n-dimensional space M~, is equivalent to an appropriate 180 o rotation .... The internal space reflection ~ can be defined as the 180 rotation in J1 s of the.
LETTEI%E AL NUOVO CII~IENTO
VOL. 34, N. 12
17 Luglio 1982
Charge C o n j u g a t i o n a n d I n t e r n a l S p a c e - T i m e S y m m e t r i e s (*). M. ~)AVSIC (**) and E. RLcA~I (***) Istituto d i .Fisiea dell' U n i v e r s i t ~ - C a t a n i a , I t a l i a Istituto , N a z i o n a l e d i F i s i c a 5"ucleare - S e z i o n e di C a t a n i a , I t a l i a
(ricevuto il 29 Aprile 1982)
S u m m a r y . - We adopt the relativistic framework in which fundamental particles arc regarded as extended objects. Then, we show that the (geometrical) operation which reflects the i n t e r n a l space-time of a particle is equivalent to the operation C which invertes the sign of all its additive charges.
I n the present letter we critically comment on the discrete transformations of Minkowski space-time, namely on the effects of space-reflection ~ and time-reversal J , by exploiting some results contained in previous papers (~-3). Our aim is to show the connection between the i n t e r n a l discrete transformations (e) and the charge conjugation operator C. We assume fundamental particles to be e x t e n d e d objects, as m a n y theoretical observations suggcst to be the case, at least in relativistic theories (a). First of all, let us recall t h a t an i ~ v e r s i o n JADe(n) of the axes x A, x B, xc, ..., in a n-dimensional space M~, is equivalent to an appropriate 180 o rotation (***) ~ A B c ( m )
(*) W o r k p a r t i a l l y s u p p o r t e d b y C . N . R . a n d M . P . I . (**) O n l e a v e of a b s e n c e f r o m J . S t e f a n I n s t i t u t e , E. K a r d e l j U n i v e r s i t y , L j u b l j a n a , Y u g o s l a v i a (permanent address). (***) A l s o : C . S . F . N . e S.d.M., C a t a n i a , I t a l i a . (1) M. PAynim: Obz. M a t . F i z . , 19, 299 (1975); E. RECAMI a n d R . MIGNANI: R i v . N u o v o Cimento, 4, 209 (1974); 1%. MmXANI a n d E. RECAMI: Left. N u o v o Cimenlo, 11, 421 (1974); N u o v o Cimento A , 24, 438 (1974); I n t . J . Theor. P h y s . , 12, 299 (1975); E. REOAMI a n d G. ZIINO: N u o v o Cimento A , 33, 205 (1976). (2) M. PAvgIO~: I n t . J . Theor. P h y s . , 9, 229 (1974). (3) E. RECA~II a n d ~V. A. RODI%IGUES: F o u n d . P h y s . , 12, 709 (1982); E . REGAMI: ill •. E i n stein 1879-1979: Relativity, Quanta and Cosmology, e d i t e d b y F. DE FINIS a n d M. PANTALEO, VO1. 2 ( N e w Y o r k , N. Y . , 1979), p. 537 ; P . CALDIROL,~ ~ucl E. RECAMI: in I t a l i a n Studies i n the P h i l o s o p h y of Science, e d i t e d b y M. DALLA CInARA ( B o s t o n , Mass., 1980), p. 249; E. I~ECAMI: F o u n d . P h y s . , 8, 329 (1978). (4) See e . g . A . J . K i L ~ A u a n d B. P. TOLEDO: N u o v o Cimento, 48, 997 (1967); A. ft. K.~LNAY: P h y s . Rev. D , 7, 1707 (1973); V. S. OLKHONSKY a n d E. RECXMI: Lett. N u o v o Cimento, 4, 1165 (1970); E . REOAMI: in Proceedings of the 1981 E r i e e I n t e r n a t i o n a l School on (, Quarks and N u c l e i ,), e d i t e d b y D. ~VILKINSON (to a p p e a r ) ; P . CALDIROLA, M. PAV~IC a n d E. RECAMI: NUOVO Cimento B , 48, 205 (1978); P h y s . Left. A , 66, 9 (1978); Lett. N u o v o Cimcnto, 24, 565 (1979); P . CALDIROLA: R i v . N u o v o Cimenlo 2, No. 13 (1979). (***) W h e n M is M i n k o w s k i a n , * r o t a t i o n * will m e a n p s e u d o - r o t a t i o n .
