{"Affiliation":[{"label":"Affiliation","value":"Science, Faculty of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."},{"label":"Affiliation","value":"Physics and Astronomy, Department of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."}],"AggregatedSourceRepository":[{"label":"AggregatedSourceRepository","value":"DSpace","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","classmap":"ore:Aggregation","property":"edm:dataProvider"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","explain":"A Europeana Data Model Property; The name or identifier of the organization who contributes data indirectly to an aggregation service (e.g. Europeana)"}],"Campus":[{"label":"Campus","value":"UBCV","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","classmap":"oc:ThesisDescription","property":"oc:degreeCampus"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","explain":"UBC Open Collections Metadata Components; Local Field; Identifies the name of the campus from which the graduate completed their degree."}],"Creator":[{"label":"Creator","value":"Mariwalla, Kishin Hariram","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/creator","classmap":"dpla:SourceResource","property":"dcterms:creator"},"iri":"http:\/\/purl.org\/dc\/terms\/creator","explain":"A Dublin Core Terms Property; An entity primarily responsible for making the resource.; Examples of a Contributor include a person, an organization, or a service."}],"DateAvailable":[{"label":"DateAvailable","value":"2012-01-18T20:36:02Z","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"edm:WebResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"DateIssued":[{"label":"DateIssued","value":"1961","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"oc:SourceResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"Degree":[{"label":"Degree","value":"Master of Science - MSc","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","classmap":"vivo:ThesisDegree","property":"vivo:relatedDegree"},"iri":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","explain":"VIVO-ISF Ontology V1.6 Property; The thesis degree; Extended Property specified by UBC, as per https:\/\/wiki.duraspace.org\/display\/VIVO\/Ontology+Editor%27s+Guide"}],"DegreeGrantor":[{"label":"DegreeGrantor","value":"University of British Columbia","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","classmap":"oc:ThesisDescription","property":"oc:degreeGrantor"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the institution where thesis was granted."}],"Description":[{"label":"Description","value":"The object of the work reported in this 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 for the operators of space inversion (\u2293) particle conjugation (\u2310) and time reversal ( T ) have been given.\r\nStarting from general considerations 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. The procedure consists in enumerating the bilinears in creation and annihilation operators. It is shown that eight symmetries are the only possible ones.\r\nIn view of the TCP - theorem and the so called non-conservation of parity in weak interactions, the product operators, such as reflection ( \u22c0 = \u2293 \u2310 ) and strong reflection (S = \u2293 \u2310 T), in addition to time reversal, should be considered as the most basic symmetries.\r\nWorking in linear momentum representation, the unitary operators \u22c0, \u2293, \u2310, E ( = identity) and the unitary factors of the antiunitary operators: S, I = \u2293 T, J = T \u2310 and T are constructed for the following free fields: (I) The non-hermitian scalar field representing, for example, kaons. (II) The electromagnetic field. (III) The four-component spinor field.\r\nThe operators for the scalar field have also been worked out in the angular momentum representation. Using the anti-commutation relations for C.O.\u2019s and A.O.\u2019s, 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.","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/description","classmap":"dpla:SourceResource","property":"dcterms:description"},"iri":"http:\/\/purl.org\/dc\/terms\/description","explain":"A Dublin Core Terms Property; An account of the resource.; Description may include but is not limited to: an abstract, a table of contents, a graphical representation, or a free-text account of the resource."}],"DigitalResourceOriginalRecord":[{"label":"DigitalResourceOriginalRecord","value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/40154?expand=metadata","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","classmap":"ore:Aggregation","property":"edm:aggregatedCHO"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","explain":"A Europeana Data Model Property; The identifier of the source object, e.g. the Mona Lisa itself. This could be a full linked open date URI or an internal identifier"}],"FullText":[{"label":"FullText","value":"REPRESENTATIONS OF DISCRETE SYMMETRY OPERATORS IN QUANTUM FIELD THEORY by KISHTN H. MARIWALLA B . S c , The U n i v e r s i t y o f Bombay, 1956 A THESIS SUBMITTED I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n t h e Department o f PHYSICS We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1961 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by t h e Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department The U n i v e r s i t y o f B r i t i s h Columbia Vancouver $, Canada. Date U^.l t'iU FACULTY OF G R A D U A T E STUDIES PROGRAMME OF T H E FINAL ORAL EXAMINATION FOR THE DEGREE OF HlSlEA&IWG) DOCTOR OF PHILOSOPHY o R A L e X A M of 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 COMMITTEE IN CHARGE Chairman: G. M. SHRUM F. A. KAEMPFFER P. HARNETTY R. BARRIE E. LEIMANIS J. GRINDLAY C. A. SW ANSON L. de SOBRINO P. RASTALL External Examiner: L. E. H. TRAINOR University of Alberta, Edmonton, Aha. F O R , HIS R E P R E S E N T A T I O N S O F D I S C R E T E S Y M M E T R Y O P E R A T O R S IN Q U A N T U M T H E O R Y OF FIELDS A B S T R A C T The object of the work reported in (his thesis was to con-struct 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 repre-sentations. Furthermore, in the work reported hitherto, only in-complete 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. The 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-conserva-tion of parity in weak interactions, the product operators, such as reflection ( A = l~l D and strong reflection (S = r! [~ T) , in addi-tion to time reversal, should be considered as the most basic sym-metries. 