particle collision lifetimes: the duration of any process is one of the main .... where K~, K~ are the total momenta before and after the collision respectively, and [[> ...
IL NUOVO CIMENTO
VOL. L X I I I A, N. 3
1o Ottobre 1969
About Collision-Process Lifetimes and Causality. V. S. OL~I~OVS~r I n s t i t u t Teoretiehesl~og F i z i k i , Al~ademiya 2gaul~ Ukrainskog SSI~ - K i e v K i e v s k y Gosudarstvennyi Universiteta i m e n i (~ T. G. Schevtehenko ~ - K i e v
E. REtAIn (*) I n s t i t u t Teoretieheslcoi ~ i z i k i , Alcademiya .~aulc U k r a i n s k o i SSI~ - K i e v Istituto di .Fisica dell' Universith - M i l a n o Istituto di .Fisiea dell'Universit5 - Catania
(riccvuto il 26 Marzo 1969)
-The authors generalize their previous results, by defining and calculating reaction and partiM-process lifetimes, to the case of twowave-packet collisions with an arbitrary number of final packets. Besides, they express the principle of causality and the new one of (~anticausality ~) in terms of interaction time shifts. Some notes about tachyons are added. Summary.
1.
-
Introduction.
Classifying and investigating reaction mechanisms includes
analysing
p a r t i c l e collision lifetimes: t h e d u r a t i o n of a n y process is one of t h e m a i n c h a r a c t e r i s t i c s of its m e c h a n i s m . I n this article (**) we generalize t h e results, o b t a i n e d iI1 a p r e v i o u s w o r k (1)~ t o t h e case of t w o - w a v e - p a c k e t collisions w i t h a n a r b i t r a r y n u m b e r of final packets. This subject is m o r e o v e r c o n n e c t e d w i t h t h e principle of c a u s a l i t y : we will consider in p a r t i c u l a r consequences t h a t t h e n e w c o n c e p t of (~a n t i c a u s a l i t y ~ p r o j e c t s onto t h e p r e d i c t i o n s for antip a r t i c l e - r e a c t i o n lifetimes. (*) On leave of absence from the Istituto di Fisica dell'Universit~ di Milano, Milano, nnder an exchange program supported by the Istituto Nazionale di Fisica Nucleare, Sezione di Torino, and by the Ukrainian Academy of Sciences, Kiev. (1) V. S. OZKtlOVSKY and E. RI~eAMI: 2gUOVO Cimento, 53A, 610 (1968). ('*) This work was first published as preprint IFT-68-82 (Kiev, l l September 1968). 814
ABOUT
COLLISION-FROCESS
LIFETIMES
AND
CAUSALITY
815
As is well known, in the initial stage (*) our two-particle (the projectile a and the target b) system m a y be described by: (1)
v~(to)-- cf~(r~, ~ , to)~b(r~, ~, to) --
]n~, nb>f dkadkbg~(k~)gb(kb) exp [-- iE~to/I~]]k~, k~> , where ]ka> = (2z)-~exp[ik~'ra] and ]n~>, ln~> are the orthonormalized functions which describe the internal states of a, b ($~, ~ being the internal coordinates). Besides~ E ~ _ E~ + E~ is the initial particle total energy (kinetic plus internal), ga(k~) and g~(k~) are square-integrable weight functions, and to m a y be the instant of the packet to, when it is still possible to neglect the packet interaction and to consider their motion as a free one: (2)
°A
v~(t) = exp [--tH~(t--to)/h]v~(to)= [n~, n~>/dk~ dkb g~(k~) gb(kb) exp [-- iE~ t/~][k~, k~} ,
H~ being the H a m i l t o n i a n of the (initial) packet kinetic and internal motion. At every instant t during the collision interaction, we will have
(3)
v(t) = exp [--i~(t--to)/li]v~(to) ,
/~ being now the t o t a l H a m i l t o n i a u (for the interaction). We m a y have the creation of a new set of particles, as a result of the interaction. Let us choose - - t o fix our i d e a s - - a particular channel/, relative to a definite set of ~ new final particles with given internal states. I t is possible to represent the total-system wave function (in the region where the initial particle flux is negligible) as follows (2): (4)
~v,(t) = In1, n~, ..., n~)fdktdk~ ... dk~gd~t(k~, k2, ..., k~). • exp [--iEft/l~]]k~, k., ..., k~)(k~, k~., ..., k ; n~, n~, ..., n~l. • exp [i/~: t/~] exp [ - - i H ( t - to)/h] exp [--iJff~ to/~]"
J where E ~ , / ~
are the final particle t o t a l energy and Hamiltonian, gdo~(kz, ..., k,) is the weight function expressing the detector properties.
