Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Representations of discrete symmetry operators in quantum field theory Mariwalla, Kishin Hariram 1961

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1961_A6_7 M27 R3.pdf [ 4.29MB ]
Metadata
JSON: 831-1.0085330.json
JSON-LD: 831-1.0085330-ld.json
RDF/XML (Pretty): 831-1.0085330-rdf.xml
RDF/JSON: 831-1.0085330-rdf.json
Turtle: 831-1.0085330-turtle.txt
N-Triples: 831-1.0085330-rdf-ntriples.txt
Original Record: 831-1.0085330-source.json
Full Text
831-1.0085330-fulltext.txt
Citation
831-1.0085330.ris

Full Text

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,  —-  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0085330/manifest

Comment

Related Items