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) The non-hermitian scalar field representing, for example, kaons. (ii) The electromagnetic field. (iii) The four-component spinor field. The operators for the scalar field have also been worked out \u2022 in the angular momentum representation. Using the anti-commu-tation 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 representa-tion 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 F. A. Kaempffer Special Relativity Theory p. Rastall Electromagetic Theory G . M . Volkoff Nuclear Physics J. B. Warren Related Studies: Dynamical Systems I E. Leimanis Differential Equations C. A . Swanson Functions of Complex Variables A . H . Cayford i i ABSTRACT The o b j e c t o f the work r e p o r t e d i n t h i s t h e s i s was to c o n s t r u c t and study the e x p l i c i t r e p r e s e n t a t i o n s o f d i s c r e t e symmetry o p e r a t o r s (D.3,0,'s) i n quantum f i e l d t h e o r y . I n s p i t e o f the c o n s i d e r a b l e importance of the D.S.O.'s i n p r e s e n t day p h y s i c s , not much has been r e p o r t e d I n the s y s t e m a t i c study o f such r e p r e s e n t a t i o n s . F u r t h e r m o r e , i n the v\/ork r e p o r t e d h i t h e r t o o n l y i n c o m p l e t e r e p r e s e n t a t i o n s f o r the o p e r a t o r s o f space i n v e r s i o n ( \\\\ ) p a r t i c l e con-j u g a t i o n ( \\~~ ) and time r e v e r s a l ( X ) have been g i v e n . 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 t r a n s f o r m a t i o n s , w i t h the a s s o c i a t e d c o n s e r v a t i o n l a w s , a s y s t e m a t i c procedure f o r c o n s t r u c t i n g the r e p r e s e n t a t i o n s of the D.S.G.'s has been f o r m u l a t e d . The procedure c o n s i s t s i n enumerating the b i l i n e a r s 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 symmetries are the o n l y p o s s i b l e ones. I n v i e w o f the T C P-theorem and t h e so c a l l e d non-c o n s e r v a t i o n o f p a r i t y i n weak i n t e r a c t i o n s , the p r o d u c t o p e r a t o r s , such as r e f l e c t i o n ( \/\\ - FIT ) and s t r o n g r e f l e c t i o n ( S \u2014 HI T ^ * a d d i t i o n t o time r e v e r s a l , s h o u l d be c o n s i d e r e d as the most b a s i c symmetries. Working i n l i n e a r momentum r e p r e s e n t a t i o n , the u n i t a r y o p e r a t o r s A, Pj > r , E ( = I d e n t i t y ) and the 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 o p e r a t o r s \"T \u2014 1 \\ , and T are c o n s t r u c t e d f o r the f o l l o w i n g f r e e f i e l d s : (1) The n.on-Hermitian s c a l a r f i e l d r e p r e s e n t i n g f o r example kaons. (2) The E l e c t r o m a g n e t i c f i e l d . -(3) The four-component s p i n o r f i e l d . The o p e r a t o r s f o r the s c a l a r f i e l d have a l s o been worked out i n the a n g u l a r momentum r e p r e s e n t a t i o n . U s i n g the a n t i -commutation r e l a t i o n s f o r \\ G \u00bb 0, 1 sT)and i A , 0 . ' s , an a l t e r n a t e c o n s t r u c t i o n o f D.S.O.'s o f the D i r a c f i e l d i s e x h i b i t e d . More t h a n one r e p r e s e n t a t i o n has been g i v e n i n each c a s e , I n a d d i t i o n a two d i m e n s i o n a l m a t r i x r e p r e s e n t a t i o n has been given., I t i s shown t h a t by an 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 , t h e s e c a n be reduced to the o r d i n a r y form. i v T A B L E O P C O N T E N T S INTRODUCTION 1 C H A P T E R I G E N E R A L CONSIDERATIONS 8 1\u00b0 Symmetry P r i n c i p l e s 8 2\u00b0 The D i s c r e t e Symmetry O p e r a t o r s ( D.S.O.'s) 12 C H A P T E R I I ' D I S C R E T E S Y M M E T R Y O P E R A T I O N S P O R T H E S C A L A R F I E L D 19 1\u00b0 D e f i n i t i o n s and N o t a t i o n s 19 2\u00b0 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 under Symmetry T r a n s f o r m a t i o n s 23 ( i ) Space I n v e r s i o n 23 ( i i ) Time R e v e r s a l 2LL ( i i i ) P a r t i c l e C o n j u g a t i o n 29 ( i v ) S t r o n g R e f l e c t i o n 30 '(v) R e f l e c t i o n 31 ( v i ) Weak R e f l e c t i o n 31 ( v i i ) Phase T r a n s f o r m a t i o n 32 ( v i i i ) I n v e r s i o n 32 3\u00b0 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 S c a l a r F i e l d 33 ( i ) T r a n s f o r m a t i o n 3U-( i i ) Space I n v e r s i o n and Time R e v e r s a l 35 ( i i i ) 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 R e f l e c t i o n if.0 ( i v ) R e f l e c t i o n and Weak R e f l e c t i o n I | l (v) C o n c l u s i o n i\\3 k\u00b0 Two Dimensional:. R e p r e s e n t a t i o n s i4.i1 5 \u00b0 The A n g u l a r Momentum, R e p r e s e n t a t i o n 52 (A) N o t a t i o n 52 (B) Symmetry O p e r a t o r s 53 (a.) Space I n v e r s i o n 53 (b) Time R e v e r s a l 5k (c) P a r t i c l e C o n j u g a t i o n 57 (d) S t r o n g R e f l e c t i o n 