How to recover causality for tachyons even in macrophysics

Download PDF
0 downloads 0 Views 948KB Size Report
Comment
(2) For a. generalized definition of equivalence (with recourse to some symbolic logic) sec ref. (1), p. 249. See also E. FAm~: Nuovo Cime~to, 14, 1130 (1959); A.
IL NIJ~)\:~} CIMENT~)

VOL. 36A, N. 2

21 Novembre 1976

How to Recover Causality for Taehyons Even in Macrophysics (*). M. l'av~t(~ (**)a, ml E. REtAIn (***) I,,tiluto di Fisiea Teorb,a, U~dversith di Cata~ia - Catania, Italia (ric,wuto il 19 Luglio 197(;)

Summary. • The postulate that negative-energy particles do not exist (travelling forward in time) leads automaticMly to the ~, Reinterpretation Principle~> by Stiickelberg and by Feynman. It has been already ~hown that such a ((principle ~>, assumed as the Third postulate of special reb~tivity, ensures the validity of the law of (retarded) causality both in standard relativity and in (extended) relativity with tachyons and with Supcrluminal inertial frames. Our Third postulate, moreover, allows predicting antiparticle existence in a purely relativistic context. In this paper we show that the Third postulate is enough to implement the law of caus;dity even in macrophysics, when usual macro-objects interact with microt,chyons and macrotachyons. To that aim, some tachyon kinematics is further developed, which e;m be useful even ill mlderstanding elementaryparticle interaction.~ (and may be hadron structm'e). Many other related pr(~l)hm~s ~t'e (Its(tossed.

1. - I n t r o d u c t i o n .

11 }ms b e e n kllowll t l m t Sl)eci:,.1 rel:~tivity (81¢) c a n be e x t e n d e d (~) t,o f a s t e r ~ h a n - l i g h t i n e r t i a l fr~mles (2) ~n(l o b j e c t s (tachyons) w i t h o u t l e a v i n g o p e n a n y (*) Wol'k partially supporl,ed by the Italian M.I'.I. (ex-art. 286) and by C.N.R. (under contract 75/00442.02). (**) On leave from the J. Stefan Institute, University of Ljubljana, Yugoslavia, supported ill part by the (( B. Kidri6 fund,). (***) Also I.N.F.N., Sezione di Ca tania, Catania, and C.S.F.N. c S.d.M., Catania, itMia. (1) E. RECAMI ~;l~l R. MI(~NAXI: RiV. Nuovo Cimento, 4, 209-290, 398 (1974), and relere.nces thereiJ,. See also L. PAICKFa: Phys. Re*,., 188, 2287 (1969). (2) For a. generalized definition of equivalence (with recourse to some symbolic logic) sec ref. (1), p. 249. See also E. FAm~: Nuovo Cime~to, 14, 1130 (1959); A. A(~om: unpublished. 171

172

M. Pxv~i~ and n. l~CA~II

causality problem. For instance, all the proposed (~) (~causality paradoxes )) have been easily solved (1,4), on the basis of the Dirac-Stfiekelberg-Feynman Reinterpretation Principle ( R I P ) (~.5). N a m e l y an Extended Relativity (~), free of a n y causal violation, can be derived from the three following postulates (6): I) Principle of relativity (PR). Laws of mechanics and electromugnetism are eovariant (--~ i n v a r i a n t in form) when passing from one inertial frame to a n o t h e r frame in uniform, straight relative motion (with speeds - - c ~ u ~ + ~ ) . I I ) Space-time is homogeneous and space isotropic (notice: light speed invariance needs not to be postulated, since it can be derived from postulates I a n d I I , as k n o w n since 1910) (7). I I I ) Third postulate. (( Negative-energy particles m o v i n g / o r w a r d in time do not exist and, for every observer, physical signals are t r a n s p o r t e d only b y the objects t h a t appear as carrying positive energy ~ (s). Careful investigation of SR, in the light of such a postulate, together with the observation t h a t we (( explore )) space-time only forward in time (i.e. in the time direction fixed b y macro-object behaviour), automatically leads us to the (5) (, R I P ,):

(a) Cf., e.g., F. A. E. PI•ANI: Phys. l~ev. D, l, 3224 (1970). (a) E. RECAMI: in Eneiclopedia EST-Mondadori, Annuario 73 (Milano, 1973), p. 85; Scie~gia, ]09, 721 (1974) ; and to appear; P. CALDII~OZAand E. RECAMI: in Proceedings o] the (( Con]erence on the Concept o] Time (S. Margherita Ligure, 1976))), to appcar; J. A. PARMENTOLAand D. D. M. YEE: Phys. Rev. D, 4, 1912 (1971); R. G. ROOT and J. S. TR]~FIL: Lett. Nnovo Cimento, 3, 412 (1970); P. L. CSONKA: Nucl. Phys., 21 B, 436 (1970); O. M. P. BILA~IUK and E. C. G. SUDARSHAN:Nature, 223, 386 (1969); E. C. G. SUDARSHAN:in Symposia on Theov. Phys. and Math., Vol. l0 (New York, N. Y., 1970), p. 129; 0. M. P. BILA/qIUK,V. K. DESHt"ANDEand E. C. G. SUDARSHAN: Amer. Journ. Phys., 30, 718 (1962). See also E. RECAMI: Giornale di Fisica, 10, 195 (1969); M. BALDO, G. FORTE and E. RECAMI: Lett..Yuovo Cimento, 4, 241 (1970). (5) P . A . M . DIRAC: Proc. Roy. Soc., A126, 360 (1930); E. C. G. STtiCK~.LBERG: Helv. Phys. Acta, 14, 32L, 588 (1941); R. P. F~YNlUAN: Phys. Rev., 76, 749, 769 (1949). See also ref. (1,4). The RIP has been given a new, more elegant formulation in five-dimensional space by E. RECAMI and G. ZIINO: Nuovo Cimento, 33 A, 205 (1976). (e) See ref. (1); and R. MmNA?¢I and E. RECAm: Report IC/75/82 (ICTP, Trieste), to appear in Phys. Lett., B; Int. Journ. Theor. Phys., 12, 299 (1975); Nuovo Cimento, 24 A, 438 (1974); Lett. Nuovo Cimento, 11, 421 (1974); E. RECAMI and E. MODICA: Lett..Nuovo Cimento, 12, 263 (1975); E. RECA~I and R. MIO~ANI: Lett. Nuovo Cimento, 8, 110 (1973); Phys..Lctt., 62B, 41 (1976). See also ref. (1). (7) W . V . IGNATOWSKI:Phys. Zeits., 21, 972 (1910); P. FRANK and H. ROTItE: Ann. der Phys., 34, 825 (1911); E. HArI~r: Arch. Math. Phys., 21, 1 (1913). (s) The meaning of this postulate is self-clear within information theory; it is supported by thermodynamics and by all physical experience. Furthermore, it allows us to predict (1)--merely from relativity--the existence of antiparticles.

