REPRESENTATIONS OF DISCRETE SYMMETRY I N QUANTUM F I E L D OPERATORS THEORY by K I S H T N H. MARIWALLA B.Sc, The U n i v e r s i t y o f Bombay, 1 9 5 6 A T H E S I S SUBMITTED I N P A R T I A L F U L F I L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF S C I E N C E i n t h e Department of PHYSICS We a c c e p t t h i s t h e s i s a s c o n f o r m i n g required to the standard THE U N I V E R S I T Y OF B R I T I S H COLUMBIA April, 1961 In presenting the this r e q u i r e m e n t s f o r an thesis in partial advanced degree a t of B r i t i s h Columbia, I agree that it freely agree that for available the f o r r e f e r e n c e and permission f o r extensive s c h o l a r l y p u r p o s e s may D e p a r t m e n t o r by be gain s h a l l not be a l l o w e d w i t h o u t my Department The U n i v e r s i t y o f B r i t i s h V a n c o u v e r $, C a n a d a . Date U^.l t'iU shall study. I Columbia the of University copying of his representatives. copying or p u b l i c a t i o n of t h i s the Library g r a n t e d by that fulfilment make further this Head o f thesis my It i s understood thesis for written financial permission. F A C U L T Y OF G R A D U A T E STUDIES PROGRAMME OF T H E FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY KISHIN HARIRAM MARIWALLA B.Sc. (Hons.) Bombay, 1956 M.Sc. (Hons.) Bombay, 1958 FRIDAY, APRIL 28, 1961 AT 4:00 P.M. IN ROOM 303, PHYSICS BUILDING IN CHARGE Chairman: G. M. SHRUM F. R. J. L. A. KAEMPFFER BARRIE GRINDLAY de SOBRINO o R A L FOR, of COMMITTEE HlSlEA&IWG) P. HARNETTY E. LEIMANIS C. A. SWANSON P. RASTALL External Examiner: L. E. H. TRAINOR University of Alberta, Edmonton, Aha. e X A HIS M REPRESENTATIONS O F DISCRETE S Y M M E T R Y OPERATORS I N Q U A N T U M T H E O R Y O F FIELDS ABSTRACT The object of the work reported in (his thesis was to construct and study the explicit representations of discrete symmetry operators (D.S.O.'s) in quantum field theory. In spite of the considerable importance of the D.S.O.'s in present day physics, not much has been reported in the systematic study of such representations. Furthermore, in the work reported hitherto, only incomplete representations of the operators of space inversion (fl ) particle conjugation (r~) and time reversal (T) have been given. Starting from general consideraiions on invariance principles and infinitesimal transformations with the associated conservation laws, a systematic procedure for constructing the representations of the D.S.O.'s has been formulated. T h e procedure consists in enumerating the bilinears in creation and annihilation operators. It is shown that eight symmetries are the only possible ones. In view of the T C P - theorem and the so-called non-conservation of parity in weak interactions, the product operators, such as reflection ( A = l~l D and strong reflection (S = r! [~ T ) , in addition to time reversal, should be considered as the most basic symmetries. Working in the linear momentum representation, the unitary operators A 11 ! > P > E ( = identity) and the unitary factors of the antiunitary operators: S, I = H T , J = T T and T are constructed for the following free fields: - (i) T h e non-hermitian scalar field representing, for example, kaons. (ii) T h e electromagnetic field. (iii) T h e four-component spinor field. The operators for the scalar field have also been worked out • in the angular momentum representation. Using the anti-commutation relation for C.O.'s and A.O.'s an alternate construction of D.S.O.'s of the Dirac field is exhibited. More than one representation has been given in each case. In addition a two dimensional matrix representation has been given. It is shown that by an appropriate unitary transformation these can be reduced to the ordinary form. G R A D U A T E STUDIES Field of Study: Quantum Field Theory Advanced Quantum Mechanics Special Relativity Theory Electromagetic Theory Nuclear Physics F. A . Kaempffer p. Rastall G . M . Volkoff J . B. Warren Related Studies: Dynamical Systems I E . Leimanis Differential Equations C. A . Swanson Functions of Complex Variables A . H . Cayford ii ABSTRACT The o b j e c t o f t h e w o r k r e p o r t e d c o n s t r u c t and s t u d y symmetry o p e r a t o r s the e x p l i c i t t h e s i s was t o representations of discrete ( D . 3 , 0 , ' s ) i n quantum f i e l d s p i t e o f the c o n s i d e r a b l e present i n this theory. i m p o r t a n c e o f t h e D.S.O.'s i n day p h y s i c s , n o t much h a s b e e n r e p o r t e d systematic I n the o f such r e p r e s e n t a t i o n s . Furthermore, i n t h e v/ork r e p o r t e d h i t h e r t o only incomplete representations for the operators o f s p a c e i n v e r s i o n ( \\ ) particle jugation study In ( \~~ ) and t i m e reversal ( X ) have b e e n congiven. S t a r t i n g from g e n e r a l c o n s i d e r a t i o n s on i n v a r i a n c e p r i n c i p l e s and i n f i n i t e s i m a l associated conservation t r a n s f o r m a t i o n s , w i t h the laws, a systematic procedure f o r c o n s t r u c t i n g t h e r e p r e s e n t a t i o n s o f t h e D.S.G.'s h a s b e e n formulated. bilinears The p r o c e d u r e c o n s i s t s i n e n u m e r a t i n g t h e i n c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s (C.OJstand: A O i ) . I t i s shown t h a t e i g h t s y m m e t r i e s a r e t h e o n l y p o s s i b l e o n e s . TCP-theorem I n view o f the conservation of parity i n weak i n t e r a c t i o n s , operators, such as r e f l e c t i o n reflection ( S should — HI be c o n s i d e r e d and t h e so c a l l e d n o n - T ^* ( /\ - FIT the product ) and s t r o n g a d d i t i o n t o time r e v e r s a l , as t h e most b a s i c symmetries. W o r k i n g i n l i n e a r momentum r e p r e s e n t a t i o n , t h e u n i t a r y operators A, Pj > r , E ( = I d e n t i t y ) and t h e u n i t a r v f a c t o r s o f the a n t i u n i t a r y "T — 1 \ , and T operators are constructed f o r the f o l l o w i n g free fields: (1) The n . o n - H e r m i t i a n s c a l a r f i e l d representing f o r example kaons. (2) The E l e c t r o m a g n e t i c (3) The f o u r - c o m p o n e n t s p i n o r The o p e r a t o r s out field.field. f o r the s c a l a r f i e l d i n the angular have a l s o b e e n w o r k e d momentum r e p r e s e n t a t i o n . Using the a n t i - c o m m u t a t i o n r e l a t i o n s f o r \ G » 0, sT)and i A , 0 . ' s , a n a l t e r n a t e 1 c o n s t r u c t i o n o f D.S.O.'s o f t h e D i r a c field More t h a n one r e p r e s e n t a t i o n h a s b e e n g i v e n I n a d d i t i o n a two d i m e n s i o n a l given., these i n each case, m a t r i x r e p r e s e n t a t i o n has been I t i s shown t h a t by a n a p p r o p r i a t e formation, i s exhibited. c a n be r e d u c e d unitary trans- to the o r d i n a r y form. iv TABLE OP CONTENTS INTRODUCTION CHAPTER 1 I GENERAL 8 CONSIDERATIONS 1° Symmetry P r i n c i p l e s 2° The D i s c r e t e Symmetry O p e r a t o r s ( D . S . O . ' s ) CHAPTER I I ' DISCRETE THE 1° 2° 3° Definitions 8 SYMMETRY SCALAR OPERATIONS 12 POR FIELD and N o t a t i o n s T r a n s f o r m a t i o n o f C r e a t i o n and A n n i h i l a t i o n O p e r a t o r s u n d e r Symmetry T r a n s f o r m a t i o n s (i) Space I n v e r s i o n (ii) Time R e v e r s a l (iii) P a r t i c l e Conjugation (iv) Strong R e f l e c t i o n '(v) Reflection (vi) Weak R e f l e c t i o n (vii) Phase T r a n s f o r m a t i o n (viii) Inversion 19 19 23 23 2LL 29 30 31 31 32 32 R e p r e s e n t a t i o n s f o r the D ^ S . Q . ' s f o r the Scalar (i) (ii) (iii) (iv) (v) 33 Field Transformation Space I n v e r s i o n and Time R e v e r s a l P a r t i c l e C o n j u g a t i o n and S t r o n g Reflection R e f l e c t i o n and Weak R e f l e c t i o n Conclusion k° Two D i m e n s i o n a l : . R e p r e s e n t a t i o n s 5° The A n g u l a r (A) (B) (a.) (b) (c) (d) Momentum, R e p r e s e n t a t i o n Notation Symmetry O p e r a t o r s Space I n v e r s i o n Time R e v e r s a l P a r t i c l e Conjugation Strong R e f l e c t i o n 3U35 if.0 I|l i\3 i4.i1 52 52 53 53 5k 57 57 V (e) (f) Reflection Weak R e f l e c t i o n 58 59 (g) Inversion 59 THE ELECTROMAGNETIC FIELD 61 CHAPTER I I I 1° Formalism 61 2° R e p r e s e n t a t i o n of D.S.O.'s 63 (A) (B) Space I n v e r s i o n and R e f l e c t i o n Time R e v e r s a l and Weak R e f l e c t i o n 63 65 (C) Other ©.S.O.'s 65 3° R e p r e s e n t a t i o n i n C i r c u l a r Components CHAPTER IV 66 THE POUR-COMPONENT SPINOR FIELD 69 1° Notation 2° 3° B i l i n e a r s of the D i r a c F i e l d T r a n s f o r m a t i o n P r o p e r t i e s of Cr Sat l o a n and A n M h i l a t i o n Operators (C.O.'s and A.O.s) Construction of Representation (i) Space I n v e r s i o n (ii) Time R e v e r s a l (iii) P a r t i c l e Conjugation (iv) Reflection (v) Weak R e f l e c t i o n (vi) Inversion 4° 69 0 (vii) Strong R e f l e c t i o n 5° One Dimensional 6° Alternate Representations BIBLIOGRAPHY Representations 71 74 77 78 79 80 81 81 82 82 83 85 90 vi ACKNOWLEDGMENTS I wish to express my g r a t i t u d e t o P r o f e s s o r Kaempffer f o r suggesting encouragement this t h e p r o b l e m and f o r P. A. continuous and g u i d a n c e t h r o u g h o u t t h e p r o g r e s s o f work. I Physics am i n d e b t e d t o t h e members o f t h e T h e o r e t i c a l G r o u p , i n p a r t i c u l a r P r o f e s s o r W. f o r h e l p f u l c r i t i c i s m and good I wish Opechowski advice. t o t h a n k many f r i e n d s , i n p a r t i c u l a r Mr. K. N i s h i k a w a and Mr. P. A„ G r i f f i n f o r c o n s i d e r a b l e i n the preparation o f t h i s I am g r a t e f u l thesis. to the N a t i o n a l Research C o u n c i l o f Canada f o r f i n a n c i a l h e l p d u r i n g I 9 6 0 and d u r i n g help t h e Summer the Winter S e s s i o n s 1960-6l Session, o INTRODUCTION The almost current literature o n quantum f i e l d theory complete l a c k o f mention o f r e p r e s e n t a t i o n s i n terms o f c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s f o r d i s c r e t e transformations. almost shows a n Indeed these e x c l u s i v e l y as symbolic symmetry t r a n s f o r m a t i o n s are employed operators. On t h e o t h e r hand r e p r e s e n t a t i o n s i n t e r m s o f c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s are g i v e n f o r the observables momentum and a n g u l a r momentum generators Such r e p r e s e n t a t i o n s g i v e p r o p e r t i e s o f the f i e l d s . The a d v a n t a g e o f h a v i n g such r e p r e s e n t a t i o n s f o r v one symmetry o p e r a t o r s i s o b v i o u s , , w a n t s t o c o n s t r u c t a many p a r t i c l e parity ^ t - P state of specific r e p r e s e n t a t i o n s f o r TT * n a n n i h i l a t i o n operators enables computation. r u l e s governing interactions of discrete s t a t e |>,the terms o f p a r t i c l e straightforward problem 1 \ G o n v e r s e l y j . f o r a g i v e n many p a r t i c l e The f o r e x a m p l e , when namely the s o l u t i o n s o f the e i g e n v a l u e TT I > mutators o f as t o the n o t i o n o f p a r t i c l e s - as c a r r i e r s o f the mechanical discrete , conceived o f u n i t a r y t r a n s f o r m a t i o n s o f d i s p l a c e m e n t s and r o t a t i o n s i n space and t i m e . substance such as e n e r g y - one t o f i n d c r e a t i o n s and the p a r i t y by F u r t h e r m o r e many selection i n v o l v e k n o w l e d g e o f com- symmetry o p e r a t o r s w i t h commutation r e l a t i o n s o f d i s c r e t e observables. symmetry operators 2- ai»e a l s o i m p o r t a n t i n connection the o b j e c t o f t h e w o r k r e p o r t e d and study w i t h t h e TCP t h e o r e m . I t ,was i n this the p r o p e r t i e s o f such r e p r e s e n t a t i o n s . In a l o c a l quantum f i e l d theory, invariance under proper L o r e n t z statistics connection, so c a l l e d (S.R.) ( s e e e . g . P a u l i 1 9 5 ^ ) . transition probabilities equal Lorentz and s p i n - one f i n d s t h e s y s t e m h a s an i n v a r i a n c e under s t r o n g r e f l e c t i o n are i f one assumes transformations a d d i t i o n a l invariance property--the reflection t h e s i s t o produce strong P h y s i c a l l y , the i m p l i e s that the f o r t h e f o l l o w i n g two processes ( G r a w e r f e t . 