57 V (e) Re f l e c t i o n 58 (f) Weak Reflection 59 (g) Inversion 59 CHAPTER III THE ELECTROMAGNETIC FIELD 61 1\u00b0 Formalism 61 2\u00b0 Representation of D.S.O.'s 63 (A) Space Inversion and Reflection 63 (B) Time Reversal and Weak Reflection 65 (C) Other \u00a9.S.O.'s 65 3\u00b0 Representation i n Cir c u l a r Components 66 CHAPTER IV THE POUR-COMPONENT SPINOR FIELD 69 1\u00b0 Notation 69 2\u00b0 B i l i n e a r s of the Dirac F i e l d 71 3\u00b0 Transformation Properties of Cr Sat l o a n and 0 AnM- 74 hi l a t ion Operators (C.O.'s and A.O.s) 4\u00b0 Construction of Representation 77 (i) Space Inversion 78 ( i i ) Time Reversal 79 ( i i i ) P a r t i c l e Conjugation 80 (iv) Reflection 81 (v) Weak Reflection 81 (vi) Inversion 82 ( v i i ) Strong Reflection 82 5\u00b0 One Dimensional Representations 83 6\u00b0 Alternate Representations 85 BIBLIOGRAPHY 90 v i ACKNOWLEDGMENTS I w i s h to e x p r e s s my g r a t i t u d e to P r o f e s s o r P. A. Kaempffer f o r s u g g e s t i n g the problem and f o r c o n t i n u o u s encouragement and guidance throughout the p r o g r e s s o f t h i s work. I am i n d e b t e d to the members o f the T h e o r e t i c a l P h y s i c s Group, i n p a r t i c u l a r P r o f e s s o r W. Opechowski f o r h e l p f u l c r i t i c i s m and good a d v i c e . I w i s h t o thank 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\u201e G r i f f i n f o r c o n s i d e r a b l e h e l p i n the p r e p a r a t i o n o f t h i s t h e s i s . I am g r a t e f u l to the N a t i o n a l R e s e a r c h 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 the Summer S e s s i o n , I960 and d u r i n g the W i n t e r S e s s i o n s 1 9 6 0 - 6 l o INTRODUCTION The c u r r e n t l i t e r a t u r e on quantum f i e l d t h e o r y shows an almost 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 of 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 symmetry t r a n s f o r m a t i o n s . Indeed these t r a n s f o r m a t i o n s are employed almost e x c l u s i v e l y as sy m b o l i c o p e r a t o r s . On the o t h e r hand 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 are g i v e n f o r the o b s e r v a b l e s such as energy-momentum and a n g u l a r momentum , c o n c e i v e d o f as g e n e r a t o r s 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 . Such r e p r e s e n t a t i o n s g i v e substance 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 m e c h a n i c a l p r o p e r t i e s o f the f i e l d s . The vadvantage 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 d i s c r e t e symmetry o p e r a t o r s i s obvious,, f o r example, when one wants to c o n s t r u c t a many p a r t i c l e s t a t e o f s p e c i f i c p a r i t y ^ t namely the s o l u t i o n s of the e i g e n v a l u e problem TT I > - P 1 \\ Gonverselyj . f o r a g i v e n many p a r t i c l e s t a t e |>,the r e p r e s e n t a t i o n s f o r TT * n terms o f p a r t i c l e c r e a t i o n s and a n n i h i l a t i o n o p e r a t o r s e n a b l e s one to f i n d the p a r i t y by s t r a i g h t f o r w a r d c o m p u t a t i o n . Furthermore many s e l e c t i o n r u l e s g o v e r n i n g i n t e r a c t i o n s i n v o l v e knowledge o f com-mu t a t o r s o f d i s c r e t e symmetry o p e r a t o r s w i t h o b s e r v a b l e s . The commutation r e l a t i o n s o f d i s c r e t e symmetry o p e r a t o r s 2-ai\u00bbe a l s o i m p o r t a n t i n c o n n e c t i o n w i t h the TCP theorem. I t ,was the o b j e c t o f the work r e p o r t e d i n t h i s t h e s i s to produce and study the p r o p e r t i e s of such r e p r e s e n t a t i o n s . I n a l o c a l quantum f i e l d t h e o r y , i f one assumes L o r e n t z i n v a r i a n c e under p r o p e r L o r e n t z t r a n s f o r m a t i o n s and s p i n -s t a t i s t i c s c o n n e c t i o n , one f i n d s the system has an a d d i t i o n a l i n v a r i a n c e p r o p e r t y - - t h e so c a l l e d s t r o n g r e f l e c t i o n (S.R.) (see e.g. P a u l i 1 9 5 ^ ) . P h y s i c a l l y , the 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 i m p l i e s t h a t the t r a n s i t i o n p r o b a b i l i t i e s f o r the f o l l o w i n g two p r o c e s s e s are e q u a l ( 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 as 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 at 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 w i t h momentum and s p i n \u2014 &jC r e a c t to g i v e a n t i p a r t i c i e s a t - x ^ w i t h momentum P-u and -s p i n - cf~, o Thus under s t r o n g r e f l e c t i o n the o r d e r o f e v e n t s i s i n t e r c h a n g e d , where i s p o s i t i v e f o r p a r t i c l e s and n e g a t i v e f o r a n t i p a r t i c l e s . The t r a n s f o r m a t i o n i n v o l v i n g r e v e r s a l o f motion ( T ) , i . e . , i s c a l l e d time r e v e r s a l , and i s known to be s t r i c t l y v a l i d . The p r o d u c t o f S.R. and T may th e n be c a l l e d \" R e f l e c t i o n \" ( A )\u201e I t i m p l i e s : P - P and Q, -Q R e f l e c t i o n i s a l s o a s t r i c t l y obeyed symmetry. A s t i l l h i g h e r symmetry i s o b t a i n e d i f one c o n c e i v e s o f r e f l e c t i o n 3 -as the pr o d u c t o f p a r t i c l e c o n j u g a t i o n ( (\"\"\" ) where (^-^-Q^and space i n v e r s i o n Tl which t r a n s f o r m s X ~ ]t and P - . Weak i n t e r a c t i o n s are not i n v a r i a n t under these two s e p a r a t e l y . S t i l l a n o t h e r c o m b i n a t i o n i s HT \u2022= 1 and i s c a l l e d I n v e r s i o n (GJJR.