357
358
~ . I'AV~I6 a n d E. I~xcA~I
in t h e h y p e r p l a n e (x ~ , x B , x c . . . . . x ~) of t h e m - d i m e n s i o n a l s p a c e Mm w i t h m > ~ n . I f t h e n u m b e r k of t h e i n v e r t e d a x e s x ~, x B, x c . . . . . is e v e n , t h e n it m a y b e m = n ; b u t if k is o d d , t h e n m ~ > n + 1. I n p a r t i c u l a r , t h e t o t a l i n v e r s i o n (of all axes) in a n - d i m e n s i o n a l s p a c e E~ c o r r e s p o n d s t o a r o t a t i o n e i t h e r in E~ (if n is even) or in a ( n - 9 1 ) - d i m e n s i o n a l s p a c e E~+~ (if n - 9 1 is even). F o r i n s t a n c e , in t h e 2 - d i m e n s i o n a I p l a n e t h e effect of t h e i n v e r s i o n S , ( 2 ) (i.e. x-*--x, w h i l s t y---*y) is e q u i v a l e n t t o t h e effect of t h e 180 ~ r o t a t i o n ~,~(3) in t h r e e dimensions around the y-axis: (1)
J~(2)=~(3)]E
~,
w h e r e t h e s u b s c r i p t E~ m e a n s t h a t eq. (1) is t r u e as f a r as we confine o u r s e l v e s to t h e effect of its r.h.s, i n t o t h e i n i t i a l 2 - d i m e n s i o n a l space. I n M i n k o w s k i s p a c e t h e r e are t h e following d i s c r e t e t r a n s f o r m a t i o n s (*) (we a d o p t t h e n o t a t i o n x ~ (x ~ x 1, x 2, x 3) ~ (t, x, y, z)): s p a c e reflection:
Jl(4)
time reversal:
Jo(4) ~ J -
~
( i n v e r s i o n of x l ) ; ( i n v e r s i o n of x ~
T h e p r o d u c t ~ ' 3 - = y - . . ~ is e q u i v a l e n t t o t h e 180 ~ ( p s e u d o ) r o t a t i o n in 3/4: (2)
3-,~ ~ Je(4) ,.r
= Y'od4) = ~o1(4).
T h o u g h t h e p r o d u c t ~ f can b e c o n s i d e r e d as a r o t a t i o n in M 4, of c o u r s e n e i t h e r n o r 3- a l o n e c a n b e r e p l a c e d b y a n y r o t a t i o n i n M4. H o w e v e r , if i n s t e a d of t h e 4 - d i m e n s i o n a l s p a c e M 4 w e c o n s i d e r t h e 5 - d i m e n s i o n a l space Ms, so t h a t a n e v e n t e is d e s c r i b e d b y t h e five c o - o r d i n a t e s e:
x "4 ~
( x ~ x 1, x 2, x 3, x 4) ,
eeMs,
t h e n t h e effect i n 21I4 of t h e r e f l e c t i o n ~ is e q u a l to t h e effect in M 5 of t h e 180 ~ r o t a t i o n a r o u n d t h e s p a c e (x o, x 2, x3), i.e. of t h e 180 ~ r o t a t i o n of M 5 , four-momentum spacc, etc. (This means that T, in particular, when acting on a four-momentum vector, will change also the sign of energy.) But let us go back to the mere c.hronotopical space. Now, if we apply P T = - - 1 from the active point of view to thc world-tube W in fig. 3a), we have to rotate it (by 180 ~ in four dimensions) into ITV(fig. 3b)). Such a ro~ation wilt effect also a reflection of the internal 3-space of a particle a, transforming i t - - a m o n g the others--into its mirror image. Analogously, from the passive point of view, if we apply P T to the space-time in fig. 3a), containing also W, we shall pass to a _PT-ed frame whose space-time derives from the complcte 180 ~ (~rotation ~) of the internal space-time. Again, this will operate also the reflection of the internal spacetime of particle a (relatively to the new observer). Then, wc extend the Reinterpretation Principle (1-3) to thc case of extended objects, i.e. we apply it (e.g. within the active point of view) to the world-tube W of fig. 3b). The world-tube W represents an (internally reflected) particle not only going backwards i n time, but also carrying negative energy. Therefore applying the reinterpretation principle (s) will rigorously transform VV into W (fig. 3e)), the anti-world-tube W representing the antiparticle ~. (*) Cf. eq. (11) i n t h e f o l l o w i n g . (r A n e w f o r m a l i z a t i o n of t h i s p r i n c i p l e (~, R I P ~)) h a s b e e n v e r y r e c e n t l y g i v e n b y C. SCHWARTZ: P h y s . Rev. D, 2 5 , 356 (1982).
