LETTERE AL NUOVO CIMENTO
Recovering
VOL. 17, N. 7
Causality for Taehyons
16 0ttobre 1976
e v e n i n M a e r o p h y s i c s (*).
M. PAV~I~ (**) and E. R]~CAm (***)
Istituto di Fisiea Teoriea dell'Universith - Catania, I t a l i a
G. ZIINO
Istituto di F i s i e a dell'Universit~ - Palermo, I t a l i a
(ricevuto 1'8 Luglio 1976)
It was already shown (1) that (( retarded causality ~) can be implemented in relativit~y
by assuming the postulate that (( negative-energy objects travelling forward in time
do not exist ~) (third postulate of special relativity) (x,2). This (~third postulate ~) is fundamentally equivalent to the well-known (( reinterpretation principle ~) (RIP) by DmAC,
STi)CKELBERG and FEYN~A~ (~). The (( RIP )) is quite enough (1) to ensure the validity of the taw o / r e t a r d e d causality also for taehyons, even if many physicists are
still unaware of this fact. In particular, it is generally believed that causal problems
with taehyons are still present in macrophysics. That situation has been very recently
emphasized, e.g., by BASANO (4).
(*) W o r k p a r t i a l l y s u p p o r t e d b y M . P . I . , o x - a r t . 286, a n d b y C N R , u n d e r c o n t r a c t 75/00442.02.
(**) O n l e a v e f r o m J . S t e f a n I a s t i t u t e , U n i v e r s i t y of L j u b l j a n a , Y u g o s l a v i a , s u p p o r t e d i n p a r t by the
B o r i s K i d r i e f u n d ~.
(***) A l s o : I s t i t u t o N a z i o n a l e 4i F i s i c a N u e l e a r e , Sezione di C a t a ~ i a ; a n d C e n t r o Siciliano di F i s i e a
N u e l e a r e e di S t r u t t u r a d e l l a M a t e r i a , C a t a n i a , I t a l i a .
(x) ]~. REOAMI a n d R . MZ(~NANI: R i v . N u o v o Cimento, 4, 209, 398 (1974), a n d r e f e r e n c e s t h e r e i n . See
a l s o , E. RECA~I: i n E~eielopedia E S T - M o n d a d o r i , A n n u a r i o 1973 (Milano, 1973), p. 85.
(2) P . GALDIROLA a n 4 E. REC~'~I: i n Proceedings o/ the 1976 Con/erence on the Co~.cept o/ T i m e , S . M a r gherita Ligure, Italia, to a p p e a r ; E . RECASII: Scientia, 109, 721 (1974); a n d t o a p p e a r ; R . MIGBY&NI
a n d E . REOXMI: R e p o r t I C / 7 5 / 8 2 , I C T P , T r i e s t e (1975), t o b e p u b l i s h e d ; I n t . J o u r n . Theor. P h y s . ,
12, 299 (1975); N u o v o Cimento, 2 4 A , 438 (1974); L e f t . N u o v o Cimento, 11, 421 (1974); E . R E C ~ I
a n d E . NIoDICA: L e f t . N u o v o Cimento, 12, 263 (1975); E . R E C A ~ a n d R . NIIGXANX: L e t t . N u o v o Cimento,
8, 110 (1973); R . G. ROOT a u d J . S. TREFIL: L e f t . N u o v o Cimento, 3, 412 (1970 ; J . A . PA/CMENTOLA
a n d D. D. H . YEE: P h y s . R e v . D , 4, 1912 (1971) ; E. C. G. SUD~SttA~r : i n S y m p o s i a on Theoretical Physics
a n d Mathematical, Vol. l 0 ( N e w Y o r k , N. Y . , 1970), p. 129; O. NI. BILANIUK a n d E. C. G. SUDARSH~i~r:
N a t u r e , 223, 386 (1962); E . R E C ~
a n d R . N I I G ~ I : P h y s . L e f t . , 62 B, 41 (1976).