-.Tr 1959). Here o n l y spins change s i g n (and o f cour s e X \u2014> -X and b~j>-\"t ). T h i s a l s o does not h o l d f o r weak i n t e r a c t i o n s . F i n a l l y one can haver\"!\" \u2014 ~J (weak r e f l e c t i o n - W.R.) * There are two types o f norm p r e s e r v i n g mappings i n H i l b e r t space, v i z . l i n e a r and a n t i - l i n e a r ma'ppingslWigijhgr 1932). The symmetry o p e r a t o r s are t h e r e f o r e n e c e s s a r i l y u n i t a r y o r a n t i - u n i t a r y ( i . e . u n i t a r y but a n t i - l i n e a r ) . Of the e i g h t o p e r a t o r s : I d e n t i t y = fc- , Tl , I , A = II T j nrT-5yI=nT j \" ^ p \"J\" and T , the f i r s t f o u r are u n i t a r y and the l a s t f o u r a n t i - u n i t a r y . S i n c e o b s e r v a b l e s i n quantum f i e l d t h e o r y are b i l i n e a r i n f i e l d f u n c t i o n s , they take a p a r t i c u l a r l y s imple form when e x p r e s s e d 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 (C . O.'s and A.O.'s) i n momentum space, where the;r\/ are b i l i n e a r i n C.O.'s and A.O.'s. I n 3\u00b0 o f the next c h a p t e r a method w i l l be o u t l i n e d f o r s e l e c t i n g b i l i n e a r s s u i t a b l e 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 papers have so f a r been w r i t t e n on these P a u l i (195#) uses the term W.R. f c r H i r e p r e s e n t a t i o n s ; 1 o Ro Go Sachs (1952); I~I and T c o n s t r u c t e d i n a n g u l a r momentum r e p r e s e n t a t i o n f o r s c a l a r f i e l d s . 2o W o l f e n s t e i n and R a v e n h a l l (1952): T f o r D i r a c f i e l d and non H e r m i t i a n s c a l a r f i e l d , u s i n g a method due to G.C\u201e Wick. 3. Watanabe (1955): TI , P , ~]~ f o r v a r i o u s f i e l d s u s i n g the methods o f ( 2 ) . l\u00b1. Nigam and F o l d y (1956) \u00ab ^~ f o r ^ -component s p i n o r f i e l d , 5. P.A. Kaempffer (1961); TI , I , ~[ f o r complex s c a l a r and \u00a3irac f i e l d s . I n r e f e r e n c e s 1, 2 and I4., the a r b i t r a r i n e s s o f phase a l l o w e d i n r and T t r a n s f o r m a t i o n s has been d i s r e g a r d e d . I n r e f e r e n c e l\u00b1t the r e p r e s e n t a t i o n s f o r [~ u n f o r t u n a t e l y c o n t a i n an e x t r a redundant f a c t o r t , so t h a t f \\ & >^ - -C J and l~\" ^ I o ) = \u2014 j 0y I n the p r e s e n t work two d i f f e r e n t t y p e s o f r e p r e s e n t -a t i o n s f o r the d i s c r e t e symmetry o p e r a t o r s are f o u n d . The f i r s t method employs the t r i c k o f an a d d i t i o n a l gauge t r a n s f o r m a t i o n o ' - U s i n g t h i s d e v i c e , more than one r e p r e s e n t -a t i o n has been found f o r each D\u201eS.Oo f o r the f o l l o w i n g f i e l d s ; (1) Non-Hermitian s c a l a r f i e l d i n l i n e a r and a n g u l a r momentum r e p r e s e n t a t i o n s (Chapter I I ) . T h i s i n c l u d e s as a s p e c i a l case the H e r m i t i a n s c a l a r f i e l d . (2) Pour component s p i n o r f i e l d 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 -U s i n g P a u l i ' s p r i n c i p l e a l t e r n a t e r e p r e s e n t a t i o n s f o r D.S.O.'s have been e x h i b i t e d . (Chapter IV) ( 3 ) E l e c t r o m a g n e t i 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 (Chapter I I I ) . The second method employs a two by two r e p r e s e n t a t i o n . I t i s shown t h a t by an 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 t h i s form can be reduced to g i v e the r e p r e s e n t a t i o n s o b t a i n e d by the f i r s t method. I n o r d e r to see how a two by two r e p r e s e n t a t i o n may be se t 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 m a n i f o l d [) C H i l b e r t space (H) onto a n o t h e r l i n e a r m a n i f o l d D_ ^ H and v i c e v e r s a , such t h a t the s c a l a r p r o d u c t < D_ | > < D _ J S H |D +) Z < D _ | S | D > - D ( Where, ( Df and D_ are assumed not to i n t e r s e c t ) . D + and D- t o g e t h e r c o n s t i t u t e the e n t i r e H \u00b0 \u00b0ne t h e r e f o r e o b t a i n s i n g e n e r a l two r e p r e s e n t a t i o n s S+ and S_ , v a l i d i n the two h a l f s p a c e s . L e t | ^ >^ and j V'-s be a s e t o f e i g e n v e c t o r s o f D-+ and \u00a3)_ , r e s p e c t i v e l y . The two s e t s w i l l be m u t u a l l y o r t h o g o n a l . C o n s i d e r the s t a t e : Under a symmetry o p e r a t i o n , one has: 5 IV, r U > = L \/ ^ > z y > where K\/ \u00bb\/ I f S + and S_ be the two r e p r e s e n t a t i o n s f o r D+ and D-r e s p e c t l v e l y , then -6-Hence e i g e n v e c t o r s i n the two h a l f spaces may be viewed as components i n a two d i m e n s i o n a l space. Thus: where T denotes the t r a n s f o r m e d e i g e n v e c t o r . I f and t>_ be the C.O.'s and A.O.'s i n the two sub-s p a c e s ; t h e n under a symmetry o p e r a t i o n S, one w r i t e s : s- \u00b0) v\u00b0 b\" J W \u00b0 ; ~ V \u2022 s.i,is: The f o l l o t i r i n g i s a summary o f the a l g e b r a i c r e l a t i o n s used ; I . I f Xi_ be a h e r m i t e a n o p e r a t o r , the S = \u00a3\u2022 i s u n i t a r y . I I . One has the i d e n t i t y ; where L\"^\"2\/P'3 \u2014 L\"^\" '^\" '^ D i s the second commutator w i t h \\S e t c . I I I . I f an o p e r a t o r (3> t r a n s f o r m s under a symmetry o p e r a t i o n a s : S p s \" ' - - ^ 7 then choose \u00a31_ such t h a t C-\u00b0-, f^] = \u00a3 and C-0-^ '] - where | ^ j - ^ t h e n -7-K ^ f o r \u00b0C = -{L - - f 3 , f o r o< = n . T a k i n g ^ = \"\"^'t, o n e S Q t s the r e q u i r e d r e s u l t . The minus s i g n f o r ot - i n d i c a t e s t h a t the double symmetry