IIOV)" TO R E C O V E R

C A U S A L I T Y FOR T A C H Y O N S E V E N I N M A C R O P I I Y S I C S

173

(~A~hv negative-energy p a r t i c l e P d o e s also t r a v e l b a c k w a r d s in t i m e ( a n d vice versa); it c a n , a n d must, be r e i n t e r p r e t e d as its a n t i p a r t i c l e P t r a v e l l i n g , w i t h p o s i t i v e e~mrgy, forw~,rd in t i m e )> ("). As s h o w n in ref. (t,4-s) a n d p a r t l y in t h i s p a p e r (~o), o u r Third postulate, i.e. t h e (( R I P ~), is e q u i v a l e n t t o p o s t u l a t i n g t h e principle o/ (retarded) causality, since i~ e n s u r e s (',~,s) t h a t every observer will ~ l w a y s see (( c:~uses ,) (~t) to h a p p e n c h r o n o l o ~ ' i c a l l y b e f o r e t h e i r o w n (( effects )) (11). T h e i m l , o r t ~ n t p o i n t is th:~t SIC r e q u i r e s o n l y physical laws (1:) ('rnd n o t (( d(,s(q, il)tions )) (t~.)) t o be c o v a r i ~ n t . F o r i n s t a n c e , in ref. (~) it h'~s b e e n s h o w n tlm~ in ( e x t e n d e d ) r e l a l i v i t y t h e (( p r i n c i p l e of ( r e t a r d e d ) caus:~lity ~)is a law, wh,'r(,as t h e a s s i f f n e m e ~ t s of t h e n a m e s (~c a u s e ,> a n d (, effect ,> ~re description details. T h e r e f o r e , a c c o r d i n g t o t h e (, I C I P , (cf. fig. 16 a t p. 239 of ref. (~)), we axe f o r e e d (',4) t o abandon t h e ohl c o n v i c t i o n tha~, t h e j u d g e m e n t a b o u t w h a t is e:~use a n d w h a t is effect is i n d e p e n d e n t of t h e o b s e r v e r (13).

2. - Macro-objects, entropy and information transmission by tachyons. \Ve w a h l t o s t r e s s h e r e t h a t o u r (~T h i r d p o s t l f l a t e ~>of SIC recovers causality [or t¢~chyons (H) n o l o n l y in t h e c~rse of e l e m e n t a r y l)rocesses--,~s m o s t l y c o n s i d er(~d, till n o w - - , b u t also in macroscopic processes (to), i n w h i c h e n t r o p y c o n s i d e r alioHs ~tr(, in o r d e r ( ~ ) ; e v e n if m a n y i ) h y s i c i s t s "~re s t i l l u n a w a r e of t h e s e f a c t s . l~et us c o n s i d e r t w o b o d i e s , A ~ n d B, for s i m p l i c i t y a t r e s t o n e w i t h r e s p e c t t o ih(~ o t h e r . Then~ l e t us c o n s i d e r - - i n t h e i r r e s t f r a m e s (i.e. in / h e lab)--the exelm~lge of ~ t ~ e h y o n T ~dong t h e p o s i t i v e x - d i r e c t i o n , w i t h s p e e d + V, f r o m A (source) to B ( d e t e c t o r ) . N o w t h e f r ~ m e w i t h s p e e d u u , , = c~-/V will see t h e t a e h y o n T l rav(,lling a t i n f i n i t e s p e e d (t)~ a n d (as i l h l s t r a l e d , e.g., in lift. 12 of ref. (~), p. 236) : m y f r a m e w i t h s p e e d u>c~-/V w o u h l ((see~) T as g o i n g 1)~l,(~kwards in tim(, :rlld b e a r i n g negati~'e e n e r g y ; t h a t is t o s a y , it will

(9) For a rigorous t rcatm(~nt of the RIP, and for details, see ref. (1), and E. RECAM! ;/11(I G. ZIINO: Nuoco Cime~*,to, 33 A, 205 (1976). See also ref. (4.6). (10) Cf. also M. t)iVgl(~. E. RECAMI and G. ZIINO: Lett. Nuovo Cimento, 17, 257 (1976). (~) For definitions of cau~(~s, effects, causal connections, see, e.g., ref. (1), pp. 261-262. (l~) Suitabh, analysis and definilions of law, description, etc., can be found in ref. (1), p. 242. (13) (:f. p. 258 in ref. (1). (~'~) Recall, however, that, the (( third postulate ~) (RIP) must be introduced cveli in st;t.ndard re|ativity in order to at?old (1,~,6) energy transmission into the past by usual parl.icles (bradyons), and in order to better understand the connection particles/antipa,rticles (and the whole subluminal physics as well). See also footnote (s). (15) See, c.ff., the recent work by L. BASANO: preprint Genova University (1976), subn~itted for publication; and Lett. Nuovo Cimento, 16, 562 (1976).

