~) _~ (ct, ix, iy , iz). Let us repeat tha t the t ransformations L will be l inear [20, 28].

(*) We choose throughout this work the metric ( q - - - - - - will be adopted when convenient. (**) See for instance ref. [29].

). Natural units (c = 1)

CLASSICAL Ttt~ORY 01~ TACttYONS 21~

3"4. Superluminal inertial ]rames. - I t is easy to convince ourselves tha t

the sign plus in eqs. (3)-(5) refers to the usual [29, 28] case of subluminM rela-

t ive speeds, whilst the sign minus has to be chosen for SuperluminM relative

speeds [23[.

The Postula tes 1, 2 allow considering f rames s and S on an equivalent (*)

footing []9, 23]. Therefore, even Superlnminal observers S are supposed to be able to fill space with meter-s t icks and (synchronized) clocks, M1 at rest

relat ive to S, %hnt is to say to build up their (( la t t icework of meter-st icks and clocks >~ [30].

F rom the requi rement tha t SuperluminM frames be physical [] 9], it follows

of course t ha t objects mus t exist, which are a t rest relative to S and tachyons

relat ive to f rames s. F r o m the fur ther fact t ha t luxons t show the same

veloci ty to any observer s or S, it can be deduced tha t a b radyon relat ive to

an S, B(S), will be a t achyon relat ive to any s, T(s), and vice versa:

(6) B(S) = T(s) , T(S) ~-- B(s) , [(S) = t~(s).

This accords [18, 23-25] wi th the dual i ty principle, that we are going to complete

by adding t h a t [23] <(frames S are supposed to have at their disposal exact ly

the same physical objects as f rames s have, and vice versa >>.

In conclusion, when f rames s, S observe the same event, (~ t imelike >> vectors

t r ans form into (( spacelike >> vectors and vice versa, in going f rom s to S, or

f rom S to s. On the contrary, it is well known tha t usual Lorentz t ransforma-

tions, f rom s~ to s2, or f rom S~ to $2, preserve the four-vector type. One is

therefore allowed to say t ha t subluminM LT's, i.e. usual Lorentz t ransforma-

t ions (LT), are expected to be such the t

(5a) c2t'~+ ( i x 'V= ÷ [c~t~+ (ix) ~] ( ~ < 1),

while (~ Superluminal Lorentz trans]ormations ~> (SLT), f rom s to S or f rom S

to s, are expected to be such t ha t

(5b) c~t':+ (ix')~ = - [c2t~-÷ (ix) ~] (fi~> 1).

3"5. Generalized relativistic trans]ormations. - The case (5a), i.e. the case

of usual LT's , is ve ry well known. Let us invest igate case (5b), i.e. the case

of SLT's . This problem has been independent ly approached, for the bidi-

mensionM ease, in ref. [18, 22] and, for the four-dimensioi~M case, in ref. [19].

In the bidimensionM space-t ime case, P A ~ E R [18] showed tha t the linear

t ransformat ions P satisfying eq. (5b), for /~2~ 1, are unique; in ref. [19] it

(*) See also Sect. 9, 10.

216 ~. RtlCAMI an4 R. MIGNANI

has been observed tha t those P m a y be s imply wri t ten [19]

x - - u t t ' t - - u x / c ~ { u~ ~ - - ~ - - \ ] ~ - ~ > 1 (7) x ' = ± @ ~ . ~ = ~ ,~,//~2 1 /~ "

By the way, let us ment ion t h a t PARKEI~ chose only the sign m i n u s [18]. At

this point , the problem of the generalized L T ' s is tha t of finding new (~ genera-

lized Lorentz t ransformat ions ~ (GLT), tha t reduce to the usual ones for f12 < 1,

and to Pa rke r ' s in the case of D2> 1 and two dimensions. This problem

has been solved in ref. [19, 23]. In the par t icular case of a boost (collinear

motion) along the x-axis, the GLT ' s read [25] (for - - o o < f i < co)

(8)

x - - ut t - - u x / c 2 t' x , = ± =

V l - - f l 2 z , = :j_ z ~ i l - - f l 2 Y'= ±Y Ii-fl~l' I1-~1

( - ~ < f l < + ~ ) .

We shall see t ha t the double sign in eqs. (8) is required b y the inver t ib i l i ty of

GLT's . I n Subsect . 4"6 we shall rewri te eqs. (8) i n a more compac t /orm, b y

subst i tut ing a suitable generalized funct ion for the double sign. Here we want

explicit ly to emphasize t h a t the GLT's , th rough their sign, do depend also on

the sign of ft. We shall come back to this point in Subsect. 4"6. I t is worth-

while to notice ~hat a pr ior i ~ v / f l 2 - - 1 = q = i V ' l - - f i ~, since ( ± i ) ~ = - - 1 . Always we consistently choose the sign m i n u s , for the reasons we shall see.

I t is as well understood tha t 4- ~ / ~ f 1 2 represents the u p p e r hMf-plane solu-

t ion for f12> 1. Let us consider only SLT's , i.e. eqs. (8) in the case fi,2 :> 1 ; they are such t ha t

(9) { c~t'2 4- ( ix ' )2= (ict)2 4- x ~ ,

(iy,)2 4- (iz,)~ = y24- z ~ '

and therefore satisfy eq. (5b), which m a y read (*)

(5b) c~t '~ 4- ( ix ' )24 - ( iy ' ) 2 4- (iz ') 2 = (ict) ~ 4- x ~" + y~" 4- z ~ •

I t is clear tha t imaginary units entering last two of eqs. (8) essent ial ly

record t ha t SLT ' s change the four-vector types so tha t in the present case

(*) A priori, we might also have a rotation of space (i.e. of axes y, z) around the motion line (axis x) when (~ crossing ~) the light-cone. Notice that we have a discontinuity at the light-cone, since the relative speed of two fl'amcs, travelling at velocities c - - s 1 and c + e~ with respect to us, cannot tend to zero, but on the contrary runs from c to infinity.

CLASSICAL THEORY OF TACHYONS 217

y , 2 = _ y2 z , ~ = _ zO.. Of course, both s and S will observe real quantities.

Analogous is the fact tha t space co-ordinates are well supposed to be ob-

served as real, even if we are writ ing a generic vector as

(10) x = (xo, x,, x.~, xa) ~ (ct, ix, iy, iz) .

Namely, every (subluminal or Sul)erluminal) observer will see his owll space- t ime as Minkowskian (pseudo-Euclidean), a.nd will use the same metric, i .e . - - for

e x a m p l e - - t h e convent ion x (ct, ix, iy, iz). However , if an observer s wants

to look at space-t ime through the observat ions of am observer S, or vice versa,

then he will have to do his calculat ions by using the mul t ip l ier i (i.e., by

using the convent ion x' (ict', x ~, y', z'), without chang ing - -however - - the cor-

respondence between norm-sig,~ and ]our-vector type): in o ther words, some i 's

ha~e to enter the relations bctween the two sets (t, x, y, z) and (f , x' , y', z')

of observ:~tions! (') Notice th:~t such considerat ions app ly to every inertial

fr:~me ]~{ I } , which hav'c all been seen to be equivalent (see also Sect. 9, 10).

I t shouht be clear theft we do not have metric changes when passing from

sublumin'~l to Sul)erluminal fr~rnes (or vice versa). Moreover, bo th observers s

and S will use the same metric when observing both B's and T's. Contrary

assumptions about this delicate point led OOLDONI, in a few very recent

works [31 ], to deal with i~tfinitely re:my metrics and buihl a theory of tachyons

essentially nonintcract ing with usual 1)arti(~les. Besides, (~ Lorentz t ransforma-

tions )) as generalized by tha t author do not form '~ group. Analogously ALA-

GAR-I~AsIANUJAM And NA3IASIVAYAM [31], t ry ing to overcome the difficulties

connected with the s~me point, have been induced- - in an otherwise nice pa-

p e r - t o make assumptions th:~t violate the relativistic coltditions (5).

We shall ver i fy tha t the GLT's , eqs. (8), do on the contrary consti tute a good generalization of l~T's.

3"6. Four-momentum. - I f we write condition (5b) in the fou r -momen tum

space

(11) E , 2 _ p , 2 _ _ ( E 2 _ p 2 ) (fl~ > 1),

one can immedia te ly see tha t SLT's operate (*) a <~symmetry~) with respect to the

dight-cone~> = p-~ 0, or be t te r t ransform (**) two-sheeted hyperboloids p~ = m~ > 0

into single-sheeted hyi)erboloids p2 = m~ < 0. Of course in the te t ra impulse

space, B ' s are characterized by tim(dike four -momenta , ~'s by lightlike four-

momen ta and T ' s by spacelike four -momenta .

(*) Note added in proo/s. - This (hdicatc p o i n t will be b e t t e r cons ide red in :R. MIG~ANI a n d E. RECA~rI: Lett. Nuovo Cimento, 9, 357 (1974). See also G. C. WICK: p r e p r i n t CERN-73-3 (Geneva , 1973); R. :PENItOSE a n d M. A. H. MAc CALLVM: Phys . Rev. 0 C, 245 (1973). (**) At the finite. ~t least.

15 - Rivisla del Nuovo Cimento.

218 ~. RECAMI and R. MIGNANI

3"7. Conservat ion laws. - F r o m Pos tu la te 1 the usual conservat ion laws (of energy, m o m e n t u m and angular m o m e n t u m ) follow for an isolated sys tem

even in generalized re la t iv i ty theory, i.e. even when Superluminal velocities

are allowed. For instance, with the help of the results of Sect. 6, the deri-

vat ions contained in the second of ref. [29] m a y be immedia te ly generalized.

Therefore, the G L T ' s (of the group G of the nex t Section) will preserve the

val idi ty, inside any isolated system, of energy, m o m e n t u m and angular-

m o m e n t u m conservat ion laws, in accord with the generalized definitions of

Subsect. 7"2.

No different fur ther results m a y be derived f rom Postula te 1 and consider-

a t ion of space- t ime translat ions. This it the reason why we are explicit ly

invest igat ing only the group of (~ space- t ime rotat ions ~.

4. - Group G of generalized Lorentz transformations (GLT).

4"1. The group elements. - The GLT ' s form a new [19, 25] group G of

t ransformat ions in four-dimensionM space-t ime. This group is the group of

all (( rotat ions )) in Minkowsky space. I n our terminology, (~ G-covariant ~) will

mean eovar iant under the whole group G.

Let us represent the t ransformat ions L b y 4744 matrices, and consider

first, for simplicity, a universe free of (~ charges ~)(*). I f we call usual proper,

or thochronous (homogeneous) L T ' s

(12a) A< ~_ A(fl z < 1 ) ,

then it m a y be realized [23] (see eqs. (8)) t h a t SLT ' s have the form

where

(125)

SLT : -4- iA> ,

A> =_ A@ > 1)

are matrices (**) formal ly identical to the A<'s , bu t correspond to values of

f5 with ]/5] > 1. I t is easy to observe t ha t

(13a) [A<(/5)] -~ ~ A<I(/~)---- A<(-- fl),

(13b) [ iA> (fl)]-~ ---- - - iA>~(fi) = -- iA> (-- ~) .

(*) In the present work, the word (~ charge ~) is used in its widest meaning. (**) Let us explicitly notice that matrices A> are complex.

Thence

but on the c o n t r a r y

[iA>(fl)]. [--iA>~(fi)] = 1 ,

[iA>(/~)]. [iA>~(~)] - - 1

and the general ized [19] g roup G conta ins as an e lement the total inversion

(or (~ s trong-ref lect ion ~)) opera tor

(14) P T _ --'1 .

Precisely, b y consider ing successive appl icat ions of G L T ' s of t y p e A< and iA>,

it is easy to realize t h a t the group G of the G L T ' s consists of four subsets

(15)

where

(15') { y i {A<},

~ o t i c e t ha t , if L c G, t h e n also -- L : ( P T ) L ~ G. I n the following, we shall

s imply denote b y L the generic e lement (GLT) of G.

All matr ices are un imodu la r [19], and

(16) det L = -~ 1 , VL ~ G .

The only proper subgroups of G are gf~ (the proper , m~hochronous Loren tz group), and & o _ ~Lf~ w ~.~.

Las t ly , let us emphas ize t h a t G is a complex- transformation group. This

accords with the fac t t h a t no real generMizat ion of L T ' s exists [32] for Super-

lumina l velocities, which satisfies condit ions (3)-(5). See also ref. [33], where

some considerat ions abou t t a c h y o n s are p u t fo r th in the case of a ]inite space-

t ime (*).

4"2. Tensors. - I t m a y be useful to check how G L T ' s ac~ on tensors.

Le t us confine our a t t e n t i o n to 4 X 4 matr ices T , which are (second-rank)

tensors (**) [34] unde r G:

(17) = (L e G) .

(*) See ref. [105] as well. (**) We also eliminated the distinction between ~ covariant ~ an (~ contravariant ~ components, by our suitable intro4uction of imaginary units in the spatiM-componen~ definitions. In eq. {17), and ~he following ones, a summation is understood over tb6 repeated indices. See, e.g., ref. [34], p. 156.

220 E. RECAMI a n d R. MIGNANI

Even if

(18) L ~ L ~ = ~: 5 ~ , L r L = :t: 1 (fi2<> 1)

(where the superscript T means transpose)--so tha t SLT's do invert the tetravector square magnitude s iga,-- the so-called tensor invariants [35], T 2,. and e ~pT T~Z (where e~.~z is the completely ant isymmetric unit 4-tensor), are actually left unchanged even by SLT's:

(19) T' ' (f12 X 1). e~veo u~ Tea = eaZro T=fl Ty~

4"3. Trascendent trans]ormations. - I t is easy to recognize tha t a correspond- enee exists between (subluminal) LT's from a frame so to a frame s moving with velocity u (0 c ) , in the sense tha t

(20) -- iA>(U, O, (p) = /~+(0 , q)).A<(c~/U, O, q)) (u, U > 0),

/~+ being a matr ix independent of the magnitudes of velocities u, U, and dependent only on their common direction (that we indicated by the two an- gles O, ~). In the case of col]inear motion, one has [23]

(20 his) -- iA>(U) = K+'A<(c~/U)

where now A < , - iA> are boosts, K+ ~-/~+(0, 0), and

~+(0, ~) = R(O, ~o).K+.R-I(O, ~) ,

(u, U > O),

the matr ix R(O, q~) being the space rotat ion which brings u, U parallel to the x-axis.

The matrices K+, K+ represent (( transcendent SLT's ~> (we may call them also (~ transluminal LT's ~) for the reasons we shall see later). In fact, for /7-->-}- c~, eqs. (20), (20 bis) become

(20') L ( + c~, 0, ~) ~ L + ~ ~ l im [-- iA>(U, O, ~ ) ] = K+(O, ~),

L ( + ~ ) ~ L+~ =_-- iA>(+ c<)) ~ K+.

This agrees with the fact that , if a taehyon moves with a velocity v ~ v~ relative to us, it will appear with divergent velocity to the observer s" having the (eollinear) critical velocity u - u~ ~-c2/v relative to us (see Fig. 3 in the following).

CLASSICAL TIt~0RY OF TACHYONS 221

For negative velocities, i.e. for - - c ( U) = K_-A<(c: /U) (u, U < 0 ) ,

w h e r e n o w

(21 bis) L ( - ~ ) - L_~ --=-- i A > ( - - ~ ) ~ K .

In the simple case of eollinear motion along the x-axis (u = u , , U ~ U~),

one gets the t ranscendent SLT's represented by the 4X 4 matrices

(22) K ± ---- = - - ( T a2 0 i(~o) ,

0 - - i 6 o

where ~0 ~ and ~2 ~ are Pauli matrices.

In ref. [18, 23] it was argued tha t K+, K should be equal, since instan- taneous motion ought not to be assigned a direction (alollg the motion line).

F rom eqs. (20), (21), it appears tha t in special relat ivi ty this seems not to be the case (cf. also Fig. 2, in the following).

In collinearity ease, the matrices K~ transform y(u), z(u) into --iy(v~/u), - - iz(c~/u) and

(23a) K+ :

x(u ) > x ' = ÷ e . t (c~ /u) ,

t(u) > t' = + x ( c V n ) C

(23b) { x ( u ) > x ' = - - c . t ( c ~ / u ) ,

1 ; = : t (u ) > t' - x (c~ /u) C '

as will be reverified when geometrically interpret ing the GLT's.

I t is apparent f rom the foregoing tha t the (~ dual i ty principle ~> is character-

ized by the fact t h a t - - w i t h reference to a f rame s0--there is a one-to-one correspondence [24, 19] between observers with speeds u and U : c~/u. Precisely, the (( symmet ry ~> (see Fig. 1) between subluminal and Superluminal frames is a part icular (( conformal mapping ~> (an inversion) [23, 25, 22, 18, 26]:

(24) u + + c 2 / u .

In such a mapping, the velocities u---- U---- c are the united ones (as required),

222 E. ~ECAM~ and R. MmNANI

a n d v e l o c i t i e s zero, i n / i n i t y c o r r e s p o n d t o e a c h o t h e r (u, U > 0 ) :

( 2 4 b i s ) u - - - - c e - ~ U = c , u = O ( - ~ U - - - - - c o .

P

B t

- - c = - ) U ~

S o

Fig. l . - Represen ta t ion of the conformal mapping (inversion) u~-~ e2/u in the sim- plified case of iner t ia l f rames v i th coll inear veloci t ies u -- u~ ~ 0 re la t ive to a f rame s o (see the text) . Since [u I <> c and we have to deal also wi th the t ranscendent f rame, we pro jec t f rom a pole P the axis u onto the circle hav ing u = -V e as d iametr ica l ly opposed points . The chords A B (where A, B refer rcspec t ive ly to a subluminal ve loc i ty u and Super luminal ve loc i ty U - c2/u, i.e. to corresponding veloci t ies in the conformal mapping) are normal to the axis u.

The latter recalls us the known fact that the divergent velocity plays for tachyons the same role played by the null velocity for bradyons [21c].

4"4. G L T ' s m a t r i c i a l / o r m . - T h e G L T ' s , i.e. eqs . (8)~ in t h e i r m a t r i c i g l f o r m

x ' = L x (25)

explicitly read

j t 01)it ) (25') l i x ' ~ ~ : - - i f l Y 7 0 ix = • ( - ~ < ~ < ~ ) , 0 0 5 iy

where the relative motion is collinear (along the x-axis), and where

~ + ( [ 1 - ~ 2 [ ) ~ and ~ + V l ~ - ~ l "

In Subsect. 4"6 we shall see that the double sign is better substituted by

( l - - t g ~ ) c o s ~ _ cos¢.6~ ~=V(V) = i i _ t g 2 v l . l c o s ¢ ] lcos Vi

C L A S S I C A L T H E O R Y O F T A C H Y O N S 223

with t g 7 ~ ft. In the case of two frames moving along ~ generic line, one has

(26) L ----

• U x . U u . Uz c c ' 7 c

C U 2 U 2 ~2

• U 2 _ _ ~ - y ~ y Uz - - i ~ ~ g U 2 U 2 ot U ~

2 • Uz ~x~z UvUz Uz

(p<> 1),

where ~ ~ 7 - - 5. In eq. (26), one should pay at tent ion aot to confuse the origin

of imaginary units (i), coming from the chosen metric, vdth that of imaginary

units (contained in 5) intrinsic to the SLT's.

4"5. Physical meaning o/ the ]our subsets o/ G. - In order to illustrate the

physical meaning of the four subsets in eq. (15), let us confine for simplicity

our ~ttention to boosts ulong the x-axis. We can represent velocities u ~ u~ X 0

(rel~t.ive to ~ frame s~) ~long ~ line: see Fig. 2. Since we h~ce to deal also with

x(-~) iA>(-U)~iA~(U) ~ iA>(U:c2/u) --iA>(--U)~-iA-l>(U) " - - . . _ _iA>(U=c2/u)

6~s ' ~3 6 ~ 2

\ / , , - , : / -,<i 17I I

A <(-u)=- A-< (u) A<(u) A<(-u)~A<)(u) x(+l-~-~ A Au) a) b)

Fig. 2. - Il lustrating the physical meaning of the four subsets (see cqs. (15)) that form the generalized Lorentz transformation group G in four d.imensions. For simplicity, we consider only franlcs with collinear velocity u ~ % relative to a framc s o. Both Figures m u s t be road only in the counterclockwise sense. Notice that A<(- -u) = A<t(u), A>(- -U) A>I(U). In a) the successive alternation of the signs in the GLT's, eqs. {8), is represented. We have a change both when by-passing the <<branch point ~> P ~ and at the <<points >> u = :j:c. In b), on the contrary, it is indicated when the moving material object or frame appears to be made of matter (M) or antimatter (M), alternatively, with respect to s o. We have a change in the apparent character M/_M when the moving object--or frame <, by-passes,> the infinite-velocity point (P~).

224 ~. ~ECAM~ and ~. ~GNAN~

divergent velocities, one m a y usefully borrow a bit f rom project ive geomet ry [21c]. :Namely, it is convenient to project the axis u f rom the p o l e / ) (see Fig. 1) onto

the circle ~ having u z ± c as diametr ical ly opposed points (notice tha t chords

A B in Fig. 1 are normal to u). E v e r y point of y represents a f rame ], in col]inear

mot ion relat ive to so. The characteris t ic feature [23] of Fig. 1 and 2 is t h a t

the circle 7 mus t be considered as <, double ~, ei ther with a <( branch point >~

at Pco(U-~ oo) or with three (~ branch points ~> at P~ and u : ~ c. I n fact ,

along 7 we pass f rom frames s ~ (e.g., with a right-handed spatial f rame) to to ta l ly

inver ted f rames sL-~ ( P T ) s ~ (with a left-handed spatial f r ame and a reversed A

t ime axis). This could have been expected [37], since the to ta l inversion P T is nothing but a par t icular <~ ro ta t ion ~) in (four-dimensional) space-t ime, and

we have seen t ha t such a ro ta t ion m a y be actual ly achieved when one does not restr ict oneself to subluminal relat ive velocities.