362
M. I~Av~I~ and E. RECAMI
b)
a)
c)
Fig. 3. - Given a world-tube (a)), we show the effect of the (antichronous) Lorentz transformation ~ = --1 before (b)) and after (c)) the application of the (, reinterpretation principle ~ (~.2,5). See the text.
I n conclusion, as far as t h e c h r o n o t o p i c a l space is concerned, the (antichronous) Lorentz transformation PT~--1 can be considered as (10)
-- 1 ~
PT
= ~E3-~I3-I
= PT~I3-
I ,
so that in p a r t i c u l a r (10')
PT
= ~3-.
A t t h i s p o i n t we h a v e to recall t h a t in ref. (1,3) we s h o w e d - - b y t a k i n g account also of t h e f o u r - m o m e n t u m space and b y a p p l y i n g t h e --that (11)
PT
= CPT
,
where C represents t h e c o n j u g a t i o n of a l l t h e a d d i t i v e charges (1,3). L e t us add, going b a c k to eq. (9), t h a t all k n o w n (relativistic) e q u a t i o n s and (relativistic) i n t e r a c t i o n s are a c t u a l l y C P T - c o v a r i a n t . F r o m eqs. (10), (11) it is i m m e d i a t e to derive t h a t (12)
~i3-x=
~-i~i = C .
W e h a v e t h u s shown the (geometrical) o p e r a t i o n of reflecting t h e i n t e r n a l spacet i m e of t h e considered p a r t i c l e to be e q u i v a l e n t to the o p e r a t i o n C which i n v e r t s t h e sign of all its a d d i t i v e charges. W e h a v e also seen t h a t t h e i n t e r n a l transforlnations ~ , 3-~ do change t h e p a r t i c l e i n t r i n s i c state. If we convene to w r i t e N i a + + = a+_; 3 - i a + + = a_+, t h c n (7) (12')
~i3-ia++ = a__,
w h e r e t h e subscripts d e n o t e t h e i n t e r n a l p a r a m e t e r s t h a t t r a n s f o r m u n d e r t h e action of ~ and ~ , r e s p e c t i v e l y ; and w h e r e a__ represents t h e intrinsic ( = internal) s t a t e of t h e a n t i p a r t i c l e ~. All w h a t precedes can be applied also w i t h i n t h e r e a l m of q u a n t u m theories. B u t let us h e r e conclude b y e m p h a s i z i n g t h a t - - i n our opinion, and for t h e results in this p a p e r and in ref. (1"3)--we should a d v a n t a g e o u s l y s u b s t i t u t e in t h e o r e t i c a l physics t h e new operations P ~ ~ and T ~ 3 - for t h e o r d i n a r y operations P , T, w h i c h are m e r e l y e x t e r n a l reflections (e.g., o n l y t h e f o r m e r do belong to t h e full L o r e n t z group). (7) M. PAynim2: M i r r o r particles a n d p a r i t y conservations, Report, University of Ljubljana (1976), unpublished.
LYETT:ERE AL NUOVO CI~{:ENT0
VOL. 35, N. 10
6 N o v e m b r e 1982
Charge Conjugation and Internal Space-Time Symmetries. iV[. PAv~Id a n d E . Rv, cA~I
I s t i t u t o di _Fisica dell' Universith - Catania, I t a l i a I s t i t u t o 2Vazionale d i F i s i c a 2Vucleare - Sezione d i Catania, I t a l i a (Lett. N u o v o Cimenta, 34, 357 (1982))
On p. 361, line 7, s u b s t i t u t e t h e w o r d s h e a r w i t h s h e a f ; on t h e s a m e p a g e , line 8 from the b o t t o m , s u b s t i t u t e the word internal with initial.
by Societk Italiana di Fisica Proprietk letteraria riservata Direttore responsabile: RENATO ANGELO RICGI Stampato in Bologna dalla Tipografla Gompositori col tipi della Tipografla Monograf Questo fascicolo ~ stato licenziato dai torchi il 2-XI-1982 352