5 a l ; 1959> T h i s w i l l be r e f e r r e d t o a s GLR 19^9) (1) p a r t i c l e s a t )ti w i t h momentum P-o and s p i n cr^ r e a c t t o g i v e p a r t i c l e s a t X ^ , w i t h momentum Pi and s p i n CTi . (2) a n t i p a r t i c l e s a t to w i t h momentum give a n t i p a r t i c i e s a t - x ^ and s p i n — &jC w i t h momentum react P-u and - s p i n - cf~, o Thus u n d e r s t r o n g the order reflection of events i s interchanged, p a r t i c l e s and n e g a t i v e for antiparticles. i n v o l v i n g r e v e r s a l o f motion valid. called -P and Q, transformation o f S.R. and T may I t implies: -Q Reflection i s also a s t r i c t l y higher The t i m e r e v e r s a l , and i s The p r o d u c t t h e n be c a l l e d " R e f l e c t i o n " ( A )„ P i s positive f o r (T), i.e., is known t o be s t r i c t l y where symmetry i s o b t a i n e d obeyed symmetry. i f one c o n c e i v e s A still of reflection 3- as the product o f p a r t i c l e space inversion Tl Weak i n t e r a c t i o n s Still which 1959). X —> -X and b~j>-"t interactions. - W.R.) ~ ]t transforms X are not invariant another combination (GJJR.-.Tr c o n j u g a t i o n ( (""" ) where (^-^-Q^and i sHT under these •= 1 - . two s e p a r a t e l y . and i s c a l l e d I n v e r s i o n Here o n l y s p i n s c h a n g e s i g n ). and P (and o f c o u r s e T h i s a l s o d o e s n o t h o l d f o r weak one c a n haver"!" — ~J Finally (weak reflection * T h e r e a r e two t y p e s o f norm p r e s e r v i n g m a p p i n g s i n Hilbert The space, v i z . l i n e a r and a n t i - l i n e a r ma'ppingslWigijhgr symmetry o p e r a t o r s a r e t h e r e f o r e n e c e s s a r i l y anti-unitary operators: j " ^ p "J" four ( i . e .unitary but a n t i - l i n e a r ) . Identity and T = fc- , T l , I , , the f i r s t Of t h e e i g h t j nrT-5 I=nT y anti-unitary. field functions, when e x p r e s s e d operators In field theory are b i l i n e a r they take a p a r t i c u l a r l y i n terms (C.O.'s are b i l i n e a r for unitary or f o u r a r e u n i t a r y and t h e l a s t S i n c e o b s e r v a b l e s i n quantum in A = II T 1932). simple form o f c r e a t i o n and a n n i h i l a t i o n and A.O.'s) i n momentum s p a c e , where the; / r i n C.O.'s and A.O.'s. 3° o f t h e n e x t c h a p t e r a method w i l l be o u t l i n e d selecting bilinears suitable f o r c o n s t r u c t i n g the symmetry o p e r a t o r s . Some f i v e Pauli papers have so f a r been w r i t t e n on t h e s e (195#) u s e s t h e t e r m W.R. fcr H i representations; 1 o Ro Go S a c h s ( 1 9 5 2 ) ; I~I and T constructed momentum r e p r e s e n t a t i o n f o r s c a l a r 2o W o l f e n s t e i n non H e r m i t i a n 3. Watanabe scalar f i e l d , ( 1 9 5 5 ) : TI , P fields. (1952): T and R a v e n h a l l i n angular for Dirac f i e l d and u s i n g a method due t o G.C„ W i c k . , ~]~ f o r various f i e l d s using the methods o f ( 2 ) . l±. Nigam and F o l d y 5. P.A. K a e m p f f e r and £irac r , ~[ TI , I transformations l± t h e r e p r e s e n t a t i o n s f o r [~ and t l~" ^ I o) In the present = — t - Using f \ & ^> - -C J jy 0 types o f r e p r e s e n t - symmetry o p e r a t o r s are found. method e m p l o y s t h e t r i c k o f a n a d d i t i o n a l transformationo' In unfortunately , so t h a t work two d i f f e r e n t a t i o n s f o r the d i s c r e t e allowed h a s been d i s r e g a r d e d . c o n t a i n an e x t r a r e d u n d a n t f a c t o r first field, f o r complex s c a l a r 1, 2 and I4., t h e a r b i t r a r i n e s s o f p h a s e and T reference (1961); ^~ f o r ^-component s p i n o r fields. In references in (1956) « The gauge t h i s d e v i c e , more t h a n one r e p r e s e n t - a t i o n h a s b e e n f o u n d f o r e a c h D„S.Oo f o r the f o l l o w i n g fields; (1) N o n - H e r m i t i a n s c a l a r f i e l d momentum r e p r e s e n t a t i o n s i n l i n e a r and a n g u l a r (Chapter s p e c i a l case the Hermitian scalar (2) P o u r component s p i n o r f i e l d II). This i n c l u d e s as a field. f o r p a r t i c l e s o f non zero mass ( D i r a c f i e l d ) i n l i n e a r momentum r e p r e s e n t a t i o n . -5- Using Pauli's principle alternate representations f o r D.S.O.'s have b e e n e x h i b i t e d . ( C h a p t e r I V ) (3) E l e c t r o m a g n e t i c ation field i n l i n e a r momentum represent- (Chapter I I I ) . The s e c o n d method e m p l o y s a two b y two r e p r e s e n t a t i o n . It transformation i s shown t h a t b y a n a p p r o p r i a t e t h i s f o r m c a n be r e d u c e d t o g i v e t h e representations obtained In order by the f i r s t method. t o s e e how a two b y two r e p r e s e n t a t i o n may be set up, r e c a l l t h a t a symmetry o p e r a t o r ( S ) i s a mapping of a l i n e a r manifold [) C l i n e a r manifold D_ ^ H and v i c e v e r s a , s c a l a r product < D_ | ( Df and D_ ( Where, and D- obtains the together will > (H) onto space <D_JS H o f D-+ another such t h a t t h e Z < D _ | S | D > - |D ) + a r e assumed n o t t o i n t e r s e c t ) . two r e p r e s e n t a t i o n s two h a l f s p a c e s . be m u t u a l l y Hilbert c o n s t i t u t e the e n t i r e H ° i n general eigenvectors unitary L e t | ^ ^> S and S_ , v a l i d i n + and j V'-s Consider + °ne t h e r e f o r e and £)_ , r e s p e c t i v e l y . orthogonal. D be a s e t o f The two s e t s the s t a t e : Under a symmetry o p e r a t i o n , one h a s : 5 IV, r U > If S + and S_ respectlvely, =L /^>z > y where K/ »/ be t h e two r e p r e s e n t a t i o n s f o r D+ and D- then D -6- Hence e i g e n v e c t o r s i n t h e two h a l f s p a c e s may be v i e w e d a s c o m p o n e n t s i n a two d i m e n s i o n a l s p a c e . where T d e n o t e s If and Thus: the transformed eigenvector. t>_ be t h e C.O.'s and A.O.'s i n t h e two s u b - s p a c e s t h e n u n d e r a symmetry o p e r a t i o n S, one w r i t e s : ; - °) v° s The " J W ° ; ~ V • s.i,is: b f o l l o t i r i n g i s a summary o f t h e a l g e b r a i c relations used ; I. I f X i _ be a h e r m i t e a n o p e r a t o r , t h e S = £• is unitary. II. One h a s t h e i d e n t i t y ; L"^" P'3 2/ where with III. as: \S i s the second commutator etc. I f an o p e r a t o r S p s " ' - - ^ and then — L"^"'^"'^ D 7 (3> t r a n s f o r m s u n d e r a symmetry o p e r a t i o n t h e n c h o o s e £1_ s u c h t h a t C--^'] 0 where | C-°-, f ^ ] = £ ^j -^ -7- ^ K Taking ^ = ""^'t, o n e S t s the r e q u i r e d o p e r a t i o n does n o t y i e l d difficulty IV. To d e t e r m i n e - O - for 3, i n d i c a t e s t h a t the d o u b l e the I d e n t i t y . o< = Thus this S - <=>, S x t h e f o l l o w i n g i d e n t i t i e s a r e used = A O , C ] V. P o r P e r m i f i e l d s , s i n c e + CA,C]& f many o p e r a t o r one c a n c o n s t r u c t a h e r m i t i a n G e ^ ^ l + G C e ^ - i ^ l t ^ n symmetry To e l i m i n a t e a second f a c t o r i s i n t r o d u c e d . 0 & , 0 VI. - f result. Q The m i n u s s i g n f o r ot - f o r °C = -{L o r Bose products fields vanish, s u c h t h a t G^= G . Then for G - - % Use h a s a l s o b e e n made o f H a u s d o r f f ' s theorem: . . CHAPTER I GENERAL CONSIDERATIONS 1°. Symmetry The Principles l a w s o f P h y s i c s a r e i n g e n e r a l g o v e r n e d b y two m a i n types o f p r i n c i p l e s , v i z . (1) t h e d y n a m i c a l describe (2) equations the dynamical principles, behaviour d e s c r i b i n g the system may a r i s e discussed o f t h e s y s t e m , and with respect to c e r t a i n The i n v a r i a n c e p r o p e r t i e s o f t h e i n the form o f d i s c r e t e i n the next t r a n s f o r m a t i o n s such as d i s p l a c e m e n t s t h e phase i n v a r i a n c e . this generator of displacements 1.1.2 and In Hamiltonian formu- i n terms o f P o i s s o n b r a c k e t s a s : 5 R - € { R,G| Thus,using i n space and t i m e , t h e t o a c o n s e r v a t i o n l a w and t h e an i n f i n i t e s i m a l change i n £ variable. group o f f o r a uniform motion corresponding constant o f the motion. l a t i o n one may e x p r e s s t o be E a c h i n v a r i a n c e u n d e r an i n f i n i t e s i m a l transformation gives rise is symmetries, s e c t i o n , o r the continuous i r r e l e v a n c e o f the s t a t e o f motion 1.1.1. which the i n v a r i a n c e p r i n c i p l e s - - i n v a r i a n c e o f the changes i n the v a r i a b l e s . system i . e . the equations = , in £ ^G where i s then said b e i n g any 6 t o be a dynamical familiar notation; S H = S t i H" , Gl J - -St ^2 dfc expresses that _ 9 the H a m i l t o n i a n (H) i s a g e n e r a t o r o f d i s p l a c e m e n t s i n time. Similarly, f S P = STL [_ P, G } 1.1.3 [ L,G\ linear of = and = -Se b& e x p and a n g u l a r momenta a r e r e s p e c t i v e l y the l i n e a r successively and a n g u l a r d i s p l a c e m e n t s . 6- e q u a l t o P , and L , r Q s s t h a t t h Q the generators I f one £ H puts vanishes, i f these a r e c o n s t a n t s o f the motion. The above c o n s i d e r a t i o n s c a n be e x t e n d e d mechanics., set t o quantum I n . quantum m e c h a n i c s one ;may._consider o f c o m m u t i n g H e r m i t i a n o p e r a t o r s o( , w h i c h taneous e i g e n v e c t o r s |«C)> . transformation I o 1 o4. oL = U. 0^ U. have U n d e r an a r b i t r a r y LL > t h e t r a n s f o r m e d a complete simul- unitary Hermitian operators, have a n i d e n t i c a l eigenvalue spectrum, i . e . where same s e t o f e i g e n v a l u e s . 