174

M. PAVSIC and E. RECAMI

a c t u a l l y o b s e r v e a n a n t i t a c h y o n 1' (~). I n o t h e r w o r d s , if we c o n s i d e r a (subl u m i n a l ) b o o s t a l o n g t h e p o s i t i v e x - d i r e c t i o n w i t h s p e e d c~/V < u ~ c, t h e n t h e n e w o b s e r v e r s' will see ( ~ , " ) - - b e c a u s e of t h e (0 Fig. 1. - The figure represents the exchange from A to B of a particle P with ~,eqative energy (and (( charges ~) and travelling backwards in time (t~ < tl). From the third postulate of special relativity ((~ negative-energy particles travelling ]orward in time do not exist )>) and from the fact t h a t we necessarily ((explore ~> Minkowski spacc-time going c. See fig. 2. And let us choose as c o m m o n origin 0 t h e e~'ent X*

/ /

///

Fig. 2. - The first apparent paradox here discussed. It seems theft a Supcrluminal observer (x', t'), while overcoming at O tlw observer (x, t) (~at rest ~), call tell the latter informations about a ]uture--for the latter event E, i] tachyon signals exist travelling at (almost) infinite speed. The paradox is immediately solved on the basis of the third postulate of SR, when one r~member~ that SR requires only laws (and not descriptim~ details) to be invariant. The important point, her(', is that causM paradoxes with tachyons can be easily solved ece.Jt, in m(~crophysics (see the text). Notice lhat tg ~ -- fl .... U/c.

17fi

~. PAV~IC a n d ~. R E C A M !

in w h i c h b o t h f r a m e s coincide. To s, t h e x-axis represents all t h e e v e n t s c o n t e m p o r a r y w i t h 0 ; whilst, t o s', all t h e e v e n t s c o n t e m p o r a r y w i t h 0 are r e p r e s e n t e d b y t h e x'-axis. I f i n f o r m a t i o n can be carried b y ( a l m o s t (17)) infinite-speed t a c h y o n s , t h e n o b s e r v e r s ' - - w h e n n e a r 0 - - c a n receive signals f r o m a n e v e n t E (see fig. 2) a n d c a n give this i n f o r m a t i o n to s. I t seems, t h e r e fore, t h a t s' c a n c o m m u n i c a t e (when n e a r 0) to s i n f o r m a t i o n a b o u t t h e possible f u t u r e of s. H o w e v e r , it is clear t h a t t h e (( R I P ~) t e a c h e s us (~.4.s) t h e following: the signal E ~+ 0 will appear to the observer s as a signal (or better ~ antisignal ~ (is)) sent by s' f r o m 0 towards E (since, with r e s p e c t t o s, it w o u l d be a (( n e g a t i v e signal g o i n g b a c k w a r d s in t i m e ~), w h i c h m u s t (4.e) be r e i n t e r p r e t e d a c c o r d i n g t o tim (~R I P ))). I n conclusion, a c c o r d i n g to s, t h e o b s e r v e r s' is p r e d i c t i n g to h i m n o t h i n g b u t w h a t c a n be actually ~ f o r e c a s t ~)--on t h e basis of our p h y s i c a l o p e r a t i o n s a n d of t h e p h y s i c a l l a w s - - ; i.e. t h e o b s e r v e r s', to s, is m e r e l y (~predict i n g ~ t h e o b v i o u s effects on E of t h e (anti)signals sent b y himself f r o m 0 tow a r d s E , to influence it p h y s i c a l l y (lO). E v e n in this case, each o b s e r v e r p u t s f o r w a r d a different description of t h e s a m e c h a i n of events, b u t t h e law of (retarded) c a u s a l i t y holds in all descriptions, for all observers. N o w let us pass t o a m o r e (( subtle ~ e x a m p l e i n v o l v i n g t a c h y o n m e c h a n ics a, little m o r e deeply, a n d s p e n d some m o r e words in solving t h e relative ( a p p a r e n t ) p a r a d o x (lo).

4. - T a e h y o n s , free w i l l and entropy.

N o w let us consider n process like t h e one in fig. 1, b u t with t h e b o d y B m o v i n g with r e s p e c t t o t h e b o d y A. L e t us, therefore, consider t w o usual, inertial observers (fig. 3) m o v i n g a l o n g t h e x-direction relative to each o t h e r with speed u < c, a n d let us suppose t h a t (according t o t h e f r a m e s ~ (x, t)) t h e o b s e r v e r s sends

(17) Infinite-speed tachyons carry zero energy (but 3-momentum magnitude equal to too), and their direction along their motion line is not definite. As (~bradyons ,) at rest are points in space, extended in time along a line, so transcendent taehyons are (~points ~ in time, extended in space along a line: cf. R. MIGNANI and E. RECAMI: Lett. Nuovo Cimento, 16, 449 (1976). Infinite-speed tachyons will be dealt with elsewhere, also due to their possible role in hadron structure (recall, e.g., the (~instantons ~), the quark (~biloeal functions ~), besides (, pomerons ~)). Cf., e.g., R. MIGNANI and E. RECA~I : Nuovo Cimento, 30 A, 533 (1975); Phys. Lett. (to appear); and quotations (87)-(89) in ref. (1). {is) From ref. (1), p. 239, it is clear that an (( antisignal ~) is related to its (~signal )) as antiparticles to their particles: an (~autisignal ~ is carried by antiparticles as the signal is carried by (their) particles. Of course, they both carry positive energy (and positive information)! See also the following. In the case of photons, they coincide with antiphotons.

HOW TO RECOVER CAUSALITYFOR TACHYONSEVEN IN ~¢[ACROPIIYSICS

177

a signal t o S'--~(x',t') b y m e a n s of a t a c h y o n i e b e a m w i t h speed V:> c2/u. A c c o r d i n g to s', h o w e v e r , t h e ~,t a e h y o n b e a m ~) w o u h t actually a p p e a r as a n a n t i t a e h y o n b e a m e m i t t e d b y s' himself t o w a r d s s.