I n the following (see Subsect. 7"3) it will be shown tha t in P~ ~ P( U--~ oo)

one must have a (, branch point ~) (see Fig. 2). On the contrary~ due to the

(~ discontinuities ~) across ~ - - - - ± c, one may a priori have ei ther two <~ branch

points ~, or no (, branch point ~>, at u ---- ± c. Such <( branch points ~ are mere ly

due to the double sign entering the GLT's , eqs. (8) or (25'): the previous sentence

mean tha t we m a y a priori get t ha t sign reversed ei ther a t P ( U : c o ) , or a t . P ( U : co) and at u---- :J= c.

I n ref. [23] the first a l ternat ive was chosen for s implici ty 's sake, in the mean-

t ime however suggesting [23] t ha t the theory could result to be clearer, smoother

and more <, continuous ~ if one adopts the second al ternat ive. To get convinced

about t h a t point, it is enough to inspect the functions I1-- fl21:~ and ]1-- f121½ , since the double sign in eqs. (25'), (8) is due to the presence of the double- valued factor 1/~/~ -- f12 I.

Here~ we shall adopt the choice with three <(branch points )~, for the

above-ment ioned reasons. See :Fig. 2.

I f A< is the LT f rom So to a f rame s ~ l moving with velocity u ~ 0 ,

then the (conformally correspondent) t ransformat ion - - i A > will connect So to

the f rame $2 - 2 moving with velocity U = c 2 / u ~ 0 with respect to so and

with infinite velocity with respect to s~ (see Subsect 7"3).

By the way, the corresponding f rames s, S, such t ha t in Fig. 2 s S ± u ,

have infinite relative velocity, as shown in the following by means of the

generalized veloci ty composit ion law. The SLT f rom s to S will be in this case

L+~ ---- K+. Thus, with reference to So, we m a y say tha t K+ operates a transi-

t ion f rom s(u) to S(c2/u); in par t icular (with self-evident symbols)

(27) SLT s(c--~) >S c~- , e

valid also in the l imit e - + 0 +. Therefore, the <( t ranscedent ~ SLT's , K± , can

be considered as transluminal: K+ operates a t ransi t ion across ~-e, and K _

C L A S S I C A L T H E O R Y OF T A C H Y O N S 2 2 ~

across -- c, in the ~bove sense (see Fig. 2). :Sow (see Fig. 3, 4), f rom S.~ we m a y

go back <~ s t ra ight ,> to t he f r ame So b y app ly ing the inverse t~r~nsformation

(-- iA>) -~ ~ + iA> ~ , in such a w a y t h a t (-- i A > ) ( + i A ~ ~) = 1. Bu t we migh t

a) b)

Fig. 3. - a) and b) correspond to the ease considered in Fig. 2 b). They must be re~d only in the counterclockwise sense, a) and b) symbolicMly, formMly depict transformations --L-l(fl) and + L-l(fl), respectively, with f l> 1 (see the texti eq. (13b); and eq. (31b) below). Notice that transformations L(O, A), L(A, B), L(A, O) correspond to transformations Z(So-+ S~), L(so-+ SB), L(so-~ So), respectively, of Fig. 6 b).

also decide to go back f rom $2 to so (~ by-pass ing ~) the transcendent ]rame Poe

(relative to So: see F ig 4 b)) ; in this case, however , we would reach the f r ame so

with all its axes reversed, (Drf)so, since (-- iA>) - ( - - iA> 1) ~ -- 1. We shall see

more clearly (Sect. 7) t h a t an ideal f rame, t h a t unde rwen t such a (( t r ip ~), would

come back wi th spat ial axes inver ted (space par i ty ) and with part icles

t rans formed into antiparticles. More generally, if LIs~)-->/) is the LT connect ing

f

X(-1)

l !b" I I 1

X(+l) O,(s o)

a) b)

Fig. 4. - a) snd b)correspond to the cssc considered in Fig. 2 b), but a.re now to be read only in the clockwise sense, a) symbolicully shows the transformation L(- - 8) = - - L-l(fl) ~ - - iA> (-- 8) ~ -- iA>t(fi) . b) symbolically shows the transformation - - L ( - - f l ) = L-~(fl) ~ + iA>(--f l) ~ + iA>~(fl). Always - - 8 < - - 1 , i.e. 8 > 1.

226

f r a m e s so ~ n d ], w e h ~ v e

L ( So -->

L(so -~

L( so -->

L(so

(28) L(So -~

L( so -~

L( so -~

L(So ->

L(So

:E, R E C A M I a n d R . M I G N A N I

1)=+A<(u),

2 ) = _ - - i A > ( c ~ / u ) ,

3 ) ~ - i A > ( - - c ~ / u ) = + i A ~ ( c 2 / u ) ,

4 ) ~ - A < ( - - u ) = - - A ~ ( u ) ,

5)~--A<(u),

6 ) ~ - + i A > ( c ~ / u ) ,

7 ) ~ - - i A ~ ( c ~ / u ) ,

S)~A~(u),

9 -=-1) - A < ( u ) .

T h e r e l a t i v e v e l o c i t y b e t w e e n sl =: 1 a n d $2 ~ 2, or b e t w e e n $7 == 7 a n d s8 ~- 8,

is i n f i n i t e , w h i l s t t h e r e l a t i v e v e l o c i t y b e t w e e n S~ ~ 2 a n d S ~ - 3, o r b e -

t w e e n ss - - 8 a n d s~ ~ 1, is

2, ( c ) (29) 1 + (u/c) 2 - 1 + (U/c) 2 U = - o .

i

- C

- -C

vr i I I

• I

. . . . l - . . . . . t I I I I

v ~ c ~ V

lg"

l \ . l

:!\

I

5

Fig. 5. - Genera l i ze4 ve loc i ty compos i t i on law, in t he case of co l l incar m o t i o n u ~ u~, t !

v --= v~, v v~. T h e p lo t r e p r e s e n t s t h e b e h a v i o u r of v as a f unc t i on of u. The ve loc i ty v is t a k e n as a f ixed positive p a r a m e t c r . L ines a) rcfer to v ~ ~ ~ c, l ines b) to v = V

e2/~> c. Sce eqs. (32) of t h e t ex t .

W h e n we cross the u-axis , we have the exchange t i m e l i k e e - + s p a c e l i k e , or

fo rmal ly a mul t ip l ica t ion b y ~ i (in the sense of Fig. 2). Precisely, at u = q- c

we get the t rans i t ion f rom { ~ A<} to ( ~ iA>} , and at u = - - c f rom { ± iA>}

to {~=A<}. W h e n we by-pass the l)ole, we get the /3~'-inversion, or be t t e r "~

mul t ip l ica t ion by -- 1. We can writ(; in an order ly w a y

The physical mean ing of this sequence, as well as the symmet r ies implied, are

shown in Fig. 2, 3, 4, 6, 11.

(M)\

A< ~ \ 8

A~ S® A< _ /~=/~(n_cp) ; , ~ (~o) , > ;/~(cp)

L

s o (~) a) b)

Fig. 6. - The sa)nc as in Fig. 2 b) when confining ourselves only to ((spatially right- handed)) frames, as seen by so, i.e. to cqs. (8) with only the positive sign. Frame s o is supposed by dciinition to ht~ve <~ right-hande4)) space axes. In a), with self-evident

~ C ~ A < . symbols, ux2 an4 L12 K+, Ust = 2u/J1 + (u/c) 2] and Lsl = 2 Of course A< ~ Lol. The point u = c~ cannot bc gone through, otherwise we could have a total invcr sion of the observe4 h'amc, b) helps understand the physical meaning of the GLT's (~ geometrical)) paramctrization, eqs. (8 bis). See (he text. See also the caption of Fig. 3.

4"6. P a r a m e t r i z a t i o n o] the e lements o] (I. - Il l the l ight of the foregoing,

eqs. (8) e:rIl be wr i t ten in ~r more c om pac t fashioi1. ~ a m e l y , the GLT~s for

boosts can be 1):rrametrized by set t ing, for /72~1,

l fi t g c f ,

(3{)) cos q> 1 tg z qv cos ~ 62

One migh t also wri te r 1 r]([3) with ~(fi) a two-vMued funct ion. Af terwards ,

eqs. ( 8 ) r e a d (/~ ;1)

(8 bis)

y ' = ( @ ) y , z ' = (nS)z

228 E. RECAMI and R. ~IGNANI

where, as previously,

/ 1 - - tg 2 - + (ll-tg v0-*.

Subluminal LT's are got for q e (~bl w ¢2), where ¢~ -- [-- ~r/4 < q < ~r/4] and

q ' ~ - [¼u < ~v < ~u]; while Superluminal ones are got for ~ c (¢3w ~b~), where ¢ 3 - - [~/4 < ~o < -~t] and (~)4 ~ [5~ < ~) < 7~.~]. At this point, we ma y write

(31) L(q~)=~(q~)5(8~)A(8)--L(~, ~,8) - - ~ < q < ~ J r .

Cf. Fig. 2, 6 and part icularly Fig. 11.

We shall give the geometrical interpretat ion of eqs. (8 bis), i.e. of GLT's, in Sect. 6. Here we want to observe the following. Due to the ansatz 8 ~ tg~ , i t holds

8) = ± v(8) ( - < 8 <

as il lustrated in Fig. 6 b). Therefore, for any L(8), we always have two

L(-- 8)'s (cf. Fig. 11). In the usual, subluminal case (-- 45 ° < q < 45°), the first one, Ls(--~v), operates re turn to the initial frame So, and is considered the actual inverse t ransformat ion; the second one, L,(Tt--~) , operates on the contrary return ¢o the frame (/ST)s 0. However, in the Superluminal case (45°< ~o < 135°), it is the second one, L~(n- -~) , tha t operates re turn to the /3T-ed frame (PT)so. Now, therefore, L s ( ~ - - ~ ) i n the actual inverse transformation: we shall call it L~(--8) tout court; the other one will of course result to be - -L~( - -8 ) . Consequently we may write

(31a) I (-- 8) ~ Ls(-- q~) ---- L-81(8) (82 < 1) ,

(31b) Ls(-- 8) -- Lz(~-- q) = L~1(8) (82 > 1) .

This does not contradict space isotropy, which requires only tha t

L ( - 8) = P L ( 8 ) ' - P - 1 = _p .L@ . p .

See also Fig. 4.

For example, we have [K+]-~= -- K_ .

By the way, remember t h a t the principles of rec iproci ty asks always t h a t

U - ~ U f ~ - - U

both for frames /, / ' with the same space-parity and for frames /, /' with di//erent spatial parities.

CLASSICAL TILEOIIY OF TACIIYONS 229

5. - General ized ve loc i ty c o m p o s i t i o n law.

Let us spend some words abou t generalization of the veloci ty composition law for speeds exceeding th:~t of light [24, 19]. I f we are working with four- velocities (as well as with four<~ccelerations), mere use of eqs. (8) is in order.

I f we insist on imrking recourse to three-velocities, then we mee t the problem

of in terpret ing the imaginary units (*). Nevertheless, f rom eqs. (8) with what-

soever sign, in the case of collinear motion (u = u~), we shall have [19, 23]

(32)

VU - - C 2 - - U V ,

, I ) ± ~ / ~ 2

I ) ± ~ C 2 C 2 - - U/~ ,

--- X

for both fi~ < 1 and /~ > 1 (see Subsect. 3"5). For instance, for the velocity magnitudes, one gets [24, 19] tha t the trans-

format ion law (GLT)

c~-v'~ (c~-v:)(c~-u~) ( 3 3 ) c ~ - (c 2 - u . v ) ~ (uz' v2' v'~ ~ c2)

holds for both B's and T ' s as seen from both f rames s and S. Equat ion (33)

is a first example of a (,G-covariant~) law. As has first been argued by

OLKHOVSKV and 1-¢EC:t~I [24], cq. (33) has the very immediate physical meaning depicted iu Table I : i) the LT ' s do not ch:~nge the character of B or T, if) the SLT's do chan?.'e it (B ~ T ) .

TABLE I. - - Extended velocity composition law /or velocity magnitudes.

n 2 < C 2 V 2 < C 2 =:~ VP2 < e 2

v 2 ~ c 2 ~ v / 2 ~_ c 2

v 2 > e 2 : > v p2 ~> e 2

~a 2 ~ e 2 ~2 > 2 ~ e 2 ~: ~ c ~ > v t2

U 2 : > C 2 V 2 < ~ c 2 : : > v ' 2 > e 2

V 2 ~ C 2 ==> V t2 ~ C 2

V 2 : > e 2 = : > v ' 2 < : c 2

Let as go back to the first of eqs. (32), confining ourselves (see Fig. 5) only to

v = v~. In the following (Sect. 7), it will be shown tha t a t aehyon T m a y al)-

(*) See the note added in proofs on p. 217.

2 3 0 n. R~.eA~I a n d R. ~IIGNAI~I

pear as a (( part icle ~) (e.g. in the ~ init ial state ~ of a certain reaction) to ~n observer s and as an (~ ant ipart ic le ~) (in the opposite (( s ta te ~, i.e. outgoing ]rom

that interaction region) to other observers s ' [9, 38, 39]. This fact is here (partly)

represented b y the behaviour of the first of eqs. (32), which yields [23] for

~ C 2 and v ~ o

(34)

!

for u ~ c~/v~ --> v x -= c~ ,

for - - c sign (v:) ~- -- sign (v) .

Another peculiar i ty of eqs. (32) is the following. Let us consider our own

rest f rame So and two eollinear Super luminal f rames S, S ' having (positive)

velocities U -- U~ ~ c and V -- V~ > c, respectively. Le t us call L(so--~ S) and

L(So--~ S') the G L T ' s connecting so to S and S' , respectively. In the case tha t S '

is fas ter t han S with respect to us, f rame S will not observe S' moving with

posit ive veloci ty! This m a y seem cont rary to common sense. Bu t t rans-

format ions L(so--~S, S') have the form - - i A > . Inspect ion of this fact, and

Lorentz comparison of space and t ime distances between couples of events, lead

to the conclusion t ha t : Wi th respect to So, the f rame S appears as possess-

ing parallel t-axis and reversed-par i ty space axes (an antiparallel x-axis , and

270°-rotated y, z axes). Such analysis is enough to satisfy intuit ion, which m a y in any ease get help also f rom the G L T geometrical in terpre ta t ion pu t for th in the next Section. Therefore, naming v ' - : v: the S ' veloci ty as seen by S,

we have [23]

[ v ' < 0 when V > U , (35) / v' > 0 when V 

frames appear. Let our rest f r ame be So and our (( t ranscendent f rame ~ be S~.

Then, if A< is the LT connecting So and the f rame Sl - -1 , t ravell ing with veloc-

i ty u ~ 0 (relative to so), the same L----A< connects S~ and the f rame

S~ ~ 2, t ravel l ing with veloci ty U = c2[u with respect to So. This is required

by the group p roper ty of G, as exploited in Sect. 4. Again, this means t h a t

S~ mus t judge $2 to move with posit ive veloci ty u, even if according to us

f rame $2 is slower t han S~ ( remember t ha t So, sl, $2, S~ are all collinear

frames). By the way L(8 -> 1) ~ A~. The above-ment ioned fact m a y be seen also f rom Fig. 7, where S ~ is a

(Superluminal) f rame travel l ing faster than S relat ive to So, i.e. to us (all f rames

are still collinear). By inspection of the components of a generical space-

t ime displacement vector associated with the mot ion of S' (relative to so),

CLASSICAL TIIEORY OF TACIIYONS 231

one realizes tha t S' appears to S as moving with negative velocity (even if

it moves with positive velocity with respect to us). Analogously, frames

s', s", S" will be seen by S as moving/orward. See also Table I I in the following.

~0 XS f

\ |'~ I / \ ¢."

I / I F ; ' ; ,

~ , ~ 'X X 0 X " ~ ~ X X 0

/ / /

Fig. 7. - Where it is seen that, when S' tr,~vcls faster than S relative to so, i.e. to us (all the frames are collinear), then S' appears to S moving with negative velocity (even if it moves with positive velocity with respect to us). See the vector corresponding to S'. Analogously, frames s', s". S" will be seen by S ,~s moving forward (even if they would be expected to be seen travelling backwards by S. according to the usual, o14 intuition).

At last, let us again, explicitly recall tha t the relative speed between two frames

s(c - ~,) , S ( c ÷ E~)

never goes to zero, ba t runs from c to ~ .

6 . - Geometrical interpretation of GLT's. Comparison of space and time intervals.

With the s tandard procedure, from the GLT's for the four-vector compo-

nents one gets the GLT's for the units of Sl)~ce (standard-rod length) and time

(time standard interval). If Axo ~md Ato are the proper intervals, the observed [23] ones result

(36) (fl~ X 1) ,

At IAtob l = Atoll/J1 -- 21

where the considerations of Subsect. 4"6 are to bere membered. In particular,

it is worth-while to explicitly notice tha t standard definitions of Ax0, Ato, Ax, At--even in subluminal re la t iv i ty - -do actually correspond to the absolute

232 E . RECAMI an~ ~. MIGNANI

value of a difference between two measurements . For simplicity~ we considered

two collinear frames. Figure 8, depicting eqs. (36), shows tha t for f i~> 1 we can have both

(~ Lorentz ~) contract ion and di latat ion of space or t ime intervals. By the way,

in Fig. 10 the fact t ha t Ax----Axo, A t = At0 for fl = ~/2 (besides f i = 0) is

geometr ical ly i l lustrated.

IAtI,IA~I t

" I '~ / / 1 \

f p ~ - ~ -1 o 1 ,/2

Fig. 8. - Dependence of the magnitudes of sp~ee length an4 time interval vs. relative velocity tic, in the case of collinear motion, for both fie X 1. Sec cqs. (36) of the text. By the way, the geometrical reason for the fact that A t - A t o, Ax= Ax o, also for f l= ~/2, besides for f l= 0, comes out from Fig. 10. Notice that in the following Ax ---IAxl, At = IAtl.

Let us now invest igate G L T ' s for space and t ime intervals by extending the usual geometrical interpretation of Lorentz t ransformat ions . For simplicity,

we shM1 il lustrate the c~se of a bidimensional space-t ime. This p rob lem is

immedia te ly solvable, since, by introducing definitions (30), i.e. b y writ ing

G L T ' s in the form (8 bis), we have a l ready paramet r ized the generalized Lo-

rentz t ransformat ions through the quan t i ty ~, which is an angle running f rom

0 to 2~. In other words, let us consider any GLT as effecting ~ t ransi t ion f rom

a fixed f rame So(X, t) to a f rame /(x', t'), i.e. set a one-to-one correspondence

between G L T ' s and f rames /(x', t'). Then, we continuously get all GLT's ,

eqs. (8 bis), by continuous, counterclockwise ro ta t ion of the x'-axis, f rom the

x-axis position, b y an angle ~ running f rom 0 to 2z (~nd by the corresponding

ro ta t ion of t'-,~xis in clockwise sense). In fact , f rom eqs. (8 bis), if we pu t

H= ~]~--- -+ | / l÷tg2~ 1--tg27 (37)

it follows immedia te ly t ha t ( 0 < q < 2 ~ )

( 3 s ) x' = H(x cos ~ - - ct sin ~) ,

t' = H( t cos ~ - - (x/c) sin ~) .

C L A S S I C A L T H E O R Y O F T A C I t Y O l q S 233

Or~ converse ly , if we set

we ob t a in

(38')

c : +Vl~_tg~ml,

l x=[C(xi e°sq~i-ct' x' sin (P) i

t = C ' t' cos q~ -k c Sin (p .

I n generalized r e l a t i v i t y eqs. (38) h a v e a m e a n i n g for all values 0 < ~ < 2 ~

(el. Fig . 1]). T h e y r ep re sen t [40] a con t inuous <,ro¢ation ~ of the re ference

f r a m e in space- t ime , where the x-axis ro t a t e s us usual [40] in the counte r -

c lockwise ( l irection and co r respond ing ly the t -sxis in t he clockwise dire(.t ion

(by an angle of the s~me magn i tude ) . F igu re 7, and pa r t i cu l a r l y Fig. 10, r ep re sen t the ease 4 5 ° < ~0 < 90 °, whi ls t

Fig. 9 refers to t he s t a n d a r d case 0 ° < ~ < 45 °. The s y m m e t r y be tween , e.g., Fig. 9 and Fig. 7~ ]0 m a y be f u r t h e r ev idenced b y wr i t ing the first two of

eqs. (8 his), for the case 45 ° <~<90 ° (~ l < f i < ~ ~ ) , as [23]

x'l# + cr t'l# + x'lc (39) * = -VJ ~ i I~ ' t = v ' V ~ l#.

t i t , / Z / . ~¢

/ . 4 >

~-bs x-,~/ % x +',

cL ~ )

Fig. 9. - The geometrical interpretation of the usual LT 's (f l< 1), for space or t ime intervals [At, A x > 0] in the case of collincar frames and fl > 0. One has length con- traction an4 t ime dilatation. Besides, making recourse to the standard definitions, one must notice tha t i) A x = [afle[, where [a[ is the time (relative to us) taken by the moving standard rod to pass <~before our eyes ~>; ii) A t = [b/fie[, where ]b I is the space travelled (in our frame) by a lighting <( lamp ~> which is switched on for a unitary t ime (this t ime being measured in the co-moving frame). Notice tha t OA ~ Ax < Ax o, OB ~ A t > Ato; OU~ = OUt.