1.1.6 1.1.7 (Jl /S T and |3 hav« t h e I f LL I s i n f i n i t e s i m a l , ^ where £ / 3 = - X L . P \ ^3 £L = Q- r and . F o r a n i n f i n i t e s i m a l c o o r d i n a t e t r a n s f o r m a t i o n , one o b t a i n s , writing X-^> and -CL. = P , -10- Xl 1.1.8 - 3 ^ - << • Momentum b e i n g t h e g e n e r a t o r o f d i s p l a c e m e n t s i n c o o r d i n a t e s , it follows that H ments displace- and, SH 1.1.9 In i s i n v a r i a n t under c o o r d i n a t e general If.H = * {. H , P} i s invariant = o [H,P] - u n d e r t h e change o f variables CO » w h i c h has f o r i t s g e n e r a t o r , t h e o b s e r v a b l e -CL- , t h e n 1.1.10 f H = ^ t h a t -O- i s conserved i ^ h - i ^ ' ^ J =0 implies and i s fa c o n s t a n t o f t h e m o t i o n . One can t h u s i m m e d i a t e l y w r i t e down t h e i n f i n i t e s i m a l t r a n s f o r m a t i o n W~ a s , 1.1.11 ) ^ (where = One can f o r m a l l y extend o arsd COi, t h e method o f transformations to cover f i n i t e changes. £L = ^ n.] =-t circle 1.1.12 of -j- (>0 To do this, the c a n be e x p a n d e d i n a power s e r i e s i n t h e convergence i O b = Z then t h i s e x p r e s s i o n remains i s r e p l a c e d by a m a t r i x within # infinitesimal assume t h e v a l i d i t y o f t h e f o l l o w i n g t h e o r e m J ^ ^ " I f function ) with A v a l i d when the s c a l a r whose c h a r a c t e r i s t i c the c i r c l e o f c o n v e r g e n c e " . In argument values l i e particular ( i ) P o r f i n i t e m a t r i c e s i t s v a l i d i t y i s g e n e r a l l y w e l l known, see' f o r e x a m p l e ; G a n t m a c h e r . T h e o r y o f M a t r i c e s , C h e l s e a P r e s s , N.Y.Porithe g e n e r a l c a s e see R e f . ( i i ) , on t h e n e x t p a g e . "11= 1.1.13 -A Hence f o r a r e a l o( and H e r m i t i a n .CL , (E i s unitary, Thus i f U - I.l.lZj. i.i.is ^ F " = j Ux 1.1.18 U then where (1.2.11) t h a t t h e d i s p l a c e m e n t and t i m e are r e s p e c t i v e l y : STx « P = ^ p = i - U t ~ y -c P ^ ^ P P° -~ --c ° 1 ~ 3 §"t- a somewhat more i n v o l v e d way one c a n c o n s t r u c t (•Iii) o p e r a t o r s f o r r o t a t i o n i n IL-D s p a c e . " infinitesimal to ' <^ I I 1.1.19 ^ a i n coordinates, angles 1.1.17 ol* E ^ / C ^ l ] ,e,+c. and C 'W=^. -I In ^ n ^ ^ " ^ 9 ] . = then f o l l o w s from generators f - Z L^-/P] = 1.1.16 It *~ Still another t r a n s f o r m a t i o n i s t h e 'gauge'.transform.a.tloh'i. w h i c h h a s do w i t h t h e c o n s e r v a t i o n o f c h a r g e . The above n o t i o n s c a n be e a s i l y e x t e n d e d t o f i e l d ( i i ) P. Hausdorff., L e i p z i g , B e r . Ges. W I s s . M a t h . B h y s . K15_8 19 ( 1 9 0 6 ) . ( i l l ) . -Kemmer (1959). -12 theories. I n extreme f i e l d t h e o r i e s , s u c h as "General fei Relativity', In local t h e c o n s e r v a t i o n l a w s e n t e r as field t h e o r i e s these c a n be identities.' d e d u c e d by Noether's theorem. The l o c a l c h a r a c t e r o f quantum f i e l d t h e o r y i s i m p l i c i t i n the r e q u i r e m e n t t h a t t f i e ^ i commute w i t h e a c h other and w i t h the f i e l d observables are r e p r e s e n t e d therefore occur one The and the dynamical dynamical by in field operators the and must v a r i a b l e s , enables into theory. connection connection remark t h a t the r e l a t i o n s - I . l l p - 1 9 m u t a t i s m u t a n d i s D i s c r e t e Symmetry The The by H e r m i t i a n as b i l i n e a r s to c a r r y over quantum f i e l d 2° variables. Operators b e t w e e n t h e d i s c r e t e symmetry i n v a r i a n c e s equations i s somewhat d i f f e r e n t f r o m b e t w e e n the c o n t i n u o u s laws. transformations and the the This i s because a d i s c r e t e t r a n s f o r m a t i o n i t s v e r y name i n d i c a t e s , t h a t i t c a n c o n t i n u o u s l y f r o m the i d e n t i t y , n o t be so t h a t the of the d i s c r e t e t r a n s f o r m a t i o n i t s e l f and generated group c o n s i s t s u n i t y : e.g. space i n v e r s i o n ; o o -I o o 1.2.1 o .(•iv). S c h r 6 ' d i n g e r (1950). o o o ) \ *4 / for -13- so t h a t the FI group c o n s i s t s of There are t h r e e t y p e s o f 1. Space I n v e r s i o n ( H ) 2_ PI and " unity. s u c h b a s i c symmetry o p e r a t i o n s ! „ Here t h e s i g n o f a l l the s p a t i a l c o o r d i n a t e s i s r e v e r s e d , t h e r e f o r e l i n e a r momenta change 2. sign. Time R e v e r s a l l i n e a r and ( T ); S i g n o f t i m e a n g u l a r momenta change s i g n . also called 'Reversal of 3. Conjugation Particle particle so t h a t a l l It i s therefore Motion'. ( P ); .A p a r t i c u l a r case i s charge s i g n of charge i s r e v e r s e d , i . e . , a c o n j u g a t i o n , where t h e positive i s reversed i s r e p l a c e d by a negative particle and j o vice versa. Since there are mesons w h i c h have d i s t i n c t concept neutral particles p a r t i c l e s and such as |\ = antiparticles, the i s more g e n e r a l . The requirement t h a t a s y s t e m be i n v a r i a n t under d i s c r e t e o p e r a t i o n s , i m p o s e s c e r t a i n c o n d i t i o n s on system which give r i s e quantum m e c h a n i c s . under these discrete assumed i m p l i c i t l y , to the selection rules in symmetry o p e r a t i o n s has always been P o r e x a m p l e , I t i s w e l l known i n e n t i r e l y e q u i v a l e n t and a time the I n c l a s s i c a l t h e o r i e s the i n v a r i a n c e m e c h a n i c s t h a t r i g h t and Similarly so c a l l e d left handed d e s c r i p t i o n s a r e a r e a mere m a t t e r reversed of convention. system i s a l s o p h y s i c a l l y p o s s i b l e , whether i n c l a s s i c a l mechanics or i n electro-magnetic as l o n g a s " t h e these system i s r e v e r s i b l e . phenomena a r e a l s o known t o be The theory, electromagnetic i n v a r i a n t under interchange -III.- of p o s i t i v e and n e g a t i v e In o r d e r t o extend charges. the concepts o f d i s c r e t e o p e r a t o r s t o quantum t h e o r y , one must f i r s t p h y s i c a l meaning. 1.2.2 <p> U n d e r space 1.2.3 < 1.2.4 this ~ \P/ _Y t / <P> ,-t r their s c-^ v) ^U/tj dr = /^ inversion > investigate F o r l i n e a r momentum one h a s rt symmetry gives, — 'C^X-yt - - < a P>v, n d u n < ^ 9 r time reversal . t S i m i l a r l y f o r a n g u l a r momentum, one h a s 1.2.5 <L>^ t = \^ i x - <\ \—^Y [-ch t ) t Y x V ) (// C l j t ) dr^ hence, 1.2.6 ^L/'y- "•7 <L> , r Similarly {_ t - - <U> + r £ but _ _ fc s p i n c h a n g e s s i g n u n d e r t i m e r e v e r s a l and c h a r g e under charge c o n j u g a t i o n . In quantum f i e l d characteristic field t h e o r y each type o f p a r t i c l e has a f u n c t i o n which c a n be s p l i t into positive and n e g a t i v e f r e q u e n c y p a r t s — i n t e r p r e t e d a s creation and a n n i h i l a t i o n o p e r a t o r s . An o b s e r v a b l e i n - 1 6 - quantum f i e l d t h e o r y i s t h e n c o n s t r u c t e d f r o m b i l i n e a r com- b i n a t i o n s o f c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s . concludes t h e r e f o r e t h a t a d i s c r e t e symmetry o p e r a t o r also c o n s i s t o f b i l i n e a r combinations. unitary, 1.2, should S i n c e i t s h o u l d be the immediate c h o i c e i s -LoLD- where X L . i s H e r m i t i a n . t l - e. Now suppose u n d e r a d i s c r e t e an o p e r a t o r symmetry t r a n s f o r m a t i o n ^S> i s t r a n s f o r m e d ^ into . 1\ IT) ^C-0 i . . 9 a pa', 2 ^ r 0^.'PJ ^2 + c 2 ' n-0 - I f one now c h o o s e s XL- s u c h Io2.10 One IL One c a n w r i t e oL 2T\ £2* + r\=0 that K\=» and 1.2.11 one o b t a i n s for oL^ for o^r^iT T h i s method c a n be u t i l i z e d to c o n s t r u c t a l l the d i s c r e t e symmetry o p e r a t o r s . proceeding Before s h o u l d be made o f t h e s p e c i a l n a t u r e transformation. is E f u r t h e r a mention o f the time reversal I n quantum m e c h a n i c s t h e o p e r a t o r o f e n e r g y l 1.2.12 i = - -t - 17 So II that i f one e x a m i n e s t h e S c h r o d i n g e r equation, t h a t under time r e v e r s a l , w h i l e t- —> - t d o e s n o t change should definite. s i g n , f o r energy Thus one i s f o r c e d a t t h e same t i m e . there exist » the Flamlltonian a l w a y s be p o s i t i v e to p e r f o r m complex Again i n the Heisenberg e n e r g y momentum o p e r a t o r s Xyu. (whose lith com- P/*- unity the t r a n s l a t i o n group ( -i, ) X j*- - and obey t h e c o m m u t a t i o n r u l e s : L / " r 1.2.13 Now conjugation representation, ponent i s e n e r g y m u l t i p l i e d by the i m a g i n a r y which are connected w i t h one f i n d s U = ( i ) under r e f l e c t i o n first ^ oV • - X , (apart from ) the 3 c o m p o n e n t s on-the.right change s i g n and t h e r e f o r e one may demand t h a t Px changes s i g n a l s o , w h i l e energy remains unchanged. (ii) the r i g h t Under the t r a n s f o r m a t i o n above c h a n g e s s i g n . perform complex c o n j u g a t i o n on t ~^ ~k , i^th component on Thus one i s a g a i n f o r the case o f time r e v e r s a l account o f p o s i t i v e d e f i n i t e n e s s o f energy. obvious from equations complex c o n j u g a t i o n l e d to This i s also l,*2}12.0ne c o n c l u d e s t h e r e f o r e i s a sufficient that c o n d i t i o n f o r time it reversal. Under complex c o n j u g a t i o n the Schrodinger equation \ 1.2.11a w db \ c b > H V .-c ^ *- goes o v e r i n t o -18- Hence t h e n e c e s s a r y c o n d i t i o n i s t h a t essentially different, a unitary operator. i.e. be u n i t a r y while and and _U.HU H are not U. where The t i m e r e v e r s a l i s t h e r e f o r e as a p r o d u c t o f a u n i t a r y conjugation, H H written f a c t o r and an o p e r a t o r o f c o m p l e x other operators linear. is s u c h as H and P will CHAPTER I I DISCRETE SYMMETRY OPERATORS FOR THE SCALAR F I E L D 1° D e f i n i t i o n s and N o t a t i o n s The complex s c a l a r f i e l d , which t r a n s f o r m s according to the one d i m e n s i o n a l r e p r e s e n t a t i o n o f t h e L o r e n t z satisfies the K l e i n - G o r d o n II.1.1 (• The equation: - If^O Cp ("*-, t) = owhere ro -mass o f t h e p a r t i c l e general feature of a l l the f i e l d s i n t e r a c t i n g p a r t i c l e s i s t h a t they c o n d i t i o n the equation I I . 1 . 2 . coordinate group (2*-) t o momentum ( F o u r i e r t r a n s f o r m o f the f i e l d d e s c r i b i n g non- s a t i s f y as necessary Hence i n a t r a n s i t i o n from ) r e p r e s e n t a t i o n , the function i s different from z e r o , o n l y when II.1.2 i.e. K \<, h a s t o be i n t h e b a c k w a r d o r f o r w a r d One c a n t h e r e f o r e a l w a y s decompose a f i e l d positive two and n e g a t i v e f r e q u e n c y light function into parts corresponding cones i n a L o r e n t z i n v a r i a n t f a s h i o n . cone. to the I n t h i s c a s e on m a k i n g t r a n s i t i o n t o quantum t h e o r y , where one r e p l a c e s complex conjugate ( ^ ) by H e r m i t i a n c o n j u g a t e obtains i n discrete details (~£ ) one l i n e a r momentum r e p r e s e n t a t i o n ( f o r see e r g * dBd.goliubov, Shirkov 1959). =20= ?I ,.3 normalized ^**%>^2M^ i n a b o x o f volume V. denote p o s i t i v e field The s u f f i c e s and n e g a t i v e f r e q u e n c y are r e s p e c t i v e l y particles II.1.5 k. x, ~ /c x - £j£ space. I I . 1.6a The 1 [^ ( / observables 11.1.7 ayiJ Momentum ^ ] ( a r e (see e.g. ? w ~ rJ^_ '+ ^ J = 2 *> a 11.1.8 Energy ? = ^ Z I I . 1.9 Charge £) = 2/2/^ ^ F o r d e t a i l s see e ;g.-.. B o g o l i u b o v i t h e e n t i r e JS. .. = t'K^-i) Bogoliubov 2 2 are; _ 6b „ and Ix e x t e n d s o v e r The c o m m u t a t i o n r e l a t i o n s creation ) and a n t i ( 7"= 2- ) w i t h wave number l a b e l summation i n I I . 1 . 3 - - The a d j o i n t dimensionless ( Y-\ annihilation operators of p a r t i c l e s II parts. and ( — ) function i s ; t The ( •+) (;v) and S h i r k o v ^ ^ C^ ~ 1959) = - tr ^ tz HA*^) £'A£(ft)electronic and S h i r k o v (1959) charge -21- U n d e r a symmetry o p e r a t i o n one o r more change s i g n . A c c o r d i n g l y they ( l ) Space i n v e r s i o n change s i g n . 11.1.10 x 11.1.11 t ~>-"k c a n be c l a s s i f i e d a s f o l l o w s ; ( /T ) and t i m e However observables (~f)t reversal Momenta 7" i s a n t i l i n e a r . xi TJ P n T -I „ - - r PT* = -~P These a r e s a t i s f i e d b y : II.1.12 ' (It being assumed (2) P a r t i c l e that conjugation Charge changes s i g n . ii.1.13 x x - II.l.iij. x-->-;*: These ^ 5 S are r e a l ) ( \~ ) and s t r o n g r e f l e c t i o n i s ^ t ) antilinear. t—^t ? (S ror - - SQS — v t "<3 and -Q. satisfy; II.1.15 < ,' (3) R e f l e c t i o n The f i r s t ( A * H T i s linear 3 N / ) and weak r e f l e c t i o n b u t the second a n t i l i n e a r . ( J - T P ) Momentum and -22- charge both II.1.16 II. (I4.) 1.17 change s i g n . joi for A Thus * J" X t — X - - - k* Phase t r a n s f o r m a t i o n Both leave factor. the o p e r a t o r s ) and I n v e r s i o n ( I unchanged e x c e p t ~ !~J T ) f o r a phase Thus 11.1.18 J" 11.1.19 ( X- —>• X . t 5 (linear) -t I II.1.20 Alternately (antilinear) 1 one d e f i n e s : ( t ) "Tr"' = 11.1.21 TT 11.1.22 T<£0/t)T~' - 3 ^ ^ ^ ' ^ >? T 11.1.23 11.1.26 j 4> c^t; + -' j ^ y\ 4> c*>t; -23- II.1.28 II.1.21, and 2 3 , 25 and 27 a r e u n i t a r y and I I . 