racah time reversal and the k 20 riddle

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LETTERE AL NUOVO CIMENTO VOL. 28, N. 7 14 Giugno 1980 Racah Time Reversal and the K2 ~ Riddle. O. COSTA DE BEAUR:EGARD Institut Henri Poincard - 11 rue P. et M. Curie, 75005 Paris (rieevuto il 17 Aprile 1980) The Raeah (1) time reversal operation T has a priori two advantages over the Wigner (3) motion operation 3". First, being a reversal of the time axis t, it is homo- geneous to the P-~ ~ reversM of the space axes x, y, z, so that the PT 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, but not k4); units such that e----- 1 and h= 1 are used throughout, i,j,k, 1---- 1,2,3,4; x 4--it. Second, the T operation is, like P, unitary, with eigenvalues =E 1. We then define the C (# ~) unitary operation by C =--PT or CPT ~ I, and show that the C, P, T scheme is a straightforward interpretation of the Feynman zigzag scheme. Considered as passive, a PT 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 CPT (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 CPT and the ~5 invariance, is interpreted in the CPT scheme exactly like the ~ invariance of weak interactions. If by definition we require that a spinning wave equation (2) be CPT invariant, the C operation can obviously be defined as changing the sign of the mass m, but 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

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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.