1 6 - R i v i s t a de l N u o v o C i m e n t o .

234 :E. I~CAMI a n d R. MIGNANI

o o > f l > 1 ~t x"(fl=~)

T - d

At = [b/#c I > Ato for 1 < fl < V'2 x" / ~ 4/

for I ~ ~ t " ( P = ~ ~ > v / 2 ; /_ ~ t,'(p=~) < Ato / / / ~

for 1 f l< ~/2 C - ~ ? ~- ! / A x = l a l ~ c l < ~ z o < , -- , , ~- -,,t,,

f o r ~ # > V2 > AXo a /

~ J B

A(-Ax,O) 0 b x

Fig. 10. - The geometrical interpretation of the 8LT'B (for fl> 1). Notice the exchanged r51c of hyperbolas, in considering space and time intervals, respectively, due to the change in the (spacelike or timelike) type of intervals that one meets when passing from frames s to frames S. Here one has still At = [b/flc I, and Ax = [aflc I, but now--according to Fig. 8--we get both Lorentz contractions and dilatations [Ax, At> 0]. In particular (for f l> l ) , we have At=At o and Ax=Ax o when fl= ~/2. Moreover, for 1 < # < ~/2, we have At>At o and Ax<Ax o, whilst, for f l> ~/2, we have At<At o and Ax>Ax o.

and by put t ing

fl - - tg F , C ~ -}- [ f l 2 1 .

We get [23]

(40) x---- (~(x' sinv 2 + ct' cos Y~i'

,

where now ~f runs ]rom 0 to 45 °. Of course, also eqs. (40) can be interpreted as in Fig. 7, 10.

Let us go back Co GLT's, eqs. (38) and (38'). The (~ geometrical >~ analysis of space and t ime unit transformations becomes easier if we remember (besides the standard definitions [30]) tha t [41] i) the Lorentz-transformed space uni t Ax may be also derived from the t ime a (relative to us) taken by the moving standard rod to (~ pass before our eyes >>: Ax = laficl; ii) the Lorentz- transformed t ime unit At may be also derived from the space b travelled (in our frame) by a lighting (( lamp >> which is switched on for a uni tary t ime (this t ime being measured in the co-moving frame): A t = [b/flc[. From Fig. 10 we can deduce what already got from Fig. 8; in particular, it is apparent tha t At ---- At0, Ax = Ax0 (for f12> 1) when fl = V'2.

Equations (8 his) correspond to eqs. (8) taken alternatively with the sign plus or m i n u s - - b e c a u s e of the effect of coefficient ~--as depicted ia Fig. 11 and 6. The (substantial) interpretat ion of the effect of the sign entering

x(-1)

x(+i)

I @2

\0.<

< ,\/-'<'<° -iA>' ~ "~

x( + i)

co<#< -1 x(-1)

Fig. ll . - Illustration of the sam(', considerations depictc4 in Fig. 2, through a general look on the geometrical interpretation of eqs. (8 bis). See the text, and cqs. (38). Cf. also Fig. 9 and 10.

eqs. (8 bis), as shown in Fig. 2 b), allows us to ~ssert t ha t eqs. (8 bis), with

n/2 < ~ < ~ , yiehl the same .~uccession of cases as yielded b y eqs. (8 bis),

with -- n/2 < 7 < ~/2, but now relative to to ta l ly inverted f rames (see Fig. 11).

7. - Ant imat ter and tachyons . A ((third postu late ~ : the Dirac-St i icke lberg- F e y n m a n - S u d a r s h a n ({ re interpretat ion principle ~ ( R I P ) .

7"1. B r a d y o n s , luxon.~', tachyons. - Let us write down eq. (3) in four-

m o m e n t m n space, i .e. for tetr '~impulse vectors:

(41) E '~ ~- ( ip ' ) z ---- -]= [E s + ( ippJ (f12 X 1) ,

where we used units such t h a t n u m e r i c a l l y c = 1.

Since in the bradyonic case

(42a) E 2 - p : - - p C - - mZo> 0 (fl~ < 1) ,

f rom the dual i ty principle it follows tha t , for a tachyon, .i.e. after a SLT, we

shall have [42]

(42b) E ' 2 - - p '2 : - p ' : = - - m ~ < 0 (flz> 1, mo real) .

Besides, it is known tha t in the luxonic case

(42c) E ~ - p~ ~ p~= m~ = 0 (#~= 1) .

236 E. RECA~X and R. ~IGNANI

Therefore, one has [15]

(43a) p~ 2 = m o > O

(43b) p2_~ 0

(43c) p ~ = - - m 2 < 0

for b r a d y o n s (case I , or t imel ike ) ,

for luxons (case I I , or l ight l ike) ,

for t a e h y o n s (case I I I , or spacel ike) .

I n f o u r - m o m e n t u m space (see Fig. 12), eqs. (42) represent respec t ive ly

i) for b radyons , a two-shee ted hyperbo lo id of r o t a t i o n a round the E -ax i s ;

(C J,

c~) I b)

f Py

Fig. 12. - :Representation of the hypersurfaces E ~ - - p 2 - p ~ , for a) bradyons, with p~ ~ m ~ > 0 (timelike case); b) luxons, with p2 ~mo2 = 0 (lightlike case); c) tachyons, with p2 ~ _ m o ~ < 0 (spaeelike case), where m0 is always real. In a), the points A ' and A" represent the particle kinematical states obtained by applying the operations CPT and CT, respectively, to the kinematical state A. In the case when we confine ourselves to subluminal frames and to usual LT's, then it happens that the (( matter ~> or (~ antimatter )> character is invariant for B's, but relative to the observer for T's. When eliminating the previous restriction, we may pass from particles to their antiparticles (through GLT's) even in the case of bradyons.

ii) for luxons, a double indefinite cone, hav ing E as axis; iii) for t achyons , a

s ingle-sheeted ro ta t ion hyperbolo id . I n all cases, we have iv[ = [p/E I. F o r

obvious reasons, in Fig. 12 on ly the th ree-d imens iona l <~ cross-space }~ pz----0

has been depicted. R e m e m b e r t h a t a n y SLT maps the <~ in ter ior ~> of the l ight-

cone p 2 = 0 in to the <~ exter ior >> (*), in accordance wi th the dua l i ty pr inciple : in

par t icular , t r a n s c e n d e n t t r ans fo rma t ions K± , K ( 0 , ? ) opera te a symmetry

with respect to the l ight -cone p~---~ 0:

(44) E,2 _ p,~. ___ _ (E: _ p2) (fl.~ > 1) .

7"2. F o u r - m o m e n t u m space. - The usual re la t ivis t ic formulae for t he ene rgy

and t r i m o m e n t u m of a free t a c h y o n t rave l l ing parallel to the x-axis wi th

(') At least, this happens at the ]inite.

CLASSICAL THEORY OF TACI:IYON8 237

velocity v ~ v~ :- tic, when fl'-' > 1, will read [15, 43, 44[

m o C 2 , m0/~c (f12 > 1, mo real) .

In Fig. 13, the relativist ic energy behaviour is represented, for a free par-

ticle (bradyon or tachyon) , vs. its veloci ty v. Figure 14 shows the behav-

c 0 c v c v

Fig. 13. Fig. 14.

Fig. 13. - Magnitude of relativistic total energy vs. velocity v for a free particle, either bradyonic (Iv] < c) or tachyonic (Ivl > c). For simplicity, we may refer to a velocity directed along the x-axis of the reference frame.

Fig. 14. - Three-momentum magnitude vs. velocity, for both bradyons and taehyons.

iour of the t h r e e - m o m e n t u m magni tude vs. veloci ty for both bradyons and

~achyons. I t m a y be noted t h a t a) the speed c preserves of course its character

of l im i t kinemat iea l p a r a m e t e r of the four-dimensional universe (even if we

know tha t such a l imit has two ~ sides ~); b) tachyons will slow down when

energy increases and accelerate when energy decreases. In part icular , d ivergent energies are needed to slow down the tachyonic

veloci~,y towards the (lower) l imit c. On the contrary, when t achyon ' s veloci ty

tends to infinity, its energy tends to zero ; this prevents violat ion of the common

postula te t ha t (( energy m:~y be t r ansmi t t ed only at ]ini te sI)eed ~, since a t achyon

shows zero energy to the same observers relat ive to whom it presents divergent

velocity. Notice t ha t a b radyon m a y have zero m o m e n t u m (and minimal energy moC"-) and :~ tachyon m a y have zero energy (and minimal m o m e n t u m

moC); however (Fig. 12), bradyons B eannot exist at zero energy, as well as

tachyons T cannot exist at zero m o m e n t u m (with respect to the observers

relat ive to whom they appear :m tachyons!) . I t is immedia te to see *hat in-

finite veloei ty belongs to taehyons corresponding to the interseet ion of the

hyperboloid in Fig. 12 c) with the plane ]A = 0.

B y the way, since t ranscendent taehyons do t ranspor t m o m e n t u m , they

allow get t ing the r ig id-body behaviour even in special re l s t iv i ty [43] ; as a

consequence, in e lementary-par t ic le physics (see the following), tachyons might a pr ior i result to be useful for int, erpret ing diffractive seat tering or the so-called

pomeron-exehange react ions [45-49].

~ :E. RECAMI a n d R. MIGNANI

Before going on, let us de]ine the negative-energy points of hyperboloids in :Fig. 12 a)-c) as representing the possible kinematical states o] the (( anti-

particle )~ (of the particle represented b y the corresponding posi t ive-energy

points). We shall see the reason for such a definition; in part icular , it will be

shown to coincide with the usual definition in the bradyonic case. By the way,

let us r emember ~hat (in the usual language) the operat ion of (( changing particles

into antiparticles ~>, and vice versa, is the ~( C/~T operat ion ~ (see the following).

7"3. The (~ reinterpretation principle ~: the third postulate. - Let us last ly

r emember tha t the k inemat ica l s ta te of a generical ]ree particle (with para-

meter m0) will be represented by a point on one of the hypersurfaces in Fig. 12. And the kinemat ical states of t ha t free particle with respect to all the sub-

luminal inertial f rames will be represented by all the points of the same sur-

face sheet. In fact, usual (subluminal) LT ' s do not effect t ransi t ions f rom a

sheet to another. A simple look at Fig. ]2 c), which shows a connected hy-

persurface, imposes the following considerations.

For tachyons, (subluminal) LT ' s will exist t ha t operate with cont inui ty

transi t ions f rom upper -semi-space points to lower-semi-space points. I n other

words, it m a y seem tha t a tachyon, regularly appear ing to observer 0 as having

posit ive energy (see point A of the upper semi-hyperboloid), will appea r to other observers 0 ' as bear ing negat ive energy (see, e.g., point A ' of the lower

hyperboloid) .

However , if a LT, eqs. (8), is such as to inver t the energy sign, the same LT, eqs. (8), will inver t the sign of any other tetraYector four th component , associated with the same observed object ; in part icular , t ha t LT will invert also the sign of t ime [15, 46, 47]. This fact is visually shown in Fig. 15. 1Namely,

if a t achyon moves, e.g., along the x-axis with posit ive velocity U with respect

to us, the above-ment ioned sign inversions happen [38, 50] for all boosts cor-

Xr

0 X

Fig. 1 5 . - World-line OT of a tachyon. With respect to observcrs 0'-----(t', x') the tachyon appears to move backwards in time (relative to the time arrow, uniquely determined by the (~ thermodynamical~) behaviour of usual macrosystems). In fact, projection of OT on the axis t' is directed towards the negative semi-axis. However, observcrs O' are the same ones to which the taehyon will show (~ negative energy ~>. Those two paradoxical occurrences compensate each other, easily allowing an orthodox physical interaction. See thc following.

CLASSICAL THEORY OF TAC~YONS 2 ~

r e spond ing to pos i t ive veloci t ies u ~ c~-/U (along the x-axis , and wi th reference

to us). I n conclusion, if a t a c h y o n is expec t ed to show negative energy re la t ive to

a ce r ta in observer , i t is also e x p e c t e d to a p p e a r to the s ame obse rve r as m o v i n g backwards in time (with r e spec t to the t i m e a r row un ivoca l ly deter-

m i n e d b y usual m a c r o s y s t e m s ' behav iour ) . I t is v e r y easy to convince ourselves

t h a t those two pa radox ica l occur rences al low a qui te orthodox r e in t e rp re t a t ion ,

when t h e y are (as t h e y ac tua l l y are) s imu l t aneous . T h a t ~ reinterpretation (or " s w i t c h i n g " ) principle ~), t h a t we shall call ~ l~I1 ) >), has been first p u t for th

b y SUDAI~SHAN and co-workers [] 5, 541, in the spir i t of p rev ious in t e rp re t a t ions

b y DIlCAC [51], ST{'TCKELBERG [52 I and FEY]NMAN [53].

~Vamely, let us suppose (see Fig. 16 a)) t h a t a par t ic le P , wi th nega t ive

ene rgy (and, e.g., ~ charge ~> - - e), and t r ave l l ing b a c k w a r d s in t ime , is e m i t t e d

[t<t r]

r-q(.¢;e>0;fi~>0 ,Vv] z ph = [ , 4 j (+~);v>0

(t,x) (tr~ xq

x x, x 2 ~?(h) r - ~ ( - q ) ; E > 0 ; ~ ; p < 0 -= = - P : L ~ (+~.){v <o

(x') ( x " ) ~ (-t,x) (-t~,x')

X Y R/P (h) [-~-](-q);E<O;T;p<O

(t ,x) (t',xO ,,,.-~(P_);~q ;__E< 0 ;Tj_P < ~

(t.x,)( A )-. . . . . . ~ B ) ( t ~ , ~ 2 ) , . . . . r~q(-q) ;E>o;Y;p>o r-vq (Q);+q;E:POi?~;P >0 ~ uPT(pn : L ~ (- , t ) ;v >0 I" [~. i I

¢z) b)

Fig. 16. - a) represents the exchange from A to B of a particle LP with negative energy (and (~ charge ~) and travelling backwards in t ime (t 2 < t~). Such a process will appear to be nothing but the exchange ]rom B to A of a (standard) particle Q with positive energy (and (( charge ~), travelling forward in time. Particle Q may be shown to be (maybe except for the helieity) the antiparticle of thc initial particle: Q = / ' . See the text. b) shows a certain phenomenon ph, i.e. the exchange from emitter A to absorber B of a certain particle, and the transform~tions on ph (and on A, B) operated respectively by 0~', by the (~ reinterpretation procedure ~ (RIP, see the text) used in case a), and by CPT.

b y A at t i m e t~ and abso rbed b y B a t t ime t., < tl. Therefore , a t t i m e t l , ob-

j ec t A (~ loses ~) nega t ive ene rgy and (( charge ~) - -e , i.e. gains ene rgy and (( charge ~

-]- e; and , .~t t i m e t2 < tl, ob j ec t B ((gains ~> nega t ive ene rgy a n d (( charge ~ - - e,

i.e. loses ene rgy and (( charge ,) -~ e. Such a phys ica l p h e n o m e n o n will of course

a p p e a r to be n o t h i n g b u t the exchange ]rom B to A of a ( s t andard) par t ic le Q, wi th posit i 'ce ene rgy (and (~ charge ~) -[- e) and t r ave l l ing fo rward in t ime .

W e have the re fo re seen t h a t Q has the oppos i t e ((charge ~) of P ; this

240 E. RECAMI an4 R. MIGNANI

means tha t <~ :RIP ~> operates, among others, a charge conjugation C. inspection of <( R I P ,) (see Fig. 16 b)) tells us tha t effectively

(46) ,, n i p >) - F ~ F ,

A closer

where by E and :~ we mean the operations of energy reversal and m o m e n t u m

reversal, respectively [38, 9, 54-56]. Notice that , in our terminology, ~ means

conjugation of all charges (e.g. also of magnet ic charge, if it exists). In the present case of tachyons, we call Q the antiparticle of P :

Q - - P (v~ > e~ ) ,

thus rending precise (in the tachyonic case) what was defined in Subsect. 7"2. In other words, let us consider a t achyon T traveUing, e.g., along the x-axis,

and a continuous series of subluminul frames s, moving collinearly with our frame so. Let us call s o the (critical) frame in which T becomes transcendent .

As we go f rom so to s~, the tachyon T appears with increasing velocities. As

we by-pass s~, with a LT tha t we shall call/~, the new frames should observe a taehyon T (still with positive <~ charge ~)) moving backwards in t ime and carrying negative energy. On the basis of the Dirac-Stf ickelberg-Feynman-Sudarshan (~ rein terpreta t ion principle >> [15], however, we shall change the previous s ta tement into the following one: (~ As we by-pass the critical frame s~, the new frames will judge the observed particle as the antitachyon (*) T (now with negative " charge") , t ravell ing in the opposite direction ~> (cf. also eqs. (34)).

In fact, let us, e.g., consider the (subluminal) LT ~ ~ making transi t ion from A to A' of Fig. ]2 c) (by the way, ~ i s the x-axis boost with positive rela-

t ive velocity u -- u~ 2V j ( 1 -~ ~ ~ V i e ), if V is the velocity of the considered

tachyon in the kinematical state A). I t is easy to see (cf. Table II) t ha t L acts kinematical ly on the observed tachyonic object T as the product ETa , where is the velocity-reversal operation. When we apply (~ ]~IP ~, we get eventual ly tha t the second frame observes--as regards the tachyon ana lysed- - the same

effect as produced (in the first frame) by a mere CT operat ion [57],

(0~p)(k2+) -> C2,

applied only to the observed object T. Such a CT-action, as shown by Fig. 16b),

does prove our previous s ta tement in quotat ion marks. The emerging fact

that , given a particle 1 ), the concept of <( antiparticle ~> P is a purely relativistic one will be soon revisited.

The previous analysis suggests us to introduce into the special-relativity theory a third postulate (besides ~he two in Sect. 3), i.e. the (~ re in te rpre ta t ion principle ~

(*) Cf. also the second of ref. [55].

CLASSICAL TtI]BORY OF TACHYONS 241

(RIP), in the form [39, 50, 23, 57]: (~ Physical signals are actually transported only by positive-energy objects (i.e. by the objects tha t appear to us as carrying positive energy and going forward in time) ,~. The meaning of such a principle within information theory is straightforward. The (( R I P ~ can be inserted har- moniously (*) into special re la t ivi ty: it is indeed necessary to the self-consistency

of generalized special relativity. For example, the generalized velocity composition law, in part icular the above-mentioned eqs. (34) of Sect. 5, hohl for the

reinterpreted objects. The same happens, e.g., for the electric charge q; tha t

is to say, a LT making a transit ion between a f rame/1, (( preceding ~ the critical

frame s~, and a frame ]~, (~ following ~ s~ (i.e. such as to (( overcome ~ the critical

velocity), will automatical ly yield the final electric charge shown by the reinterpreted objects. In fact, tha t LT, being such as to invert the fourth-components ' sign, will inveI~ also the sign of charge density (, and therefore of the particle (total) (.h~rge q:

(47) 0 - ~ - ~ , q - > - q,

where we have defined

(47,) q +feldVI.

This accords to the fact tha t the above-considered LT transforms tachyons into ant i tachyons, and vice versa. We shall come back to the R I P in Sub-

sect. 8"5.

7"4. Antimatter and matter. - The R I P is appropriate (*) in tile bradyonic case as well [46, 47, 8]. Namely, a particle P in the kinematical state corresponding to a point of the lower hyperboloid (Fig. 12 a)) will be shown to appear as the antiparticle P of P in the usual sense.

The fact is quite interesting that , once the notion of particle is introduced (as is usually done in special relativity), merely from special relat ivi ty itself the concept of antiparticle follows [23, 46, 47, 8]. Precisely, since 1905, on the basis o] the double sign entering the relation

(48) E = ± v~p 2 ÷ m~,

the existence, /or any particle, o] its antiparticle could have been expected, provided the third postulate (RIP) had been used.

Moreover, let us emphasize that---when we limit ourselves to subluminal

(*) It is indeed necessary, even in usuM special relativity, to avoid transmission of information into the past! Deprived from this (~third postulate,), standard relativity would allow sending signals into t, he. past (~ts observed, e.g., by L. ]~ANTAPPI~).

242 E . R E C A M I a n 4 R. M I G N A N I

]rames--the clean separation between matter and antimatter is confined only to bradyons, owing to the fact t ha t the hyperboloid in Fig. 12 a) consists (*) of

two disconnected sheets [9, 46]. On the contrary, in the case of tachyons, the character ma t t e r / an t ima t t e r is no longer absolute, but relative to the (sub-

luminal) observer [9, 46]. However, if we consider also Superluminal / tames, since the product of two

suitable SLT's may yield a GLT of the type -- A< ~ (/ST)A<, we may get, by

means of a GLT, the transi t ion mat te r ~- an t imat te r (see the definition in Sub- sect. 7"2) even/or bradyons. Let us consider the particular (nonorthochronous) LT tha t is usually called /5~:

- / i < ~ P 2 (/I< ~ 1).