1 . 2 2 , 28 a n t i u n i t a r y . w i l l be t r e a t e d 2 i | , 26 I n the next s e c t i o n each o f the cases separately. 2°Transformatiort o f C r e a t i o n and A n n i h i l a t i o n O p e r a t o r s u n d e r Symmetry Transformations, ( i ) Space Inversion. Consider II.2.la Since has Cpir*,*) summation o v e r on i n t e r c h a n g i n g - tf K e x t e n d s o v e r t h e e n t i r e JC - s p a c e one JC and —J* on the r i g h t : II.2.lb Using one t h e d e f i n i t i o n I I . 1 . 2 1 and c o m p a r i n g t h e two s i d e s obtains; (a) TT ^t^TTr \ k 0 ) k and II.2.2 Since two s u c c e s s i v e i n v e r s i o n s should be a u n i t , o p e r a t i o n , -21+- one must have ii.2.3 I (ii) The i.e. Time R e v e r s a l . * ] (Reversal r - of d i r e c t i o n of t'ime r e v e r s a l t r a n s f o r m a t i o n , l i k e member o f the full L o r e n t z group. ' space i n v e r s i o n ^ i s a The two differ proper or r e s t r i c t e d L o r e n t z group In that Is equal to (i.e. 71 and antiunitary The —j . ~f However the using a l i n e a r operator i s that s i g n o f momenta b u t field also T were u n i t a r y a p a r t i c l e with serious one only be inversion changes The to chapter. time the reversed f i n d s that c an a state = w i s transformed a n t i p a r t i c l e of l a b e l j ~ /J [ & the . a n n i h i l a t i o n o p e r a t o r c o n s e q u e n c e s , f o r an Into for a l l 2 i . wave number l a b e l K, ( \ l » y II.2.5 formed i t not has first performs time that of charge. Jf - c r e a t i o n operator of has i n the two function i s : 4>t**> If I f one the t h e i r determinant operator T reasons explained immediate d i f f i c u l t y from s i m i l a r i t y b e t w e e n the ) e n d s h e r e , f o r the f o r the motion) . into a This -particle state, <J)] -=— lf\'.p-0 ( I t i s shown i n t h e 1 0> t-K*^ next ia | tr.,.- = section that 0 It of -25- i s not p o s s i b l e to c o n s t r u c t s a t i s f y i n g t h e above a unitary transformation transformation law). c o n t r a d i c t s the commutation r e l a t i o n s . Furthermore i t The commutation r e l a t i o n s are (a) EM*) L , b^CD = °Sr,r' II.2.6 (b) Hfoo , One t^^]- s = n o t i c e s t h a t under a u n i t a r y l i n e a r the c o m m u t a t o r s ( b under time i n v e r s i o n , while transformation s i g n , because D ) change the l e f t , I ) ( x changes " ; sign hand s i d e r e m a i n s invariant. (2) t h e c o m m u t a t o r s (3. ) change changed w h i l e go o v e r i n t o [> s t h a t the o r d e r o f o p e r a t o r s The is above o b s e r v a t i o n s i g n , as bS <S r e m a i n s un- and v i c e v e r s a , +/ so interchanged. s u g g e s t s two ways t o remedy t h e situa tion. (A) To p e r f o r m i n a d d i t i o n t o t h e u n i t a r y the o p e r a t i o n ( b o f complex c o n j u g a t i o n , ) commutators, b e i n g that exp(iKX) that s i s transformed are transformed t h e R.H.8. o f r e m a i n unchanged Such an o p e r a t i o n into commutators are a l s o p r e s e r v e d . ii.1.22 so t h a t pure imaginary, under the combined o p e r a t i o n . transformation, T + CJL^-HT" = 1 into b_£ s <?tfC-ikK) a implies , so nd therefore Thus d e f i n e ; 4>art) CV) -26- II.1.22A complex T - VlL , Where U. i s u n i t a r y and I L_ s conjugation. T h i s i s t h e d e f i n i t i o n u s e d by S i g n e r , L u d e r s and o t h e r s . (B) The'above i s t h e s o l u t i o n f r o m t h e p o i n t o f v i e w o f (k ) commutators. (a) The r e v e r s a l o f t h e o r d e r o f o p e r a t o r s the time r e v e r s e d complex conjugate together with operators. w.iere follows that, ii. 2. Y ~ transposition--' 1 ~ 4>t-t)<t> (-t). + suggested by S c h w i n g e r , t h i s d e f i n i t i o n i s used (195V) Watanabe (1955), J a u c h and R o h r l i c h the o p e r a t i o n o f complex c o n j u g a t i o n linear operator, the product of a unitary operator (1955;). i s an a n t i - i t f o l l o w s t h a t the time r e v e r s a l operator a n d an a n t i l i n e a r That t h e o p e r a t i o n o f t r a n s p o s i t i o n i s a l s o an a n t i l i n e a r operator, an 0* U k , 1 .Since operator. + e^+j^c+Je" = u ^ a ' a ^ a m " Originally is t r a n s p o s i t i o n o f a l l the 1 It s f u n c t i o n by i t s e+cxtje"* * % 4> t*,-f), (b) Pauli field Thus; (a) I I . 2.7 by ( S c h w i n g e r 1951) • c o m m u t a t o r s s u g g e s t s a n o t h e r method Here o n e r e p l a c e s i n the i s c l e a r , f o r , an a d j o i n t o p e r a t i o n i s i n v a r i a n t o p e r a t i o n under a u n i t a r y transformation. 27- Incidentally this equivalent, formally shows t h a t li- so t h a t one can I n the sense see t h i s as exactly follows. o f Wignar^the time r e v e r s a l t r a n s f o r m a t i o n i s : 1 Since the adjoint operation a unitary transformation, II.2.7 or 1 T = 1 <i + t * . - t ) . i s an i n v a r i a n t o p e r a t i o n one UK4>ti/t) Thus is are i n b o t h the c a s e s i s . t h e . .s.ame.. More T+U/HT" = u ^ u u j u " II.i.22 The the two o p e r a t i o n s under has: r'lt » 1 = 1 D <t> c^,-t) + and d i s t i n c t i o n between the l i n e a r the and a r e a l s o t h e .same antilinear operators summarized i n the f o l l o w i n g . G i v e n c o m p l e x numbers X> Q. H l i n e a r manifold l i n e a r manifold Linear, D'& 3, b H i f ( [JL ) 2. U»T, ; 3 „ on t o Antilinear, U3 x T X , y of a another is 1. Its U i vectors , a mapping o f P 1. lL(ax+by)= aux t blLy 2. and 3 . i f (T ) Tfcx+by) = a* T x + b*Ty Ti T a - U.' *T lt,T, If Ux Ue i f l f 3X X» , a also e v x X ij,. I f T X * , then •28Therefore t h e concept . o f e i g e n v a l u e makes no s e n s e (x,Uy) = 5o t ^ X . y ) 5. (Tx.yJ = ( U x , y ) •- ( X , l t y ) + Linear w.r.t. A n t i l i n e a r w.r.t, 6. U n i t a r y : It U.L X X Linear w.r.t, y 6. U, - U+ Antiunitary: T Antihermitian: T ; T,,T*,- ( lls^U*, Thus complex J I f E ds the u n i t o p e r a t o r , Under a u n i t a r y CL- Xj then y transformation 5 I n an e n t i r e l y c a n he w r i t t e n The i m p o r t a n t p o i n t c o n s t i t u t e a group . the T unitary factor of (IS. a n a l o g o u s manner one c a n summarize t h e r u l e s f o r ttoei'o.parator ©SUK,operating i n the a d j o i n t t o remember i s t h a t order o f f a c t o r s i n H i l b e r t space. T X conjugation. transforms a s ; of ) f o r m a group where, L 4 w.r.t. ) b e i n g the subgroup. (Ui,Uz, as Linear 1 i s obvious that [T^y.x) y = Lt" Hermitian: (y,T*x) (*,Ty) - 8 space. reverses the Now u s i n g the p r o p e r t i e s and t h e d e f i n i t i o n o f t i m e r e v e r s a l i n ( I I . 1 . 2 2 ) , one obtains 4v Comparison is {uo c - yields; t II.2.9 -29- are real, Ubgti) LL" 1 ii.2.10 % • T% o)T" Particle , except that the a r b i t r a r y , on account o f the Thus *= T { n b ^ u ) J T - H ^ T b . c O T 2 1 f c (iii) T i s completely antilinearity of T ii.2.ii 1 b.^O) t o t h e r e l a t i o n s f o r f~| These a r e s i m i l a r phase f a c t o r s. 1 T = conjugation This i s a g e n e r a l i z a t i o n o f the concept o f charge c o n j u g a t i o n , which transforms negative c h a r g e and v i c e v e r s a . jugation transforms versa. a p o s i t i v e charge i n t o a I t should a particle I n t o an a n t i p a r t i c l e be remembered t h a t p a r t i c l e s i s not confined meson i s an e x a m p l e . Unlike a member o f t h e L o r e n t z the n o t i o n o f a n t i - particle c o n j u g a t i o n i s n o t even group i n the e n l a r g e d ). to the p r o p e r L o r e n t z spin-statistics connection), i n v i e w o f the above. the moment. Using group sense, though I t may be t h a t the i n v a r i a n c e under r e f l e c t i o n i s a l w a y s assumed, i f t h e t h e o r y with respect the K- s p a c e i n v e r s i o n and t i m e Is unitary ( remarked i n p a s s i n g con- and v i c e to charged p a r t i c l e s a l o n e , reversal transformations, the o p e r a t o r In general, particle strong i sinvariant (together with w h i c h i s somewhat s u r p r i s i n g However t h i s the d e f i n i t i o n does n o t c o n c e r n us a t (II.1.23) one o b t a i n s o n -30- comparison of the coefficients: and I I . 2.12 r = rb cz) A i r bjcco . Hence "2 2 r bj^ct) r ii.2.13 Thus the phase f a c t o r (iv) Strong = is arbitrary, Reflection Using the d e f i n i t i o n is antilinear, 11.2.11; (II.1.21+) and remembering that one o b t a i n s S bfcCOS" ' bjcci). = 1 on comparing 5 coefficients: and 1, b * ( a ) Therefore, s" S b*Ci) 2 ii.2.15 In terms of , and p o s s i b i l i t i e s for s l e a d i n g to the K$ ~ I . condition fl » f = Is V T 7$ 7f , there are the following » depending upon the o r d e r i n which occur. Trn riTr mr mi nrT ni If 1 T _3i- I f one assumes t h a t II.2.16 lir I f " (v) R e f l e c t i o n ( A *i ¥ = , one g e t s t h e c o n d i t i o n + - = HT ) Prom 1 ° C h a p t e r I I , r e f l e c t i o n h a s f o r i t s d e f i n i n g equation; C h a n g i n g t h e summation f r o m K to in JL+ <p ( ~~A - f / )» y one h a s , C o m p a r i n g c o e f f i c i e n t s , one o b t a i n s 1 1 0 2 . 1 A b , u ) A~* + 7 <btjcCO and A 11.2.18 1^ so t h a t a b*Ci) A~^ = bj^co i s arbitrary. ( v i ) Weak R e f l e c t i o n ( J As e x p l a i n e d = T!~~ ) i n 1° Chapter I I , 0" t r a n s f o r m s t h e o b s e r v a b l e s and c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s i n a way to analogous reflection. A However d e f i n i t i o n 1 1 . 1 , 2 6 , one o b t a i n s J i s antilinear. Using the -32- 1j b-fcU) II.2.19 and J^b cO T II„2o20 Thus i f I s " t h e 1 II.2.21 (vii) = 1 K IjijcO), Identity operation = ±| ^ V cr Phase t r a n s f o r m a t i o n Prom d e f i n i t i o n I I . 1 . 2 7 , if II.2.22 bfcCi) f one must have s = ( ^ This ) i t i s obvious that =• 1|. b^^-O 1 and t h e r e f o r e so t h a t (viii) T^T. ( I Inversion transformation i s the. I d e n t i t y operation, ) i n absence o f s p i n i n t o an a n t i l i n e a r phase t r a n s f o r m a t i o n degenerates and I s not. t h e . I d e n t i t y , Thus II.2.23 X bjcCD I" i b* fe) i I* Mo I" r TT n b|cco r Condi tt'Or? on 7i 0 = 1 2 1 x - % = bjc T b O) £ u) bt and ci) . I Wo A lib-gW s fi A S n b co <b- 0) in i=i J in H t B -33- 3 Representations Scalar f o r t h e D i s c r e t e Symmetry O p e r a t o r s f o r t h e Field. I t was s u g g e s t e d constructed i n C h a p t e r I t h a t D.S.O.'s may be i n the form o f a u n i t a r y •=. e The with o< r e a l and Sl hermitian. identity, ii.3.2 = A CB,c3. + CAB,C]_ e n a b l e s one t o c o n s t r u c t XL c r e a t i o n and a n n i h i l a t i o n b i l i n e a r expression operators, f o r e n e r g y , momentum e t c . where j f L i s so c h o s e n C i L Ca«.p3-+11 ^ l . RP'R" = - 1*P 1 C-v^f From (l*^->^ f " " Eiu 0 4cr 3 ) o{= M TV that then f o l l o w s t h a t , II.3.5 B i n analogy to the = n.i.k TA/Cj as b i l i n e a r c o m b i n a t i o n s o f = I g£ "•3-3 It operator = , U , f ] - = 1*P ; i IH -34- The d i f f e r e n c e between I I .3.3 difficulties. The .and . I I .3 . 