(a) P . A . NI. DII~AO: P r o c . R o y . S o c . , 126 A , 360 (1930); E. C. G. ST(~CKELBERO: H e l v . P h y s . A c t a , 14,
3 2 L , 588 (1941); R . P . FEYNM~N: P h y s . R e v . , 7 6 , 749, 769 (1949). See a l s o , O. 1K. P . B n ~ N r U K ,
V. K . DESHPANDE a n d E. C. G. SUDARSHAN: A m e r . J o u r n . P h y s . , 30, 718 (1962); E . RECAMI: Giornale
di F i s i c a , 10, 195 (1969). T h e * R I P * h a s b e e a g i v e n a n e w , e l e g a n t f o r m u l a t i o n i a 5 - d i m e a s i o n a l s p a c e :
see, E . R E C ~ I I a n d G. ZIINO: Nuo~o Cimento, & (in press).
(a) L. BASANO: L e f t . N u o v o Cimento, 16, 562 (1976).
257
258
M. pAV~I~, E. RECAMI a n d G. ZlINO
W e w a n t here to stress, on t h e c o n t r a r y , t h a t our (~ t h i r d p o s t u l a t e ~ r e c o v e r s causality
for t a c h y o n s n o t o n l y in t h e case of e l e m e n t a r y p r o c e s s e s - - a s m o s t l y c o n s i d e r e d ,
till n o w - - , b u t also in m a c r o s c o p i c processes, w h e r e e n t r o p y c o n s i d e r a t i o n occurs.
L e t us c o n s i d e r t w o b o d i e s , A a n d B , for s i m p l i c i t y at rest one w i t h r e s p e c t to the
o t h e r . T h e n , let us c o n s i d e r - - i n t h e i r rest f r a m e s (i.e., in t h e l a b ) - - t h e e x c h a n g e of
a t a c h y o n T along 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, from A (source) to B
( d e t e c t o r ) . Now, t h e f r a m e w i t h s p e e d u ~ u= = c 2 / V will see t h e t a c h y o n T t r a v e l l i n g
a t infinite speed (~) a n d (as i l l u s t r a t e d e.g. in fig. 12 of ref. (~), p. 236) a n y f r a m e
w i t h s p e e d u > e e / V w o u l d * s e e , T as g o i n g b a c k w a r d s in t i m e a n d b e a r i n g n e g a t i v e
e n e r g y : t h a t is to say t h a t it will a c t u a l l y o b s e r v e an a n t i t a c h y o n T (~). I n o t h e r
w o r d s , if w e c o n s i d e r a (subluminal) b o o s t along 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
c2/V<u<c,
t h e n t h e n e w o b s e r v e r s' will see (~,5)--because of (~RIP)~--an antit a c h y o n T e m i t t e d b y B (source) a n d a b s o r b e d b y A (detector), a n d carrying, of course,
p o s i t i v e e n e r g y f o r w a r d in t i m e . See fig. 1.
x
X1 x~=~]Z[
(x') (x',)'~
y~
[t,>t2j
,....~ (P);_~q ;E<O ;~i;p_<0
_.~.
Fig. l. - The figure represents the exchange from A to B of a particle P with negative energy (and
, charges *) and travelling backwards in time (t= < tL). From the Third Postulate of special relativity
(* aogative-energy particles travelling forward in time do not exist *) and from the fact that we necessarily
explore * l~Iiukowski space-time going *forward in time *, it follows that such a process appears as
the exchange from B to A of a particle Q endowed with positive energy (and * charges *) and moving forward
~'n tim~. Particle Q results then t,~ be--see the text--the antiparticle of the initial particle (except
for the hetieity, ia this ease): Q ~P. &s is well-knows, the * third postulate * of SR--whiic solving in n~ece
all the (apparent) paradoxes, which seem to imply an information--transfer into the past, both with usual
particles (bradyons)' and with tachyons--furthermorc predicts, for any particle, the existence of its
antiparticle.
The p o i n t is t h a t , while s o b s e r v e s e.g. t h e e n e r g y of A d e c r e a s i n g in t h a t process,
on t h e c o n t r a r y s" o b s e r v e s t h e e n e r g y of A i n c r e a s i n g in t h e s a m e process. H o w e v e r ,
for each o b s e r v e r , t h e l a w of e n e r g y c o n s e r v a t i o n is s a t i s f i e d in t h e o b s e r v e d process.