Here we meet the impor tan t point tha t follows: I n a universe with (~ charges ~,

the generalized Lorentz t ransformat ion tha t we called P T does effectively produce the exchange particle ~-ant ipar t ic le , and must be actually considered a

C P T operation [37]:

(49) A A A

--A<------I, -- I =--OPT,

where, as before, ~ means (~ inversion of all (additive) "charges" )). We shall deepen such considerations in the following. Let us remember

tha t the transi t ion from a particle P to its antiparticle P (in the same kinematical stnte, as seen by the same observer, and with emitter and absorber interchanged), is performed by the ¢ /3~ operation: see Fig. 16 b). Str ict ly spe~king,

in Fig. 12 a) the true antiparticle state of A is A', and in Fig. 12 c) it is A". Since (~/5~ is known to change the helicity sign, the (( true antiparticle state ,) of a

particle with helieity + 2 has helicity - - L This operation may reduce to ~/5 or to C when the interactions tha t P and P undergo are T or P T eovariant , respectively. An example of the last case is tha t of (purely) electromagnetic

interactions.

8 . - ~ / 5 ~ a n d (, R I P )).

8"1. Case o] bradyons. - We have just seen that , since -- I is a chronotopical

<, rotat ion )>, i.e. a GLT, relativistic physical laws are expected to be covariant

under the C P T symmetry . Since the ~( to ta l inversion ~) is a subluminal LT (corresponding to fl ~ 0 but ~ ~ ~), the previous s ta tement should refer also to usual relativistic laws, even if not ye t wri t ten in G-covariant form. This

agrees with what was found on a ]ormal basis, e.g. in ref. [37].

(*) At the finite, at least.

CLASSICAL TIIEORY OF TACIIYONS 2 ~

I t is impor tan t to notice tha t , as we saw in the last Subsection, in the case

of brady(ms the t ransi t ion f rom an ((upper ~ point A to a (~lower ~) point A ' of Fig. 12 a) is actual ly per formed by a nonorthochronous LT of ~hc type

-- A<, e.g. by the (( strong reflection )> -- 1 itself. Bu t the G L T - - - - - 1, t ha t we initially called /3~, does change the sign

not only of t and x, but also of all other four-vector compoaents , and in partic-

ular of energy E and m o m e n t u m p of any considered object:

(49 bis)

Consequently, if we r emember eq. (46), i.e. t ha t (( R I P ~>~- CE~, we m a y draw

~he conehlsion tha t strong reflection and R I P yield [9, 38, 54-58]

(50)

Since eq. (50) obviously holds not only for B's but also for T 's we have thus proved

eq. (49). See also Fig. 16. Notice tha t , iu our theory , C~/3T is a linear oper-

ator [57 [, as well as all GLT's . In such considerations (eL also Subsect. 7"3), when assuming R I P , it is

always possible to write the relation

which is very interesting for the physical unders tanding of charge conjugation.

By the way, notice tha t also in the case of b r a d y o n s - - b y means of i) suitable G L T ' s of the type - -A<, ii) their k inemat ical effects (see Table I I ) on ~n ob-

served object and iii) the :RIP- -one m a y eventual ly get tha t the second ]rame observes (as regards the analysed bradyon) the s~me. effects as produced (in the ]irst ]rame) by :~ CT ol)eration [571 applied to the observed bradyon. As in the

case of tachyons, moreove| ' , the generalized veloci ty composit ion law does

hold for the objects re in terpre ted by RI lL Analogously for the (,electric

charge ~> density, t ransformed by the GLT considered. I~ is immediate to conclude t h a t - - s i n c e - -1 :: ~/5~/, both for bradyons

and t a c h y o n s - - u n d e r the GLT (, strong reflection ~), particles (B or T) in the

initial st:~te of an interact ion process will be t | ' ansformed into antiparticles

(B or T) in thc final s tate of the same iuteraction process, and vice versa

[9, 38, 54-581. For instance, the two reaetions

(5~) ad-b-->c÷d,

~÷~-~ +~

are the two difi'erent descriptions of the same phenomenon us seen by the two

different inertial f rames ,% and --,% (CPr[')so, respectively. By the way, from the foregoing it follows tha t usual symbols like /5 ~nd

244 ~. ~v.CAM~ and R. MIGNANI

h a v e too restricted a m e a n i n g [57]; one ough t on t h e c o n t r a r y t o i n t r o d u c e s y m b o l s

m e a n i n g t h e s ign inve r s ion p r o d u c e d b y a G L T in all the t e t r a v e c t o r s ' f o u r t h

c o m p o n e n t s (e.g. t h e s y m b o l T) a n d f irst t h r e e c o m p o n e n t s (e.g. P), a n d so on.

( W e a l r e a d y speci f ied t h a t C ~ ~ m e a n s t h e s ign inve r s ion of a l l t h e a d d i t i v e

(~ charges ~.) I t ho lds of course t h a t

(49ter) { T ~ : T~'""p~(RIP) ~ , . -~ =~-~P "" '

8"2. Case o/tachyons. - L e t us veri]y e x p l i c i t l y in t h e case of T ' s t he conc lu-

s ions a b o v e (Subsec t . 8"1).

F i r s t of all , f r o m Tab le I I i t can be d e d u c e d t h a t , b y m e a n s of s u i t a b l e

t r a n s f o r m a t i o n s of t h e t y p e - - A < , of t h e i r k i n e m a t i c a l effects a n d R I P , we

TABL~ I I . - E]lect o] GLT's on the sign o] various ]our-vector components o/an observed object, in the case o] coil±near motion aTong the x-axis. Both subluminal (u ~ % ) and Superluminal (U ~ U~) relat ive velocities are considered. Analogously, both bradyons (having velocities v relat ive to the ]irst. unprimed frame) and taehyons (having velocities V relat ive to the unprimed frame) are as well considered. For simplicity, only the cases + A<(fl2< c 2) and - - iA>( f l 2> c 2) are considered, as well as only the eases v > 0 and V > 0. Notice explici t ly tha t only x-components of v or V are effective, in this context. Last ly, for compactness ' sake, it is assumed V~ > c.

tce]~- lull ~ M < c, l u l l ~ IUI > c luxl ~ lul < c, rive V= > c 0 < v, < c

velocity 0 < v= < c V~ > c

u > 0 u < 0 U > 0 U < 0 U > 0 U < 0 u > 0 u < 0

sign x' + + + - - ± - - 4- +

sign x for U ~ V, for u > < v~

sign t' i + ± - -

sign t for e2/V~>< V~ for u >< v,

sign E ' ± + 4- - -

s i g n E forc2/V,~u for U><c=/v, + - - + +

sign v~ ± + ! - - ± - - + +

sign v, for c21v,~ u for ~ c~lvx for u < > v~ for u >< v.

sign j~' + + + - - ± - - ± +

sign j , for U ~ V~ for u~v,

sign Q" ± + ± - -

sign Q for c2/Vx~ u for U ~ c2/% + - - + +

CLASSICAL TII~OI~Y OF TACHYONS 245

m a y finally get t ha t the second f rame observes the same effects as produced (in the first f rame) b y the (~/5~ operation. In the part icular case when - - A <

is precisely the strong reflection, then we get obviously the true ODT, as s~id above, i.e. the t ransi t ion f rom a tachyonic phenomenon to the C/5£P-ed one

(both seen now in the same /tame~). Let us recm~sider the explicit i l lustration of a concrete example. Consider a

tachyon T and a succession of subluminal f rames (all moving collinearly with T,

for simplicity). Let s~ be, as before, the f rame observing T with divergent veloc-

ity. I f a f rame moving slower thart s~ sees T travell ing in a certain direction,

then any f rame moving faster t han ,% will actual ly observe T as an anti-

t aehyon (e.g. with the opposite electric charge) t ravell ing in the reversed direc-

t ion [9, 38, 39, 57 I. Therefore by-passing f rame s~ (in the above sense) implies

charge conjugation. With reference to Fig. 2 b), if we observe f rom our

f rame So ~ succession of Superluminal f rames (or vice versa), when we by-pass the (~ t ranscendent f rame ~ S ~ - - i n o~her words, for trans/inite trans/ormations-- we get the C-symmet ry . Thus, when we operate a ((rotation ~ (in the four-

dimensional space-t ime: see Fig. 2) a imed to reach the to ta l ly inverted f rame

(PT)so, actual ly we reach the f rame ((~iO~P)so.

At this point , we complete the considerations pu t for th in Sect. 5 by giving

the effects of G L T ' s (both subluminal and Superluminal : d-A<(fi~<c 2) and

- - i A > ( f i ' - : > c~')) on various physical quant i t ies ' signs: see Table I I .

8"3. Case o/ luxons. - The GLT ' s mapp ing the upper light-cone into the lower light-cone, and vice versa, arc all the (subluminal, nonm%hochronous)

LT ' s of the tyl)e - A < ( f l X 0 ) and all the SLT ' s of the type d-iA>(D>c) or

- i A > ( f i < - c). Let us consider in part icular the tot .d inversion ~ /5~ = _ A<(fi = 0); we

shall assume in this case too t ha t it t ransforms a particle state into its anti-

part icle state. I f the luxon bears a ((charge ,> (as neutrinos, that bear (~ lepto-

nic charge ~), since 0/55P changes the helicity sign, then a luxon (neutrino) with helicity 2 = -- 1 will be t rans formed into its (( ant i luxon )) [46] (antineutrino) with hclicity 2 = @ 1. Cf. Fig. 12 b).

I n the ease when luxons do not seem to bear any <( charge ~) (e.g. photons),

the distinction be tween ((photons ~) and (, ant iphotons ~ [461 becomes moot.

We m a y only say tha t , i/ a photon has helicity 2 = @ 1, then its antiphotou will have 2 = - 1 (with source and detector interchanged).

8"4. About P, T and CDT nonconservation. - Le t us make some fur ther

considerations :

i) Briefly, Subsect. 8"1 and 8"2 tell us tha t : (( The right way/or doing P T is doing O_PT, [23, 57] (*). The 0 /3T covariance, as a l ready mentioned, is

(*) See also ref. [74].

24,6 ~. RECAMI and R. MIGNANI

required by our mere <~ extended PR ~ (when we do not restrict ourselves arbitrari ly to subluminal relative velocities) [19].

In the cases when T covariance is supposed to hold, we get as a corollary

tha t : <~ The right way ]or doing P is doing CP ~, which expresses the essential teaching of LEE and Y A ~ [59]. In fact, in the case considered, re la t ivi ty says

tha t we can <, safely ~> (i.e. covariantly) reflect space only if we simultaneously

apply C, so as to have particles changed into antiparticles.

ii) At this point, let us add the observation tha t ~/5~ covariance (like the other symmetries we shall meet in the following) is imposed by special re la t iv i ty on the physical laws of mechanics and electromagnetism only. Exten- sion of the PR also to nuclear and subnucleur phenomena (i.e. for strong and weak interactions) would possibly lend to a new wider theory, just as extension to electromagnetism led from Galilei-Newton's theory to Einstein's. There-

fore, were, e.g., a CPT covariance violation found, it could mean tha t Lorentz t ransformations are no longer precise enough to be used in building up

strong and weak fie](] theory.

iii) Lastly, let us spend some words on T nonconservation [60a]. (This par t may be skipped in a first reading).

I t is well known that , in classical physics, elementary processes are supposed to be t ime reversible, and irreversibility is expected to appear only in macroscopic processes: i.e. tha t only statistical ((~ large numbers ~)) laws assign an arrow to time.

Even if we saw tha t extended relat ivi ty requires only some symmetries, such as C/ST covariance, but no pure T covariance, we shall ten ta t ive ly extrapolate the ~bove-mentioned result outside its theoretical domain. :Namely, as a working hypothesis, we shall assume tha t [60a]: <( In special re la t iv i ty (and even in quantum mechanics), t ru ly elementary processes (among jew, elementary objects) are symmetrical under T, and ¢ / 5 operations. Possible violations are due to statistical contributions by large numbers of e lementary objects ~. Here the word (, e lementary ~> roughly means wi thout parts and without inner structure

(or without inner-structure intervent ion in the considered process) (*). The conclusion of the foregoing is tha t T-violat ing (~ e lementary processes ~>

--i.e. more generally the so-called super-weak interactions--are not to be actually

considered as elementary. On the contrary, in those processes the very interior of the interacting particles will be concerned, so tha t ~' covariance may

result to be violated. And such an (~ interior )> should consist of a very large number of consti tuents (maybe of the order of Avogadro's number?) . These consti tuents will be called by us (, partons ~, following FEYN~IAN.

Previous assumptions are substant ia ted b y the following analogy [60 b].

(*) In the particular case of (~-symmctrical processes, some interesting consequences may be easily derived.


I n nuclear physics reactions m a y be roughly divided in two classes: the p romp t processes, corresponding to (( direct reactions ~ (with lifetimes of the order 10 -~4

to 10 -~-~ s) and the delayed ones, corresponding to fornRrtion and decay of a (( compound nucleus ~) (10 20 to 10 -~4 s, and sometimes even much more). The

(~ d i r e c t , processes are in terpre ted as nuclear interactions, with few degrees

of f reedom tak ing par t , whilst the (~ compound nucleus ~) ones are interpreted

as a consequence of the par t ic ipat ion of very m a n y degrees of freedom. In

e lementary-par t ic le I)hysics, we have (( strong processes ~) ((10 24--10-22) s) and

(~ weak interactions )) (_~ 1 0 ~ s, and sometimes e~-en much more). We are led

to suppose tha t in (( strong )~ interactions only few internal degrees of freedom

are involved, while in the ((weak)~ ones (*) :r lot of inter~tal degrees of freedom

take pa r t in the reactions. At present, we do not pay a t ten t ion to the (~ inter-

mediate~> processes, i.e. to electromagnet ic interactions ((10 ~ s ~ 1 0 - ~ " ) s ) .

Therefore we can imagine tha t :

In strong interactions, particles interact (( coherently )) (or a lmost coher-

ently), as if made up of one (or a few) objects. For example, the baryon

A(1236) generally reacts as if it were a (, submolecule)) nueleon-pion.

I n weak interactions (particularly the ~P-violating ones), the particles

- - b a r y o n s , mesons - - in t e r ac t also through direct, ((incoherent)) part icipat ion

of their internal cons t i tuen ts - -which may be however grouped in (( subshclls ~)

or more general ly in subgroups (as ((coagula ~, or similar) with various sta-

bil i ty degrees.

F nally, let us ment ion another amdogy, support ing our assumption. In

ref. [60cl, DtiI~I~ has noticed t ha t the e lementary-par t ic le spect rum has str iking

similarities with the energy spectrum of systems in~olving many consti-

tuents (atoms).

8"5. Again about G L T ' s plus I~IP - ()lose inspection of cases dealt with

in Subsect. 8"1 and 8"2 reveals tha t their meaning vanishes (e.g. because of the intervening exchange of emission and absorpt ion processes) if we cannot

refer our B 's oi" T ' s to some interaction (space-time) regions. For example,

when a t achyon overcomes the divergent velocity, it I)asses fronl being, e.g., a t achyon T entering (a certain interact ion region) to being an ant i tachyon

leaving ( that interact ion reg ion) [9 , 54, 5(;[. In conclusion, the third postulate ( R I P ) w i l l be completed by s tat ing that [551: (( Under a "trans- critical" GLT, when, e.g., the r61es of enlissions amt absorptions happen to be

interchanged, any negative-energy object in the i~itial (]inal) ~state" corresponds physically to its positive-energy antiobject in the ]inal (initial) "state", and vice versa ~, in the above sense.

(*) Particularly in the supra'weak ones.

248 r. r:~CAMI an4 ~. MIG~ANI

Of course the th i rd pos tu la te - - in order to be used for reinterpret ing

the GLT's effect--requires considering processes with both initial and final

(~ states ~> [9, 55]. Therefore, we can see that extended special re la t ivi ty strongly

suggests dealing with interactions, and not with objects (in quantum-mechanical

terminology, dealing with (~ amplitudes ~) ra ther than with (( states ~ [55]). The

same philosophical indication will arise while considering Sect. 10 as well as

causali ty (Sect. 12). At last, let us explicitly ment ion tha t it is actually the (( ]~IP ~> tha t allows

us [57] to apply directly the nonorthochronous LT 's to four -momentum vec-

tors (as well as to the other four-vectors).

9. - The tachyonizat ion rule.

Let us assume tha t we know, besides the class z¢ of usual physical laws

(of mechanics and electromagnetism) for bradyons and ant ibradyons, also the class ~ of the physical laws for tachyons and anti tachyons. When we pass from a subluminal f rame s to a Superluminal one S, class ~ /wi l l have of course to t ransform into class ~ , and vice versa [23, 25, 18]. In this sense, the to ta l i ty of physical laws ( ~ / u ~ ) will be covariant under the whole group G, i.e. G-covariant [19]. And in this sense inertial frames (with relative velocities

]u I ~ c) are all equivalent (cf. also Subsect. 3"2). Moreover, we shall see tha t physical laws (of special relativity) may be

wri t ten in a (universal) form valid for both B's and T's, a form obviously coin- ciding with the usual one in the bradyonic case. For example, we shall soon

show tha t the G-covariant expression

m0 m . . . . . . . (f12>< 1, m 0 real)

in such a form has (~ universal ~> validity. F rom previous considerations, the ((rule o] tachyonization (RER))~ im-

mediate ly follows, extending Parker ' s principle [18] to the four-dimensional space-time [25, 23]: (( The relativistic laws (of mechanics and electromagnetism,

at least) for tachyons follow by applying a SLT--e.g. the t ranscendent trans-

formation K + - - t o the corresponding laws for bradyons )). Such a rule ma y

be also named the ((rule of ex tended re la t iv i ty ~) (RER).

10. - Description of Nature, physical laws and GLT's.

10"1. Interactions and objects. - We have already seen that , e.g. through suitable generalized boosts, a particle moving along the positive x-axis (as seen in the first frame) may t ransform into its antiparticle (as seen in the second

CLASSICAL TH]~01~Y OF TACHYONS 249

frame) travell ing along the negat ive x-axis. Therefore, we argued (Subsect. 8"5)

t ha t language abou t Na tu re should refer to global interaction processes ra ther

t han to (( objects )~. I n fact , f rom such a viewpoint the initial assertion has been

given ~ precise meaning b y writ ing t h a t (( a particle in the ini t ial (final) state

m a y appear as its antipart icle in the f inal (initial) state, when observer is

changed (by means of GLT's)~ .

At this point, it is worth-while to r emark t ha t re la t iv i ty does not require

a t all t h a t two different observers give the s~me description of the same

phenomenon. I t does require only t h a t they find t ha t phenomenon to be ruled

by the same physical laws (generally speaking, conservation laws).

10"2. Descript ion and laws. - I n fact, let us choose [41] a set ~ of certain,

well-defined reference f rames r, the set ~ of the phenomena p of mechanics

and electromagnet ism, and the set ~ of the descriptions d (of phertomena

p c P f rom frames r ~ ) . All observers r are supposed to possess the same

instruments , bo th physico-exper imental snd mathemat ico- theore t ica l (i.e. the

same <~ theory )>, too). Strict ly speaking, one has to deal wi th the ~( tr iads ~ dpr,

elements of the set ~ X ~ × ~ , the Cartesian produc t of the three sets considered.

As depicted in Fig. 17, to the same p there correspond two descriptions d~, d~

, ~ ctescr

T ,'X" J T ph I / I

Fig. 17. - To the s&me phenomenon p there correspond two different descriptions dl, d, by two different observers r~, r2, and so on; given any two elements out of d,p, r, the third one will be uniquely fixed [41] (see the text). Notice that this Figure has only the t~sk of supporting a preliminary intuition.

in two frames r~, r2, and so on: given any two elements out of d, p, r, the correspondence between elements dpr must be assumed to be such t ha t the third

one is univocal ly fixed. We m a y write [411]

(53) rl~.r2

dlprl * a~pr., , i.e. {rl--~ r~, p --~ p ~ dl--~ d2} ,

dp l r l , ,e.~) dp2r~ , i.e. {r l -+r~, d - + d ~ p i~+p2} .

Let us define the subsets A , c ~ :

(54) d~zJ , <:>dpr ~ 4 R ~ - - ~@X~ X ~ .

Then, following AGO])I [41], we shM1 say t ha t two frames r~, r~ are equivalent

1 7 - Rivis ta del Nuovo Cimento.

2 5 0 ~. I~CAMI a n d 1~. MIGNAI~I

( - - ) if A, is mapped onto itsel] when passing f rom rl to r2 (notice tha t the sketch

Fig. 17 is no longer helpful to our intui t ion!) :

(55) r l - - r~c=>zJ ~ A , ~ V d e L J ~ d ~ A and V d ' ~ A , ~ d ' ~ A , .

Such a condition operates an exhaust ive par t i t ion [41] of se~ ~ into subsets

of equivalent frames. Conversely, given a f rame r and a set ~ of phenomena,

it is possible to build up the set ~ of f rames equivalent to r. I t is easy to ver ify

tha t , given the set of mechanical and electromagnet ic phenomena and an

inertial f rame, two classes of equivalent f rames are respect ively the set of s tandard (subluminal) inertial f rames and the set J of our inertial frames, bo th

subluminal and Superluminal (cf. Subsect. 3"2). Le t us choose ~ _ ~ Y . I n our

case, ~ ~ ~ ~) ~ s , where ~ n ~s ~ 0, the sets ~, and A s being the class of

subluminal inert ial f rames s and of Super luminal inertial f rames S, respect ively:

:Now, moreover, 9---- A~-- A, V r e ~ .