5 Pleads. to.. some d i f f e r e n c e i n s i g n c a n be easily com- with ^-J\ p e n s a t e d by u s i n g an a d d i t i o n a l o p e r a t o r I n case the phase f a c t o r i s a l s o r e q u i r e d t o be one an c a n use R" different operator - e ^" e' ^ = 9 where 1 and II.3.6 = I n g e n e r a l one f r o m the Four o f fore The C.O.'s and these can unitary can form i n a l l 8 types o f K l i n e a r s A.O.'s; operators. used t o f o r m r e p r e s e n t a t i o n s f o r f o u r Thus one o b t a i n s f o u r u n i t a r y and i n t e r e s t i n g connection 'commutators o f the b i l i n e a r s ' h a n d , and other. bilinears This enables the g i v e n four operators. There i s a v e r y one there- used f o r c o n s t r u c t i n g u n i t a r y r e p r e s e n t a t i o n s . f o u r c a n be antiunitary etc. i n v o l v e o n l y C.O.'s o r o n l y A.O.'s and n o t be other ^(3 one and and the the To p r o p e r t i e s o f the b i l i n e a r s w i l l observables the on symmetry o p e r a t o r s , on t o s e l e c t the symmetry o p e r a t o r . between see be proper how this the the bilinear for i s done, the enumerated In the following notation; (i) Particle number o p e r a t o r : one has the f o l l o w i n g r e l a t i o n s ; -35- N (r) 11.3.7 11.3.8 This = b^ri^Cr) t [ Njccn, b ^ i r ) } _ = j - i s p r e c i s e l y theproperty M^Cr) cr) ; required o f -N^tr) f^tr), XI." i n b T(r/)_ k II.3.6. C fc-fcs. Let, 11.3.9 R lf) = 11.3.10 H=N t + Then, f o r a r e a l 11.3.11 + M „ - ^ KUr) { N (r) 2 , and - b R (ii) Define; represents B*CO fe a pure phase - k^co b^or) All other ii.3.ik C& kCr) , b ± : t i jr')^ commutators v a n i s h . fie = z ' N^cr) ^ 2. , k*>o Mr. efcf>(jS~N ) Crj ^ H ^ ) ^ ^ ^ ^ ^ ) . transformation, t B^cr) - b ^ (\r) T h e i r commutation r e l a t i o n s with 11.3.13 - •+ hum j 4 g^Cr)^-. Thus 11.3.12 Z ' N^r) = tr) . are: _ b^Cr) ^ One f u r t h e r o b t a i n s : [ Bfe ( r ) , B * ( r ) ] ^ N ^ W - f ^ c r ) and -36- P =• ^ ' K Pfc 11.3.15 where prime i n d i c a t e s ~" summation over a hemispace, say IIo3«13 shows t h a t used to c o n s t r u c t into b..^ . Bk^ a r the b i l i n e a r s that c a n be e a u n i t a r y o p e r a t o r that w i l l II.3.11, w i l l anticommute with are 5 >0. transform fc>£_ then shows that t h i s u n i t a r y operator ?_ B'S . I n f a c t the b i l i n e a r s the only b i l i n e a r s which stand i n this peculiar r e l a t i o n s h i p to the anticommuting observable and the .D..S.O. T h i s circumstance enables one to choose i n a unique manner the b i l i n e a r c h a r a c t e r i z i n g a D.S.O. I t may be remarked that and T • Pj< Hence one can use of these t r a n s f o r m a t i o n s . changes s i g n under both f l Bn's f o r the c o n s t r u c t i o n To see how t h i s can be accomplished, \it 11.3.16 X L = 2, * Cl 8&~*Z where the p a r t i c l e label T b\ J be H e r m i t i a n and U| -I has been dropped f o r convenience. Then 11.3.17 One U,^}. then v e r i f i e s 11.3.18 R = --iH>-* that, i f €xp (1 5 Sl) , C n , k - B ] - - i4*J» £ , -37- Comparison with the table above ( 1 1 . 2 ° ) shows that these are indeed analogous to the transformations f o r f ~ l and \Ji excepting f o r the c o n f l i c t i n g phases, which can be e a s i l y r e c t i f i e d by using the device of equation I I . 3 . 6 , 2 There are cases. n = »t*. (1) Let II.3.19 .a N + then (2) When ^ = H * , table (II.2°) shows that transform with conjugate 1 I - 3 - 2 = «p [ i f Z' i [ t 1 U ii.3.22 where S w+ = e w l i A f ' ( t g^) B b^CT) KfcCO- has been so chosen that Therefore, l e t , f6 co - B r To compensate f o r the phases i n phases. » WD) o n e c a n u s e |=«f{UTr-2S(W (i)- e ^ — 1 j1 produces the required change i n sign. and b f c O ) Thus e g . + and |j* -38- The u n i t a r y f a c t o r o f time y r e v e r s a l then i s : U= U Ut N + n.3.23 = ^p| II.3.23A _ l Y T r . - ) 2 / t ^ c . ) - N^fe)]* ^ One - ve r i f,1 e s" t h a t i * 11.3.23B z & ~ complex c o n j u g a t i o n . ' * t TT^ULL+a " = 1 uu"" = 1 U.4tlUa^(C H. Similarly 11.3.23c T - uu*= cor e -* )N+ <?*+ - e e Where u s e h a s b e e n made o f t h e f a c t However U x %| N It follows b LlJ + - from, 2 n that a n d N h f r o m I:c -3.l8 U*= U + U + l l ^ U ^ U - , " K II.3.23D Therefore, U ' ^ yof = Hf U U * MH+U+U^U^ ( T T H Bjc x inspection that introduces a factor or i n t r o d u c e s a f a c t o r (—I ) , so t h a t ^ \l o p e r a t i n g on any , while z on t h e whole Uf gives -39- a factor ^ or double o p e r a t i o n formation (iii) ty* . I' } I f one p u t s yields identity, 1 so t h a t t h e the r e s u l t i n g t r a n s J~] . i s space i n v e r s i o n o p e r a t o r Define II.3.21+ II.3.25 Cj< = b^Cl) = b / l l ) W U> 7 , then ' v All these hold again i f K i s replaced by — J L everywhere. Also, II.3.26 It i s c l e a r t h a t one c a n u s e CK_ i n the c o n s t r u c t i o n o f ^ Cjc J j n=€ ' T~~, Consider, iii = II.3.27 ?/ ^ - (• ta,>^')L = I I . 3.28 aM l2a then ^ f b , W ; L " ^ . i U f c O ) X similarly Hence. r, bfc CO T, I I . 3.29 - bjc(i) «( which are the c o r r e c t transformation properties of f , -i+0- However for one o b t a i n s : bg(2.) II.3.30 and which d i f f e r i n s i g n from the r e q u i r e d t r a n s f o r m a t i o n . To c o m p e n s a t e f o r t h e s i g n , we a g a i n make u s e o f (II.3.5") and write . r= r , r , = c^N„w .-f i.-ctV-WKJ N € e II.3.31 where e ^ involves only particle number o p e r a t o r s and (2.) a n d b ^ C * ) b u t g i v e s a t h e r e f o r e commutes w i t h •f- f a c t o r o f m i n u s one when o p e r a t i n g o n Since i n strong r e f l e c t i o n also only C'' and Q~&--Gl t h i s c a s e i s z e r o ) , one c a n use is again. a n a n t i u n i t a r y o p e r a t o r , t o be w r i t t e n S s SL II.3.32 where S b ^ Cl) (for, spin i n However, S 3S i s unitary Then « < ? * p { u f , %co) s =s ,s, H II.3•33 * e e I f one i m p o s e s t h e c o n d i t i o n , H.3.3U -2 S^MS- * 1 Is ' I b*(r>, one must have -la- dy) Define: A = bjcd) \>Xw t a n ^ 0 ) W A*' d 2 ) , a . Therefore II.3.36 a l l other commutators v a n i s h . = CA n.3.37 t / A + h Prom above f o l l o w s t h a t constructing of TT °f • b^'s a representation PJ" f J ; . Ab^o>A = where II.3.40 .From 3.62 I I . 3.1+1 the o p e r a t o r o f f~n Consider one h a s : a n a t c a n be' u s e d f o r o r the u n i t a r y down t h e factor transformations These a r e s e e n t o b e f o r t h e u n i t a r y /\L H II.3.39 N C.) l = N-tW- To do t h i s , one must w r i t e under factor of One f u r t h e r h a s , ^= 1 A W ^ 1r1r. ^1 A For I { ^ - W 2 M H tfl>^) - ~ f |~"=/\L >°ne w i l l } and have A, - l ? / * ^ . e'^^ 1 -42- T h u s , t h i s Sl\ However d o e s have t h e c o r r e c t p r o p e r t i e s f o r t h i s g i v e s the wrong p r o b l e m may sign f o r (^^.^(z) "' The II„3»5« once more be s o l v e d b y u s i n g Thus: . . II.3.42 1A - where ? T ?r* ; II.'^43 f AL^ f T ' Howe ve r , not i c e , AN, AI AM, AI = ii.3.44 which i s a u n i t On t h e o t h e r AN, (AHTAIAN, ) operator. hand, (AL) = AA^ - AM,Ai AM, Af X = AT A* Thus t h e r e Ai~/UA|A, - ^ Uhit A ,A AJAT N ( ob€ir«for t i s no s e n s e i n c o n s i d e r i n g an o p e r a t o r , s u c h as whose d o u b l e a p p l i c a t i o n d o e s n o t l e a d t o an i d e n t i t y formation. Thus i f one demands t h a t as a symmetry o p e r a t o r constitute I T i n the sense t h a t a g r o u p , t h e n one o b t a i n s r = * l T , or 7 1~ T trans- considered and t h e a c o n d i t i o n on t h e phases: v i z . 1 be a l s o = T~T identity AN, (v) Finally creation operators construct up one h a s t h e b i l i n e a r s from or only their t o an i n f i n i t e creation versa. operator However formulated. into II.3.4-5 W bilinears ( either unitary One c a n Hamiltonian, c a n be below: 0 bjcU) Further, I I . 3.1+7 II.3,1+8 Hence H - *p£ S -? ^ " do n o t g i v e and rise to any i s invariant transformation. Define I I . 3.1+9 M(r) = W W ~\ci-r) b II.3.50 II.3.51 II.3.52 Similarly II.3.53 one h a s ffc (|) ^ HFjcCo, F b^CDb^Cx) - ^ o ] . _ ~ Ni^o)-t (2.) M_ u)-t| k a and v i c e transformation a r e enumerated only c a n change an a n n i h i l a t i o n o p e r a t o r properties = the observable A l l these no c o n s i s t e n t Their involve annihilation operators. commutators factor. which -44 LFjcO.), F 11.3.54 C0]_ = N 0)+ K/-^2-)+\ t t f f c t O , b j t o ) } . ^ t_ t>) Here 11.3.55 4 f e . etc. t Finally, n . 3 . 5 6 fi^Cr, = II.3.57 [>,ar), ^(t)] II.3.58T Here One thus constructed b > ) 4_ ( N ^ i O + i ] ^ [fr^tl) b ^ O j = bjc ® # concludes N]c , from • t h a t a l l p o s s i b l e DoS.C.'s c a n be C-k_ /|jc as done so f a r . and R, P/ T» A , I , Thus t h e o p e r a t o r t c .S ^T, are the only p o s s i b l e D . S . O . ' s w i t h i n the frame work o f t h e t h e o r y . 4° Two D i m e n s i o n a l Representations. I t was f o u n d i n 3 bilinear characteristic the t h a t , though there o f each type i s one u n i q u e o f symmetry e x p l i c i t representation constructed operator, from i t gives c o r r e c t t r a n s f o r m a t i o n p r o p e r t i e s o n l y up t o a p h a s e f a c t o r s One is therefore f o r c e d t o apply volving particle difficulty a phase t r a n s f o r m a t i o n I n - number o p e r a t o r s . i s contained symmetry o p e r a t o r . i n the very The germ o f t h i s d e f i n i t i o n o f the One c a n l o o k upon a symmetry { S ) as a mapping o f a l i n e a r m a n i f o l d space (M ) on t o another l i n e a r m a n i f o l d that the s c a l a r product operator X>+- o f H i l b e r t 3?_ o f H o f the e i g e n v e c t o r s , , such spanning the -k-5- two m a n i f o l d s , 11.4.1 vanishes, i . e . <Mo.>= <P.|S |P->= <MsD+> - o H where i t h a s b e e n assumed t h a t t h e two m a n i f o l d s i n t e r s e c t , and t h e two t o g e t h e r w i t h entire P do n o t constitute the 0 H ' „ S J P > =- ( W > 11.4.2 0 As a n e x a m p l e , c o n s i d e r >lr>/ The o p e r a t i o n an Yi- p a r t i c l e s t a t e o f momentum K_ : ~ p gives so [n 1^) - 0 . T h i s i s a l w a y s s o , u n l e s s t h e s t a t e h a s momentum Thus that n h t ru> = zero. v>, I n g e n e r a l , t h e r e f o r e one o b t a i n s two r e p r e s e n t a t i o n s , and f l - o p e r a t i n g on the s t a t e spaces say, convenient Q and JC5 < 0 (Hk^ ° I t i s therefore t o go o v e r t o a t w o - d i m e n s i o n a l Further, i ti s clear that I n t h e two h a l f space. very Thus, 46- f \o>\ |o>/ _ Therefore f o r the p u r p o s e s o f t t B . ' J S - d i r a e n s i o n a l r e p r e s e n t a t i o n , one d e f i n e s : In will the f o l l o w i n g , r e p r e s e n t a t i o n s f o r v a r i o u s be f o u n d i n two d i m e n s i o n a l f o r m and i t w i l l how one c a n g e n e r a t e v a r i o u s o n e - d i m e n s i o n a l f r o m t h i s by a p p r o p r i a t e (i) unitary operators be shown representations transformations. Space i n v e r s i o n From 3° t u } ii.4.6 f l ^ t This Z L 1 % Lr)- - *PH ?I g i v e s a wrong s i g n f o r II.4-7 ° Therefore _- H' '«CB «rt- n.'e \ n.4.e 'xi i iiA rj- = n+ - nf+ , BfeJ K (r)Jj one c a n w r i t e B*<(rj] , i.e., + Therefore II.4.9 In the order t o g e n e r a t e a one-dimensionab. r e p r e s e n t a t i o n , unitary IIo4»10 transformation 10 n* \ M consider -47- One canwrite : o rw\/o n^Uo nut w**° II.4.11 \( 0 nnAlo 4 ^- j/o n f Wow because >» 7 rim C Pfc >— ) n„* fr = - { B^t) - = ^ f l u t . B j Also Thus, u s i n g II.3.20, PI * n .n H< II.4.12 + ^ n+flM. These are f a i r Still another representation I s 11.4.12 (ii) flr one d i m e n s i o n a l H^ir)-te lV- tx|>{-i$f % Time Inversion Here i f one u s e s b_^Cf) representations transform L (J[± d e f i n e d with a wrong L ^ O ) U-t + = for f l. <L6*cr>]} by I I . 3 . 2 1 one f i n d s phase. Thus However, Furthermore bjc 0) that -k.8- Hence a two-dimensional representation i s : U-f. \ 0 U. - 11.i4.oi3 ( £T+ 0 - W T 0 Further, I I •4.14 T ^ Kl{ - ^ ^ 0 U J |j a s required . To o b t a i n f u r t h e r r e p r e s e n t a t i o n s , one c a n p e r f o r m a unitary transformation 11.4.15 Llxf using a n < 3 UM-~£ s and g i v e n by I I . 3 . 2 2 . u=5+ 11.4.16 liw_ where tf^ i s The f i n a l r e s u l t i s MM+ = ^UN> UU,.WH.