Only t h e d e s c r i p t i o n ~cd e t a i l s , are n o t i n v a r i a n t , in full accord w i t h special r e l a t i v i t y (SR)
t h a t does n o t r e q u i r e t h e i r i n v a r i a n c e . F o r s u i t a b l e definition a n d analysis of (~law )),
* d e s c r i p t i o n ,, etc., see e.g., p. 248 of ref. (1). To clarify t h e s i t u a t i o n , let us r e m e m b e r
t h a t S R (and D o p p l e r ) t a u g h t us t h a t , for e x a m p l e , t h e colour of an o b j e c t is n o t L o r e n t z
i n v a r i a n t (even if n o t h i n g p r e v e n t s a n y o b s e r v e r , w h o k n o w s SR, from (~c a l c u l a t i n g
o u t t h e , r e a l , colour, i.e., t h e i n t r i n s i c colour d i s p l a y e d by t h e o b j e c t in its rest frame).
T h e n e c e s s a r y r e q u i r e m e n t of S R is t h a t each o b s e r v e r f o r w a r d s a s e l f . c o n s i s t e n t description c o n t a i n i n g t h e s a m e ( c o v a r i a n t ) p h y s i c a l l a w s (1).
(6)
E. R E c ~ x
and R . MIGN&NI: Riv. NuoVO Cimento, 4, 209 (1974)~ See also ref. (2).
R~.COVF.RING CAUSALITY FOR TACHYONS EVEN IN MACROPHYSICS
2~9
A t t h i s p o i n t , w e h a v e n o t h i n g t o d o b u t n o t i c e t h a t in fig. 1 w e can h a v e ( i n s t e a d
of a tachyon exchange) a tachyon signal transmission by means of a modulated taehyon
beam. Then, entropy changes and information exchanges behave--when passing from
o n e f r a m e t o o t h e r o n e s - - e x a c t l y a s e n e r g y e x c h a n g e s b e h a v e d in t h e p r e v i o u s e x a m p l e
(fig. 1); so t h a t , e.g., a n i n f o r m a t i o n i n c r e a s e o b s e r v e d b y s c a n b e c o m e a n i n f o r m a t i o n
d e c r e a s e a s r e g a r d e d b y s', a n d so o n .
We shall illustrate the arising, possible problems by considering two examples,
i.e. b y p r o p o s i n g t w o a p p a r e n t p a r a d o x e s , a n d t h e n b y s o l v i n g t h e m .
F i r s t , l e t u s c o n s i d e r t h e c a s e o f a s u b l u m i n a l f l a m e (x, t) a n d o f a S u p e r l u m i n a l
f r a m e (~) (x ~, t') b o o s t e d 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 r e l a t i v e s p e e d U ) c .
See fig. 2. A n d lot u s c h o o s e a s c o m m o n o r i g i n , O , t h e e v e n t w h e n b o t h f r a m e s c o i n c i d e .
T o s, t h e x - a x i s r e p r e s e n t s all t h e e v e n t s c o n t e m p o r a n e o u s w i t h O ; w h i l s t , t o s', ~ll
the events contemporaneous with O are represented by the x'-axis. If information can
b e c a r r i e d b y ( a l m o s t (~)) i n f i n i t e - s p e e d t a e h y o n s , t h e n o b s e r v e r s ' - - w h e n is n e a r O - c a n r e c e i v e s i g n a l s f r o m e v e n t E (see fig. 2) a n d c a n t e l l s u c h i n f o r m a t i o n to s. I t seems,
t h e r e f o r e , t h a t s' c a n c o m m u n i c a t e ( w h e n is n e a r O) ~ o s i n f o r m a t i o n a b o u t t h e p o s s i b l e ]uture o f s .
/ x~
~
.//
/./'/
Fig. 2. - The first apparant paradox here discussed. It seems that a Superluminal observer (x',t'),
while overcoming at O the observer (x, t) * at rest *, can tell the latter informa'Aon about a ]uture--for
the latter--event E, i! taehyon signals exist travelling at (almost) infinite-speed. The paradox is
immediately solved on the basis of the third postulate of SR, when one remembers that SR requires only
laws (and not description dermis) to be invariant. The important point, here, is that causal paradoxes
with $achyons can be easily solved even in macrophysics, tg ~ = fl m U/c.