o p e r a t i o n does not y i e l d the I d e n t i t y . To e l i m i n a t e t h i s d i f f i c u l t y a second f a c t o r i s i n t r o d u c e d . Thus S - <=>, S x . IV. To determine - O - the f o l l o w i n g i d e n t i t i e s are used 0 & , 0 = A O , C ] + C A , C ] & f o r Bose f i e l d s V. Por Permi f i e l d s , s i n c e many o p e r a t o r p r o d u c t s v a n i s h , one can c o n s t r u c t a h e r m i t i a n G such t h a t G^= G . Then e ^ ^ l + G C e ^ - i ^ l t ^ f o r G - - % V I . Use has a l s o been made o f H a u s d o r f f ' s theorem: CHAPTER I GENERAL CONSIDERATIONS 1\u00b0. Symmetry P r i n c i p l e s The laws o f P h y s i c s are i n g e n e r a l governed by two main ty p e s o f p r i n c i p l e s , v i z . (1) the d y n a m i c a l p r i n c i p l e s , i . e . the e q u a t i o n s which d e s c r i b e the d y n a m i c a l b e h a v i o u r o f the system, and (2) 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 e q u a t i o n s d e s c r i b i n g the system w i t h r e s p e c t to c e r t a i n changes i n the v a r i a b l e s . The i n v a r i a n c e p r o p e r t i e s o f the system may a r i s e i n the form o f d i s c r e t e symmetries, t o be d i s c u s s e d i n the next s e c t i o n , o r the c o n t i n u o u s group o f 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 i n space and t i m e , the i r r e l e v a n c e o f the s t a t e o f mot i o n f o r a u n i f o r m motion and the phase i n v a r i a n c e . Each i n v a r i a n c e under an 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 g i v e s r i s e t o a c o n s e r v a t i o n law and the c o r r e s p o n d i n g c o n s t a n t o f the mo t i o n . I n H a m i l t o n i a n formu-l a t i o n one may e x p r e s s t h i s i n terms o f P o i s s o n b r a c k e t s a s : 1.1.1. 5 R - \u20ac { R,G| = - where 6 i s an i n f i n i t e s i m a l change i n \u00a3 , i s then s a i d to be 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 \u00a3 ^ G b e i n g any d y n a m i c a l v a r i a b l e . T h u s , u s i n g f a m i l i a r n o t a t i o n ; 1.1.2 S H = S t i H\" , Gl - - S t ^ 2 e x p r e s s e s t h a t J dfc _ 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. S i m i l a r l y , 1 . 1 . 3 f S P = STL [_ P, G } = and [ L,G\\ = -Se b& e x p r Q s s t h a t t h Q l i n e a r and a n g u l a r momenta are r e s p e c t i v e l y the g e n e r a t o r s o f the l i n e a r and a n g u l a r d i s p l a c e m e n t s . I f one put s s u c c e s s i v e l y 6- e q u a l to P , and L , \u00a3 H v a n i s h e s , i f these are c o n s t a n t s o f the m o t i o n . The above c o n s i d e r a t i o n s can be extended to quantum mechanics., I n . quantum mechanics one ;may._consider a complete se t o f commuting H e r m i t i a n o p e r a t o r s o( , which have s i m u l -taneous e i g e n v e c t o r s |\u00abC)> . Under an a r b i t r a r y u n i t a r y t r a n s f o r m a t i o n LL > the t r a n s f o r m e d H e r m i t i a n o p e r a t o r s , I o 1 o4. oL = U. 0^ U. have an i d e n t i c a l e i g e n v a l u e spectrum, i . e . where \/ST and |3 hav\u00ab the same set o f e i g e n v a l u e s . I f LL I s i n f i n i t e s i m a l , 1 . 1 . 6 (Jl ^ where \u00a3L = Q-r and 1 . 1 . 7 \u00a3 \/ 3 = - X L . P \\ ^3 . For an 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 , w r i t i n g X-^> and -CL. = P , -10-1.1.8 X l 3 - ^ - << \u2022 Momentum b e i n g the 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 , i t f o l l o w s t h a t H i s i n v a r i a n t under c o o r d i n a t e d i s p l a c e -ments and, 1.1.9 S H = * {. H , P} [ H , P ] = o -I n g e n e r a l I f . H i s i n v a r i a n t under the change o f v a r i a b l e s CO \u00bb w h i c h has f o r i t s g e n e r a t o r , the o b s e r v a b l e -CL- , t h e n 1.1.10 f H = ^ i ^ h - i ^ ' ^ J = 0 i m p l i e s t h a t -O- i s conserved and i s fa c o n s t a n t o f the m o t i o n . One can thus i m m e d i a t e l y w r i t e down the 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 \u00a3L = ^ ) w i t h = o arsd COi, n.] =-t # One can f o r m a l l y extend the method of 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 s to c o v e r f i n i t e changes. To do t h i s , assume the v a l i d i t y o f the f o l l o w i n g t h e o r e m J ^ ^ \" I f the f u n c t i o n -j- (>0 can be expanded i n a power s e r i e s i n the c i r c l e o f convergence 1.1.12 i O b = Z then t h i s e x p r e s s i o n remains v a l i d when the s c a l a r argument i s r e p l a c e d by a m a t r i x A whose c h a r a c t e r i s t i c v a l u e s l i e w i t h i n the c i r c l e o f convergence\". I n p a r t i c u l a r ( i ) Por 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 example; Gantmacher. Theory 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 case see Ref. ( i i ) , on the n e x t page. \"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 u n i t a r y , Thus i f U - *~ f ^ o l * ^ I . l . l Z j . - Z n j ' then i . i . i s ^ F \" = ^ ^ \" ^ 9 ] . = where 1.1.16 L^-\/P] = E^\/C^l] ,e,+c. and Ca'W=^. I t t h e n f o l l o w s from (1.2.11) t h a t the d i s p l a c e m e n t g e n e r a t o r s i n c o o r d i n a t e s , a n g l e s and time are r e s p e c t i v e l y : -I STx \u00ab P 1.1.17 U x = ^ - p = i y 1.1.18 U<^ - ~ I I - P ^ ^ P \u00b0 P - --c 3 1.1.19 U t ~ c 1 \u00b0 ~ \u00a7\"t-I n a somewhat more i n v o l v e d way one can c o n s t r u c t (\u2022Ii i ) 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 . \" S t i l l a n o t h e r 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 i s the 'gauge'.transform.a.tloh'i. which has to do w i t h the 