(56)

where

I t is immedia te to see t ha t

~ / p e ~ ~ 9 = 9 , w 9 s , wi th 9 . n 9 s : 0 ,

d 9... [{s), { s ) ] o r = [,, s ] .

:Notice tha t , in passing f rom an s to an S, we have t ha t 9 is m a p p e d - - a s

required--onto itself, bu t in such a way t h a t 98 goes onto 9 s , and vice versa. Given a phenomenon p, if dl and d2 are its descriptions in the f rames rl , r2

respectively, and if the t rans format ion L is such t ha t

(57) Lrl ~ r~ ,

we shall consequent ly use the convention of writ ing

(57 bis) . L d I : d 2 .

Let us suppose we have a criterion C for ~ given description d to belong to the

set 9 of the descriptions of phenomena p e ~ f rom the f rame r e ~ ; we write

(58) C(d) verified ~=>d e 9 ;

we shall call C a <~ good criterion ~) if it holds for any d:

(59) Vd e 9 ~ C(d) verified <=> d ~ 9 .

CLASSICAL TIIEORY OF TACIIYONS 251

I t follows tha t C is covariant in form under any L:

(5~r) C( Ld) verified <:~ L d + ~ .

We shall b y definition call (: (or be t te r the union of the various, possible

(~ good criteria >> C~, C.~, ...) the ensemble o/ physical laws of phenomemt p ~

~ts seen by f rames r c ~ . Conversely, ~t proposit ion will be considered ,~ physical

law if it is '~ p~rt of C. I n other words, given ~ ,~nd ~ , we define (~ physical

l:~w ~) any proposi t ion regarding a p ~ :2 which is covaritmt within ~ .

10"3. A n example. - Going back to Subsect. 10"1, let us for instance em-

phasize ttu~t electric ch:~rge--:~s well as e~lergy, ~nd so on - -o f ~n isolated

sys tem in not required to be :t rebttivistic invariant , bu t only to be (;onstant

(as seen by any observer) during the sys tem transform:~tions.

a) ~ b)

to

Fig. 18. - In a) and b) two different descriptions are shown of the samc phenomenon corresponding to two different inertial nbscrvers. 01 and O~ respectively (sec the text). According to 0~, a positively charged particle a 4ccays into a neutral particle c and another positively charged particle b. According to 02, on the contrary, a positivcly charged particle a combines with a negatively charged particle b in a process of (( resonance ~> formation (or of (( annihilation )>). Both 4escriptions are consistent with special- relativity theory. In particular, the total electric charge Q is conserw~d during the reaction, in every /tame of reference (but it is not necessarily Lorentz invariant: see the text; in these examples, Q changes in fact, its value under the GLT considered). Notice that the total number of particles (initial ones pbts firml ones) particip:lting in the reaction is Lorcntz invariant.

I t is instruct ive to analyse :m explicit example (Fig. 18). Le t us consider

a posit ively charged particle a tha t , with respect to a first, observer 01, decays

into a neutra l particle c and ~mother posit ively (;harged particle b having in

general different velocity (Fig. 18 a), where the t ime arrow is represented too).

I t is possible to find another observer, 02, with respect to whom e.g. the out-

going particle b behaves as an incoming antipart iclc b bearing a negative charge.

Observer O~ will judge the process as a <(resom~ncc ~ fornn~tion (*) (Fig. 18 b)).

Fr,~me 01 observes a tot~fi electric charge + 1 , whilst 0., ~ to ta l charge zero.

(*) Or as an ~< annihilation ~> process.

252 E. RECAMI and R. MIGNANI

Both observers, however, will agree tha t the electric charge conservation law is verified in the observed processes. Moreover, before interaction, 01 will see one particle, while 02 sees two particles. Therefore, the very number of particles (e.g. of taehyons), at a certain instant of time, is not Lorentz invariant [16, 9]. However, the total number o/particles (e.g. of tachyons) participating in the reaction (in both initial and final states) is Lorentz invariant, due to the very features of R I P [9]. Again, we are encouraged to build the physical theory in terms of <~ reaction processes ~> rather than of << objects ~. Such a suggestion is of para- mount philosophical meaning.

Before closing this Subsection, let us underline tha t the usual proofs of electric-charge invariancc hold only for subluminM, orthochronous LT's applied

to bradyons [61]. We have already shown (Sect. 7 and 8) tha t the charge of a particle may

change sign under GLT's.

11. - Crossing relations, OT, ~15~ and all that.

11"1. G-invariant amplitudes in special relativity. - Let us consider, e.g., a large number of two-body (macroscopic) scatterings

(60) A + B -+ everything

in a certain frame r0. Observer ro will measure the probability A W(0e, ~%) of the process

(61) A + B - - + C + D

for C contained in a certain (small) solid angle around direction 0e, % by calculating the ratio between the number of the good events, (61), and the total number N of events, (60).

When going from ro to r', the quant i ty AW remains of course invariant (for conservation o] events, and the RIP behaviour); it will now give the probability of the transformed process (i.e. of (61) as seen by r), with the transformed C contained in the transjormed solid angle:

(62) ! !

~x w(oo, ~ , ...) = ± w(o~, ~o, ...).

In other words, if da/d$2 a(O, q2) is the ff good ~ process) di]]erential, invariant [62] cross-section, our (invariunt) differential probability dW coincides with da ~ a(O, qg).dtg, except for an invariant flux normalization. Of course, da is as well invariant [62, 63]:

~(0, q~).dD ~ &(O', q~') .d~' ,

CLASSICAL Ttt:EORY OF TACHYONS 253

or be t t e r

(63) da(O, 9, ...) = da(O', q/, ...).

The quan t i ty da m a y be wri t ten [62] as the product of a (invariant) kinematical

factor t imes another factor I , invar iant ~s well, which specifies the interac- t ion (~ dynamics )).

Let us now pass to the microscopic physics. W h a t was previously said is still

valid, under the only assumpt ion tha t the e lementa ry interact ion is relativistic-

ally covariant. Strictly speaking, this is known to happen only ]or electromagnetic interactions (Maxwell equations will be generalized so as to be G-eovariant). For relat ivist ically eovar iant interactions, the above-ment ioned factor I m a y be

wri t ten [62] as the square modulus of an invariant, complex function A of all

qu 'mti t ies characterizing initial an(] final states:

(64) I - = IAI 2 .

The quan t i ty A is known as the invariant amplitude of the process when its

variables are explicit ly chosen so ~s to be ~t least invar iant [62] under

subluminal LT ' s (in such ~ c~se, they will change sign under SLT's : see the

following). When considering a t w o - b o d y - t o - t w o - b o d y reaction, besides the

(usual) conservat ion laws, one meets the new law (i.e. the speci]ic law of t ha t process)

(65) d W - - dW(Oc, %, . . .) ,

as clarified in Subsect. 10"2. By tile way, eq. (65) expresses also the G-

invariance of tile global number of particles par t ic ipat ing in an interact ion

process (in the initial plus final st~tes). In the microphysics case, eq. (65) is be t te r subs t i tu ted by the law

(65') A = A(,~', t, ...),

where s, t, ... represent all the so-called (( invar iant variables ,) on which A

mus t depend. In par t icular s and t are the ((M~mdelstam variables ~, i.e. [64]

(66) I s : : (p~ ÷ PBV -= (Pc + P~) ' ,

t (P~-- Pc)'-' ~ (P~-- P~P;

these quant i t ies (being four-vector magni tudes squared) ~re actual ly invariant under subluminal LT ' s (~-A<), and change sign under SLT's ( ± iA>):

(67) (SLT)s = -- s , (SLT)t = - - t .

Sometimes [64] it happens tha t A(s, t, ...) _: A(t, s, ...).

254 ~. RECAMI and R. MIGNANI

11"2. Effects of GLT's on reaction process descriptions. - When the same interaction process p originates different descriptions from different observers, i.e. appears as different scattering processes d~(p) and d2(p) in different frames r~ and r2, then the PR requires tha t d~(p) and d2(p) are ruled by the same dynamical law (65'). Therefore, because of relativistic covariance, processes

d~ and d2 present the same scattering ampli tude A = A(s , t, ...), provided tha t the physical meanings (and possibly the signs) of the invariant variables s, t, ...

are accordingly changed [641]. Remember tha t GLT's and R I P do automati- cally save the validity of the usual conservation laws as well.

Now, let us consider sublumina] and Superluminal boosts along the x-direction. We shall first consider only T's having [V~ I > c . I t is then easy to observe (e.g. from Table II) tha t :

1) A subluminal boos;, L = :J: A<, applied to an interaction among B's (and/or luxons), may essentially make transition only from a certain process p either to p itself or to the totally C P T - e d one: see Subsect. 8"1.

2) A subluminal boost, L = :t: A<, applied to an interaction among T's (and/or luxons), allows transition [9] from a certain process p either to p itself or to i) any scattering p' obtained by CT-ing [57] one or more particles, ii) any scattering p" obtained by /5- ing no, one or more particles and (?/ST-ing all the remaining particles, provided that processes p' and p" are kinematicalty allowed (or, better, satisfy the conservation laws of energy, momentum, angular momentum and all (( charges ~>). Scatterings p" are nothing but the C/ST-ed ones of scatterings p'. Besides:

3) A Superluminal boost, L = ~ iA>, app!icd to an interaction among T's (and/or luxons), acts as in point 1), while simulCa, l~.eously transforming T's into B's (i.e. changing -- s, -- t, ... into -~ s, ~- t, . . . ) .

4) A Superluminal boost, L = =J= iA>, applied to an interaction among B's (and/or luxons), allows transition from a certain process p either to p itself or to i) any scattering p' obtained by CT-ing [57] one or more B's with Vx > 0

and 0PS~-ing all B's with v~<0 or vice versa, ii) any scattering p" obtained by either /5-ing or ~PT- ing all B's with v~ > 0 (and leaving unaffected all B's with v~<0) or vice versa, provided that processes p ' and p" satisfy the con- se:vation laws of energy, momentum, angular momentum and all (, charges ~). Moreover, the SupcrluminM boost will t ransform B's into T's (i.e. will change s, t, ... into - - s , - - t , ...). As before, scatterings p" are the C/3T-ed ones of scatterings p'.

One could analogously consider the effect of GLT's on (( mixed ~> interactions among both T's and B's (or T's having IV~[ < c). Those interactions are interesting also for the questions in following Subsect. 14"4.

CLASSICAL THEORY OF TACHYONS 2 ~

11"3. Case of interactions among tachyons (with tV~I > c). - Let us confine for simplicity our a t tent ion to interaction processes where the sum of initial and final particle numbers is /our. From point 2) of the previous Subsection~ it follows--]or interactions between tachyons (with IV~[ > c ) - - t ha t [54, 55, 9] the same scattering ampli tude governing the process

(61) A(p~, q~, 2,) 4- B(pB, qB, '~B) -+ C(Pc, qc, "~e) 4- D(p~, q~, ,~),

or the process

(61')

~ A ( m , - q~,- X~)+ B(P.,- q . , - ~),

where p, q, 2 are t r imomentum, (( charge ~> and helicity respectively, is required by extended relat ivi ty to govern also processes (as well as their total ly CPT-ed versions) like

(6s) A 4- C(-- Pc, -- qc, 4- 2c) -+/~(-- PB, -- qB, 4- 2e) 4- D

(and similar partic]e permutat ions) and also processes like

(69a)

(69b)

A -+ /~ ( - -PB, - qB, 4- 2~) + C4- D ,

A + B 4- C(-- pc , - - qc, 4- Xo)-~ D

(and similar particle permutations), provided tha t they are kinematically allowed. Notice that , e.g. in eq. (68), the t r imomenta of A and D are the t ransformed ones of p ~ , p s by means of the subluminal boost L under consideration. Besides, trimomenta appearing in C(-- Pc, ...), B(-- P~, ...), as well as in the /ollowing, record only the versus o/ the transjormed trimomenta la~, p~ with respect to the original ones Pc, PB, ra ther than assigning them a precise value.

Moreover, it is easy to find out relations between a reaction p among B's and reactions among the corresponding T's (i.e. reactions obtained by applyimg

a SLT to reaction p). For instance, let us choose the first observer in the c.m.s. of reaction (61) among four B's. The previous point 4) tells us that , if A = A(s, t, ...) is the scattering ampli tude of

(61) A(p~, q~, ~ ) + B(p~, qB, 2~) -~ C(po, q~, 2c) 4- D(p~, q~, 2~),

then the same function A = A ( - - s , - t, ...) will be the scattering amplitude o] the ]ollowing processes (and of their similar particle permutat ions, as well as of all their C/ST-ed versions), now considered as reactions among T's~ pro-

2 5 6 E. P~ECAMI a n d R. MIGNANI

vided that they are kinematically allowed:

(70) A + C(po, -- qo, - 2~)-+ B(pB, -- qB, -- ,~) + D ,

(71) C(-p~ , - q~, + ~) + D(p~,- q~,- ~) -+

~ ~ ( p ~ , - q ~ , - L) + ~ ( - p~, - q~, + ~ ) ,

(72) C(pc, - - qc, -- ~v) --> A(-- p.~, -- q~, -}- .~A) -}-.B(pB,-- q~, -- ,~B)-~ D ,

(73) A ~ - C ( p o , - q o , - ~)-4- D ( - p ~ , - q ~ , ~- ~ ) - ~ B ( p B , - q ~ , - ~ ) ,

where--as before--the (~ trimoment~ ~) appearing inside the brackets record only the versus of She transformed (tachyonic) trimomenta with respect to the original (bradyonic) ones.

11"4. Case o[ interactions among bradyons. - I t is noticeable that, by using SLT's, we may get results holding [or bradyons [65]. In fact, if two processes among B's (e.g. an interaction and the crossed one [64]) are two different reactions Pl, P~ as seen by us, but they are seen as the same interaction d~ : d ~ ~ d 2 (among T's) by two different Superluminal observers $1, $2 (cf. point 4) of Subsect. 11"2), then we may conclude the following. We may get the scattering amplitude of p~, i.e. A(pl), by applying the SLT(SI-+ so)=-L~ to the amplitude A~(d~) found by S~ when observing scattering pl:

A(pl) : Zl[Al(dl)];

conversely, we may get the scattering amplitude of P2, i.e. A(p2), by applying the SLT(S2-~ So)~ L2 to the amplitude A2(d2) found by $2 when observing scattering p~:

A(p2) = L2[A2(d2)].

But, since by hypothesis

(74)

it follows that

(75)

A l ( d l ) • A2(d~) : A ( d ~ ) ,

A(pl) = A(p2)

for all reactions among B's satisfying the initial hypothesis. From point 4) of Subsect. 11"2 and from the previous Subsection (eqs. (61),

(70)-(73)), it follows that extended relativity requires the scattering amplitude A(s, t, ...) to be given by the same [unction o/ the kinematical variables [or the

CLASSICAL TI-I.EOI~Y OF TACHYONS 257

/ollowing reactions (we l imit ourselves, as before, for s implici ty, to f o u r - b o d y

processes):

i) t he process

(61 A(pa, qA, 2a) -V B(pB, %, ~ ) -+ C(Pc, qc, )'c) ~- D(p~, q~, ~ ) ,

ii) and t he to t a l ly CP~/'-ed one (see eq. (61')),

iii) and t he crossed processes like eq. (70),

iv) and the p a r t l y C/ST-ed and pa r t l y CT-ed processes like eq. (71);

v) t h e << decay processes >> of the t y p e of cq. (72), when ,~llowed;

vi) t h e (( f o r m a t i o n processes )> of the t y p e of eq. (73), when allowed.

Of course, the kinemutic~fl vari:~bles s, t, ... will have for the different

processes the different Ine~mings and values pe r t a in ing to t h e m for the new

processes (*).

We conclude t h a t (cf. also Sect. 14"4):

]) We have der ived crossing relations [61], even for B's , f rom mere ex-

t ended re la t iv i ty .

2) ~ e w <( crossing-type ~ relations are requi red by P g : such relat ions m a y

well serve a~ "~ tes t for relat ivis t ic covar iance of ((force fields ~> like (< s t rong

in terac t ions ~> and p~r t icular ly (, weak in terac t ions >) or possible, new (( iI~ter~c-

t ion fields ~ (which a priori :~rc not re la t ivis t ical ly cov~riant) .

3) Extende( l r e la t iv i ty itself requires t h a t the s~me func t ion A(s, t, ...) gives the sca t te r ing anipl i tudes of different processes (as channels s, t, u, ...

of a f o u r - b r a d y o n re:~ction) in correspondence wi th their physical domains

o f s, t, . . . :

Therefore, in this f r amework , (( analyticity ~> [64] is unnecessary, and better substituted by the G-covariance requirement.

12. - Causality and tachyons [66].

Two qui te different s t a t emen t s are k n o w n under the n a m e of ~ causa l i ty

principle ~> in s t a n d a r d li ter~ture.

The first s t a t e m e n t be~rs t h a t n a m e v e r y improper ly , since it mere ly re-

(*) Since GLT's may change also the observed channel, in order to avoid confusion it is better to consider the action of GLT's on physical quantities as the (~ observed total entering four-momentum ~> or the (~ obserced transferred four-momentum ~>, rather than on their ]ormal expressions according to a certain observer (i.e. relative to a certain channel). In fact. such expressions for the abov(~-mentioncd physical quantities may well change when changing observer (i.e. when the observed channel changes). See also Sect. 14"4.

2 ~ :F. RECAMI a n d m M I G N A N I

quires nonexistence of Super luminal signals. Obviously we gave up such an a rb i t ra ry assumption.

The second s ta tement asserts t ha t causes must chronologically precede

their own e]]ects. This second statement will be adopted [66] by us as the de]ini-

tion o/ (~ causal i ty ,) (or be t te r of retarded causality [67[).

Some authors [68] believed tha t t achyons would violate causality, since,

as we know, any t achyon T, t ha t appears in the labora tory f rame s~ as emi t ted

by A and absorbed by B, will appear to a class of new inertial observers s' as

follows: i) wi~h negat ive energy, ii) moving backwards in time.

But we saw t h a t the re in terpre ta t ion principle or (( bhird pos tu la te ~

(asserting t ha t ]or every observer ((signals ~ are transported only by objects

showing positive energy) does easily [69, 39, 66] eliminate the (~ informat ion

transfer)) backwards in t ime. However , this success is obta ined a t the price

of abandoning the old conviction that judgement about what is (( cause ~ and what

is (~ e]]ect )~ is independent o] the observer. In fact , in the case under exami-

nation, observer So will judge the event at A as causing the event at B. Con-

versely, any observer s' ( interpreting the phenomenon as exchange of an

an t i t achyon T f rom B to A) will judge ~he event a t B as the cause of the

event at A [17]. Nevertheless, all observers will always see the cause to precede chronologically

its own e//ect. Once again, the law (of (( re ta rded causali ty ~) is relat ivist ically eovariant ,

and holds for all inertial observers, both subluminal and Superluminal . On the other hand, we have not to expect at ~11 covariance of the phenomenon de-

scription and of the (( description details ~): in the present case, of the ass ignment of (( cause )) and (( effect ~ names [21c]. Moreover, we shall see [19, 66, 67, 70, 71]

t ha t two events (~ causally connected ~ according to observer s~ (e.g. th rough a tachyonic exchange) may even appear as totally uncorrelated to other observers

s.,; so t ha t not even the very existence of a causal correlation is re la t ivis t ical ly

covariant . l~elat ivi ty of judgement about cause and effect, and even more of existence

of a (~ causal correlation ~), led to a series of apparen t (( causal paradoxes ~) [68, 72]

t h a t - - e v e n if easily solvable [66, 69, 39, 70, 7 t ] - - g a v e rise to some perplexi ty .

We shall ~ta[e and resolve here only the causal pa radox which seems to

be the most sophisticated. I t was proposed by PIm~NI [72] in 1970 and

substant ia l ly solved b y PAI~)[E~TOLA ,~nd YEE [71[ in 1971, on the basis

of ref. [69, 67, 39]. Let us consider four observers A, B, C, D having some given [72] velocities

in the plane (x, y) with respect to ,~ fifth observer so. Le t us suppose t ha t ~he four observers are given in advance the instruct ion to emit a t aehyon as soon

as they receive a t achyon f rom another observer, so t ha t the following chain

oJ events takes place. Observer A initiates the exper iment by sending t achyon 1 to B; observer B immedia te ly emits t achyon 2 towards C; observer C sends

CLASSICAL THEORY OF TACtIYONS 2 ~ 9

t achyon 3 to D, and observer D sends t achyon 4 back to A, with the result [72] tha t A receives t achyon 4 (event A~) before having initiated the experiment b y

emit t ing t'~chyon 1 (event A~). The sketch of this (<gedanken e x p e r i m e n t , is in Fig. 19, where oblique vectors represent observer velocities relat ive to so

and lines parallel to the Cartesian axes represent the t achyon paths.

x (D) "3 (c) .

2 C

A(A~)~I4 I= .¢(@ 4

Fig. 19. Pirani's pam~dox, solwsble on the basis of the (<reinterpretation principle, (the (( third postulate ~). in the sense that each observer will see the law of (~ retarded causality ~> (which requires (, causes ~) to precede chronologically their own (¢ effects ~)) satisfied iu its ~rame.

I t is impor tan t to notice tha t Fig. 19 does not represent the process actual

descril)tion by any observer! [66]: in fact , the arrow of each taehyonie line

s imply denotes its motion direction with respect to the observer t h a t emi t ted

tha t part icular tachyon. (By the way, t achyon and observer velocities are

supposed to have been chosen 172] in such a way t ha t , wi th respect to observer so, all t achyons effectively appear ones indicated in Fig. 1.9.)

