IL NUOVO CIMENTO VOL 33 A, N. 2 21 Maggio 1976

Parity Current and Parity Representation (*).

D . M . F~ADKI~ ~

Department o/ .Physics, Wayne State University - Detroit, Mich. 48202

(ricevuto il 29 Gennaio 1976)

S u m m a r y . - - We introduce the concept of par i ty current and relate its derived ~(charge ~) to an explicit representat ion of the par i ty trans- formation (in terms of field variables) in an analogous way that the (( charge ~) of a current arising from a continuous transformation of the fields is related to an explicit representation of a continuous transformation.

I t is wel l k n o w n (1) t h a t , in t h e c o n t e x t of L a g r a n g i a n q u a n t u m field t h e o r y ,

a l oca l ly v a r y i n g in f in i t e s imul t r u n s f o r m a t i o n of t h e f ields def ines an a s s o c i a t e d

c u r r e n t a n d f u r t h e r m o r e t h u t th i s c u r r e n t is d ive rgence le s s if t h e L a g r a n g i a n

d e n s i t y is i n v a r i a n t to a c o r r e s p o n d i n g c o n s t u n t i n f i n i t e s ima l t r a n s f o r n m t i o n .

F o r c o n t i n u o u s t r a n s f o r m a t i o n s , c o n s e r v e d c u r r e n t s t h u s ref lec t t h e sym-

m e t r y p r o p e r t i e s of t h e L u g r a n g i a n , a n d i n d e e d for ~ l ineur t r a n s f o r m a t i o n

t h e c o n s t a n t d e r i v e d (~ charge ~) m a y be used to c o n s t r u c t a n exp l i c i t r ep re sen -

t a t i o n of t h e t r u n s f o r m u t i o n in t e r m s of field va r i a b l e s . A d i sc re t e t r a n s f o r -

m a t i o n such as p a r i t y c a n n o t be t r e a t e d w i t h a s imi l a r f o r m a l i s m to o b t a i n an

a s soc in t ed c u r r e n t . ~Nevertheless, t h e i n v a r i a n c e of a. t h e o r y u n d e r p a r i t y is

c o n n e c t e d w i th a c o n s e r v e d q u a n t i t y , a n d an exp l i c i t r e p r e s e n t a t i o n of p a r i t y

in t e r m s of field v a r i a b l e s does ex i s t (~). One m a y ques t i on w h e t h e r a n as-

(*) To speed up publication, the author of this paper has agreed to not receive the proofs for correction. (1) Cf. S. L. ADLER and R. P. DASH~'~: Curre~t Algebras (New York, N .Y. , 1968), p. 15. (~) P. G. F]~DERBUS~I and M. T. GRISARU: N~OVO Cimento, 9, 890 (1958).

233

234 D.M. FRADKIN

sociated conserved current exists (3), and if so, wha t is the connection be tween its <~ charge )) and an explicit representa t ion of par i ty . We answer these questions in this note.

For concreteness, let us consider a free Dirac field sat isfying the usual

canonical an t i commuta t ion relations. (One m a y easily ex tend these a rguments

to free Bose fields, and also b y s tandard methods (4) to the case of in terac t ing fields.) A continuous linear t r ans fo rmat ion m a y be wri t ten

(1) T ' ( x ) = exp [-- iS(x)] T ( x ) ,

and leads b y the usual infinitesimal methods to a current whose associated

(~ charge ~ is

T ( y ) d~y .

Here S(y) is a ma t r ix possibly containing differential operators in space (but

not in t ime) t h a t is independent of fields, i.e. a c-number. For equal t imes

we use the ident i ty

(3) exp [i$f] ~ ( x ) e x p [-- i5 P] =-

= T(x) + i[~f, T(x)] + ... + ~ .

and the field an t i commuta t ion relat ion to show tha t exp [i~q~] is an explicit representa t ion of the t ransformat ion , i.e.

(4) exp [igf] T(x) exp [ - i ~ ] =- exp [ - iS(x)] T ( x ) .

Similar algebraic reasoning shows tha t , if one s tar ts with a quan t i ty

(5) 0~ = / T + ( - y , t) Q ( y ) T ( y , t) dSy ,

where the c-number m a t r i x Q(y) is even in y (,~nd not differential in t),

(6) exp [i-~] T ( x , t) exp [-- i~] = {cos Q(x)} T ( x , t) - i {sin Q(x)} T ( - x, t) .

~ o w let us tu rn our a t t en t ion to par i ty . The invar iance of the Dirac equa-

(z) Some authors, cf. J. BERNSTEIN: Elementary .Particles and Their Currents (San Francisco, Cal., 1968), 1 o. 12, statc that the discrete conserved quantities cannot be related to currents. However, if one expands the usual definition of a current to include not only four-vector currents but also nonlocal tensor currents, then (as we shall show) a conserved current in this scnsc can be constructed. (~) Cf. J. D. BJORKEN and S. D. D~ELL: Relativistic Quantum Fields (New York, N.Y., 1965), 10. 111.

PARITY CURRENT AND PARITY R E P R E S E N T A T I O N 2 3 5

t ion and the an t i commuta t ion relat ions to space reflection imply (5) t h a t

(7) ~{T~(- x, t)y# T(x, t)} = o ,

as m a y be verified b y explicit use of the equat ions of motion. We m a y call the quan t i ty in bracke ts { } the pa r i t y current . A priori, one would not expect it to be a t rue four-vector (8) and one readily sees t ha t it ac tual ly t ransforms

like the 4#-th componen t of a tensor. I n any Lorentz f rame the pa r i ty current is uniquely prescr ibed b y eq. (7) and leads to a conservat ion law.

The conserved (( charge ~) derived f rom this current is

(8) P = f T*(-- x, t) 74 kP(x, t) d3x.

F rom it is const ructed the pa r i t y operator

(9) ~ = exp [-- ½ i~P],

which by eq. (6) has the desired p rope r ty

(10)

In te rms of a normal-ordered plane-wave decomposit ion, P m a y be wr i t ten

(11) P = ~ d 3 p {b~(-V, ~)b(v, ~ )+ X(-V, ~)a(v, ~)}.

The opera tor given here is simpler t h a n t ha t obta ined by FEDEI~BUSH and GRISARU (2) because we have chosen the phase factor giving i instead of 1 in eq. (10). This choice relates P direct ly to a conserved charge ins tead of to a conserved charge plus the to ta l num be r opera tor for particles and antipart icles.

As expected, the in teract ion of the pa r i ty opera tor with the inhomogeneous Lorentz group of t rans format ions is

(12)

.~U(O, a ) ~ -x = U(O, -- a) = U - I ( O , a ) ,

~u(o , a,) 2-~ = V(o, a,),

~:~U(Asy . . . . . ta t lon ' O) ~ - 1 ~-- u(AsI~ . . . . . tation, 0 ) 9

~U(A~oo~, 0 ) ~ - 1 = U-~(Aboo~, 0),

where the U's are const ructed f rom their associated currents as indicated in eqs. (2) and (4).

(5) D. M. FRADKIN: Journ. Math. Phys., 6, 880 (1965). (8) See eq. (12) for the effect of parity on the inhomogeneous Lorentz group.

236 D . M . FRADKIN

• R I A S S U N T 0 (*)

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zione espl ic i ta di una t ras fo rmaz ione cont inua .

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