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 can be e a s i l y extended to f i e l d ( i i ) P. Hausdorff., L e i p z i g , B e r . Ges. WIss. Math. Bhys. K15_8 19 (1906). ( i l l ) . -Kemmer (1959). -12 t h e o r i e s . I n extreme f i e l d t h e o r i e s , such as \"G e n e r a l fei R e l a t i v i t y ' , the c o n s e r v a t i o n laws e n t e r as i d e n t i t i e s . ' I n l o c a l f i e l d t h e o r i e s these can be deduced 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 o t h e r and w i t h the f i e l d v a r i a b l e s . The remark t h a t the o b s e r v a b l e s are r e p r e s e n t e d by H e r m i t i a n o p e r a t o r s and must t h e r e f o r e o c c u r as b i l i n e a r s i n f i e l d v a r i a b l e s , e n a b l e s one t o c a r r y o v e r the r e l a t i o n s - I . l l p - 1 9 m u t a t i s mutandis i n t o quantum f i e l d t h e o r y . 2\u00b0 The D i s c r e t e Symmetry O p e r a t o r s The c o n n e c t i o n between the d i s c r e t e symmetry i n v a r i a n c e s and the d y n a m i c a l e q u a t i o n s i s somewhat d i f f e r e n t from the c o n n e c t i o n between the c o n t i n u o u s t r a n s f o r m a t i o n s and the d y n a m i c a l l a w s . T h i s 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 by i t s v e r y name i n d i c a t e s , t h a t i t can not be generated c o n t i n u o u s l y from the i d e n t i t y , so t h a t the group c o n s i s t s o f 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 u n i t y : e.g. f o r space i n v e r s i o n ; 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 each 1.2.1 o -I o o ) o o o o o \\ *4 \/ .(\u2022iv). Schr6'dinger (1950). -13-2_ so t h a t the group c o n s i s t s o f FI and PI \" u n i t y . There are t h r e e t y p e s o f such b a s i c symmetry o p e r a t i o n s ! 1. Space I n v e r s i o n ( H ) \u201e Here the 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 s i g n . 2. Time R e v e r s a l ( T ); S i g n o f time i s r e v e r s e d so t h a t a l l l i n e a r and a n g u l a r momenta change s i g n . I t i s t h e r e f o r e a l s o c a l l e d ' R e v e r s a l o f M o t i o n ' . 3. P a r t i c l e C o n j u g a t i o n ( P ); .A p a r t i c u l a r case i s charge c o n j u g a t i o n , where the s i g n o f charge i s r e v e r s e d , i . e . , a p o s i t i v e p a r t i c l e i s r e p l a c e d by a n e g a t i v e p a r t i c l e and j o v i c e v e r s a . Since t h e r e are n e u t r a l p a r t i c l e s such as |\\ = mesons which have d i s t i n c t p a r t i c l e s and a n t i p a r t i c l e s , the concept i s more g e n e r a l . The r e q u i r e m e n t t h a t a system be i n v a r i a n t under these d i s c r e t e o p e r a t i o n s , imposes c e r t a i n c o n d i t i o n s on the system which g i v e r i s e to the so c a l l e d s e l e c t i o n r u l e s i n quantum mechanics. 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 under these d i s c r e t e symmetry o p e r a t i o n s has always been assumed i m p l i c i t l y , Por example, I t i s w e l l known i n mechanics t h a t r i g h t and l e f t handed d e s c r i p t i o n s are e n t i r e l y e q u i v a l e n t and are a mere m a t t e r o f c o n v e n t i o n . S i m i l a r l y a time r e v e r s e d 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 o r i n e l e c t r o - m a g n e t i c t h e o r y , as l o n g a s \" t h e system i s r e v e r s i b l e . The e l e c t r o m a g n e t i c phenomena are a l s o known to be i n v a r i a n t under i n t e r c h a n g e -III.-o f p o s i t i v e and n e g a t i v e c h a r g e s . I n o r d e r to extend the co n c e p t s o f d i s c r e t e symmetry o p e r a t o r s to quantum t h e o r y , one must f i r s t i n v e s t i g a t e t h e i r p h y s i c a l meaning. F o r l i n e a r momentum one h a s s 1 . 2 . 2
rt = \/ ^ c-^ v) ^U\/tj dr Under space i n v e r s i o n t h i s g i v e s , 1 . 2 . 3 < \\ P \/ > _ Y \/ t ~ \u2014 'C^X-yt a n d u n < ^ 9 r time r e v e r s a l 1.2.4 < P > r , - t - - < P>v, t . S i m i l a r l y f o r a n g u l a r momentum, one has 1.2.5 < L > ^ t = \\ ^ i x t t ) [-ch Y x V ) (\/\/ C l j t ) dr^ hence, 1.2.6 ^ L \/ ' y - {_ - <\\ \\\u2014^Y + \u00a3 but \"\u20227 ^ ) \" \u2022 3 - 3 = I g\u00a3 Ca\u00ab.p3-+11 0 f \" \" Eiu = C-v^f3 4cr o{= M TV where j f L i s so chosen t h a t n.i.k C i L ^ l . = , U , f ] - = ;1*P i IH I t t h e n f o l l o w s t h a t , I I . 3 . 5 RP'R\"1 = - 1*P -34-The d i f f e r e n c e between I I . 3 . 3 .and . I I . 3 . 5 Pleads. to.. some d i f f i c u l t i e s . The d i f f e r e n c e i n s i g n can be e a s i l y com-pensated 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 w i t h ^-J\\ 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 to be d i f f e r e n t one can use an o p e r a t o r R\" - e 1^\" where e' ^ = 9 and I I . 3 . 6 = ^ ( 3 e t c . I n g e n e r a l one can form i n a l l 8 types o f K l i n e a r s from the C.O.'s and A.O.'s; Four o f these i n v o l v e o n l y C.O.'s o r o n l y A.O.'s and t h e r e -f o r e can not be 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 . The o t h e r f o u r can be used t o form r e p r e s e n t a t i o n s f o r f o u r u n i t a r y o p e r a t o r s . Thus one o b t a i n s f o u r u n i t a r y and f o u r a n t i u n i t a r y o p e r a t o r s . There i s a v e r y i n t e r e s t i n g c o n n e c t i o n between the 'commutators of the b i l i n e a r s ' and the o b s e r v a b l e s on the one hand, and b i l i n e a r s and the symmetry o p e r a t o r s , on the o t h e r . T h i s e n a b l e s one to s e l e c t the p r o p e r b i l i n e a r f o r the g i v e n symmetry o p e r a t o r . To see how t h i s i s done, the 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 be enumerated I n the f o l l o w i n g n o t a t i o n ; ( i ) P a r t i c l e 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 ; -35-1 1 . 