The above paradoxical situation

to move in directions opposite to the

does actually arise by mixing together ob- servations by four diHerent observers [71, 66]. On the contrary, it is necessary to invest igate how each observer describes th~ event chain.

Following ref. [71], let us pass, for this end, to 5I inkowsky space and s tudy

the space-t ime description given, e.g., by observer A. From a dynamical

viewpoint , the other observers m a y be replaced by external forc~ fields tha t

scat ter the tachyons (or by :~toms, able to absorb and emi t tachyons). In Fig. 20 it is clearly shown t h a t the absorpt ion of 4 happens bejore the emis-

sion of 1. I t might seem t h a t one can send signals into the past of A. However , observer A will effectively see the sequence of events as follows: event D

consists in the creation of pair 3 and t by the external field; taehyon 4 is then

absorbed at A1, while 3 is scat tered at C ( t ransforming into t achyon 2); event

A2 is the emission, by A itself, of t aehyon 1 t ha t annihilates at B with tachyon 2.

Therefore, according to A, one has essentially an initial pair creation a t D,

and a final pair annihi lat ion at B; and tachyons 1, 4 do not appear (causally)

correlate(1 at all [71,661. In other words, according to A the emission of 1

does not init iate any chain of events t ha t brings to the absorpt ion of 4, and we

are not in the presence of any effect preceding its own cause. Analogous, orthodox descriptions (i.e. the descriptions pu t for th b y the

2 6 0 ~ . n]~c~M~ a n d ~ . M m S A N I

c

D ~

ct ~ B

Fig. 2 0 . - Space-time description of the sequence of events constituting Pirani's gedanken experiment, according to observer A (whose world-line coincides with axis ct). Events A 1 and A 2 represent absorption and emission of tachyons 4 and 1 respectively. At D one has the emission of the two tachyons 4 and 3 (initial state). Afterwards, tachyon 3 is seen to interact (at C) and transform into tachyon 2. Finally in the final state, taehyons 1 and 2 appear to be absorbed at B.

remaining observers) may be obtained by Lorentz- t ransforming the above

description by A. Let us now reformulate the same <~ paradox )> in its <( strong )) version [71, 66].

Let us suppose tha t t achyon 4, when absorbed at At by A, blows up the whole laboratory of A, eliminating the physical possibility tha t tachyon 1 (believed to be the sequence starter) is subsequently emit ted (at A2), and thus originating a supposedly contradic tory state of th ings--according to the paradox terms. Following ROOT and TREI~IL [70], we can see, on the contrary, how, e.g., observers so and A will really describe the phenomenon.

In particular, So will observe the laboratory of A blown up after emission (at A1) of tuchyon 4 towards D. According to So, therefore, t achyon 1 emit ted by B will proceed beyond A (since it is not observed st A2) out to infinity, and will eventual ly be absorbed ut some remote sink point U of the Uni- verse [70, 66]. By means of a LT, start ing from the description by So, we can

obtain the description given by A [66]. Observer A, af ter having absorbed at At t achyon 4 (emit ted a~ 2) together

with 3), will record his own laboratory explosion. At A~, however, A will

realize t h a t he has been by-passed by a tachyonic cosmic ray 1 (coming from

the remote source U), which will annihilate at B with tachyon 3 scat tered at

C (i.e. with tachyon 2) [70, 66].

1 3 . - D i g r e s s i o n (*).

We have seen that no problem about the G covariance o] the (~ retarded causality law ~> is le]t with tachyons. ~qevertheless, we want to add the following (~ digres-

(*) This Section may be skipped, in a first reading.

CLASSICAL TIt]~ORY OF TACI~IYONS 2~1

sion ~) without any re/erence to what precedes. L e t us r e m e m b e r t h a t t he (( ~r row )7

to ( r e t a rded ) c a u s a l i t y is re~son,~bly g iven on ly b y t h e s t a t i s t i c a l b e h ~ v i o u r

of m a c r o s y s t e m s [73-76, 67]. On t h e c o n t r a r y , in a m d y s i n g one e l e m e n t a r y

process , i t is n o t e a s y a t al l to choose which is cause ~nd which ef fec t : for e v e r y

m i c r o p h e n o m e n o n , i t seems poss ib le to buihI u p a (( second )) desc r ip t ion , which

s~tisfies t h e law of (, advanced causali ty ~ [67, 73-76, 66].

A c c o r d i n g to some ;~uthors [67, 74-76], even if in m a e r o p h y s i c s t he a r row

of ( r e t a rded ) c:~.us~flity is well d e t e r m i n e d , ill m ic rophys i c s t he causa l connec t ion

is ,~ s y m m e t r i c correl '~t ion.

I t might hapl)en, however , t h a t i~ f u t u r e microphysics one wil l f ind e i the r ~

(( ful l c a u s a l i t y ~) [74] l aw to ho ld (where an effect m a y b o t h fol low ~nd precede

i t s et~usc); or ;~ part icular T - v i o l a t i o n to t a k e p lace in a c e r t a i n mic rophe -

n o m e n o n , such that (i.e. (fin such " d i r e c t i o n " t h a t ~)) send ing a s ignal ((into

the p a s t )7 I)ecomc r ea l l y poss ib le t h r o u g h t h a t p h e n o m e n o n (*). O t h e r poss ib i l -

i t ies in t h a t s e ~ s e - - i m p l y i n g t h n t f u n d a m e n t a l ( ( symme t r i e s ~7 of N;~ture, or

of t he ,~ccepted (( logic )7, a re b r o k e n - - h ~ v c been cons ide red in rcf. [77-79].

H e r e we w a n t m e r e l y to show 1;hat t h e p o s s i b i l i t y of send ing a s ignal

(~ in to t h e l)a,s~ )) wouht no t bc logically c o n t r a d i c t o r y [77-80, 66]: i) Le t us

suppose theft a statist , ical correlatio~t ex is t s b e t w e e n two series of events , in

t h e sense t h a t , e.g., each second-ser ies e v e n t h a p p e n s :~bout l second before

a f i rs t -ser ies e v e n t (see Fig . 21). Such a s t a t i s t i c a l cor re la t io t t will be ca l led

/ / ///_ oi- - t

t2 t - e I t c -

a) v)

Fig. 21. - In a), an example ia shown of st;i.tistical corrclation bctwcen ~, cause ~)- events and (( effect ))-events: ~irst-series events h~ppen at t imc instants recorded on a x i s t l , and second-series events ~t instants recordc4 on axis t 2 (the two axes having the s~unc ((zero)7). [n b), the distr ibution in tim(~, supposed to be s Gaussi,~n sroun4 point t~, is given of cv{~nts t~ with rcspcct, to t.ln~ corresponding events t 1 (whose hap- pening insl;mts all formally coin(ride at t~).

a (( causal connection )7. ii) L e t us now SUl)l)ose theft l i r s t - se r ics ev(mts ~re

t h e (, i n d e p e n d e n t ~7 ones, in t h e sense tht~t we m a k e t h e m occur, e.g., a t in-

s t a n t s chosen b y consu l t i ng r a n d o m - v a l u e ~,ables ( m a y b e p r o d u c e d b y a r e m o t e

c o m p u t e r , w i t h o u t a n y r ea sonab le r e l a t i o n w i th t he even t s cons idered) . Such

even t s will be s;ud t h e ((causes)7. iii) The second-ser ies even t s wil l be t h e n

(*) ()r, b(dt.~r, tim (~ third postulate ~7 (RIP) might happen to be violated by some m;~cro- or micro-phunontenon (not(~ that such ,~ phenomenon would then be (~×pected to vioh~t(~ ;flso Of '~ cov:~ri~ncc), with ttm resull of allowin~ information to be s(~nt into i.hc l)~l,sl.

2 6 2 E. RECAMI a n 4 R. MIGNANI

called the (( dependent ~) ones in the causul correlation defined ~t point i). They

will be said to be t.he(~ effects ~). iv) One m a y therefore conclude, f rom the

above definitions, t ha t in this case effects do chronologically precede their own

causes (Fig. 21). Le t us r emember tha t WHEELER and FEY:N~AN [75] did meet a <~ causal

p rob lem )) of this t ype since 1949, when considering both (~retarded ~) and

(( advanced ~ solutions of Maxwell equations. See also ref. [74, 80].

To conclude the present (( digression ~ (which has no relation also with what ]ollows), in order to shed some light on the na ture of our difficulties in con-

ceiving effects chronologically preceding their causes, let us repor t f rom ref. [74]

the following anecdo te - -which does not involve present prejudices.

For ancient Egypt ians , who knew only the ~i le and its t r ibutaries, which

all flow f rom South to ~North, the meaning of the word ~( South ~ coincided

with the one of (( up-s t ream ~), and the meaning of the word ~ ~North ~) coincided

with the one of (( down-s t ream )>. When Egypt ians discovered the Euphra tes ,

which unfor tuna te ly happens to flow f rom ~North to South, they passed th rough

such a crisis tha t it is ment ioned in the stele of Tuthmosis I , which tells us

abou t (( that inver ted water which goes down-s t ream (i.e. towards the ~Nor~h)

in going up - s t r eam )~ [74].

1 4 . - T a c h y o n m e c h a n i c s .

14"1. Rest mass and proper quantities. - The first noticeable observat ion

t ha t we can draw f rom the REt¢ (Sect. 9), in the field of dynamics, is the fol-

lowing. E v e r y t achyon results actual ly at rest with respect to a certain

class of Super luminal f rames So. Relat ive to any f rame •o, the t achyon

will behave exact ly as a b radyon at rest with respect to us; in par{:icular

it will show a real inert ial (~ rest mass )> m0 (and not an imaginary proper

mass [18, 19, 22, 23, 66, 31]). :Now, if we want to know which <~ relat ivist ic mass ~> m the t achyon will

show to us, it is enough to app ly the SLT making the txansition f rom So to

us [19] [too real]:

mo mo i?~o mo (76) ~= m = ~/]1--f121 - - i V ~ - - f l 2 ~¢/1 _ ~ 2 ~ - 1

where the double sign refers to tachyons and ant i tachyons respect ively (see

Subsect. 7"3). Cf. also Sect. 9. I n eq. (76), we chose ~ / f i 2 - - 1 : - - i ~ l - - f i 2 (cf. Subsect. 3"5) in order

t h a t a t aehyon T (not an ant i taehyon) , mo~cing with positive velocity V~ > c

with respect to us, appears to us as u t achyon (not an an t i taehyon) :

( 7 7 ) m - _=~>I .

~ / 1 1 - # ' 1 = ~/#'- ~

C L A S S I C A L T I I E O R Y OF T A C I I Y O N S 263

This is done consistently with the geometrical interpretat ion of GLT's (Sect. 6) and with Fig. 7, 9, 10. Notice that this choice is equivalent to taking (as we did) L = - - i A > ( / ? ) as the SLT from us to the rest frame of T. Notice, moreover,

tha t m----mo for fi----~/2, consistently with Sect. 6 and Fig. 10. Let us underline tha t the imaginary unit i entering ec t. (76) is due to the

SLT, and not at all to the fact tha t tachyons must be actually a t t r ibuted '~

(( pure imaginary ~> [15] proper mass (see eqs. (43)) ! One may well ~voi(l completely the use of any imaginary unit, as in eq. (77).

The same holds of course for the other 1)roper quantities, such as proper t ime and proper length. For example, since (lro ~ ~ - ( 1 T ~ / l l - fl21, when passing from the rest system of the above-considered tachyon T to our frame So, one h~s

dvo i(lro dro (78) d~= + V l l _ ~ I ~ /1-~ ~/~-.L ( ~ > ~ ) '

where obviously dr , d~o are both real. Again, it is d~----d~o for ~ = ~/2.

14"2. Generalized relativistic Newton's law. - As is well known, in the

bradyonic case the fundamenta l equat ion of dynamics reads [40]

d ( d~'.~ (79a ) T , ~- c (i s m o c - d ~ ] (fl: < 1) .

:By applying a SLT (in pnrt icular the (( t ranscendent t ransformation ~> K+ of

Subsect. 4"3) ~o eq. (79a), we get immedia te ly- -according to Sect. 9 - - t h e

]undamental equation o/ tachyon dynamics:

d ( dx , \ (79b) .F, ~- --( ' ( is m°C-ds ] (f12> 1),

where, as expected, in the case of reetiline~r motion the '~eceleration direction is opposite to the force direction (cf. Fig. 13). In fact, tachyons must decel- erate in order to incre~se their energy, and vice versa.

By remembering tha t for tachyons d s = J= icdr0, where of course (lr0 is a G-eovariant quant i ty , it is immediate to write the relat ivist ic Newton's law in G-covariant ]orm :

(80)

valid for both bradyons and tachyons. Notice explicitly tha t dx/ds is a four- vector only with respect to the group ~fl of usual LT's, whilst dx/dTo is a /our- vector with respect to the whole group G (and the latter represents tile actual, ex tended /our-velocity).

2 6 4 v.. RECAMI a n d R. M I G N A ~

:Explicit calculations have been done also by YEZHOV [81], who, in the part icular case of particle velocities along the x-axis, found (+~ ~: dv=/dt)

(Sl)

dp: m~): 2'= ~ -- ÷ - - for b r a d y o n s ,

d~o tl - -~ : l 2 dp: m~):

/ ~ ~ -- for t aehyons . d~o I1 --fl : l ~

14"3. A n application. - The fact tha t for a free tachyon (see eqs. (42), (43))

(42b) E ~ ---- p~-- m~ (f12 ;> 1, mo real)

has some unusual consequences [55, 16].

For instance, a pro ton p - - w h e n absorbing, e.g., a taehyon t or an anti- t aehyon t (coming from a nearby emit ter or from cosmic rad ia t ion) - -may t ransform into itsel]:

(82) p-+- t -+p , p-+- t - + p ,

as can be kinematically verified in the global c.m.s. In the same frame (the

global c.m.s.) it is similarly s traightforward to verify that , on the contrary, the decay of a proton is not kinematical ly allowed.

However, if we pass f rom the global e.m.s, to another (subluminal) inertial frame, moving collinearly, e.g., with positive speed u = u x ~ c2/V~: (where V: is the velocity x-component of ¢ or t , and where it is assumed tha t V: ~ c), we know from Subsect. 8"5 tha t the tachyon t entering the first reaction of (82) will appear as an escaping ant i tachyon; and the ant i tachyon t entering the second reaction of (82) will appear as an outgoing tachyon. I n the new ]tame,

therefore, the following reactions are kinematical ly allowed, according to our crossing relations, Subsect. 11"3:

(83) p --+ p~-~, p -+ p-+-t (only in ]light).

In other words, a proton in ]light (but not at rest!) may a priori [10] be seen to

decay into itself plus a t achyon or aut i tachyon (which will either be absorbed

nearby or proceed out to cosmic distances). By comparing the couples of reactions (82) and (83), we can see t ha t - - s t r i c t l y

speaking- - the second of eqs. (83) is not the t ime-reversed of the first of eqs. (82), owing to the intervening kinematical boost. However, on the other hand, extended relat ivi ty requires nothing but that, for tachyonie kinematics (and not ((pure ~ T-reversibili ty). In the case of tachyons, therefore, we may still speak of (kinematical) T-reversal, bu t with ~he above-ment ioned

clarification.

CL:&SSICA.L T H E O R Y OF T A C I t Y O N S 2 6 ~

14"4. (, Virtual particles ~> and tachyons. - Let us now briefly invade a field

usually reserved to quantum mechanics, i.e. tha t of models for elementary-

particle strong interactions, confining ourse lves- -however- - to their relativistic (high-energy) beh,uviour.

Since 1968, essentially within the framework of (( peripheral models ~ (e.g. one-particle exchange models), RECAMI [79] has suggested the possibility tha t the usual ((virtual particles ))[37b] actually be considered as tachyons. In

fact (in the s-channel) they generally bear the negative squared four-momentum [64~ 82,451

(8~) p~ t < 0 .

(Tile results in ref. [79] appeared also in ref. [46, 47].)

Let us remember tha t in a two-body- to - two-body scattering the above

quant i ty t is known both to become positive and ~o change its meaning (e.g. from ~( momentum transfer squared ~> to (( total energy squared ~>) when passing from

the s-channel to the t-channel. This accords with the fact (Sect. 11"2) tha t a SLT may transform a reactiou (among bradyons) into the crossed one (among tachyons).

These points help clarify as well why passing through two SLT's (e.g. one SLT changing channel s for B's into channel t for T's, and one SLT only changing T's into B's, without affecting the channel) is needed for going from an interaction among bradyons to the crossed interaction, still among bradyons (cf. Sec~. 11"4).

In the previously mentioned framework of the peripheral model (with

~( ~bsorption )> [8311), a very rough tes t - -meaningfu l only within tha t mode l - - has been hE,de [79, 46, 47] for virtual particlc velocities~ which just supported the Superluminal velocity hypothesis. In model ref. [83a], to calculate the effect of ~, absorptive ~ chammls on the (( one-particle exchange model ~), one cuts out t, he low 1)artial wa-~-es from the Born amplitude. Precisely, an impact parameter (Fourier-Bessel) expansion of the Born amplitudes is used, and a sharI) cut-off (step function) is adopted at a radius R, which is wLried to fit the experimental data. Now, while considering--for example--di f ferent cases of pD reactions with K-meson exchange, in ref. [83a] values of R ranging from 0.9 to

1.1 fm have been found, i.e. radius values much greater than the K-meson

(( Compton wavelength ~>. In ref. [83cl, the previous model (with form factors, at

a few GeV/e) has been also applied to pion-nueleon reactions with ~ production,

via p exchange, on the basis of ref. [83b]; and analogously a value has been found

even for 1he p-meson of R ~ 0.8 fro, much greater than the p Compton wavelength, h t tha t model, therefore, one shouhl deem-- f rom [84] the Heisenberg

uncer ta inty pr inei l ) le-- that (,-,~irtual ~ K and p mesons of the proton (( c loud , [841 t ravel faster than light [851. For instance [79, 46, 471, in the

2 first case, for t - -- m K, one would find <[?: ?z 1.75.

18 - Rivis fa del Nuovo Cimento.

266 ~. RECA~I an4 R. MmSANI

The most important observation (essentially due to SUDAI~SI-IAN) is tha t the actual presence of a tachyon exchange would produce [38, 57, 9, 10] a resonance peak in the scattering amplitude as a function of the (( momen tum transfer )~ t. Namely, i t would produce a (, negative t enhancement ~)[82, 45], fixed when s varies, and possible to be found also in different, similar processes [38, 55-57, 9]. For instance, in ref. [9] a t tent ion has been called to such kind of results as the ones in ref. [86]. Let us also mention ref. [87]. Recently, the first interesting theoretical evidence in this direction appeared [88, 89], mainly due to GLEES051 et al. We simply refer here to ref. [88, 89].

15. - Tachyons in the gravitational field.

15":l. Introduction. - Firs t of all, let us remember that , even in the Gati-

lean relat ivi ty case, the universal gravitat ion law may be wri t ten [90] in a form similar to tha t of Einstein's gravitational equations. Namely, given a certain space-time, let us call F~,(x) the (~ affinity ~) (or afiine connection) components defined on tha t space-time [90]. The Galilei-Newton space-time has a very particular affine geometry: the (,flat afline geometry ~), where F~',(x) - - / ~ vanishes at every point of the space-time manifold.

Then, ~he ~ewtonian equations of motion of a particle experiencing a gravitational force F~ may read [90]

(85) m0 d2X~dt- ~ + 1~.-~-° dxq dXad~ ~ /~" (Newton's case) .

15"2. Taehyons and gravitational ]ield in special relativity. - Also in special

relativity theory (Minkowsky space-time), it is possible to introduce the gravitational field [90], and to build up a theory explaining all observed gravitat ional phenomena (advances of planetary perihelia, bending of light rays, and so on), and deducible by means of a variational principle [90]. We are referring essentially to the formulation by BELI~FA~TE [91] and by FIERZ [91]. B y the way, such a theory may be improved [92], so to lead to the same field equations of general relativity.

In Belinfante 's theory [90, 9J] (which, e.g., succeeds in being associated with a definite spin value, namely 2), the gravitational field is described by a sym metric second-rank tensor h~. The equations of motion ]or a bradyon turn out essentially to be [90, 92, 29]

d~x" dxq dx " _ 0 (ds t imelike), (86) ds - ~ + / ' ~ ds ds

where the quantities F~, are the components of a certain (~ symmetric affinity ~) [90], i.e. are the Christoffel symbols formed from a certain tensor [90] ]~, (built up

C L A S S I C A L T H E O R Y OF T A C I I Y O N S 267

by means of the Minkowskian metric tensor g,. and of h ) . For our purposes, it is noticeable tha t Christoffel symbols behave like (third-rank) tensors with respect (only) to linear transformations of the co-ordinates. This is our ease, since all GLT's ~re line~r transformations of Minkowsky space-time.

We are of course assuming tha t the gravitational interaction is relativ- istically covariant. Eqm~tion (86) tells us tha t the considered bradyon suffers the gravitational 4-force:

dx~ dx ~ (86 bis) ~ ' ~ -- moF~ ds ds (fl~< :1).