It

/t ~

~/

Fig. 3. - The second apparent paradox here discussed; it deals, as the previous one, with macrophysics with tachyons. It seems that, if the (( at rest )) observer (x, t) sends information from A by a modulated tachyon beam to another subluminal observer (x', t'), boosted along the positive x-direction with speed u ~ c, then heavy paradoxes arise when the beam tachyons have speed V .-V~> c~/u (in fact, in this case the latter observer should see the tachyon beam emitted by B, i.e. by himself). The paradox is solved by the fact that, in macrophysics, tachyon mechanics allows B to absorb only ¢aehyons with speed V~c2/u, in which case no paradoxes arise, (where the above speeds are all measured in the A rest frame). Notice that t g ~ = f l < u/c.

Now, we c a n well i m a g i n e t h a t w h e n o v e r c o m i n g s' at O - - t h e o b s e r v e r s t o l d s' a b o u t his i n t e n t i o n of t r a n s m i t t i n g t o h i m (i.e. to s') a t a e h y o n i c signal (with speed V > c2/u) at t i m e t, whose L o r e n t z - t r a n s f o r m e d v a l u e is t i m e t'. Then, it seems t h a t t h e o b s e r v e r s' will be compelled--at a c e r t a i n i n s t a n t t ' - - A t ' - - t o e m i t a n (anti)signal t o w a r d s .% in o r d e r ... t o save t h e v a l i d i t y of t h e classical t h e o r y of t a c h y o n s in f o u r dimensions. B u t this w o u l d of course v i o l a t e t h e (~free will ~>of t h e o b s e r v e r s', who can, on t h e c o n t r a r y , decide to send no signal (0i" antisignal) t o w a r d s s. A ]ortiori, t h e o b s e r v e r s' c a n be in t h e impossibility of s e n d i n g that signal, b e i n g l a c k i n g in i n f o r m a t i o n a b o u t it (for instance, s t o l d s ' - - a t O - - t h a t , at t i m e t, he, s, will send t o s' one piece of m u s i c b y a t a c h y o n b e a m w i t h V > c2/u; a n d s' m a y h a v e no r e c o r d of t h a t music, so t h a t he cannot send to s a n y a n t i m u s i c , i.e. a n y m u s i c b y a n t i t a c h y o n beams). The s i t u a t i o n seems s e l f - c o n t r a d i c t o r y . The solution of t h e a p p a r e n t p a r a d o x is m e r e l y in t h e - - p r e v i o u s l y i g n o r e d - - f a c t t h a t mechanical laws o] tachyons do not allow the observer s' (the m o v i n g b o d y B) to absorb any tachyon with speed V > c2/u. I n o t h e r words, s' c a n get i n f o r m a t i o n f r o m s t h r o u g h such process ( t a c h y o n b e a m emission b y A a n d t a c h y o n b e a m a b s o r p t i o n b y B) only b y m e a n s of t a c h y o n s w i t h speed V smaller t h a n c2/u (in w h i c h case no causal p r o b l e m s arise) (~0). W e e x p l a i n this f a c t in t h e following section. 12

-

II N a o v o C i ~ c n l o A.

178

M. PAV~I~ and ~. RECAMI

5. - Some (further) kinematics for tachyons. L e t us fu'st emphasize t h a t a b o d y cannot emit or absorb a n y w h a t e v e r tachyon m o m e n t u m , b u t only definite values of it. For instance, a b o d y A at rest (with respect to the observer 0) c a n n o t emit or absorb tachyons endowed with in/inite speed (with respect to 0), unless it suitably decreases its rest mass. This is a trivial consequence of f o u r - m o m e n t u m conservation (17). F o r simplicity, we shall start b y considering (moving) bodies B t h a t do not change their rest mass in the process of t a e h y o n absorption. Namely, in the beginning, i) if B is a nuclear or subnuclear particle, we shall assume t h a t the absorbed t a c h y o n does not carry enough energy for allowing B to shift to a n o t h e r (discrete) rest mass level; ii) if B is a macroscopic b o d y (as in the following), t h e n we shah assume t h a t the m o d u l a t e d t a c h y o n beams are associated--as usually assumed in the information-transmission processes--to energies v e r y small as compared to the rest energy of the body (detector) B ; so t h a t the possible change in the rest mass of B is completely negligible. Assumption ii) is equivalent to neglecting also the possible internal energy associated to t a e h y o n absorption or (inelastic) scattering. I t must be stressed t h a t in this paper we are essentially considering only t a e h y o n emission or absorption, i.e. we are investigating only information transmission b y t a c h y o n beams t h a t are merely either e m i t t e d (by the (( app a r e n t ~) source) or absorbed (by the (~a p p a r e n t ~) detector), and n o t scattered. ] n fact, we still do not even know how t a c h y o n s behave when interacting with m a t t e r (19) even if e x t e n d e d relativity can help us to guess it. For instance, would the p h e n o m e n o n corresponding to (( b r a d y o n termalization ~ be the one of having tachyons going a p p r o x i m a t e l y to infinite speed? Now let us consider a macro-object B, with rest mass Ms, moving with (subluminal) speed u in the positive x-direction and absorbing tachyons moving (with respect to the same reference f r a m e so) along the positive x-axis, as well. Let the tachyons be e m i t t e d b y a b o d y A, with rest mass M~, at rest (see the following) in the origin of so. Then, in the process of t a c h y o n absorption b y B we shall have (in natural units) (1)

~/p~-- m~ + ~

M~ = ~/(p + g ) ~ + M~,

(19) We warn the reader from drawing naive conclusions when analysing information transmission by tachyon beam scattering.