3 . 7 N t ( r ) = b^ri^Cr) j M^Cr) -N^tr) 11.3.8 [ Njccn, b ^ i r ) } _ = - cr) ; f ^ t r ) , b k T ( r \/ ) _ T h i s i s p r e c i s e l y the p r o p e r t y r e q u i r e d o f XI.\" i n I I.3.6. C fc-fcs. L e t , 11.3.9 R t l f ) = Z ' N^r ) K U r ) - z ' N ^ c r ) , k*>o 11.3.10 H = N + + M \u201e - 2 { N4(r) \u2022+ hum j ^ 2. M r . Then, f o r a r e a l ^ , and - efcf>(jS~N ) 11.3.11 g ^ C r ) ^ - . bfe Crj ^ H ^ ) ^ ^ ^ ^ ^ ) . Thus R r e p r e s e n t s a pure phase t r a n s f o r m a t i o n , ( i i ) D e f i n e ; 1 1 . 3 . 1 2 B*CO - k^co b^or) t B^cr) - tr) . T h e i r commutation r e l a t i o n s w i t h b ^ (\\r) a r e : 11.3.13 C & \u00b1 k C r ) , b : t i j r ' ) ^ = _ b ^ C r ) ^ A l l o t h e r commutators v a n i s h . One f u r t h e r o b t a i n s : i i . 3 . i k fie = [ Bfe ( r ) , B*(r)]^ N ^ W - f ^ c r ) and -36-11.3.15 P =\u2022 ^ ' K Pfc where prime indicates ~\" summation over a hemispace, say >0. IIo3\u00ab13 shows that Bk^5 a r e the b i l i n e a r s that can be used to construct a unitary operator that w i l l transform fc>\u00a3_ into b..^ . II.3.11, then shows that this unitary operator w i l l anticommute with ?_ . In fact the b i l i n e a r s B'S are 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. This circumstance enables one to choose i n a unique manner the b i l i n e a r characterizing a D.S.O. It may be remarked that Pj< changes sign under both f l and T \u2022 Hence one can use Bn's for the construction of these transformations. To see how thi s can be accomplished, \\it 11.3.16 XL= 2, * Cl 8&~*Z b\\ J be Hermitian and U| - I where the p a r t i c l e l a b e l T has been dropped for convenience. Then 11.3.17 U , ^ } . - - i H > - * , C n , k - B ] - - i 4 *J\u00bb \u00a3 , One then v e r i f i e s that, i f 11.3.18 R = \u20acxp (1 5 Sl) - 3 7 -Comparison with the table above ( 1 1 . 2 \u00b0 ) shows that these are indeed analogous to the transformations for f ~ l and \\Ji excepting for the conflicting phases, which can be easily rectified by using the device of equation I I . 3 . 6 , There are 2 cases. (1) n = \u00bbt*. Let II.3 .19 .aN + then ( 2 ) When ^ = H * , t a b l e ( I I . 2 \u00b0 ) shows that b f c O ) and transform with conjugate phases. Therefore, l e t , 1 I - 3 - 2 1 = \u00ab p [ i f Z ' i [ t r f6 tco - B B g^) To compensate for the phases in b^CT) \u00bb o n e c a n u s e i i . 3 . 2 2 U w + = e w l i A f ' ( K f c C O - WD) | = \u00ab f { U T r - 2 S ( W + ( i ) -where S has been so chosen that e 1^\u2014 1 j- and |j* produces the required change in sign. Thus e g . -38-The u n i t a r y f a c t o r o f time r e v e r s a l then i s : y U = U N + U t n . 3 . 2 3 = ^ p | l Y T r . z & - ) 2 \/ t ^ c . ) - N^fe)]* II . 3.23A _ ^ ~ complex c o n j u g a t i o n . One - ve r i f,1 e s\" t h a t i * ' * t 11 .3.23B TT ^ U L L + a 1 \" = uu\"1\" = U.4tlUa^(C H . S i m i l a r l y 11.3.23c T - uu*= e c o r -* ) N + e*+ e - ( T T ' 2 n ^ Where use has been made o f the f a c t t h a t However U x %| y o f U * MH+U+U^U^ = Hf U N h a n d f r o m I : c-3.l8 II.3 .23D T h e r e f o r e , U * = U + U H + l l ^ U ^ U - , \" K U N LlJ+ - Bjc x I t f o l l o w s from, i n s p e c t i o n t h a t ^ o p e r a t i n g on any b i n t r o d u c e s a f a c t o r o r - , w h i l e Uf i n t r o d u c e s a f a c t o r (\u2014I ), so t h a t \\lz on the whole g i v e s -39-I I . 3 . 2 5 ' a f a c t o r ^ o r ty* . I f one p u t s I'1} so t h a t the double o p e r a t i o n y i e l d s i d e n t i t y , the r e s u l t i n g t r a n s -f o r m a t i o n i s space i n v e r s i o n o p e r a t o r J~] . ( i i i ) D e f i n e II.3.21+ Cj< = b^Cl) 7 = b\/ll) W U> , t h e n v A l l t hese h o l d a g a i n i f K i s r e p l a c e d by \u2014 J L everywhere. A l s o , II.3 .26 I t i s c l e a r t h a t one can use CK_ i n the c o n s t r u c t i o n o f T~~, C o n s i d e r , II.3.27 iii = ? \/ ^ (\u2022 - ^ Cjc J aMj n=\u20ac l2a' t h e n I I . 3.28 ta,>^')L = ^ f b , W ; L \" ^ . i U f c O ) Hence. r, bfc CO T, - X bjc(i) s i m i l a r l y I I . 3.29 \u00ab( w h i c h are the 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 f f , -i+0-However f o r bg (2 . ) one o b t a i n s : II.3.30 and w hich d i f f e r i n s i g n f rom the r e q u i r e d t r a n s f o r m a t i o n . To compensate f o r the s i g n , we a g a i n make use o f (II.3.5\") and w r i t e . r= r N , r , = \u20acc^N\u201ew e.-f i.-ctV-WKJ II.3.31 - e ^ where i n v o l v e s o n l y p a r t i c l e number o p e r a t o r s and t h e r e f o r e commutes w i t h (2.) and b^ C * ) but g i v e s a \u2022f-f a c t o r o f minus one when o p e r a t i n g on C'' and b^ Cl) Si n c e i n s t r o n g r e f l e c t i o n a l s o o n l y Q~&--Gl ( f o r , s p i n i n t h i s case i s z e r o ) , one can use a g a i n . However, S i s an 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 3S II.3.32 S s S L where S i s u n i t a r y Then s = s H , s , \u00ab < ? * p { u f , %co) II.3\u202233 * e e I f one imposes the c o n d i t i o n , -2 H . 3 . 3 U S ^ M S - 1 * b*(r>, Is ' I one must have -la-d y ) D e f i n e : A t = bjcd) \\>Xw a n d A * ' ^ 0 ) W 2 ) , a. T h e r e f o r e II.3 .36 a l l o t h e r commutators v a n i s h . One f u r t h e r has, n.3.37 = C A t \/ A h + l = N - t W - N t C . ) a n a Prom above f o l l o w s t h a t the o p e r a t o r can be' used f o r c o n s t r u c t i n g a r e p r e s e n t a t i o n o f f~n o r the u n i t a r y f a c t o r o f T T \u2022 To do t h i s , one must w r i t e down the t r a n s f o r m a t i o n s \u00b0f b^'s under PJ\" . These are seen to be f o r the u n i t a r y f a c t o r o f f J ; \/\\L II.3 .39 Ab^o>AH= 1 A W ^ A W 2 M H - tfl>^) where ^ = 1 r 1 r . F o r ~ f |~\"=\/\\L >\u00b0ne w i l l have l ? \/ * ^ . II. 3 .40 C o n s i d e r ^1 I { ^ - } and A, - e ' ^ ^ 1 .From 3.62 one h a s : I I . 