From the (~ tachyonization rule ~) (Sect. 9), we may immediately derive the gravitatiomfl 4-force experienced by a tachyon, by applying a SLT (e.g. the transcendent transformation K+; el. Subsec$. 4"3) to eq. (86 bis). We get

dx'q dx 'a (87) F ' " = ÷ moF~ ds , ds' (fl2>1),

where now me is the (real) proper mass of the tachyon considered. In other words, a t~chyon is seen to experience a gravitational repulsion

(and not at tract ion !). However, since the fund~mental equation (79b) of tachyon dynamics brings about ~nother sign change, the equations of mo¢ion for a tachyon in a gr~vitutional field will still read

(86') d2X"ds'" + I'Q,-'' dx'Q~ dx '~ - - d s ~ 0 (ds' spacelike) .

Therefore, eqs. (86)-(86') are already G-covariant, and hold for both bradyons and taehyons. We may (still G-covuriuntly) write

(88) a" + V g u % ° - o (fl~ ~ 1 ) ,

where a and u ~ e four-aceleration ~nd four-velocity, respectively. Notice again that , whilst dx[ds is a four-vector only with respect to the group ~ , on the eontr~ry u == dx/dTo is ~ G-four-vector.

As reg~rds tachyons, let us, e.g., consider ~ tachyon going towards a gravitational-field source. Since it will experience, a repulsive force, its energy will decrease (contrary to the bradyonic case); however (see Fig. 13) its velocity will increase. Therefore, we spoke about (~ gravitational repulsion )? since the gravitational force applied to the tachyon is centrifugal (and ~lot centripetal); however, as a result, the tachyon will bend towards the gravitational-field source (simib~rly to ~ bradyon) [93b].

In conclusion, under our hypotheses it follows tha t :

1) From the dynamical (~nd em;rgetic~l) point of view, taehyons ~l)pear as gravitationally repulsed. They may well be considered to couple (( neg~-

268 ~. Rv~CA~I and R. MIGNANI

t ively ~ to the gravitat ional field (the source velocity being inessential !), i.e. to

be the (( antigravitational particles ~.

2) F rom the kinematical viewpoint, however, tachyons appear as ((fal- ling down ~ towards [93b] the gravitational-field source (the source velocity being inessential, just like for the electromagnetic field) (*).

Let us explicit ly remember tha t usual antiparticles do (( positively ~ couple

to the gravi tat ional field. There is no gravitat ional difference between objects

and their antiobjects, in accordance with the (~/~IP ~>. For tachyons, in the nonrelativistic limit (speed near cx/2) we would find

for the three-force magnitude no longer Newton's law F------ Gmomojr2 , but

= + G mo, mo, r 2

(fl~> 1 ; too,, mo, rea l ) ,

since when passing to tachyons G transforms into - -G. All the foregoing accords with the principle of duality. In fact, given an ob-

server O, all objects t ha t appear as bradyons to him will also appear to couple

(~ positively )) to the gravitat ional field; on the contrary, all objects tha t appear as tachyons to him will also appear to couple (( negatively ~ to the grav-

i tational field. In any case, bradyons, tachyons and luxons will all bend to-

wards the gravitational /ield sources (even if with different curvatures) .

13"3. On the so-called (( gravitational Cerenkov radiation ~ and tachyons. - All tha t was previously said holds under the assumption of negligible gravita-

t ional wave radiation. If we improved eq. (86), for bradyons in a gravitat ional field, by allowing

gravitational-wave emission, the tachyonizat ion rule (in the spirit of the dual i ty principle) would still permit us to derive immediately the corresponding l a w / o r taehyons (now allowed to emit gravitat ional waves when interact ing

with that gravitat ional field). But here we want only to deal with the problem of the so-called (~ gravita-

t ional ~erenkov radiat ion ~. A usual bradyon (e.g. without charges, but of course massive) will gen-

erally radiate gravitational waves [93] as follows:

A) Firs t of all, when (and i/) it happens to enter a sensu lato medium (i.e. a force field), with speed larger than gravitational-wave speed in tha t field, then it is expected to radiate a cone of gravitational radiat ion (some- what similarly to the sound-wave cone emi t ted by an ultrasonic airplane in

(*) Thc problem of tachyon trajectories in a Schwarzschild field has been considered, e.g., in ref. [93b], within general-relativity theory.

CL2LSSICAL THEORY OF T~CHYONS 269

air). Such a cone should not be con]used with (~erenkov gravitat ional radia-

tion, which oil the contrary has the following features [94]:

i) it is emit ted by the bodies (or particles) of the material medium, and

not by the travell ing object itself;

ii) it is a function of the velocity (and not of the acceleration) of the

travelling object.

B) Secondly, when the considered bradyonic object (e.g. a galaxy) B happens to enter a cluster of uniformly dis tr ibuted galaxies, with a speed larger than the gravitat ional-wave speed inside tha t (( material medium ~ (the cluster), then B will cause the cluster galaxies to emit gravitat ional waves

such tha t possibly they will coherently sum to form a gravitational Cerenkov

cone [95, 95].

Now, we want to investigate if and when tachyonic objects can emit cones of gravitat ional waves, and in particular cause emission of gravitational-radia-

tion ~erenkov cones:

1) The first result is tha t ]ree tachyons in vacuum will emit neither gravitational (Jerenkov radiation, nor gravitational-wave cones, nor any gravitational radiation. This result may be immediate ly obtained (cf. Sect. 16) by using the tachyonizat ion rule (Sect. 9), i.e. by applying a SLT to the behav-

iour of a ]ree bradyon (e.g. at rest) in vacuum.

2) The previous case A) being substantially not different from case B) [90, 61], we shall confine our a t ten t ion to studying Cerenkov radiation cones

caused by tachyons. I t is easy to recognize tha t the relative discussion reported in Subsect. 16"2 below for the electromagnetic case holds also in the present ease. Therefore, by comparing Fig. 22, we may get tha t i) the gray-

t/

tl

Fig. 2 2 . - (* Drag effect ~) extension: generalization of the formula for the re]faction index N of a medium vs. its relative velocity u. The mcdimn is supposed to move collinearly with respect to the observer. N o is the proper refraction index. See the text.

270 ~. Rv.CAMI an4 R. ~m~ANI

i tat ional-wave speed v in a bradyonie medium appears to a bradyonic observer 0 always slower than c; ii) the gravitat ional-wave speed in a hypothetical, ideal (~ luminal ~> (u ~ c) medium would appear to 0 as always equal to c; iii) the gravitation~l-w~ve speed in a tachyonic medium will always

appear to 0 as larger than c.

Wi th use of the duali ty principle, it is s traightforward to deduce tha t

(cf. Subsect. 16"2) tachyons will emit gravitat ional (~erenkov cones only in

tachyonie media, when they happen to t ravel there with speed slower than the

gravitat ional-wave speed in the (tachyonic) medium considered. See also the

next Section.

16. - About electromagnetic Cerenkov radiation (ECR) from tachyons.

16"1. Tachyons do not emit ECR i n vacuum. - The problem is to investigate if and when taehyons will appear to emit electromagnetic Cerenkov radia-

t ion (ECR). Many authors (*), on the basis of an uncritical extension of the usual Ccrenkov-

effect formulae made by S0~ERFELD [4] in pre-relativistic times, proposed tha t tachyons would always emit ECR even in vacuum at constant velocity.

We want to show, among other things, tha t this is not the case [96, 97]. I t is necessary not to confuse ECR with the radiat ion tha t a particle can indeed

emit in vacuum when it is accelerated. First ly, let us again emphasize (cf. Subsect. 15"3) tha t EC]~ comes out f rom the excited electrons of the material medium, and not from the <( radiat ing ~> particle itself [94]. Therefore, in the usual context , the very expression ((ECR in vacuum ~> appears meaning- less [96, 98], unless one simultaneously provides an improbable, suitable theory about a possible <, vacuum structure ~>. Secondly, let us consider a particle tha t appears to us (frame So) as a T moving at a constant velocity in vacuum. Such a particle will appear as a B with respect (e.g.) to its rest frame S;

according to f rame S, therefore, the (bradyonie) differential energy loss through

~erenkov radiat ion in vacuum is obviously zero [90]:

dE (89) d--~ = 0 (f12 < 1 ) .

If we t ransform such a law by means of the SLT from S to so (and remembering the vaeuum-covariance postulate), we get the differential energy loss through

(~erenkov radiat ion for a tachyon in vacuum [23, 96]:

dE ' (90) ds' - - 0 ( /~ > 1 ) .

(*) See references in ref. [23, 96, 97].

CLASSICAL T H E O R Y OF TACtIYONS 271

We can therefore conclude that, irr extended relativity theory, tachyons are

not expected at all to emit ECR in vacuum [23, 96-98]. Moreover, a particle uniformly moving in vacuum does not emit any radiation both in the subluminal case and in the Superluminal one, as required by (extended) relativity.

16"2. The general problem. - The principle of duality, together with the tuchyonization rule, allows us to solve also the general problem of ECR from tachyons.

Let us first generalize (Fig. 22) the (, drag effect ~)[27] for Superluminal velocities, i.e. extend the calculation of the apparent velocity v of light in a moving medium, with respect to a given observer 0, for tachyonic media. Figure 22 just illustrates the (extended) <( refraction index ~>, N, for moving media, vs. the medium speed u, derived [23] from mere consideration of the generalized velocity composition law (Sect. 5):

2~oC -~ U (91) N(u) =-- N = (u 2 ~ c2);

c + N o u

the quantity v = c/5 r will give the light speed in the medium considered, with respect, to observer 0.

As anticipated in Subsect. 15"3, it is immediate to observe that i) the light speed v in a bradyonic medium appears to a bradyonic observer 0 always slower

than c; ii) the light speed v in an hypothetical, ideal <~ luminal medium ~ would appear to 0 as always equal to c; iii) the light speed in a tachyonic medium will always appear to 0 as larger than c.

I t is immediate to deduce that:

1) A bradyon B can emit (3erenkov radiation only in a subluminal medium (when it happens to travel with speed w larger than the apparent light speed v in that bradyonic medium).

From duality and the t.achyonization rule, it is then straightforward to get that:

2) A tachyon T can emit Cerenkov radiation only in a Superluminal me-

dium, and namely when it happens to travel in the (taehyonic) medium with speed w slower than the apparent light speed v in that medium. Therefore, a tachyon will never emit Cerenkov radiation in usual, subluminal media. Conversely, it is possible that a taehyon appears to move in a tachyonic medium with velocity slower than the apparent light velocity there: see Fig. 22.

From the analytical point of view, the formula of the standard differential energy loss (for unit path length) through ECt~ I tem a bradyon in a medium

272 E. UECAMI and u. MIGNANI

reads [95]

dl - - e 2 1 v d v f l ~ e ' w2 < c'~ '

[fl/qo>l]

where now f l - w/c, the quant i ty w being %he bradyon speed relative to O, and No is the proper refraction index of the medium. In particular, for No ~ 1 (i.e. for the vacuum), one gets

dW) = O (w ~ < e ~) (93a) d l - o "

Then, by applying the trascendent Superluminal Lorentz transformation K+ to eq. (92a), we find the analogous formula of the differential energy loss (for uni t path length), through ECR, /rom a tachyon in a medium:

- - 1 - - ~dv f l ~ c ~ w 2 > c 2

[~'o/fl> I ]

where No is still the proper refraction index of the medium, and w the tachyon speed (relative to 0). In particular, for N o > l (i.e. for the vacuum and for bradyonic media), one gets

( d W ) = 0 (w2>e 2) (93b) ~ o "

The (~ Cerenkov relation ~>, giving the cone angle in the bradyonic case [95]

1 (94a) cos0 ---- - - (f12< 1),

flNo

transforms under the same SLT into the following relation, giving the cone angle in the tachyonic case:

(945) cos0 ---- ----fl (f12> 1). No

All previous experiments [17, 99-102] looking for tachyons, or better looking for ECI~ from tachyons, failcd, being based upon a wrong assumption, originally due to SUDAICUItAN himself [15]. We have seen, on the eontrary~ tha t tachyons

do not emit ECI~ in bradyonic media.

17. - Dopp ler effect for S u p e r l u m i n a l a s t rophys i ca l sources .

Extended relativity allows in particular generalizing [23] the Doppler-effect formula [27]. In fi~ct, from the time interval extended transformation law

C L A S S I C A L T H : B O R ¥ OF T A C t t Y O N 8 273

(Sect. 6), it is immedi~te t,o get the Doppler-effect formula for Superluminal

sources [103]. Namely , in the c~se of rel~tive mot ion parallel to the x-~xis,

we have in both subluminal and Superluminal eases [23, 103l

(95) ~ ~°1 ~-/?cos~ (u~X c~),

where u ~ u, =:/3c is the relat ive speed and c¢ ~ ul, the vector l being directed

f rom the observer to the source.

In the par t icular c~se of relatiw~ motion str ict ly along the observat ion

r~y, since

[sign (u)[ × [sign (cos ~)] = / ~ - - corresponds to approach, ÷ corresponds to recession,

we obtain the behaviour shown in Fig. 23, where the dashed curve refers to

<~ approach ~ ~nd the solid one to <, recession ~>.

i

I I

/ - /

/ /

1 I

#*

f - -

/ /

/ !

!

Fig. 23. - Doppler-e//ect extension: observed frequency vs. relative velocity for motion along the x-axis. The sign minus refers to approach (4ashe4 line) and plus to recession (solid line). The. interpretation of the negative wflues appearing for the Superluminal approach is given in Fig. 24.

In the part icular case sin g = 0, formula (95) h~s been independent ly

derived also in ref. [104], through merely heuristical considerations.

The firs¢ important, point we want to stress is the following [23, 103]. In

Fig. 23, t, he two solid curves (recession) :~re one the conformal correspondent

of the other, as expected, in the sense th,~t the same frequency wilt be obtained

274 ~. R]~C~kMI and R. MIGNANI

]or both u~--v ~ e and U : c2/v~ c. Therefore~ an astrophysical source re-

ceding with Superluminal veloci ty U is expected to exhibit a Doppler effect

identical to t ha t of a usual source t ravel l ing at veloci ty u----c~/U.

The second result to be emphasized is the following. The above-ment ioned

conformal correspondence holds even for the two dashed curves (approach),

except for the sign. Precisely, in the case of Superluminal approach, eq. (95)

yields a negat ive sign [23, 104] (Fig. 23). Such an occurrence represents the

fact tha t a (subluminal) observer receives the radio-emission of an approach-

ing Super luminal source in the reversed chronological order [23, 103], as is made

clear in Fig. 24.

//

X / t3-- ~ ~ ~ iuper'lurn,~c~l

Fig. 2 4 . - The radio-emission of a Superluminal source, approaching the observer along the x-axis, will be received in a rcversed chronological order. This is the meaning of the negative frequencies entering Fig. 23. The line S is the Superluminal world-line.

Therefore~ if a macroscopic phenomenon is known to produce a radio-

emission obeying a certain chronological law, and one happens to detect the

reversed radio-emission~ the observed source should be considered as a Super-

luminal, approaching object.

Let us then examine~ as a th i rd point, an observer s~ and the Minkowsky

space-t ime as (( seen ~ b y him. Namely , let us s tudy [19] the relative position of the world-lines of bo th bradyons and tachyons, and of the light-cones asso-

ciated with one or more events of their history. Notice [19] t h a t - - i n a three-

dimensional spaee-time~ for s imp l i c i t y - - the light-cones springing f rom a bra-

dyonic world-line are str ict ly one inside the other, since the locus of the vert ices

passes inside the cones. This is not t rue for the light-cones springing f rom a

tachyonic world-line. In this second case, their envelope is const i tuted b y two

planes: the <, re tarded light-cones ~) occupy ent irely one dihedron with angle

larger t han 90 °. We can easily get wha t follows [19]:

A) Case o] a subIuminal receiver B. The detector B m a y receive radio

signals (I~S) f rom both bradyonic and tachyonic emitters.


B) Case o/ a S u p e r l u m i n a l receiver T.

1) The detector T will receive RS ]rom a bradyon only up to a certain

instant (i.e. T will no lont,'er receive any RS from a bradyon af ter a certain instant) . Precisely, ;/' will first receive the radiotransmission in the same se-

quence of emission, and af terwards in the reverse order.

2) T will receive RS ]rom another tachyon t, moving along a parallel

pa th (with respect to a sublumin'f l f rame!) , in the following way:

a) I f ~(T)----v(t), then T and t will be either always, or never in

radio contact .

b) I f v ( T ) ~ v(t), then T will receive (( (retarded) RS }) up to a cer-

ta in ins tant ; af terwards the contact will break.

c) I f v ( T ) ~ v(t), then the previous sequence will be reversed. For

example, T will s tar t receiving RS ' s only af ter a certain instant.

3) T will receive, f rom a second, (~ unparal le l ~ t achyon t, its RS

either never, or up ~o a certain instant , or af ter a certain instant, according

to their positions and velocities (relative to the subluminal frame!). In general,

the sequence of emission will be highly mixed up at the reception.

18. - Genera l i za t ion o f Maxwel l equat ions .

Very interesting, but ra ther delicate and troublesome, is the problem of

e lect romagnet ism and tachyons, since now the applicat ion of the tachyonizat ion

rule is not s t ra ightforward. In fact (*), the usual [29a] electromagnetic tensor

- - i E ~ 0 - - H ~ H~ (96) (F,~) = ~ - - i E ~ H~ 0 - - (tt, r = O, 1, 2, 3)

\ - - i E ~ - - H ~ H ,

a pr ior i is not expected to be a G-tensor, i.e. to behave as a tensor also under Superluminal Lorentz t ransformat ions .

Therefore, the problem arises of generalizing usual Lorentz t ransformat ions

of the electric ]ield three-vector E and of the magnet ic ]ield (**) H for the case

of SLT's , just as we had initially to do for LT's .

('/ Remember again that we are using fhe metric (q- ------) , by writing, however, thc genericM vector as x =--(et, ix, iy, iz), i.e. x o = et, x 1 = ix, x 2 = iy, xa= iz. We have thus no difference between ~ covariant 7) and (~ contr~variant ~> components. See also Subseet. 4" 2. (**) As usual, we shall indicate the magnetic field by H, even if it actually is a three- dimensional, second-rank antisymmetric tensor.

276 E. I~CAMI and R. MIGNAN~

Of course the generalized electromagnetism must satisfy the extended relativity requirements: in particulur the duali ty principle (which ruled our Sect. 16).

18"1. Autodual electromagnetic tensor. - Let us remember tha t standard Maxwell equations read [29a]

(97) ~_P~---- O,

where _P, is the dual tensor of / ~ , and Jr ~ (G,J) is the electric current density four-vector. :Notice tha t we defined (*)[105-107]

(98) ~ _]_ 1 (a, fl, y, ~ = O, 1, 2, 3) ,

where e ~ is the real, completely antisymmetric Ricci tensor (normalized so tha t soas= 1); definition (98) should not be confused with some previous ones (cf., e.g., ref. [107]). :Note tha t

(98 bis) G . = G ~ "

Moreover, the present <~ duali ty ~) effects the exchanges

(98 ter) E + i . ,

H - + - - iE .

Let us now introduce the autodual (electromagnetic) tensor [106]

(99) r . - C,+ G , G = r ,

where (/~ v ---- 0, 1~ 2, 3)

(96bis) (T~,,) --

i 0 iE~--H= iE~--H, i E z - - H ~ \ Hx-- iEx 0 iE, - - H, H~-- iE~] .

\ H = - - i E , iE~--H~ H~--iE= 0

In terms of T , the standard equations (97) can be writ ten together in an

(*) A summation is understood over the repeated indices.

CLASSICAL T H E O R Y OF TACHYONS 277

essentially single equation

(lOO)

The previous formulations (97), (100) of M'~xwell equations, as is well known, may be writ ten [109-112,108[ in a completely symmetr ic form by introducing

the magnetic-current density four-vector g~, (~,~,g). Equations (97) become

{ ~ F = i., (97') ~ / ~ = ig, (v2 < c2)'

and eqs. (100) then read

~ = j,, + % , (v~ < c~). (loo') ~ r = ~

For an extensive bibliography about magnetic monopoles [109-111], see, e.g., ref. [108, 112].

18"2. Magnetic monopoIes and tachyons. - Let us outline the following a

priori possibilities for generalizing Maxwell equ,~tions to the ease of Super- luminal sources. Let us limit ourselves only to boosts ~long the x-,~xis.

A) I] one wishes eqs. (97) to be G-cow~riant, i.e. the quant i ty F to be a G-tensor, one should postulate tha t E and H transform, under a SLT = -- iA>(fl), as follows (fl~ > 1 ) :

{ .E~ : -- E ,

(101) H ' - - - - H , E~ ~- iy(E - flHz) ,

U~ = + iy(H + ~E ),

E: = + i r ( ~ + ~ ) , H I

where y _- ~- ([1 -- f121)-~ ~- (/~-- 1) -~. l~otice tha t Superluminal transformations (101) allow G-eovariance also of eqs. (97') and (100'), valid for both

electric and magnetic subluminal sources.

B) I] one wishes Maxwell equations to be G-covariant in their form (100) under a S L T = - iA>(fi), one m~y choose, besides eqs. (101), also the following transformations of E, H (fl~> 1):

(Jo2) E' -- iH ,

H : - + iE ,

E! = i y ( E - f i l l ) ,

H' = iy(H + f i E ) , z - -

where, as before, y _: ~- ( l l - - fl*])-t :_ ~- ( f 1 2 1)-~. Superluminal transforma-

278 E. RECAMI and R. MIONANI

t ions (102) ( and (101)) a l low G-eova r i ancc also of cqs. (10O'), for b o t h e lec t r ic

a n d m a g n e t i c subluminal sources.