HOM TO R E C O V E R C A U S A L I T Y F O R TACHYONS E V E N I N I~I&CROPIIYSICS

179

where m0, p are t h e t a c h y o n p r o p e r mass a n d 3 - m o m e n t u m , respectively, a n d where P is b o d y B initial 3 - m o m e n t u m . One i m m e d i a t e l y gets p as a f u n c t i o n of m. ~¢nd P :

(2)

mo

lPl = oUN-:;[ tP]m°

~M~

+ "v/iP~+ M§)(m~ 4- 4M~)],

so t h a t , for e v e r y m0, t h e b o d y B c a n a b s o r b o n l y t a c h y o n s w i t h a definite (discrete) v~lue of p . A f t e r h'~ving considered t h e a b s o r p t i o n process (by B), let us t u r n o u r a t t e n t i o n to t h e e m i s s i o n process b y A. T h e t a c h y o n m e c h a n i c s is such t h a t t h e b o d y A (at rest in So) c a n emit t a c h y o n s (of a n y p r o p e r mass) o n l y b y lowering its rest m a s s (20). Thus, in this case, we cannot neglect AM~ ~- M'. -- M~ ; some trivial k i n e m a t i c s tells us t h a l , in this case, A(M~) ~ M'a2 - M ] < - - m~. I f A(M~) is not qua,ntized (a.s we m:~y e x p e c t w h e n dealing w i t h a m a c r o o b j e c t A), t h e e n e r g y b a l a n c e for t h e emission process,

(3) will not yield a n y definite c o n s t r a i n t on t h e v a l u e of mo (in t e r m s of p2 a n d MA)(21). Therefore, u n d e r t h e p r e v i o u s a s s u m p t i o n s , we are left w i t h eq. (2), c o r r e s p o n d i n g to fig. 4a), b). F r o m eq. (2), it is s t r a i g h t f o r w a r d to derive t h e i m p o r t a n t relation

(4)

V < c2/u,

(20) See ref. (1), p. 264; and R. MmNANI and E. RECAMI: to appear in Phys. Lett., B. Here and in the following natural units (c = 1) will be used when convenient. (21) However, we will immediately get such a constraint when A is ~ subnuclear or nuclear particle, e.g. when considering the process A3a(1232)-+p+t, in the rest frame of the Aa3-resonancc. Such considerations are useful if we consider the probable role of tachyons in elementary-particle interactions ((~ dual theories >), (~strings ,), Higgs mechanism, besides (( virtual particles ~)) (20,22). Moreover, hadrons might be considered as (~strong black holes ~)(23), so that t a c h y o n s - - b y crossing their horizon---might transform into bradyons, and vice versa. Tachyons, then, may even be the strong-field quanta (compare e.g. the electromagnetic excited-atom decay A * - ~ A + , ( with the previous, possible process A3a-~p+t). (22) See also R. MIGNANI and E. RECAMI: NUOVO Oimento, 30 A, 533 (1975), pp. 538, 539; ref. (1), pp. 265, 266. (ca) A. SALAM: in Developments in High-Energy Physics (New York, N. Y., 1972); K. P. SINHA and C. SIVARAM"Lett. Nuovo Cimento, 8, 324 (1973); E. RECAMI and P. CASTORINA: Lett. 2guo~o Cinlento, 15, 347 (1976); P. CA:LDIROLA,~¢[. PAV~I~ and E. RECAMI: in preparation.

180

~t. PXV~I6 and E. RECAMI

ip[

m0

a)

IPI=

m°

Fig. 4. - a) Relation IP] vs. m o holding for the tachyons that--due to their kinematics-can actually be absorbed by a body B moving with various speeds along the x-axis (where 0 < u 1< u s < u. < c) and having a fixed rest mass M n (i.e. that does not change its rest mass in the tachyon absorption process), b) The same case as before, but showing now IPl vs. IPI, and using now mo as parameter. The quantity [P] is the three-momentum magnitude of body B.

where we should recall t h a t u ------u= is the relative speed between A a n d B , and V-----V~ is the b e a m t a c h y o n speed. As a conclusion, if A wants to t r a n s m i t information to B b y t a c h y o n beams, he cannot use t a c h y o n s with speed V > c~/u, i.e. he can use only t a c h y o n s t h a t will appear even to B as t r a v e l l i n g / t o m A to B. Now, this conclusion will be generalized to the case in which also the detector B is allowed to change its own rest mass. Let us notice t h a t t a c h y o n mechanics is such t h a t a b o d y at rest can absorb t a c h y o n s both when changing its mass and when conserving it. Here we are going to assume that, if a moving, macroscopic b o d y B does change its rest mass while absorbing signal b e a m t a c h y o n s , then it increases its rest mass. I n other words, we are assuming t h a t t a c h y o n s absorbed b y B have rest mass mo small in comparison with the rest mass M of the macroobject B, a n d are endowed (with respect to s, that is to the rest frame of A) with not too small energies--i.e, with speeds n o t almost infinite--, so as to be actually able to carry information (17). F u r t h e r m o r e , let us refer to fig. 3: if we w a n t (as usual in maerophysics) t h a t b o d y B does not appreciably change its speed when absorbing the t a c h y o n beam, t h e n its rest mass must logically be assumed to increase (and, if this increase is negligible, we are left with the case already considered earlier). Then, for macro-objects B , we have (see fig. 5)

M'B>M,

Mi>0,

a n d now eq. (2) reads

(2bis)

[vl = 2 ~ [molPI+ ~/(P~ + M~)(m~ + 4Mi) +

~//2Z~2] ,

HOVe T O R E C O V E R

(JAUSALITY FOR

TACttYONS

EVEN

IN

MACROPHY$ICS

181

VJh

c-

IB

m0

Fig. 5. - The tachyons that (owing to their mechanics) can actually be absorbed by a body B, with initial rest inass Mz and moving with a ]ixed specd u u~ along the positive x-axis, must satisfy the relation V vs. m o shown in the figure. The quantities V~V~, and mo ~re speed ~uld rest mass of tachyons, respectively. Now body B is allowed to change its rcst mass during tachyon absorption: line 1 refers to A-~:- M~2 M ~ = 0 , and line 2 to A2 >0. Remember that u is fixed.

thus yielding a IPl value larger t h a n the one yielded by eq. (2). This meuns that, even in this more general case, the speed V of beam t a c h y o n s possibly exch:mged from A to B is, a/ortiori, constrained to values

V < c~/u ,

(4)

(in the initial rest frame of A), in which case no causal p~radoxes arise. We can ~gain conclude that every time theft (the macro-object) A transmits infornmtion b y a modul,~ted t~tchyon beam to the (moving) macro-object B, which ~bsorbs the b e a m (either b y conserving its rest mass, or by incre.~sing it), then t:~chyon dyna, mics ensures us tht~t also B will always see the beam ;rs ~'oing from A to B. So t h a t neither the free will of observer B will be jeopardized, nor we shall ~etually meet a n y c o n t r a d i c t o r y situation (of the kind hypothetically proposed in sect. 4) (15). h , connection with eq. (4), let us notice, t h a t - - i n four dimension.~ the relev:mt q u a n t i t y will be, inste:rd of V, the t a c h y o n speed componenl :along the direction B - A . I n connection with eq. (2bis), let us notiee that, for every mo, the b o d y B can absorb only t a e h y o n s with discrete values of ]p] whenever A2 can ~lssume only discre~(~ values (or~ in pa, rlicula.r, when it is zero) (~-23).