- a ~ H + . = W He nc e , U. ^ 4 Both are Still M^U+ ' or 2 ^- numbers and u n e q u a l t o a u n i t o p e r a t o r . another representation i s : II.i4_.17a where Uv= f the f i r s t where a n d t h e phase U 2. i s : €. with l- € ; f a c t o r r e s t o r e s t h e s i g n and t h e s e c o n d f a c t o r c h a n g e s t h e phase f r o m the Mi^x_ Ui - £ compensating f a c t o r 11.4.17b U^iAd c a s e may b e . \ to One t h e n v e r i f i e s that o r v i c e v e r s a as (iii) Particle Conjugation The r e p r e s e n t a t i o n f o r p a r t i c l e half space o f p a r t i c l e operators conjugation i s dbtaine'd f o r the from 3° • r^e' II ,3.27 T h i s r e p r e s e n t a t i o n g i v e s a wrong s i g n f o r a n t i p a r t i c l e operators \)£ * 11.3.27a \>^-) • T h e r e f o r e 2 P2 ~ a n < ^ ^he t w o - d i m e n s i o n a l representation is, II.lj..lb 0 -J"-=. TV „ j bicO) - / 11.^.19 and 0 M V °\ * iJ > ) D k.ct^) - To o b t a i n a l t e r n a t e one d i m e n s i o n a l apply the unitary transformation r / 0 1 I and i s obtained ("^ n.u.21 r - - (r, One t h e r e f o r e n.ii.22 r _ \ M i II.ij.o20 r where by r e p l a c i n g o'')(£, obtains.the o ] l = ( N i s g i v e n by 1 1 . 3 . 3 1 Mii(l) b^j I representations; , f r, = r, r * fir Mt representations, u Ml r»Jl Nijc. f -) . 2 50- These c a n be f u r t h e r v e r i f i e d by u s i n g t h e r e l a t i o n s t ii.4.23 ine r,r r, = o*.d r , t i K finally N verifies r - rjj ^ — R e f l e c t i o n s . ( S ). Strong (3°) that the r e p r e s e n t a t i o n o f from that o f of etc M unit operator „ another a l t e r n a t e c o n s t r u c t i o n i s : (iv) for r, t that Hij ~ ^2- k Still -f- f~ S f" i n being I t was p o i n t e d 5 i n If,however one must have ^ S"' i s r e q u i r e d 2 ; J5 — space differs A l l the r e p r e s e n t a t i o n s antilinear. g i v e n above h o l d a l s o f o r S . out e a r l i e r the u n i t a r y f a c t o r t o be a u n i t operator, real. (v) R e f l e c t i o n s ( A ) R e f e r r i n g t o 3° ( i v ) one c a n I m m e d i a t e l y w r i t e the two d i m e n s i o n a l ( 2-D) r e p r e s e n t a t i o n a s ; n.u.25 n = where i s g i v e n b y I I . 3 . 3 7 and A) vention f o r ( A , b^'/l DM f\ _ ~ f\^~ ? a r e g i v e n by I I . 4 . 1 9 . unitary transformation II.4.26 / J down ®*-) ,tfo* . The conApply the one o b t a i n s -51- A-A A, - A*A, ^ (U n.i^.27 Nl N Ml = ^^A, where u s e was made o f ^ ftt ^ - - Ak_ 11.4.28 Still e t c . and Al^jJO/lf- etc . another c o n s t r u c t i o n i s : A " e 11.4.29 ' One t h e n v e r i f i e s t h a t i n a l l c a s e s , A"-I . ( v i ) Weak r e f l e c t i o n (r~T J^, = The r e p r e s e n t a t i o n o f d i f f e r s from /\ , i n b e i n g antiunitary. demands t h a t J" he a u n i t o p e r a t o r , Thus a l l t h e r e p r e s e n t a t i o n s f o r f\ J~ F u r t h e r m o r e i f one one must have' - T/ . are also the represent- a t i o n f o r J" > t h e u n i t a r y f a c t o r o f iX . (vii) Inversion linear, ( X ) and I d e n t i t y . The f i r s t i s anti- and i t s r e p r e s e n t a t i o n i s 11.4.30 Obviously L = C where e ~ (n 'j =52 5° The A n g u l a r Momentum (A) Representation Notation In a t r a n s i t i o n t o angular momentum representations one o b t a i n s . ( F o r d e t a i l s , , s e ^ H a T m S t o T b ^ S " ° | ) - -tot ,-»• ^ f r ) r , A II.5.1 normalized relations i n a volume o f r a d i u s ir=l,2. where . The c o m m u t a t i o n are 5 QPCMWS), I I . 5 . 2 ^ 5**' t> (fcli»iV)^„_ — + Sift ' , s t a n d s f o r p a r t i c l e s and a n t i p a r t i c l e s respectively. s p a c e , and K, 6 V, , T 5 are polar coordinates cX , ^3 s h a l l s t a n d in x - f o r polar coordinates i n ^- space. tj,(*)= II.s.3 9 a«"c*.p) =i^Hatmn) ana t h e s p h e r i c a l h a r m o n i c s have t h e f o l l o w i n g w e l l y known p r o p e r t i e s ; U.S.* fe,9J = 11.5.5 = I f one demands t h a t 1 1 . 5 . 6 and ^ " c T r - e , ir+?)= j?^ Cff^ and - £-0 H)% (6,f)= h) y u) M fo(k$WI'V^ bCktfmar) l n e are r e a l s then -53- One f u r t h e r h a s t h e p r e s c r i p t i o n . ana where the clXl(^,(? ) = " S ^ ^ c L a J ^ f ^ i n t e g r a t i o n over h a l f i n the appendix X ii.5.7 and t h e p r i m e j£— space o n l y . indicates I t i s t h e n shown that 2 WfcCtr) = yiOcjfcwTr) (B) Symmetry o p e r a t o r s (a) Space Inversion: De f i ne ii.5*8 Using one n<W*, + ) H' * 1 1srT(-^l:i . (~S-»S\Q and c o m p a r i n g c o e f f i c i e n t s on t h e two s i d e s , obtains: 11,5,9 nb^Mbsltf 1 - *Jir H ) * K*Clc£»iir) ^ where i t h a s b e e n assumed t h a t *\_ ^ =• \ ^ ~ ±| -5k- 1 = (—l) One c a n a l s o w r i t e and c o r r e s p o n d s r e s p e c t i v e l y t o t h e s c a l a r and p s e u d o s c a l a r case. is and t h e r e f o r e Since t h e o n l y change t h a t F] effects i n t o change i t ' s f a c t o r , one c a n i m m e d i a t e l y e x p l i c i t representation using a p a r t i c l e 11.5.11 l> ^ " 2 2 . ^ ^ 2 Thus one c a n see t h a t a n y i - p a r t i c l e vanishing total In angular b (ti?sn2S w r i t e down t h e number o p e r a t o r : N ^ state V j ' ) o f non= l i n e a r momentum w i l l be a n e i g e n s t a t e momentum r e p r e s e n t a t i o n . o f \~\ I t does not f o l l o w h o w e v e r t h a t t h e i n d i v i d u a l p a r t i c l e s o r t h e s t a t e )4^> a r e necessarily orbital In a r e l a t i v i s t i c angular angular theory momentum e l g e n s t a t e s . the e l g e n s t a t e s momenta a r e d i f f e r e n t . In fact o f t o t a l and o r b i t a l Nevertheless f o r a given s y s t e m o f p a r t i c l e s one c a n a l w a y s c a l c u l a t e t h e number L ~- 2. (see e . g . Roman, T h e o r y o f e l e m e n t a r y p a r t i c l e s p . 251) (b) Time Reversal Define ii.5.12 T9(i t)T" / i = \ T 9 C l r i ) -55- S u b s t i t u t i n g f o r <p summation II.5-13 , u s i n g I I . 5 . 4 nc3 i n t e r c h a n g i n g t h e a from - i n to kirn N 2i0 I on t h e r i g h t , one h a s : y = 2 V^'l-o™ j bw-»or Comparing, II.5.14 Using '6T)e ; o t + bV«-«) tfce?)*** coefficients: Tbfc(w IjT ' 1 II.5.6, these IT kft&ML l ) and become. The r e p r e s e n t a t i o n f o r ( \JL ) can be immediately written down, u s i n g p a r t i c l e number o p e r a t o r ? . II.5.16 ©^{-iUiri-Si^^^f^m^jL T^UL^ where _ r| and Thus |__ = complex In order t o f i n d the representation f o r U notice ^ that I I . 5 . 1 4 I s analogous t o time momentum r e p r e s e n t a t i o n . one c a n w r i t e : Therefore conjugation. i n II.5•14 reversal f o r linear with appropriate changes •56= 11.5.17 ^^'^ ^ i m Si^= 1 )L [ B C W w i ) - B t W w i ) ] ~ 1 L B V ^ i ) - 6 «Qv»>)}} f Where p r i m e i n d i c a t e s t h a t t h e summation i s o n l y o v e r B(W«T)= v a l u e s o f vn , and 11.5.18 Since b ut IL-r - pos'i+fve kftklwr) b Clc£^w If ) . T h n d e e J =- [ J l ^ b CkV*u'|) t (2m+i)£ ( I c ' t W l ) u+bo&vou;* a - -I^C-Q^bCtfe'*.'!) . Introducing 'H.5» n U M + = C on® h a s U II.5.20 N + ll+L. I n a n a l o g y t o t h e c a s e o f l i n e a r momentum r e p r e s e n t a t i o n , one c a n w r i t e o t h e r r e p r e s e n t a t i o n s ^ = U+ Uu II.5-21 + II.5.22 - A l s o , one h a s : = tf+tijsU UN-^+ G>cf>{iClT-2^)£. U-UiU^, ( (J. = «*P 0? -Sf XX 5 23 ^ = - for I X as; where 0 ^ - » 0 - N CkC-w 2.) where ^^irjll^xp [ « 2 i^W/mdrX 2 ^ . 3) n -57- (c) P a r t i c l e Conjugation I t f o l l o w s from 11.5.21). ii.5.25 Cl ( t-3 HM) ~ -2. e ) that ^ - MCK^VMT) [ N(W j - N C « * i ) ] " e a ( w2 t ^ X where II.5.26 with It i s therefore clear that the t r a n s f o r m a t i o n momentum r e p r e s e n t a t i o n w i l l be a l m o s t e n t i r e l y t h a t i n l i n e a r momentum r e p r e s e n t a t i o n . f~ I n angular analogous t o Thus (TT 5_ Nfremi) 9 P O Similarly, i f ^NJX x ii.5.28 the other r e p r e s e n t a t i o n s a r e : r = n r - rfr - r r,t Mz Ui W2 Furthermore (d) S t r o n g Reflection From t h e s t a n d 4" C f l p o i n t II.3.3-4 t h e u n i t a r y f a c t o r s" o f $ h a s t h e same r e p r e s e n t a t i o n a s |~~ » i f one makes t h e 58 replacement \ ~? *l r ( 2.w +-2-£ f-Q s a unit operator. a unit operator Hence S being u n l e s s one Notice that f~ antiunitary will demands <| X is not be I. = s (e) R e f l e c t i o n Reflection i s defined particle in a left antiparticle handed frame I s t r a n s f o r m e d in a right o f s p a c e i n v e r s i o n and II.5.30 It as an o p e r a t i o n u n d e r w h i c h an handed f r a m e , i . e . I t i s the particle conjugation. product Hence one has: /MoCWroO i s c l e a r that a representation f o r only into a A will differ from I n a s m a l l d e t a i l o f a phase t r a n s f o r m a t i o n .since ii.5.32 t = 11,5*33 B)^ A c N(M^O~) ^ € A = &f{ Similarly,| = II.5.35 A= [iS r g-^ Also, writing 11.5.3k *" therefore I^2, - { ^ (i+ AN, Aj . AiAjla. - At + ^.) ^L^ CWm)-hlCfk£+ ^ NCt*w<tf j j^ [ i f 1 h*c*CM») -59- II.5.36 Further, A = others since itr Na g^. ••=- (H)^" . , "Silt ^(20-ft) m»r) ij^stfi] , II.5.37 (4+ F * * ) NCWwlT)]. therefore lew*)] lfrta«)+ and ll.5.38/]-f 4 i r l NftWi «*o F i n a l l y , one c h e c k s i that = From t h e d e f i n i t i o n o f ) | the trans&rmatxonnproperti&s o f and ~f~ one c a n w r i t e bCkl'WT) These a r e the same as., u n d e r JV . .However has A 1 ( f ) Weak R e f l e c t i o n ( J - |~ J One c o n c l u d e s t h e r e f o r e _ » ZMM)-left**)] f g f J* a s under 3" s follows i santiunitary. that the u n i t a r y f a c t o r t h e same r e p r e s e n t a t i o n as ^ down but i f J" t o be a u n i t o p e r a t o r , t h e n t h e p h a s e f a c t o r 2 Q~ o f J~ i s required ^ must be real„ (g) I n v e r s i o n (X~ X L ) I n v e r s i o n i s the t o t a l i n v e r s i o n o f a l l the space coordinates. Therefore w r i t i n g Y[- •=*] obtains, II.5.39 X t o l M } ) l + - ^bOtwi) T | ^ one time , -60- iT b CW.wi 2.J Xfc(^m2.JI = t 11.5.40 Hence, % _ X- 11.5.41 "P > WniT where y p _n v ~" ^ Alternately i f one a d o p t s (II.5.14) a s the correct (w. trans- f o r m a t i o n f o r J " , one h a s : ikCUZmi)! " 1 11.5.42 In = Ix. analogy w i t h o p e r a t o r s i n ( b bCk^-w 1) ) one c a n i m m e d i a t e l y write down: X - 11.5.44 x where - i ^ e m L=-<jOWJ>leX e t CcJifyU^drfTtoi^ whe r e 11.5 Similarly t h e o t h e r r e p r e s e n t a t i o n s f o r PC c a n be c a r r i e d o v e r f o r use a s I i f one makes t h e r e p l a c e m e n t CHAPTER I I I THE ELECTROMAGNETIC FIELD 1° F o r m a l i s m Maxwell's equations i n terms o f p o t e n t i a l s , ( @L , A 0 ) with III.1.1 III.1.2v Hj = CuAA. and -^ffcc/As- E_= Ao = Af-~ 0 w h e r e ^ ra,rj A.M-C M l ), Ajx^ CL\,ib*) obtains f o re x p l i c i t expressions of and 6 one Ao ^ With the h e l p o f t h e s e , a f t e r making c o n v e n t i o n s and are: tt°. Introducing the Lorentz c o n d i t i o n ( L X . HI.14 , ] D A - iWfJiv.A r-Aol = . i w h e r 9 i n . 1.3 ~ , the f o l l o w i n g t a b l e i s c o n s t r u c t e d : for J -62- n r e JL — — — -f- — — — t- — - — • — IH — — — — -f- •t- time charge space-time transformations. change s i g n u n d e r formations. -f- — — . — >.— — — — has b e e n c o n s i d e r e d as a s c a l a r under However as f a r a s M a x w e l l ' s c h a r g e c a n as w e l l be c o n s i d e r e d t o space i n v e r s i o n and t i m e r e v e r s a l t r a n s - I n the f o l l o w i n g t h e r e f o r e r e p r e s e n t a t i o n s f o r t h s . P . S . O . ' s w i l l be Assuming the u s u a l of the theory w i t h M a n d l 1958) — •— For a long theory i s concerned i - r v r s-nrT •4- Ao — all A=RF T given. t r e a t m e n t o f t h e gauge the help o f i n d e f i n i t e invariance metric, (see e.g. one c a n w r i t e T I I I . 1.5 Ap. ~ where III.1.7 ) 60- I S,,C4,S) = Ifel ( o • , w ' 1 9 r 9 T n e ^f* m a y D e polarization vectors I ) satisfy relations; expanded as the f o l l o w i n g -63- 111.1.8 ^ (A,K) ^ h U ' , *) = S»y A= 3 A - 4- , -6> - t£o The commutation r e l a t i o n s a r e : 111.1.9 Lb^(A) ; 2° R e p r e s e n t a t i o n s (A) bt'O'O - , o f d i s c r e t e symmetry Space i n v e r s i o n ( [~| ) Space i n v e r s i o n i s d e f i n e d operators. reflection ( A ) by the r e l a t i o n s : m.2.1 na«(.