C) The t h i r d pos s ib i l i t y , go ing b a c k ¢o t h e b e g i n n i n g of Sect . 18, is --more consistently--trying to generalize the (subluminaI) trans]ormations

(103)

E~ = E'~ , E~ -- E; ÷ fill: E, -- E: --flU~ , ,

H= = H ' , =

@< i)

/or Superluminal velocities, in a n a l o g y to L T ex t ens ion (Subsect . 4"6, a n d

Sect . 6). This is the way chosen by us [96, 106].

I t is e~sy to recogn ize t h a t ~he couple E , , / / ~ i n eqs. (103) behaves like t h e

couple x, t for s u b l u m i n a l t r a n s f o r m a t i o n s : c o m p a r e F ig . 9, in wh ich axes x, t

a re to be r e p l a c e d b y E ~ , H ~ r e s p e c t i v e l y . I t is s t r a i g h t f o r w a r d to gene ra l i ze

E ' H~ to r o t a t e w i t h c o n t i n u i t y t h e p rev ious t r a n s f o r m a t i o n s , b y a l l owing axes ~,

even beyond 45 ° (un t i l E : co inc ides w i t h H a n d H : w i t h / ~ , for f l -+ oo). See

F ig . 25. The re fo re we sha l l w r i t e [106] for b o t h s u b l u m i n a l and Superluminal

. +,>l / / / / I

_i i I / / " _ 0 (Eo) 7 Ey(q~p (Eo)y,z 0

a)

Erl z

iE x ~ (Ho) X

Ey(q~'~ (E~)y (Eo)×, z -

J~

l iE'#---H, [(.12)-~o"]

E, (Eo) ~

c)

Fig. 25. - Consider an unprimed][frame, So, supposed at rest, and another inert ial frame, f, to which we shall a t t r ibute, every time, all possible collinear velocities u, both subluminal (frames / ') and Superluminal (frames ]"), along the posit ive x-axis (i.e. u ~ u, > 0). Suppose tha t in ] we have a mliform electrostat ic field E parallel to the y or z axis (i.e. either E=--Ev or E=--Ez). a), b) represent how s o will see the moving fLeld, according to our eqs. (104a), (104b). For O < u < c , we shall have the usual case a), with 0 < ~ < 45 °. For 0 < u < oo, we shall have the case b), with 0 < ~ < 90 ° : in part icular , for c < u < oo, the anglc ~ will run from 45 ° to 90 °. We would have met an analogous si tuation when star t ing from a uniform magnetostat ic ~eld, parallel to either the y-axis or the z-axis. See the text. Notice tha t here the spaces (E~, H~) and (E~, H~) are pseudo-Euclidean, analogously to space (x, t). On the contrary, c) is only a symbolic graph, meaning tha t for 0 < u < c we have E,(qp') = E'~, and tha t for c < u < oo we have E=(~v") = - - l ~ ~ iE: =~H". Notice tha t we are understanding tha t H, = E,, in Gaussian units. Moreover (H0) ~ = i~,. Case c) refers for simplici ty only to the case 0 < ¢ < z t / 2 . See the text .

CLASSICAL THEORY OF TACIIYONS 279

velocities (see eqs. (38'))

(104a) {

where

= cos v + H: sin V),

H' E~ H = C ( , c o s ~ + sin~v)

~ /1 + t g ' ~ (0 <~0<2~). fl =--tg~, C -: + I i _ t g ~ I

By analogous (~ geometricM >~ extr~polation (see Sect. 6~, we may write [106], for both subluminal and Superluminal velocities,

{ E = C(E: cos cf - - / /~ sin (p), (104b) E~ (f12 ~1) ;

H = C(H: cos ~v -- sin ~v)

now, in the Superluminal case, E. , H~ beh:~vc like x, t under the transformation + iA>(fl).

Regarding E~ and H~, as suggested by Subsect. 3"5 and 4"6, we shall write for Superluminal velocities [106] (')

E iE : , H = i l l : ( f l ~ > l ) ;

in fact, as y, z components behaved like co-ordinates x, t, we assumed x-components to behave like co-ordinate y (or z). For both subluminM and Superluminal velocities we have (*) (see Fig. 25 c))

{ E = E':6 , (104c) (fl' <> 1)

where ~ + "/I/1Ufl~ .

Notice explicitly, from eqs. (100'), that multiplication by i turns quantities due

(*) In the following, we shall for simplicity confine ourselves to - -n/2 < 9 < ~/2. Notice since now that, due to invarianec of our autodual tensor Tv~ under (~ duality ,>, we may consider iE: ~ --H~ (where H~ = E~, it, Gaussian units) and iH: =-- E, (where E~ = H,, in Gaussian units).

9.80 E. REC~kMI and R. MIGNANI

to the electric current into quantities due to the magnetic current (*). Our trans- format ions (104) may be derived also by considering the complex vector

• ~ - - E ~ - i H

and suitable rotst ions (ef., e.g., ref. [29]) (*). With the transformations (104), as guessed in ref. [18, 105, 102, 23, 66], we

get, when t ransforming eqs. (100'), the interest ing result [105b] tha t under SLT's

electric and magnetic currents are changed one into the other. In other words,

Superluminal (~ electric charges ~) contribute to the electromagnetic field as subtu- minal magnetic monopoles would, ,~nd vice versa. If (bradyonic) monopoles do no~ exist, then (~ electrically charged ~ tachyons will bring into the (generalized) field equations contributions of magnetic monopole type. In fact, let us consider a SLT, e.g. the t ranscendent one K+ (from the unpr imed frame to the primed

frame) (*'). For example, s tart ing from eqs. (100), one gets [106]

(105) div E'---- 0 , div H'---- -- @' (fl~> 1) ,

and star t ing from eqs. (100')

(105') div E ' ~- ' ---- ' ~m, div H ' _ ~, (f12 > 1) .

I t is easy to verify that , wi~h our t ransformations (104), the (Maxwell)

field equ,~tions (100') can be wri t ten [106] in a G-covariant form as follows:

{ ~,T,~ : [j,(s) ÷ g,(S)] ~ - i [ g ~ ( s ) - j , (S) ] , (106) T,~ : T~, (v2 ~ c2)'

(*) Therefore, when passing from a Superluminal frame (with speed corresponding to ~8) to a collinear sublmninal frame (with speed corresponding to ~s), we can better write

{ E~(~,) = iE" = iE=(~' = --H=(~s ) , @ > 1)

where, in Gaussian units, H~ = E~ and Ex = H~. By adopting a di]]erent formalism, one can for example write (for f12> l)

E" ,s~T)>--H=,

H ; (SLT)). E~,

where now, in Gaussian units, H~ = E: and E~ : H:. These considerations do agree with the well-known invariance of the electromagnetic tensor T~ under the (~ duality exchanges ,. (**) By the way, the transformation K+ works so that E~=H~, H~----E~, E s = - - H~ = Ey, as heuristically forecast in ref. [18].

where j,(s) and g~,(S) are the subluminal electric current and the Superluminal

magnetic current respectively, and so on. I f (subluminal) magnetic monopoles are assumed not to exist, t hen eqs. (106)

will read [106]

{ ~ T,,, = j,,(s) - - ij,,(S) , (]07) ~?~ = T,,~ (v~<> c~).

:Notice tha t the quan t i ty T,, , which is a tensor under the s tandard Lorentz

group, is no longer a usual tensor under the group G, since T under a SLT behaves like a s tandard tensor, except--however--lot a fac tor i. Tha t is to say, T is no longer supposed to be a s tandard tensor in eqs. (106), (107). However , we are assuming j~ Qou~, where ~o is the proper charge densi ty

and u : d x , / d v o is the G-four-velocity, so tha t j , is a G-four-vector. Le t us now for s implici ty reduce to sublmninal LT ' s : in this case eqs. (107) are equivalent to the following ones:

V. D O(s) ,

V . B : - - ~ ( S ) ,

~ B (]08) V A E = j ( S ) - - c~t '

~D V A H = j ( s ) +

[v2~ c~; s = subluminal; S = Super luminal ] ,

whose physical meaning is clear. All such problems will be dealt with by the present authors in other pa-

pers, with more details and a t tent ion to the experimental consequences too. Let us here observe the following, with reference to eqs. (107). Since Super-

luminal << electric charges ~> (as predicted by relat ivi ty) are enough to get the (generalized) Maxwell equations in a completely symmetric form, the theo-

retical basis for expec t ing[109,111] on the contrary existence of (subluminal) magnetic monopoles becomes very much weaker [106].

19. - Note on tachyon quantum field theory.

Without the basis of a complete classical theory of tachyons, till now the quantum field theory of tachyons had no very convincing developments [66].

Among the contributions, let us ment ion the ones by :FEINt~ERG [16], A~0~'S

and SUDHARSHAN [54], ~DHAR and S U D ~ S H A ~ [55], SUD]=[AI~S:HAN [57, 38, 37C],

SUDHAItSHA~ and MUKUNDA []]3], ~BALDO and ]~ECA:~[I [58, 8], BALDO, FONTE a n d ~ E C A M I [ 9 ] , CA:~IENZI]ND[21C]~ E C K E R [ I ] 4 ] , BLUDMAiN a n d ~UDEI~-

19 - Rivis ta del Nuovo Cimento.

2 8 2 ~ . m ~ c A M I a n 4 R. MIONANI

MAN [115], KAMOI aud KA~EFUCttI [55], SO]~OEI~ [116] and STI~EIT and KLAU- DEI% [117]. Other contr ibut ions are as well listed in ref. [118].

S tar t ing f rom the extended re la t iv i ty theory, i t is l ikely tha t , in the future,

t achyon QFT will have a more consistent development .

~ow, let us only r emember here t h a t ~ s i n c e spacelike representat ions of

the Poincar6 group do exist [42J--general ly quantum-rela t iv is t ic field equa-

tions present a mass spec t rum with negat ive masses squared too. This is the

case, for instance, of inf ini te-component Majorana [119] equat ions: cf., e.g.,

ref. [113, ]20]. I n fact , the existence of spacelike components seems a na tura l ~nd perhaps

unavoidable fea ture of interacting fields [120]. I t has indeed been proved [121], e.g., tha t , if the Four ier t rans form of a local field vanishes in a domain of space-

like vectors in m o m e n t u m space, then the field is a generalized ]ree field [120].

2 0 . - M i s c e l l a n e o u s r e m a r k s .

I n connection with the last Section, let us ment ion t ha t use of quaternions

seems useful when dealing with tachyons [105, 122].

Moreover, f rom consideration of tachyons, SUDARSgA~ and co-workers [118]

have been led to s tudy the possibil i ty of complex-mass particles and a quan-

t u m field theory with indefinite metric. At this poin L let us r em ark t ha t appearance of t ranscendent k inemat ica l

s ta tes in extended re la t iv i ty supports the impression [19, 23] t ha t Minkowsky space is no longer suitable [21c], and favours a t t emp t s to use <( special re la t iv i ty ~>

with curved space-t imes [123], as for instance the so-called <( project ive rela-

t i v i t y ~> [123]. Moreover, SCUlL]) [124] pointed out t ha t existence of fas ter- than- l ight

signals m a y help the problem of specifying initial da ta in relat ivist ic theory.

At last, i t is interest ing t h a t - - i n a b idimensional space-t ime, or in the case of pure ly collinear m o t i o n s - - i t is possible [27] to define <( rapid i ty ~) the quau t i ty R~ ~ c-nr tgh fl~, so as to have an additive <~ rg~pidity composi t ion law ~>. I t

t hen results t ha t R = 0 ~ for fl = 0 ± and R ---> ± c~ for fi -+ ± c ~=. However ,

this cannot be meaningful ly done in four dimensions (*).

(*) Note added in proo/s. - Let us add a few words about the papers which we knew about after the completion of this work. i) A. RAeHMAN and R. DUTHEIL put forth a few papers (e.g., Lett. Nuovo Cimento, 8, 611 (1973); they seem to us to be incorrect) supposedly about SLT's: however, because of an initial slip, they appear to have done nothing but a different parametrizatiou of usual, subluminal LT's. ii) A. F. ANTIPPA and A. E. EVER]~TT (Phys. Rev., 8, 2352 (1973)) and A. F. ANTIePA (preprint, UQTR-TH-7, Qu6bec University {1973)) tried to avoid the difficulties [19] connected with the asym- metry between the numbers of time and of space axes {cf. also the (~ note added in proofs~) to Subsect. 3"5 and 3"6) by introducing a preferred spatial direction for tachyons. In such a way, they build a theory rotationally and Lorentz noninvariant, leading to


21. - Br ie f d i scuss ion o f the exper iments .

Unfor tunate ly the experiments performed by ALVid~ER et al. [17, 99, 100], ]:~A~A~A MUI~THY [101] and BAI~TLET~ a~d LAttANA [102] are not conclusive

- - a s we already saw--since t hey are b~sed on the assumption tha t faster-thou- light particles emit C.erenkov radiat ion in vacuum or in ~ bradyonie medium. Such an assumption is wrong [36, 23] in the light of the classical theory of

taehyons (see Sect. 16 and 18). As well, the exper iment in ref. [125] is not conclusive, because it was based

on ~ theoretical assumption ((~ actual existence of negative-energy particles, travell ing forward in t ime )~) tha t violates the th i rd postulate, i.e. the Dirac- Stf ickelberg-Feynman-Sudarshan ((reinterpretation principle ~), which has been recognized to be a necessary par t of special re la t ivi ty (see Subsect. 7"3). For

instance, an e lementary reaction such as eq. (1.3) of ref. [125]

p -> p + t ( - ) ,

where p means proton and t(-) is a (( negative-energy tachyon ~), must be ~ctually

re interpreted (Sect. 7 and Subsect. 14"3) as follows:

p +t(+) -~ p ,

where t (+) in the (positive-energy) antitachyon. Such reactions have been already

analysed in Subsect. 14"3. The two remaining known experiments [126, 127] appear a priori based

on more reasonable theoretical grounds. However, ref. [126] ~ssumes (*) the hypothesis tha t (~ churged tachyons of a given momentum follow curved paths

in u magnetic field just as singly charged ordinary particles of the same momen tum do ~. That ,~ssumption is not compatible with the generalized electromagnetism, tha t has been derived (Sect. 18) from special relat ivity. The present authors will deal more iu detail with the kinematical behaviour of tuchyons

in electromagnetic fields in the forthcoming papers. Therefore, only the (negative) result of the e~perimeat described in ref. [127]

is to be taken into account when building up the tachyon quantum mechanics.

thc nonconserv~tion of tmgular momentunl and the violation of the PR. We do not agree [23] with th,~ir philosophy, iii) On tile contrary--in the lille of previous papers (G. C. W1CK: Ann. o] Phys., 18, 65 (1962); Y:~. S~0RODINSKY: Fortschr. Phys., 13, 157 (1965)), A. Y,~cc.~I~i~[ ht~s published inter(~sting papers (Can. Journ. Phys., 51, 1304 (1973); (to ~t)pe~r); ~md (subnlitted to)). perh~.ps useful to improve the theory of extended rela~tivity (cf. Subsect. 3"5). (*) See assuinption (1) of t e l [126], p. 54.

1 9 " - Rivista del Nuovo Ctmento

2 8 4 E. RECAMI a n d R. MIGNAN1

Namely, ]3ALTAY et al. [127] have found no evidence of single, uncharged tachyons produced by stopped K- or p in interaction with protons.

In conclusion, new experiments are needed, particularly founded on a sound theoretical basis as the (extended) special-relativity theory. The present authors hope to be able to suggest the details of some such experiments in the ,~bove- mentioned forthcoming papers, particularly with regards to (~ tachyon monopoles >> [106].

Further attention is merited also by the question [88, 89] of (~ virtual particles >> and tachyons (see Subsect. 14"4), since it may well give indications on the existence of taehyons.

2 2 . - C o n c l u s i o n s .

The very (< special-relativity theory >> has revealed itself suitable for straightforward extensioh to Superluminal inertial frames and to tachyons.

I t has been therefore possible to build up a classical theory of taehyons in a self-consistent way. For example, classical, relativistic mechanics and Maxwell equations have been generalized for faster-than-light objects.

To causality problems actually arise, and all the proposed paradoxes are easily solvable.

By use of the new group G of the geI~eralized Lorentz transformations (in- cluding also the Superluminal ones), starting from bradyonie physical laws it has been possible to predict a number of quite reasonable laws, which the taehyons are expected to obey. For example, th~ following h'~ve been clarified: the behaviour of tachyons in the gravitational field, their 0erenkov radiation, both electromagnetic and gravitational, their Doppler effect and so on. By the way, tachyons seem to experience a gravitational repulsion, but, owing to their dynamics, they happen eventually to bend towards the gravitational source, similarly to usual particles.

A very interesting link betwee~ magnetic monopoles and tachyonie (( electric charges ~> has been shown. Since Superluminal (( electric charges ~> (as predicted by relativity) are enough for getting the (generalized) Maxwell equations in a completely symmetric form, the theoretical basis for expecting on the contrary existence of (snbluminal) magnetic monopoles becomes very much weaker [106].

:New experiments are needed, since almost all previous searches for tachyons were based on theoretical assumptions not consistent with (extended) special relativity. Our philosophy is synthesized in Sudarshan's known statement that (( if tachyons exist, they ought to be found. If they do not exist, we ought to be able to say why ~>. And till now no serious objection has been found against tachyon existence; on the contrary, all of relativity theory, both classical and possibly quantistic, does suggest ~heir existence, as has been shown.

CLASSIC&L TIIEORY OF TACIIYONS 2 ~

In any case, we want to stress tha t ~he present generalization of speciM re la t iv i ty is interest ing both in itself, and for the fact ~h~t it has already

~llowed progress in the physical unders tanding t~lso o] usual (bradyonic) matter.

For instance, it even shed more light oa the m~tter /~mtim~tter connection, it yielded a l)hysie~l meaning to ~he well-known C/~T eovari~mce and to the

operation, ~md it pe rmi t ted the demonst ra t ion of the (~crossing relations ~ in usual e lementary-par t ic le physics.

The authors wish ~o t h a n k Profs. A. AGODI, M. BALDO, IV[. CIN[ and V. S.

OLKI~OVSKY for interest and maI~y discussions. ThaI~ks are also due to

Profs. E~ A~ALDI, C. BERNAgDINI~ ~ . CA:BIBBO~ P. CALDIROLA~ A. GIGLI~ I. F. QUERCIA~ G. SALVINI, G. SCHIFFI~ER, G. TAGLIAFEP~RI~ M. VEgDE~ G. V.

WATAGHIN for useful talks, ~ad to Drs. g . BALDINI~ ]~. MACORINI~ S. R0-

DONb for kind collaboral~iou. For the rem~ining ~cknowledgements, see ref. [23] ,~nd references therein.

The authors ~re gr~teful as well to Mrs. G. GIUFFRIDA for her p~tient, per-

severat ing tyl)ing, :rod to Mr. F. ARRIV~ for his skilful figure drawing.

At last, one of the ,~uthors (E.R.) th~nks his own relatives L. ]~ECAlV[I SA~-

S0~I, T. RO]]EgTO RECA~[I and Dr. U. RECA:~I, who r~ised by their lo~ns

the univers i ty ~lms-p~y ~s ~ professor of physics f rom the I ta l ian Educat ion

Ministry (MPI), thus helping him to survive ,~nd, therefore, indirectly ~llow-

ing the present work to be done.

R E F E R E N C E S

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286 v.. R:ECAMI and R. MmNANI

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CLASSICAL THEORY OF TACHYONS 2 8 7

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2 8 8 E. RECAMI a n d R. MIGNANI

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C L A S S I C A L T H E O R Y OF T A C t f Y O N S 289

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290 E. RECAMI and R. MIGNAN!

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[118] S.

[119] E. [120] D.

I{IVISTA. DEL NUOV0 CIMENTO VOL. 4, ~. 3 Luglio-Settembre 1974

Classical Theory of Tachyons (Special Relativity Extended to Superlnmlnal Frames and Objects).

E. RECAMI

Istituto di tJisica Teorica dell' Universith - Catania Istituto Nazionale di ~is ica JOC, acleare . Sezione di Catania Centro Siciliano di ffisica N#cleare e di Struttura della Materia - Catania

:R. MIGNANI

Istituto di .Fisica dell' Universit~ . Roma

(Rivista det .37uovo Cimento, 4, 209 (1974))

Owing to an edi torial error, the Fig. 25 appeared in an incorrect version. I t must be subst i tuted by the following one:

(Ho)z,y

H/~p ,

0

/ hJ,y / / / / (Ho%,,,

, ~ ' y , z (H°)z~/EY

I J/<. (Eo)y E/(TI) (Eo)y, z

a)

Ey(T") (Eo)y (Eo)y,z 0 ~E x (Eo)x b) c)

Besides, in its caption (last-but-one line) the relat ion (He) ~ = H~ must be corrected so as to read (H0): ~ / ~ .

M o r e o v e r :

p. 228, line 18: p. 231, line 3 from bot tom: p. 238, last line: p. 252, line 8 from bot tom:

E r r a t a Corrige

P ~ - e d frame (P@')s o initiM frame s o bere membered be remembered interact ion in terpre ta t ion T ~,.t

At last , let us take advantage of the present occasion for point ing out tha t in Sect. 2 (Historical remarlcs) we forgot mentioning the pioneering work about faster-than- l ight iner t ia l frames by H. AaZELI~S: Compt. Rend., 245, 2698 (1957). We thank Prof. G. AnclI)I~CO~O for having signalled us this point.

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classical theory of tachyons (special relativity extended ...dinamico2.unibg.it/recami/erasmo...

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