6. -

Some

science

fiction.

I n the previous section, we have been considering macro-bradyons essenti~dly absorbing 1)earns made of micro-taehyons. Here we w a n t to give ~ hin~ about

182

M. PAVgI~ and E. RECAm

how proceeding when dealing also with macro-tachyons. At this point, let us t a k e the liberty of using some science fiction. Namely, let us assume (24) t h a t a reader /~1 seats on a chair and t h a t an a u t h o r A1 switches on, a r o u n d him, a (rotating) K e r r or K e r r - N e w m a n black hole (25) /(1, so t h a t R~ finds himself inside the ergosphere, near its internal b o u n d a r y horizon. Then, the chair starts receiving angular m o m e n t u m , so t h a t /~ crosses the external ergosphere b o u n d a r y , transforming into a t a c h y o n (cf. p. 879 of the first of ref. (2~)). Afterwards, R1 travels e.g. along the positive x-direction with speed V ~ c until he r e a c h e s - - i n a v e r y short t i m e - - a n o t h e r , v e r y far planet, where a pioneer (author A.~) had already arrived and switched on a second, analogous K e r r (or Kerr-Newman) black hole K2. Then, K2 captures R~ who, crossing the e x t e r n a l limit of K~ ergosphere, transforms again into a bradyon. At last, the R~-chair angular m o m e n t u m is slowed down until R~ approaches the internal horizon of K~ ergosphere, and eventually A2 switches off K2 (so t h a t R1 can leave his chair and walk on the new planet). Of course, K~ and K2 are practically considered at rest one with respect to the othei• (for simplicity). At this point, if we consider a second reader R~, travelling (along the positive x-direction) with subluminal speed u ~ c2/V, he will not see the reader / ~ going from KI to K., but, on the c o n t r a r y , the a n t i r e a d c r / ~ , going from K~ to K~. Or, better, he will observe K= creating a pair R~, R~ (the fact t h a t in this case the traveller /~1 is taehyonic and the walking-on-the-new-planet R~ is bradyonie is not essential in this respect: it is due to the internal action of K2). You are r e a d y to accept micro-object pair creation (provided some e n e r g y is supplied), b u t perhaps you are not eager to accept macro-object (reader-antireader) pair creation. Therefore, if you know (extended) relativity, we have a simple recipe (for o r t h o d o x persons): through the suitable Lorentz transform,~tion, go from y o u r description to the description in the K2 rest f r a m e ; and y o u shall find a more o r t h o d o x description (i.e. the reception of taehyonic R~, his slowing down and the emission of bradyonie R~). 5Taturally enough, however, we expect t h a t the reader R2 will operate an analogous Lorentz transformation whenever he observes the dress colour of the (subluminal) reader R~ as shifted b y the standard (( Doppler effect ,). Fiction aside, let us r e m e m b e r that always the descriptions will become more o r t h o d o x in some suitable frames (e.g. in the detector or in some rest frames, etc.). The same applies, for instance, also when the body B of fig. 3 has a complicated internal structure, so as to contain moving constituent objects:

(24) It will be soon clear that our relevant assumption on this respect is tha~ the set (R} of readers contains n >~2 elements. (25) See, e.g., C. W. MISNER, K. S. THOR~IE and J. A. WH~]~ZER: Gravitation (San Francisco, Cal., 1973). See also S. W. HAWKING: Comm. Math. Phys., 43, 199 (1975); F. SALTZ~2~N and G. SALTZM~N: T~ett. Nuovo Cimento, 1, 859 (1969).

HO%V TO RFCOVER CAUSALITY FOR TACHYONS EVEN IN M~tCI¢OPIIYSICS

153

in t h i s case, B c~m p r a c t i c : f l l y a b s o r b e v e n t a c h y o n s w i t h t h e ( ( p r o h i b i t e d ) ) s p e e d s V ~ c 2 / u in the overall c.m. o/ B ( w h e r e u is t h e c.m. s p e e d of B).

7. - M i s c e l l a n e o u s , further r e m a r k s .