H o w e v e r , it is c l e a r t h a t t h e , R I P , t e a c h e s u s ( ~ ) t h e f o l l o w i n g : T h e s i g n a l E - ~ 0
will a p p e a r t o o b s e r v e r s as a s i g n a l ( o r b e t t e r ~ a n t i - s i g n a l ~>(s)) s e n t b y s ' f r o m 0
t o w a r d s E ( s i n c e , w i t h r e s p e c t to s, i t w o u l d b e a , n e g a t i v e s i g n a l 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 (~.5) b e r e i n t e r p r e t e d a c c o r d i n g t o * R I P ,~). I n c o n c l u s i o n , a c c o r d i n g
(') Infinite-speed tachyons carry zero-energy (but 3-momentum magnitude equal to me), and their
direction along their motion-line has no definite senee. As * bradyons * at rest are points in space,
extended in time along a line, so trascendent tachyons are r points * in time, extended in space along
a line: cf., R. ~IGICANI and E. RECAMI: Left. Yuovo Cime~to (in press). Infinite-speed tachyons will
be dealt with elsewhere, also due to their possible rSle in hadron structure (remember, e.g., Susskind's
, lnstantons ,, Preparata's r bilocal functions *. besides * virtual particles *). Cf., e.g., It. I~IGNANI and
E. REOAMI: N~ovo C~imen~o. 30A, 533 (1975); Phys. Lett. (to appear); and the quotations (87-89) in
ref. (l).
(~) See ref. (') and p. 239 of ref. (~).
(') An * anti-signal, is carried by anti-particles, so as t h e signalis carried by (their) particles. Of course,
bo~h carry p~itive energy. S e e a l s o t h e following.
260
~.
PAu
~.
I:tI~CAMI and G. ZlI~O
to s, observer s' is predicting to him n o t h i n g b u t w h a t can be a c t u a l l y <~forecast ~>
- - o n the basis of our p h y s i c a l operations and of the p h y s i c a l l a w s ~ . e . , observer s',
to s, is merely <~predicting ~ t h e obvious effects on E of t h e (anti)-signals sent b y himself
from O towards E , so to physically influence it.
E v e n in this case, each observer forwards a different description of t h e same cha.n
of events, b u t the law of (retarded) causality holds in all descriptions, for all observerls.
L e t us now pass to a more subtle e x a m p l e , i n v o l v i n g a little more deeply t h e t a c h y o n
mechanics, and spend some more words in solving the r e l a t i v e (apparent) p a r a d o x .
L e t us now consider a process like t h a t in fig. l, but w i t h t h e b o d y B m o v i n g w i t h
respect to t h e body A. L e t us, therefore, consider two usual, inertial observers (fig. 3),
m o v i n g along the x-direction r e l a t i v e to each other, w i t h speed u < c, and let us
suppose t h a t (according to the frame s ~ (x, t)) observer s sends a signal to s'~- (x', t')
b y means of a t a c h y o n i e b e a m w i t h speed V > c2/~. According to s', however, t h e
<~t a c h y o n b e a m ,> w o u l d a c t u a l l y a p p e a r as an a n t i t a c h y o n - b e a m e m i t t e d by s' himself
towards s.
It
/f'
A~ B
+/
/ ///
~
F i g . 3. - T h e second1 a p p a r e n t p a r a d . o x h e r e d i s c u s s e d ; i t d e a l s , a s t h e p r e v i o u s o n e , w i t h m a c r o p h y s i c s with taehyons. I t seems t h a t , i f t h e ~ a t r e s t ,> o b s e r v e r (x, t) s e n d s i n f o r m a t i o n f r o m A by"
a m o d n l a t e d t a c h y o a b e a m t o a n o t h e r s u b l u m i n a l o b s e r v e r (x', t'), b o o s t e d 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 u < c, t h e n h e a v y p a r a d o x e s a r i s e w h e n t h e b e a m t a c h y o n s h a v e s p e e d V ~ V~ > c~lu
( i n f a c t , i n t h i s e a s e t h e l a t t e r o b s e r v e r s h o u l d s e e t h e t a e h y o n - b e a m e m i t t e d b y B , i.e. by" h i m h i m s e l f ) .
The paradox is solved by the fact that, in macrophysies, taehyon mechanics allows g to absorb only
t a c h y o n s w i t h s p e e d s V < c2/u, ill w h i c h c a s e n o p a r a d o x e s a r i s e ( w h e r e t h e s p e e d s a r e a l w a y s m e a s u r e d
in the A rest-frame), tg~--~--u/c.