and
(') When the two initial particles are very distant and practically free. (2) M. L. GOLDB~GE~ and K. M. WATSOn: Collision Theory (New York, 1963).
816
v . S . OLKHOVSKYand :~. RECAMI B y using the definition of the evolution operator U, we can write
(5)
~ ( t , t o ) = t'T~(t, o) ~ ( o , to) = == exp [i/~f tin] exp [ - - i B [ ( t - to)/n] exp [ - - i ; ~ to/hi,
a n d rewrite eq. (4)
(6)
w~(t) = In~, n~, ..., n~>;dl, ldk~
... dk, gdot(kt, k~, ..., k~).
• exp [-- iEf t/n]lk~, k2, ...,
k,>fdko dkbg~(ka)gb(k~).
• . Taking off the c.m. motion, the m a t r i x elements appearing in eq. (6) become (7)
=~(K~--K~) d p i 2p6,_ '
ki¢ --> p i j -- ~o°Jpi - - P6,P~
+
where po ~ [P~ + p a c k e t (4).
m~]½, P
'
being the m e a n m o m e n t u m averaged over the whole
4. - C a u s a l i t y a n d ~ a n t i c a u s a l i t y ~.
I n t h e case of usual particles, it is n a t u r a l to f o r m u l a t e t h e c o m m o n ~ macrocausality principle ~ in t e r m s of the effective i n s t a n t s of i n t e r a c t i o n s t a r t and finish, in the following (self-evident) w a y : in the nonrelativitic case:
(22)
tZ-t£>o,
(j
1, 2,
...,
,
besides, i n the relativistic one:
(23a)
] ( r ~ > , - ,l < c ( t ~ - t~),
(~ = 1, 2, ..., v; s = a, b ) .
B e f o r e all, in r e l a t i o n (23a), t h e sign (( e q u a l )~ o b v i o u s l y refers to p h o t o n s . B u t , if also (~t a c h y o n s ~ (5-16) existed, t h e m a c r o c a u s a l i t y principle for f a s t e r - t h a n (3) S. S. SC~W]~BER: A n Introduction to Relativistic Quantum l~ield Theory (Evanston, Ill., 1961). (4) C. L. HAMMER and T. A. W]~B~: Journ. Math. Phys., 8, 494 (1967). (~) 0. M. P. BILANIU~:, V. K. DESm'AZ~D~ and E. C. G. SUDA~S~AN: Am. Journ. Phys., 30, 718 (1962). (6) YA. 1~. TERLETSKY: Paradoxes in the Theory o] Relativity (New York, 1968). (7) G. FEINBERO: Phys. Rev., 159, 1089 (1967). See also: S. TANAKA: Progv. Theor. Phys., 24, 171 (1960). (a) R. G. NEWTON: Phys. Rev., 162, 1274 (1967). (D) M. E. A~oNs and E. C. G. SUDARSHAN: Phys. Rev., 173, 1622 (1968). (1o) J. DHAR and E. C. G. SUI)ARSHAN: Phys. Rev., 174, 1808 (1968).
8~2
V. S. O L K H O V S K Y a n d
E. RECAMI
light particles (*) (1~.~3) ought to be expressed as (235)
I(rj>~-- ~l > e(t~ - - t:).
B u t here we want to deal with another question, and it is necessary to open a large parenthesis. (*) One possibility for their existence may be perhaps seen in the following (~working hypothesis >>. I t is known that t a c h y o n s - - i f they e x i s t ~ h a v e a pure imaginary conventional >. I t is also well known that in the field of elementary-particle strong interactions (at high energies) virtual particles--which are usually supposed to be exchanged in