i) A n y stand:t,rd (e.g. e l e c t r o n - p o s i t r o n ) p a i r c r e a t i o n c a n a,ppear to Superluminal frame S as a unique, tr~¥elling tachyon-electron (or tachyonpositon), which emits or absorbs a couple of photons at a certain time instant. S i n c e , for S, o n e law of t h e very particular p h e n o m e n o n under examination s e e m s t o b e t h a t (( r e s t m a s s is a n ~ d d i t i v e , c o n s e r v e d c h a r g e ~), t h e n t h e s a m e l a w (~e) s h o u l d h o l d for s u b l u m i n a l o b s e r v e r s s ( o b s e r v i n g t h e p a i r c r e a t i o n ) . W e c o n c l u d e t h a t mo(e ~) ------ mo(e-), as a h ' e a d y c l a i m e d in ref. (9) a n d r e f e r e n c e s t h e r e i n ; t h a t is t o s a y , a n t i p a r t i c l e s - - / n the usual /ormalism ( 2 6 ) - - o u g h t t o b e f o r m a l l y a s s o c i a t e d t o n e g a t i v e r e s t m a s s e s ( b u t t o positive r e l a t i v i s t i c m a s s e s a n d energies~ of c o u r s e ! ) ( 9 ) . ii) I n f i n i t e - s p e e d t a c h y o n s (~7) carry no energy ( e v e n if t h e y carry a m i n i m a l m o m e n t u m moc) a n d s h o u l d not b e a b l e t o t r a n s m i t i n f o r m a t i o n . B u t , t h e n , w h y is it not p o s s i b l e t o m o d u l a t e t h e i r s p i n s (or helicities), t o e n a b l e t h e m to transmit i n f o r m a t i o n ? (27) L e t us r e c a l l , first, t h a t ( ( t r a n s c e n d e n t ~) t a c h y o n s a r e n o t a s s o c i a t e d t o a n y d e f i n i t e d i r e c t i o n a l o n g t h e i r m o t i o n l i n e : t h e y c a n b e c o n s i d e r e d (2s) as (, o u t g o i n g ,) t a c h y o n s as well as (( i n c o m i n g )) a n t i t a c h y o n s ; or e v e n we c a n c o n s i d e r t h e m a s p a i r created (:s,~9) in a n y i n t e r m e d i a t e p o i n t b e t w e e n A a n d B : see fig. 1. I1~ t h e q u a n t u m - m e c h a n i c a l f o r m a l i s m , we c o u l d w r i t e , /or example, (5)

!t(Iv I = c~)} = alt(v = -t- ~ ) } Jr blt(v = - -

~)},

la12+ Ibl 2 = 1 .

F o r t h e c o n s i s t e n c y of o u r d e s c r i p t i o n s , h o w e ~ e r , w h e n we p a s s f r o m cons i d e r i n g , e.g., A - ~ B tachyons to c o n s i d e r i n g B--> A antitachyons, t h e a s s o c i a t e d , transpor~'ed q u a n t i t i e s m u s t either c h a n g e sign, or b e zero. F o r i n s t a n c e , t h e

(26) Tile stated assertion is of course a (( law }>of that particular phenomenon (~2) in the sense t h a t it remains true under a certain, large class of (subluminal) Lorentz transformations (e.g. from S to suitable S') ; if we assume t h a t it is actually a (~G-covariant ~)(16) law, then wc get as a consequence t h a t antiparticles mus~ be formally a t t r i b u t e d a negative rest mass. But nothing will prevent us from reformulating (( extended relativity~> in a slightly different way (so as to define, for instance, a (( rest mass )~, t h a t changes sign when going from particles to antiparticles, and a (~proper mass ~), which, on the contrary, is G.invariant). (aT) W. ZAeHARIAS and J. GRODSKY: private communication. (2s) See R. MIGNANI and E. R~CAIdI: second of ref. (~o). (29) Cf. also J. D. EDMONDS jr.: Found. Phys., 6, 33 (1976).

184

~ . PAV~i~ a n d E. RI~CAMI

b + t ,

where t is an a n t i t a c h y o n . Then, u n d e r a suitable Lorentz t r a n s f o r m a t i o n (LT), a new observer can describe the same process as (1) (7)

a ~ t->b.

If, in eq. (6), t was e m i t t e d a n d t h e n h a d travelled until it has b e e n absorbed (32) b y a (near or jar) (32) detector, t h e n in eq. (7) t m u s t be of course considered as e m i t t e d b y a (near of /at) source.

(a0) We have shown, however, that an (electric) charge, when going at infinite speed, builds up only a magnetic field: see, e.g., ref. (1), and E. R~CAMI and R. MIGNANI: Lett. .Nuovo Cimento, 9, 367 (1974). (31) b~oticc that taehyons (17) can only take energy from the scatterer. Therefore, if B (for instance at rest) scatters transcendent taehyons, it should utilize its rest mass energy to supply energy both to tachyons and to itself (for its recoil motion). This would be contrary to the assumptions, about the rest mass behaviour of macro-objects, adopted in this paper. (as) Our philosophy--similar to the one by WI~]~]~L]~Rand F~YNYIAN(33)---i~ that no object can be emitted if detectors do not exist, in the Universe, that will be able to absorb it soon or later. In fact, this philosophy is a must (4,1) in (extended) relativity with tachyons (1), since tachyon physics cannot be done without taking always into account

HO~"

T O Io*ECOVER C A U S A L I T Y

FOR

TACtIYONS

EVEN

IN

MACROPItYSICS

185

I t s h o u l d t h e n be clear t h a t t h e m a t t e r emission p o w e r of t a c h y o n s n m s t be c o n n e c t e d w i t h t h e tachyon cosmic ]lux (as e x p e c t e d also f r o m other, o b v i o u s considerations). F o r instance, if A~ is t h e m e a n life of particles a for t h e d e c a y in eq. (6), t h e n (8)

[Az]-~ oc ~(t)a(a, b),

where ~(t) is t h e t a c h y o n (cosmic) ]lux density a n d a(a, b) t h e i n v a r i a n t erosssection of process (7). I n o t h e r words, Av is t h e L o r e n t z - t r a n s f o r m e d value of t h e a v e r a g e t i m e Av' t h a t p a r t i c l e a has t o s p e n d before b e i n g s t r u c k b y a cosmic t a c h y o n t a n d t r a n s f o r m i n g i n t o b:

(S')

[AT]-1=

[ L ( A v ~ ) ] -~ .

iv) F i n a l l y , let us m e n t i o n t h a t one of us (M.P.) t h i n k s t h a t causal p a r a doxes could e v e n b e t t e r be solved in t h e c o n t e x t of 5 - d i m e n s i o n a l r e l a t i v i t y (~4). I n a n y case, c o n s i d e r i n g S R in five dimensions a l r e a d y r e s u l t e d to be a useful tool for a t t r i b u t i n g t h e c o r r e c t role to p r o p e r m a s s a n d p r o p e r t i m e a n d for g i v i n g e x t e n d e d r e l a t i v i t y ~ m o r e e l e g a n t f o r m (34,35).

