racah time reversal and the k 20 riddle
TRANSCRIPT
LETTERE AL NUOVO CIMENTO VOL. 28, N. 7 14 Giugno 1980
Racah Time Reversal and the K2 ~ Riddle.
O. COSTA DE BEAUR:EGARD
I n s t i t u t H e n r i Poincard - 11 rue P. et M. Curie, 75005 P a r i s
(rieevuto il 17 Aprile 1980)
The Raeah (1) time reversal operation T has a pr ior i two advantages over the Wigner (3) motion operation 3". First, being a reversal of the time axis t, i t is homo- geneous to the P - ~ ~ reversM of the space axes x, y, z, so tha t the P T reversal is geometrically covariant (whereas the ~ .~ one is not). For example it reverses all four components of a momentum-energy k ~ (whereas the ~ operation reverses k, bu t not k4); units such that e----- 1 and h = 1 are used throughout, i , j , k , 1---- 1 , 2 , 3 , 4 ; x 4 - - i t . Second, the T operation is, like P, unitary, with eigenvalues =E 1.
We then def ine the C ( # ~) uni tary operation by C =- -PT or C P T ~ I, and show that the C, P, T scheme is a straightforward interpretation of the Feynman zigzag scheme. Considered as passive, a P T symmetry reverses all the Feynman arrows so that, in order to restore the original situation, the signposts must be reversed, which means exchanging particles and antiparticles.
So the C P T (but not ~ Y - ' ) invariance is a straightforward 4-dimensional genera- lization of the 3-dimensional ~ invariance of most weak interactions. The detailed balance theorem
(1) A + B + . . . ~ C + D + . . . . ~ + D + . . . ~ X + B + .. . .
which follows (of course) from both the C P T and the ~ 5 invariance, is interpreted in the C P T scheme exactly like the ~ invariance of weak interactions.
If by definition we require that a spinning wave equation
(2)
be C P T invariant , the C operation can obviously be def ined as changing the sign of the mass m, bu t not of the electric (or any other) charge e.
Thus, if A ~ 0, the C operation exchanges the labels of the two sheets of the mass shell (which is again consistent with Feynman 's interpretation of antiparticles).
(:) G. RAC~: Nuovo Ufmento, 14, 322 (1927). (8) E. WIt~NER: GStt. Nachr., 31, 546 (1932).
237
238 O. COSTA ]DE BEAU:REGARD
In t he Dirac case, as is well known,
(3)
where by definit ion 7 sj'' ~ y~rJ", if all indices are different. N o w we show t h a t the (~ K~ r iddle )) is much less upse t t ing in the C P T scheme
t h a n i t is in the ~ # 3 - one. Being ant ipart icles to each other, the K ~ and I~ ~ are P T
associated. Therefore
(4) V/2 K~ ~ K ~ + K ~ , V/2 K~ ~ K ~ ./~o,
are by de] in i t ion P T eigenstates with, respect ively, e igenvalues + 1 and - - 1 , so t h a t i t is qu i te possible t h a t there are C P violat ions in the K~ decay.
Being spin-0 particles, the K ~ and ]~o obey (in the 4-frequency representat ion) e i ther
the (~ t rue ~) scalar and vec tor equat ions
(5) b A ~ ~ k i A , k i A i := - - k A ,
or the (( pseudo ~) scalar and vec tor equat ions
circ
W e assume , for solving the K~ riddle, tha t the K ~ and ]~o wave funct ions comprise all the four types of components A, A ~, A E12~, A tjk~j, whence, by ( C P T invar ian t ) de- f ini t ion
(7) K ~ = KO ~ ~A[1234]/ A [1234]
and ( P T = + I and P T = - 1 eigenstates)
(8) 'il, \AEJ~J/
I n th is t heo ry there is one ad jus tab le pa rame te r ~ = • Atl*34J/A. As a p r io r i the P and T symmet r i e s are noncovar ian t , we discuss first t he K~
p rob lem in the rest f rame, which by definition is such t h a t k = 0. I n it, the only non- zero componen t s of the w a v e func t ion are A [laa], w i th par i t ies P = - - 1 and T ---- + I, and A a, w i t h pari t ies P = + 1 and T = - - 1. Therefore i t is the (large) A E12aj com- ponen t t h a t decays into three u 's , and the (small) A 4 componen t which decays in to tWO ~ ' S .
RAeAI~ TIMn R~V~RSAL AND THE K~ RIDDLE 239
Then going back to the general f rame, we s ta te covar ian t ly tha t i t is the A T M
contr ibut ion which decays into three 7:'s, and the A t cont r ibut ion which decays in to two T:'s. A n d of course this holds in bo th the CPT and the ~ 3 - schemes.
I t remains to be verified if by wr i t ing down in te rac t ion Lagrangians (comprising both t rue and pseudoscalar contr ibut ions) the whole K phenomenology can be repro- duced.
Remark. Afte r WATANABE (3), JAuCH and ROHRLICH (4) have shown tha t the CPT scheme can be used consistently. The lat ter , however , h a v e discarded it, s ta t ing t h a t in i t (~ the to ta l charge is no t an absolute invar ian t under t ime reversal ,. Bu t this s tems f rom: i) an inappropr ia te definit ion of the charge, ii) an inappropr ia te definition of par t ic le -ant ipar t ic le exchange (changing the sign of the charges and not of the rest m a s s ) .
The t ru ly covar ian t definition of the charge is
(9) (4 !) 8 iik~ dQ : ~idlxJd2xkdax~,
where e ijk~ denotes the Levi -Civ i ta object and ji (4-current), dl xj, d2 xk, dax ~ (line elements) four l inear ly independent 4-vectors. Thus defined Q is a t rue scalar (like e in eq. (2)). I n o ther words JAUCH and ROHRLICH have failed to notice t h a t under t ime reversal both ]4 and e 1234 change sign.
(a) S. WATANABE" Phys. Rev., 84, 1008 (1951). (4) J . M . JAUCH and F. ROHRLICH: The Theory o] Photons and Electrons (Read ing , Mass., 1955), p. 88.