Now, we can well imagine t h a t - - w h e n o v e r c o m i n g s' at O - - o b s e r v e r s told s' about
his intention of t r a n s m i t t i n g to him (i.e., to s') a t a e h y o n i e 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 is t i m e t'. Then, it seems t h a t
observer s' will be co~mpeUed--at a certain i n s t a n t t ' - - A t ' - - t o emit an (anti)-signal
towards s, in order ... to save t h e v a l i d i t y of the classical t h e o r y of t a c h y o n s in fourdimensions. B u t this would, of course, violate t h e <~free will >>of observer s' !, who can,
on the contrary, decide to send no signal (or anti-signal) towards s. A ]ortiori, observer s'
can be in t h e impossibility of sending that signal, lacking information a b o u t it (for
instance, s told s ' - - a t 0 - - t h a t at t i m e t he, s, will send to s' one piece of music b y a
t a c h y o n b e a m w i t h V > c2/u; and s' m a y h a v e no record of t h a t music, so t h a t he
cannot send to s any antimusic, i.e. any music by a n t i t a e h y o n s beams). The situation
seems self-contradictory. The solution of the 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 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 other words,
s' can get i n f o r m a t i o n from s t h r o u g h such process (tachyon-beam emission by A
RECOVERING CAUSALITY FOR TACHYONS EVEN IN MACROPHYSICS
261
and tachyon-beam absorption by B ) o n l y by beams of tachyons with speed V s m a l l e r
than cg/u (in which case no causal problem arises).
Lot us first remember t h a t tachyon mechanics (1) is such that no body at rest
(so as A in its rest-frame) can omit t a c h y o n s - - o f any proper m a s s - - w i t h o u t l o w e r i n g
its rest mass (9). On the contrary, a body at rest'can absorb tachyons both when changing
its rest mass and when conserving its rest mass. Here we are going to assume that a
moving, m a c r o s c o p i c body B, when absorbs signal beam tachyons, will only either
conserve its rest mass, or increase it. Here we are assuming, in other words, that the
tachyons absorbed by B havo rest mass m o small in comparison with the rost mass M
of the macro-object B and a r e endowed (with respect to s, that is to the rest frame of A)
with not too small energies--i.e, with speeds not almost infinite--so to be a c t u a l l y
able to transmit information. Remember that for tachyons, in natural units,
Po = ( P ~ - - mo2)I"
Then, in the process of beam-tachyon absorption by B we shall have
%/p-~-~ § %/~
(1)
M 2 = %/(V § p ) 9 § M,2,
where P is the body-B initial 3-momentum. One gets immediately [ A ~
IPl -- ~
(2)
M ' ~ - M2]:
[mo IPI + V ( P 2 + M2)(mo2 + 4M z) + M 2" A2],
so that, for every too, the body B can absorb only tachyons with definite values of tPl
(which are discrete if A ~ M ' ~ - M ~ can assume only discrete values, or it is in particular zero). From eq. (2) it is straightforward to derivo the important relation
(3)
V < cZ/u,
which allows us to solvo the previous paradox too. In the four-dimensional case the
relevant q u a n t i t y will be, instead of V, the tachyon-speed component along the direction A B .
F u r t h e r details will appear elsewhere. Here let us finally observe that one of us (M.P.)
thinks t h a t causal paradoxes could better be solved in the context of 5-dimensional
relativity (lo). In any case, considering SR in five dimensions already resulted to be
a useful tool for attributing the correct rSle to proper mass and to proper time, and
for giving extended relativity a more elegant form (11).
$$*
The authors are grateful to Prof. R. I~IONANI for very useful discussions.
(,) See p. 264 of ref. (1); and R. MieNANIand E. REc~x: Report IC/75/82 (Trieste), to appear in
Phys. L e f t .
(LQ) M. PAvh[~: submitted for publication; and in preparation.
(") See, e.g., E. RECAMIand G. ZIINO: Nuovo Cimento, A (in press); P. CALDIROLAaIld E. REC'A~II:
Proceedizbgs o! the 1976 Con/erence on the Concept el Time, S. Margherila Ligure, Italia; P. CALDIROLA:
L e t t . Nuovo Cimento, 15,468 (1976); and in press; Suppl. Nuovo Cimento, 3, 297 (1956); Nuovo Cimento ,
119, 25 (1942); Rend. Accad. Italia, 7, 19 (1939). See also ref. (~o).