The a u t h o r s are g r a t e f u l to Profs. P. CALDIROLA~ V. DE SABBATA, R. MIGNA~qI, W . ZACHAI~IAS, G. ZIINO for v e r y useful discussions a n d to A. AGODI~ L. ~Fo:NDA~ G. ~FURLAN for t h e i r i n t e r e s t . O n e of t h e a u t h o r s (E.I~.) a c k n o w l e d g e s also a n I~NFIq/ICTP g r a n t a n d t h a n k s t h e P u b l i s h i n g D e p a r t m e n t cf I C T P , where this w o r k '~,ppeared as p r e p r i n t IC/76/96 (Trieste, ]976). the proper sources and detectors (because of the fact that the roles of tachyon sources and tachyon detectors can invert under a standard LT). Any couple of bodies connected through tachyon exchange are realizing a symmetrical, instantaneous connection according to suitable observers. And it is not without meaning that in ref. (3s) the same philosophy was shown to be adoptable in the limiting case of photons. (as) j . A. WHEELER and R. P. FEYN~AN: Rev. Mod. Phys., 17, 157 (1945); 21, 425 (1949). (34) M. PAVSI(~: in preparation; and Left. Nuovo Cimento, 17, 44 (1976). (ss) See, e.g., E. RECA~I and G. ZIINO: -~uovo Cimento, A (in press); P. CALDmOLA and E. R~CAMI: ref. (4); p. CALDIROLA: Left. Nuovo Cimento, 15, 468 (1976); 16, 151 (1976}; Suppl. Nuovo Cimento, 3, 297 (1956); Nuovo Cimento, 19, 25 (1942); Rend. Ace. d'Italia, 7, 19 (1939).

•

RIASSU~qTO

I1 postulato che non esistano particelle ad energia negativa (che viaggino avanti Eel tempo) conduce automaticamcnte al (, con taehioni e con sistemi inerziali superluminali. I n o l t r e il nostro terzo postulato p e r m e t t e di p r e v e d e r e l'esis~enza delle a n t i p a r t i e e l l e in un contesto puramente relativistieo. I n questo l a v o r o si m o s t r a ehe il terzo postutato ~ suffieiente a s a l v a g u a r d a r e 1~ v a l i d i t ~ della legge di eausalit~ (ritardata) anche in maerofisiea, q u a n d o gli usuali m a c r o o g g e t t i interagiscono con m i e r o t a e h i o n i o con maerotachioni. A questo seopo si s v i l u p p a u l t e r i o r m e n t e la einem a t i e a dci taehioni, la quale p o t r e b b e essere utile anche p e r la eomprensione delle interazioni t r a particelle e l e m e n t a r i (e ]arse della s t r u t t u r a degli adroni). Si diseutono infine anehe m o l t i altri p r o b l e m i eonnessi.

l(al~ BDeCTaHOBHTb IlpHqHHHOCWb ~JIIl TaXHOHOB ~aace s Magpoc~H3mce.

Pe3mMe ( * ) . - l'Iocryaaw, ~tro ~tacrrtt~ht c oTprtttaTeabno~ a a e p r a e ~ He cyt~eCTBylOT (KOTOpble ~BH:)KyTC~ B IIp~IMOM Ha~paBneHm4 iio apeMeHn) aByoMaTn~ecl, npnnaMaeMblI~ Kar Tpemu~ Iloemy4am c~peuHanbHOfl v e o p a a OTHOCnTenbHOCTn, oSeclie~naaeT ClllOaBe~x~ItiBOCTba a r o n a (3alIa3ZJ,iBa~ome/~) npnqnHaOCTn, r a t a o6~,mno~ Teopnn OTHOCnTenbHOCTIL Tar n B (npoT~l~ettno~) Teopnl40THOCHTe.rIbHOCTH C TaxnoHaM~4 ~ C cyrrepnlOMHHa~bHblMn auepl~nanbHI, tMn cncTeMaMn. H a m Tpemu~ Ilocmydam, KpoMe Toro, IIO3aOJIJteT irpe~craaar~, cyIIleCTBOa a n n e anT~i~iacTni~ a ~ucmo pe/IflTHBHCTCI~OMrOnTerCTe. B aTO~ CTaTbe Mbt IlOra3blBaeM, ~TO Tpemu~ IIocmyzam ~saflewc~ ~OCTaTOqHUM, ~TO6bI o6ecne~nTb a ~ m o a n e n n e 3 a r o a a IlpnqnnnOCTn ~ a ~ e s Marpoqbnanre, x o r ~ a O6blqnb/e Marpo-o6~erT~i aaaaMo~e~cTayIOT c Mnxpo-TaXUOHaMn n Marpo-TaxuonaMn. C aTO~ u e ~ , m paaanaaeTca rnneMaTnra Taxnonoa, ~ o r o p a s MO>reT 6~IT~ IlOaeano~ ~ a s nOnnManna aaanMo~e~cTanil aaeMe~Tapu~tX ~acTnU (n, Mox~eT 6~tT~, c r p y x T y p u a~poaoa). O6cyx 0.) Notice, however, that the physical conclusions--and actually the whole paper--remain totally unchanged. Even fig. 5 still holds, since the physics was not affected at all by that miscalculation. Let us take advantage of the present occasion for repeating t h a t - - a s already (1) remarked--lines 14 and 15 (and the first eight words of line 16) ought to be eliminated at page 178. For further details about the subjects of this paper, see for instance ref. (1,2).

The authors are grateful to Prof. R. G. GLASSER for the kind collaboration.

(1)

E. RECAm: Zett. Nuovo Cimento, 21, 208 (1978), and to appear.

(3) P. CALDIROLA and E. RECAP[I: to appear in Boston Studies in the Philosophy o] Sciences, edited by R. S. COHEN and M. DALLA CHIARA (Dordrecht); E. R~cAm: Lett. _~uovo Cimento, 18, 501 (1977); E. RECAMI, Editor: Tachyons, Monopotes, and Related Topics (Amsterdam, 1978). See also V. DE SXBBATA, M. PArdI5 and E. RECAm: Lett. Nuovo Cimento, 19, 441 (1977). 298