> rr = - e ' " ^ Ac 1 Using the expansion 111.2.2 the 3 representation h± (>>/!*•) = -e |A. o f one o b t a i n s bfc(A) , laws f o r n^u^^n^ Where t h e s u b s c r i p t and w r i t i n g : III.1.6 b£(X,K)- ^ A ( * , £ ) transformation 111.2. + 1 1 7 ^ " 4 as: Ur), V\\i i n d i c a t e s t h a t i n t e r m s o f k>_£(x,jO . •the r e p r e s e n t a t i o n of :53' f o r the. s c a l a r f i e l d III.2.k n M n^= R + V\\i i s a I n analogy with one c a n w r i t e i ^ ^ i ^ j ^ * * * * L B ^ N -g <x t -64- where 111.2.5 and Equation of so _ ^ ( A h >M ^v), . I I I . 2 . 3 does n o t f i x the t r a n s f o r m a t i o n £ ^ (A,£ ) 111.2.6 B^a.io- 0 ^ , 1 0 = and bfc fx) , Therefore making the ^ ^U,^) = C-l/g^CX^-Js) properties convention, one o b t a i n s that the r e p r e s e n t a t i o n i s : in.2.8 Using n^n . ru = e K + t h e method V * v o u t l i n e d i n Chapter I I f o r the s c a l a r field^ one c a n w r i t e down t h e t w o - d i m e n s i o n a l r e p r e s e n t a t i o n a s : 0 These o p e r a t e r e s p e c t i v e l y o n , 111.2.10(a) U (X»H f ^ ( ' X H bjc^) = ° ) 6 b^C*/ and 111.2.10(b) ^ j . fk-tW o-v t> and t h e i r ad j o i n t s . I k f c W E | | (0 o MA) (K >0) 3 -65= The will (This transformation properties of d i f f e r from those under i s obvious from the t a b l e (B) rV = At 1 a Time r e v e r s a l One c a n d e f i n e a n only i n t h e f a c t o r —| . + t\\\ {/\) Therefore there transformations, ~ A . weak r e f l e c t i o n ( X d - - i T r e and u s i n g ) ^ A|. Assuming the a n t i l i n e a r nature o f the time r e v e r s a l formation exists t i m e r e v e r s a l by TA^rr' m . 2 . 1 2 n ^ ( ""J" ) under III.l). a s i m p l e r e l a t i o n s h i p b e t w e e n t h e two III.2.11 \>A trans- t h e e x p a n s i o n I I I . 1 . 6 and t h e r e l a t i o n I I I . 2 . 2 , one h a s Tbfc(Xh)T III.2.13 same as f o r [~I • linear operator T= finds that (C) £ b^jc^'f*) , Hence t h e u n i t a r y f a c t o r <y factor reflection. ( S c a n be s e e n f r o m t a b l e of Therefore . Since o f the a n t i - J" i s the. .same a s f o r j\, III.l that [~~ However the u n i t a r y f a c t o r only ( X ) , and ). f o r m t h e p o t e n t i a l s i n t h e same way. same a s P ii ( T~" ) , i n v e r s i o n P a r t i c l e conjugation antilinear. which are the U.U i s t h e same a s f o r \"\ . S i m i l a r l y one the u n i t a r y strong It - 5s and S ^ ^ I of S changes the s i g n o f transs will /\jx , be t h e -66 b^(^/K) the ii. t r a n s f o r m as rb£(>,r*) 2.HI. j"~ will is^thus, phase f a c t o r r* = - b cx,h). £ s i m p l y a phase t r a n s f o r m a t i o n , p r o d u c i n g —| . a I t s r e p r e s e n t a t i o n c a n t h e r e f o r e be w r i t t e n down i n t e r m s o f p a r t i c l e number o p e r a t o r s Mg_(A,H^ as: I in.2.15 - t ~ A,M a l s o be a r e p r e s e n t a t i o n f o r s This w i l l as e x p l a i n e d earlier. The u n i t a r y f a c t o r for X It of A\A. i n t o transforms 3° R e p r e s e n t a t i o n s X I will be s i m p l y tHe:.identity itself. i n C i r c u l a r Components . i sphysically instructive t o w r i t e down a l l t h e above t r a n s f o r m a t i o n s i n t e r m s o f c i r c u l a r c o m p o n e n t s . do t h i s , I I I . 3.1 define ecKA) =. ~ bjc(L) ^ in.3.2 III.1.6 ni.3.3( To ^ {.£&,») -r-iite,-.)] = £*a/fO | bjctO - i Wf2) ] - b/cR). then becomes: a ) A IM ;= z ^[f «*cs,u £cm- f •j^ H e r m i t i a n conjugate e?i£40lk«) 1 IKK 1 " Q I zr^ j e' + ej + k -67- js I tV ty'c)^! iu.3.3(b) Using the convention 4) f o r m r i g h t handed formation and and l e f t handed properties of bfcd) that bk(Rj f and £ ( K ^ ) (fl(=L,R,3) Hence Using 111.3.7 (3) The t r a n s f o r m a t i o n s f o r A (=01") d i f f e r from these 111.3.8 (k) r o f phase ~| in.3.9 (5) (6) will = k . only f (L) »>-*<'>. and i n t h e phase f a c t o r a phase J " [= & I Tj) —| . transformation , lbj frJi' = from eHs,3\ the trans- H t a g a i n be s i m p l y differs — rb (r)T = > ikjcCm^ S 3 J - transform as: (2) Tb^CL) + y I n t h e t a b l e one f i n d s ( t=3,^-) £ T triads. A^- l i s t e d and b ( f ) C K , respectively 111.3.6 will . I I I . 2 . 6 , one h a s i ( X L ) =- £ III.3.4 e + b£we J ilw w i CD . t i n that i t introduces a phase f a c t o r — \ The ' r e p r e s e n t a t i o n s may t h e n be w r i t t e n down, u s i n g t h e method explained Let L and ,R V i n Chapter I I , as f o l l o w s . , and T~ 3 s 4- „ Also l e tthe subscripts on t h e D . S . O . ' s denote r e p r e s e n t a t i o n s i n terms £ -68- of b^CjM and , respectively. 111.3.10 ( l ) 111.3.11 in.3.12 III.3.13 where A* = b j < ( L ) N*:!^ k * U ) Jb_t(R) n= ^ i l ? f , ( X ' ^ ~ ^ ;;jj r B (r) = b^(rJk^a) t U ^ n r r N^fr)«• J£fr) bj<cr) and , ni.3.i6(3) A=n= nln?" m . 3 . 1 7 (4) r= f (5) I where £ [f^ ) =«xp{-cf + (R) = factor T0 + t j=ti/tf + r , N*o?))j # ff,f[^Mfe(r)-C^J-ClCr>^IJ Q ^ U S a n d cr (6) The u n i t a r y I I I . 3.20 where e ^ H i f J l N f c / W - g^ftoj/ i n . 3.15 III.3.19 etc. ^ (r 111.3.1U (2) III.3.is J ^ O . ) S a n d CfeCO = b ( 3 ) b / « - ^ E of S i s ; CHAPTER IV THE POUR-COMPONENT SPINOR FIELD 1° Notation. The Dirac equation i s (for d e t a i l s , see e.g. Bogoliubov, 1959 K IV.1.1 tf"|A * ^ ~ / 0 where W is the mass of the p a r t i c l e and 'r* i s the f i e l d function. The following representation f o r the Dirac matrices w i l l be used iv. i . 2 <H = IV.1.3 The equation iv.1.5 The t (? *1 = , !|) = (<1-J. adjoint to the Dirac equation i s : Tf^ - IV. 1.4 ( o '^)^ 1TK= Wi" jjf, - = , 0 where „ solution of the Dirac equation can be written i n the form iv.i.6 -fCv)^ff l ^ normalized C s ) a i n a box of volume ^ ( S J ) e + ^CsWjT^e J -70- are respectively ( T - | particles J_ ^ S - | ") Ufc (S) r (a) e (S=2). « •2 * u r e cl X{ii(SJ(7cs) (b) (o) One f u r t h e r r e < and a n n i h i l a t i o n o p e r a t o r s o f ) and a n t i p a r t i c l e s ( \~2. and a iv.i.7 creation t° s a t i s f y t h e f o l l o w i n g VCS:)7(S)) - fe 2U (s)I (s) = M U^.Cs) The s p i n o r s V. (r)= VtV) U/cf) if = (unit V. tr> = t s5 (b) V^, I.S) repre i s adopted. i \ 0 IV. 1.9 Ufcfl)= 0 For the adjoint art 7 e q u a t i o n , one h a s \ matrix) *•].,(. condition: t following explicit and o -35 [ £ M K ~ imposes t h e n o r m a l i z a t i o n spin relations: | (a) tik&>wis')v2cs)\/)<cso =• r / IV.1.8 The ) with 0 J or -71- i > tu IV.1.10 (2) - 14V) T 4 - o hi H O o,l,- _ aw , 1 2.CJ The a d j o i n t wave f u n c t i o n i s 2° B i l i n e a r s o f t h e D i r a c The D i r a c field i s c h a r a c t e r i s e d by t h e f o l l o w i n g c r e a t i o n and a n n i h i l a t i o n IV.2.1 4 Prom t h e s e will ±fe tu), field % operators: 0,--) a t ± (2,i) contain bothVannihilation operators. into 8 types. construct a unitary operator. there bilinears , d±£i*.i). one c a n f o r m 120 b i l i n e a r s o u t o f w h i c h o n l y c a n be g r o u p e d that ; Each o f these and t h e o b s e r v a b l e s These 61+ b i l i n e a r s c a n be u s e d t o I t was shown i n C h a p t e r e x i s t s an i n t e r e s t i n g c o n n e c t i o n 61+ II.3 between the w h i c h e n a b l e s one t o s e l e c t C =72- the a p p r o p r i a t e b i l i n e a r s f o r t h e symmetry (1) As a n example,consider, fcfcts,r)s iv.2.2 I t s commutator [BK C5,r^Bs"(s,r)]_- so t h a t t h e momentum o p e r a t o r = J- s i g n u n d e r a symmetry r example ^ t h e r e f o r e H ?_ I n an e n t i r e l y ? that i s said In fact BjL^r) £ I nthe t o be t h e c h a r a c t e r i s t i c h a s the f o l l o w i n g ^Js(S, T) : flfcttrj^-a^r-) and [> (s,rJ,ajc^n]j= fe a n a l o g o u s manner one f i n d s t h e o t h e r whose p r o p e r t i e s a r e l i s t e d tfJ^T). bilinears below: (2) I d e n t i t y t r a n s f o r m a t i o n : Yl^^S.T)— d^C^r) changes symmetry s p a c e i n v e r s i o n ( f ] )» commutation r e l a t i o n s w i t h [Bfc&rt, CSA)~ /?_<ts.r) > Q , only ^2 9 t r a n s f o r m a t i o n , then t . \ k-Pjc ( s . r ) . transformation i s called iv.2.k . i s given by; I f out o f the three observables b i l i n e a r of P_^Cs,r) gives; £ p - a^Cs,r)^Cs,-r) p Cs,r)= iv.2.3.a iv.2.3.b operators. fl^tS,-r) The c h a r a c t e r i s t i c b i l i n e a r i s : =• particle number operator. C o m m u t a t i o n r e l a t i o n w i t h C.O.'s and A.O.'s a r e : iv.2.5 LM Cs,r) a ts,rjL=-afeCs.r; ; t > l£ ( / l ^ r ) , a?0,r)]_= fl£(*rj -73- (3) Time r e v e r s a l t r a n s f o r m a t i o n . s i g n a r e _T and JP . The o p e r a t o r s t h a t change These c a n be f o r m e d f r o m t h e commutators o f , ^ -A have t h e p r o p e r t y (I).) P a r t i c l e appropriate conjugation. Q Here changes s i g n . The bilinear i s : C K CS,r) - d£CS, rj Ojc (S, r') = C f c U , r ' ) and has , t h e pro p e r t i e s : iv.2.9 a£is,r)~L=-a£ts,f), Ic^r), (5) R e f l e c t i o n ( A= R T the appropriate iv.2.io A^s iv.2.11 £ S/ and Qs b o t h change s i g n , fe A^cs.rO, Cs,rO= have t h e f o l l o w i n g c o m m u t a t o r s w i t h [ A ^ l The b i l i n e a r M A ) ! L = -4fcCs.ro S X - fl T a/c^r; being, Ak( r)- a£ f s , r ) < l (6) I n v e r s i o n ( iv.2.12 bilinear ). lc&(s.r), o^ts^'i]- )• ? Only s p i n [ o/s\ A ^ l changes s i g n . therefore i s x \ i s , r ) ^ a&ls.rja^cs'x r) - PjjzLs'jr). r')]_= a / ^ r i Its commutators w i t h c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s ( CJ - T T (7) Weak r e f l e c t i o n Q , a l l change sign. are: In t h i s case JL , JP )« and The b i l i n e a r a p p r o p r i a t e f o r c o n s t r u c t i n g the u n i t a r y f a c t o r of the a n t i l i n e a r operator is: <Sfc(s,r)= IV.2.13 The commutators w i t h IV.2.ii+ (8) %*cs,rj % 6s,r) sign. ( a\(s,r) (s,r) T h i s has the properties: 3° ). = ; Here o n l y a^u'v") = [F tf4V ' ^-=- fc( ' ' , s r a s £ r ) (i) iv.3.1 iv.3.2 symmetry t r a n s f o r m a t i o n s . F^ts',*') 1 (GLR-.- n^ti+)n= n*c^)n -(-*,+) ( + Q . %V,rO]_= a/(s,r). 's under . 1959) Space i n v e r s i o n ( [ ~ l ) Is defined by the f and is; C^ ' ^ 5 £ The t r a n s f o r m a t i o n p r o p e r t i e s o f C . O . ' . s and A . O . discrete It S^flTT [ % ( s ; r ) d £ c 5 ' , r ' ) ] = a£cs,r). The c h a r a c t e r i s t i c b i l i n e a r iv.2.15 iv.2.16 are J afe(S/0].= - ^ ( S i V ^ I^SA), Strong r e f l e c t i o n change -L^cs',*-') relations I*** 1T4.. i s then the consequence o f commutation r e l a t i o n s that |1TT|~! . -75- One further requires f o r physical reasons that the ..double operation yields the. i d e n t i t y . .Thus On account of the well known ambiguity i n the sign of a spinor the above identity transformation corresponds to iv.3.k 1* = * l % - ^' > ^ a m { i n 3.1 and using Ci. ,*t) Substituting the expansion f o r 1 the relations IV.3«5 one obtains on comparing iv.3.6 coefficients, nQjs ^n'= M-*fro , ( i i ) Time reversal ( T ) i s defined by iv.3.8 T^U,-t)T = IT ? + iv.3.9 T f U i l f 1*<F(1 " t l The n%cs^)iT=-^^( ts H 1 4- by 4" matrix , X A N D has the following properties: iv.3.10 V l O O D ' 0 0 0 -t | 0 O i o X ^ - —i o o o oo V o o o | - io -76- The s p i n o r components transform as f o l l o w s : i n IV.3.8 and 9 and remembering that ~ p On s u b s t i t u t i n g these i s a n t i u n i t a r y one o b t a i n s : iv.3.12 T a j c C f t O T " * l W\^',0, TaXtsA)f^\*^)\^2) 1 T (iii) ( f~ ) i s d e f i n e d by P a r t i c l e conjugation IV.3.13 _ I t may be remarked that since i n s p i n o r space 'if- i s a column v e c t o r and l j - a row v e c t o r , the a p p r o p r i a t e trans- p o s i t i o n o f a l l s p i n o r s on one side o f the equations Is i m p l i c i t l y assumed. dtf IV.3.14a C H The r e p r e s e n t a t i o n f o r Cl C iv.3.14b = If2,5a * - and V transform a s : C = -(T is: 0 \ / 0 0 0 —I 0 0 i 0 0 o x> 0 —i LI Cl i s d e f i n e d by, The matrix = — Yj*. , IV.3.13 o and C = + <f' . -77- iv.3.15 Using ^ these ^ i n t h e d e f i n i t i o n I V . 3 . 1 3 one o b t a i n s : r^or^=v * a iv.3.16 It i s obvious that without the double o p e r a t i o n cs 2) > ra±M any r e s t r i c t i o n o n t h e phase i s an i d e n t i t y CS„ f) under 1r $ transformation. (iv) Product transformations. properties of r* = T/^C-,!). The t r a n s f o r m a t i o n [\ , J " , X a n d ST c a n be w r i t t e n down a s : iv.3.is Jat(s.i) j 1 ^ V ^-K^! > )S 2) ^ ^ J ^ ^ M ^ a i o iv.3.19 xa^c^Oi" = c-if 7j a^Cs^s), ifl cs,2ji = e i f ? / ^ ' ^ iv.3.20 Sa^aDs"^ f 7 1 t H s M^Uj , Sdtts^js^eif'Zs^ ''). 5 I t may be r e m a r k e d t h a t t h e d o u b l e o p e r a t i o n to . t h e . i d e n t i t y but i n case o f A A A" 2 , T a factor 2 , X i n c a s e o f le'ads and X , i t i n introduces H -I - I n c o n t r a s t ..for the' s c a l a r f i e l d and I " a r e a l l u n i t 2 operators. 4° C o n s t r u c t i o n o f r e p r e s e n t a t i o n s . I n 3° t h e t r a n s f o r m a t i o n p r o p e r t i e s o f c r e a t i o n and t—'i | , a n n i h i l a t i o n operators have b e e n w r i t t e n down. b a s i c b i l i n e a r s f o r a l l B . S . O . ' s were f o u n d . of the r e p r e s e n t a t i o n s The I n 2° t h e The c o n s t r u c t i o n i s t h e r e f o r e the next step. identity IV. k 1 utilize L'BC,A]_S B t c , f t | - C ^ A l + C e n a b l e s one t o + t h e methods o f c o n s t r u c t i o n o f r e p r e s e n t a t i o n s f o r boson f i e l d s . The p r o c e d u r e i s t h e r e f o r e t h e same. present s e c t i o n two-dimensional representations given. I n 5° o n e - d i m e n s i o n a l r e p r e s e n t a t i o n s (I) w i l l be will be o b t a i n e d . Space i n v e r s i o n ( (~| ) The t r a n s f o r m a t i o n p r o p e r t i e s o f $£_(S;lr) IV.3»6 and t h e b i l i n e a r b y I V . 2 . 2 . representation f o r ^l^CS,,!) such I n the a r e g i v e n by I n order to geta one must, have a H e r m i t i a n -XI that iv.4.2 [n, M s ^ . ^ - i ^ f l - f c t s , ! ) , [ J U 4 ( s j ; ] _ = . a f e cs,n. y £ Similarly for 0|c(S,i) s The e x p r e s s i o n f o r jQ_ c a n be w r i t t e n down i n a n a l o g y with the boson case a s : 4 s'fc' L A representation f o r I I J/ — then i s -79- where t h e s u b s c r i p t ( -T") i n (~j +representation i s valid i n d i c a t e s t h a t the (iS r) for y t L j i O W ) or *| (k >o 3 (f= 2_ ) . ) .I t IV.I4..2 o r I V . 1 + . 3 that 3 i s c l e a r from second o f the e q u a t i o n s for f o r o n l y fc >0 t h e phase- f a c t o r o b t a i n e d A representation f o r i s - l * (f^\) CL^CS^r) r there- fore obviously i s 1 These c a n be combined i n a t w o - d i m e n s i o n a l IVJ + 0 and li (n.o/ c 6 OM&0 = lo form a s : %<Xrjj, 0_^(^rj^| 0 f Then IV n-ru 1, I D 1 1 |~| ^ (1- There a r e two c a s e s i n w h i c h (a) I f U = (b) I f 'ITT = ± (ii) The t i m e = f|- fl V , t t L < r e v e r s a l t r a n s f o r m a t i o n c a n be w r i t t e n a s r a product o f a u n i t a r y f a c t o r and a f a c t o r o f complex conjugation. A glance Thus a t the equation p r o p e r t i e s o f QL^l},^) Using I V . 3 . 1 2 shoii/s t h a t t h e t r a n s f o r m a t i o n f o r 5 ~\ and X, t h e r e l a t i o n s I V . 2 . 6 one o b t a i n s f o r are d i f f e r e n t . S —f -80- It i s c l e a r that Xlj does not give the c o r r e c t property for 2. IV.4.8a U,= € ' ' ( . Writing , 2 o n e Hence the r e p r e s e n t a t i o n f o r S = 2 The r e q u i r e d IV.4.11 f i n d s that w i l l be two-dimensional r e p r e s e n t a t i o n ii= f ^ U f | 0 * ( UU l o transformation then i s ) . / o and Cl^(SJ) i y, are defined as a^J. [o Then (iii) UJ* / — P a r t i c l e conjugation. o f IV.2.8 one gets a Using ^ - number, the b i l i n e a r s the r e p r e s e n t a t i o n f o r &tQSsl) as £ ^ fs,J~) -81- 1 «k to This gives the wrong s i g n fl^ ( S , 2.) ation for S=2. IV.4.13b ~ <~"|^~ iv.4.14 r e p r e s e n t ar t= i o n /I ° s: t so that the two dimensional f ^ and ( i v ) R e f l e c t i o n ( A )• Q£.l$ f) Hence the r e p r e s e n t - is; r | of . /' The t r a n s f o r m a t i o n 0 properties under r e f l e c t i o n are given by IV.3.17 and the f r e l e v a n t b i l i n e a r by IV.2.10. Xl_ can be e a s i l y seen to be IV.I .is ) -"-~s6 t The two by two r e p r e s e n t a t i o n i s , (v) Weak r e f l e c t i o n ( J " )• J " heing a n t i u n i t a r y may be w r i t t e n as IV.4.17a J ^ l L where Using the b i l i n e a r s IV.4.17b 0 = € G"^( >0 s J i s the u n i t a r y f a c t o r . Q$[.2.|3) one f i n d s , -83- The two-dimensional r e p r e s e n t a t i o n - f o s l is: A 4-is Cs>r) 1 0 J 5° One dimensional 0 aj<(s r) x representations. By a p p l y i n g a p p r o p r i a t e u n i t a r y t r a n s f o r m a t i o n s two dimensional form can be d i a g o n a l i z e d such that d i a g o n a l elements are the' same.One thus obtains dimensional r e p r e s e n t a t i o n . entirely To analogous the two a one- The procedure f o r doing t h i s to that followed illustrate the f o r the scalar the method the c a l c u l a t i o n s i field. f o r space i n v e r s i o n t r a n s f o r m a t i o n are given i n d e t a i l . Consider the u n i t a r y t r a n s f o r m a t i o n / 0 IV.5.2 One can w r i t e , IV.5.3 ln-0 = [o n+y o a £ ^ r ) ) 'fit o HM \ / + y o nl\ 0 p-K^D o wo a\/b + ni -Qk- One then shows that Thus one obtains f~J as n = n+iv = n~ru , i v . 5 . 5 Since the o r d i n a r y r e p r e s e n t a t i o n f o r + ^ - rf | , ^ t one obtains, for the r e p r e s e n t a t i o n iv„5.6 n= \^ - -f I f t ~ One then checks that U =- The o r d i n a r y r e p r e s e n t a t i o n f o r other d i s c r e t e symmetry operators are l i s t e d a r Ml where i s given by IV.l+.l3a r -Ni e " r e f l e c t i o n operator - X, ( M 2 i s given by IV.i+.19a. The and € i s given by IV„i+„l5 and r e f l e c t i o n o p e r a t o r by IV.I4..17. where ~ ^ =. \ ^ iv.s.8 The i n the f o l l o w i n g '„ r r, - r, w iv.5.7 identity. the weak i n v e r s i o n operator i s -85- The u n i t a r y f a c t o r o f the a n t i l i n e a r o p e r a t o r o f s t r o n g r e f l e c t i o n i s given by I V . l | . , 2 0 . 6° A l t e r n a t e Representations: Because o f anticommution o f f i e l d o p e r a t o r s f o r Fermi f i e l d s , the squares o f C O . ' s and A.O.'s v a n i s h identically- T h i s opens up the p o s s i b i l i t y o f r e p l a c i n g e x p o n e n t i a l r e p r e s e n t a t i o n s f o r D.S.O/.'s by a f i n i t e bilinears. I n t h i s r e s p e c t the f o l l o w i n g theorem i s v e r y u s e f u l : " I f an o p e r a t o r K i s such t h a t exV.tKH)=KKC-Xfr.ci<)-0 where In p a r t i c u l a r f o r K - ^ f r and iv.6.1 sum o f terms o f exV».C+\5) Notice, that - v * - n - C\-*-^-^ — \ A K™— K i s real". °(=±T[ w^-tw implies , then one o b t a i n s c = ~ ^ - _n2T — - a—n- Furthermore, one can w r i t e IV.6.2 ex^-O - Z. A - 0 = T exV.0!.J"-O - T O-v-"-^ 15 In the f o l l o w i n g t h i s method w i l l be used f o r c o n s t r u c t i n g some o f the D . S . O J . ' S . ( i ) Space i n v e r s i o n . Consider ^he u n i t a r y operators IV.6.3 ^^^WlO-^CS.T^Ti^Ls.r^T^^cs,,) V \ * B \ c s , 0 --^S^ 86- Under t h i s u n i t a r y t r a n s f o r m a t i o n the o p e r a t o r s j are r e s p e c t i v e l y transformed into and the o p e r a t o r s are respectively IV.6.2j.B transformed There In are two into c\v <s,0, *\ y-AO^v. *0 » -""l t h i s case the operator i s v a l i d f o r both symmetry o p e r a t o r IV.6.5 < <s cases? and the t r a n s f o r m a t i o n formulae Tl-l'l r\ e a s i l y checks • "T* that <Xts,Y) and A.-* '*> ts v are the same as f o r the Thus U-nvcti.^-^ts^-^CavtW.ri where the prime on One c ^^(-4,^)5 i n d i c a t e s the product f o r a l l V > x fe <, W~ =• \ „ Hence, u s i n g IV. 6.2, one o b t a i n s : iv.6.6 (2) V H tV (n jcsir)-'n^f)*^ 1l?b » ^ r ts 1 . d i f f e r i n sign. the operator Then -<\ r ' 3). Ci r) so that IV.6.4 (A) and In order to r e s t o r e the s i g n one can u s e (B) A l s o , Let IV.6.7 Then - ^ -^k ts,r) - T L ^ r r ) - * . x T i ^ l i . r ) ^ . ^ is, f) ^ iv.6.8 ^ T ' ^ n C s,r) One then checks that IV.6.9 n^T^fc-MLTV '^ 5 such that iv.6.io VC~ Cv^tSir)^" -4 - ( i i ) P a r t i c l e conjugation Here one IV.6.11 ( \~ ) obtains tS) =• where is»r) -0 l and VV\ _ J 1 ^ L S ) - _ z A-»c (.j) -I Hence IV.6.12 r - I I (\-^CS,i^1\ ts^)^ ^ K one then checks that IV.6.13 ^ \ > so t h a t C T" "" 1 £ $ i 0 + \, ^ C t e C t i ^ -88- ( i i i ) Reflection ( A ) I f one w r i t e s ^ iv.6.ii*. one o b t a i n s iv.6.15 _ \ for nr from IV.I4..15 ^--I Jl^^,v^«.o »J« .J.1m , + 1 ftllt4 ,^ l t „ * $ ( i v ) Strong R e f l e c t i o n ( S ) Prom IV.i|..20 one has (v) Weak R e f l e c t i o n ( X )* b e i n g antiunitary,may be w r i t t e n as, IV.6.17 To f i n d one 3" \_ T ^ T where 3 i s the u n i t a r y one can use the b i l i n e a r s & K <-S|T) factor. o f IV.2? gets, iv.6.18 - H - - L t^jtkt t ^ - G u c \ , 0 ) ^ (vi) Inversion ( 7X) ° Since i n v e r s i o n i s an a n t i u n i t a r y o p e r a t o r , i t may be w r i t t e n a s : IV.6.19 X ~ X ^- where X i s the u n i t a r y factor. .89- The c h a r a c t e r i s t i c b i l i n e a r f o r c o n s t r u c t i n g to be l SiT) One f o r S ~ \ same T ; . i n t h i s case one gets two r e p r e s e n t a t i o n s , , the o t h e r f o r tSiO a n d ts',r3 S - 2- , because f o r the have both the same phase f a c t o r whereas i n forming a n -O- , i f e f f i c i e n t "\ then that o f the t r a n s f o r m a t i o n "X_ was found property T> k <-S,Y-} has co- is'.v) w i l l be ' " l * " , so t h a t f o r O.^ isS-r) w i l l be wrong. One o b t a i n s f o r S — \ , IV.6.19 H \ C ^ C i A ) - K ^ ' 0 ) ^ C b u ^ O - ^ v ^ L ^ BIBLIOGHAPKY 1« ,.:2. B l a t t , 3.n* and tfeiaskopf F.S. Theoretical Muelear Physics* Jofaa Wiley and Sons, Kew York* 1952. Bogollubov, a n a Shirkov, 0»V« Introduction to th© Theory o f Quantised F i e l d s , Set* York, Interseience, 1959• 3. Poldy ( see Nigem). 2j.o Geistraacher, P.H, Theory o f Matrices, Chelsea Press, Rett York, 1959. 5. Srawert, X«8ders, G., Ho U n l i t , B.. (OLR-1959) Fort s c a r c e Ber Physifc X , 201-328 (1959). 6* Hamilton Clarendon 9 9 The Theory o f Elementary P a r t i c l e s * Oxford; 1959. 7» Hauadorff P., L e i p s i g , Ber. Ctoa* Miss, l a t h . Phys. So Jauch, and Holirlich, F. The Theory o f Photons asd Electrons, Cambridge, Mass.§ Addisosa-ttfealey, 1955. 9. Kaempffer«, P.A. K l . £8, 19 (1906). 0an<> <T# Phys. 22 (1961). 10. Kenimer, I . , Polklnghorne, J.C», Pursey, X>.L. Heporte on Progress i n Physics, 2 2 , 368 (1959) londora. 11 o riaders, Annals o f Phys», 2 , Qrawert) • 12. Bland 1, P« 1, (1957) (see also Introduction to Quantum F i e l d Theory, Hew York, Interscience, 1959c 13. Nigon, P. and Fold, U L . Ii}.. 15o Park, David. Am* J . Phys 26, 215 (1950). P a u l i , W. I l e l e Bohr and the Development o f Physics, Fergamon Press, London, 1955. 16* Havenhall, (see Wslfenateiia). 1?. Soman* P# Theory o f Elementary P a r t i c l e s , SbrthBollaad Publishing Company, Amsterdam, 1959. l8« Sachs, 1*0,, Phys-. Ho v. 8 7 , Phys* Rev. 102, lljlO (1956). e 100. (1952). -91- 19. SchrSdingor* S 20* Schwlngei?, 21. Sehwinger, 22. tfetanabe, S* 23. Hatanabe 2l,u WX&a8r, S<« Gdttlngoz* JSaohrlchtan £ 1 , 5Jj.6* (1932). 25. • 9 0 Spac© fim© StmctupOj, Cesibi»ldg8, Phys* J* HQV. 8 2 , 91k Phys* Rev. £1, He v. Mod* Fhys, a 2£, 40 (1955). 9 L.„ and Ravenball, D. 279 (1952).. (1951). 7X3 (1953). S«, Phys* flev. 8ig. 1008 telfonsteln, 1950. (1951K Phys. Hov* 88, —-
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Representations of discrete symmetry operators in quantum field theory Mariwalla, Kishin Hariram 1961
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