3.1+1 - 4 2 -Thus, t h i s Sl\\ does have the c o r r e c t p r o p e r t i e s f o r However t h i s g i v e s the wrong s i g n f o r (^^.^(z) . The problem may once more be s o l v e d by u s i n g II \u201e 3 \u00bb 5 \u00ab Thus: \"' . I I . 3 . 4 2 where 1A - ? T ?r* ; f AL^ f T I I . ' ^ 4 3 ' Howe ve r , not i c e , ii.3. 4 4 AN, AI AM, AI = AN, (AHTAIAN, ) Ai~\/UA|A, - AN, which i s a u n i t o p e r a t o r . On the o t h e r hand, (AL)X= AA^ - AM,Ai AM, Af - A N , A ( A J AT = AT A* ^ Uhit ob\u20acir\u00abfor t Thus th e r e i s no sense i n c o n s i d e r i n g an o p e r a t o r , such as T~T whose double a p p l i c a t i o n does not l e a d t o an i d e n t i t y t r a n s -f o r m a t i o n . Thus i f one demands t h a t I T be a l s o c o n s i d e r e d as a symmetry o p e r a t o r i n the sense t h a t 1~ T and the i d e n t i t y c o n s t i t u t e a group, then one o b t a i n s a c o n d i t i o n on the p h a s e s : v i z . 1 r = * l T , or 7 = (v) F i n a l l y one has the b i l i n e a r s w h i c h i n v o l v e e i t h e r o n l y c r e a t i o n o p e r a t o r s o r o n l y a n n i h i l a t i o n o p e r a t o r s . One c a n c o n s t r u c t f r o m t h e i r commutators the o b s e r v a b l e H a m i l t o n i a n , up t o an i n f i n i t e f a c t o r . A l l t h e s e b i l i n e a r s c a n change a c r e a t i o n o p e r a t o r i n t o an a n n i h i l a t i o n o p e r a t o r and v i c e v e r s a . However no c o n s i s t e n t u n i t a r y t r a n s f o r m a t i o n c a n be f o r m u l a t e d . T h e i r p r o p e r t i e s a r e enumerated b e l o w : I I . 3 . 4 - 5 = W ( 0 bjcU) F u r t h e r , I I . 3.1+7 II.3,1+8 H - -? ^ \" and i s i n v a r i a n t Hence *p\u00a3 S do not g i v e r i s e to any t r a n s f o r m a t i o n . D e f i n e I I . 3.1+9 I I . 3 . 5 0 I I . 3 . 5 1 II.3 .52 M ( r ) = W W b~\\ci-r) S i m i l a r l y one has I I . 3 . 5 3 ffc (|) ^ b^CDb^Cx) - (2.) HFjcCo, F ^ o ] . _ ~ Ni^o)-t M _ k u ) - t | - 4 4 1 1 . 3 . 5 4 LFjcO.) , F f e 4 C 0 ] _ = N t 0 ) + K \/ - ^ 2 - ) + \\ . 1 1 . 3 . 5 5 Here t f f c t O , bjto)}.^ t_tt>) e t c . Fina l ly , n . 3 . 5 6 fi^Cr, = b > ) I I . 3 . 5 7 [>,ar), ^ ( t ) ] 4_ ( N ^ i O + i ] ^ II.3.58T Here [ f r ^ t l ) # b ^ O j = b j c \u00ae t c \u2022 One thus c o n c l u d e s t h a t a l l p o s s i b l e DoS.C.'s can be c o n s t r u c t e d f rom N]c , C-k_ and \/|jc as done so f a r . Thus the o p e r a t o r R, P\/ T\u00bb A , I , ^T, .S are the o n l y p o s s i b l e D.S . O.'s w i t h i n the frame work o f the t h e o r y . 4 \u00b0 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 . I t was found i n 3 t h a t , though t h e r e i s one unique b i l i n e a r c h a r a c t e r i s t i c of each type o f symmetry o p e r a t o r , the e x p l i c i t r e p r e s e n t a t i o n c o n s t r u c t e d from i t g i v e s 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 to a phase f a c t o r s One is t h e r e f o r e f o r c e d t o appl y a phase t r a n s f o r m a t i o n I n -v o l v i n g p a r t i c l e number o p e r a t o r s . The germ o f t h i s d i f f i c u l t y i s c o n t a i n e d i n the v e r y d e f i n i t i o n o f the symmetry o p e r a t o r . One can l o o k upon a symmetry o p e r a t o r { S ) as a mapping o f a l i n e a r m a n i f o l d X>+- o f H i l b e r t space ( M ) on to an o t h e r l i n e a r m a n i f o l d 3?_ of H , such t h a t the s c a l a r p r o d u c t of the e i g e n v e c t o r s , spanning the -k-5-two m a n i f o l d s , v a n i s h e s , i . e . 11 .4 . 1 )}} Where prime i n d i c a t e s t h a t the summation i s o n l y o v e r pos'i+fve v a l u e s o f vn , and B ( W \u00ab T ) = kftklwr) b Clc\u00a3^w If ) . T h e n 1 1 .5. 1 8 IL-r - e S i n c e [ J l ^ b CkV*u ' | ) J = - t (2m+i )\u00a3 ( I c ' t W l ) a n d b u t u+bo&vou;* - -I^C-Q^bCtfe'*.'!) . I n t r o d u c i n g 'H.5\u00bb n U M + = C on\u00ae has I I . 5 . 2 0 U N + l l + L . I n analogy to the case o f l i n e a r momentum r e p r e s e n t a t i o n , one can 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 f o r I X a s ; I I . 5 - 2 1 ^ = U+ Uu+ = tf+tijsU = UN-^+ where II.5. 2 2 - G>cf>{iClT-2^)\u00a3. 0 ^ - \u00bb 0 - N CkC-w 2.) 3) A l s o , one h a s : U-UiU^, where ( (J. = \u00ab*P 0? -Sf ^ ^ i r j l l ^ x p [ 2 \u00ab 2 i ^ W \/ m d r X XX 5 23 ^ - ^ . -57-(c) P a r t i c l e C o n j u g a t i o n I t f o l l o w s from ( t-3 ) t h a t 11.5.21). HM) ~ ^ - M C K ^ V M T ) i i . 5 . 2 5 C l - 2 . e [ N(Ww 2j - N C \u00ab * i ) ] \" e a ( t ^ X where I I . 5 . 2 6 w i t h I t i s t h e r e f o r e c l e a r t h a t the t r a n s f o r m a t i o n f~ I n a n g u l a r momentum r e p r e s e n t a t i o n w i l l be almost e n t i r e l y analogous t o 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 . Thus (TT 5_ Nfremi) 9 P O S i m i l a r l y , i f ^NJX x the o t h e r r e p r e s e n t a t i o n s a r e : ii.5.28 r = n rMz - rfrU i - rW2r, Furthermore t (d) S t r o n g R e f l e c t i o n 4\" Cfl From the stand p o i n t II.3.3-4 the u n i t a r y f a c t o r s\" o f $ has the same r e p r e s e n t a t i o n as |~~ \u00bb i f one makes the 58 replacement \\ r ~? *l s ( 2.w +-2-\u00a3 f-Q N o t i c e t h a t f ~ X i s a u n i t o p e r a t o r . Hence S b e i n g a n t i u n i t a r y w i l l not be a u n i t o p e r a t o r u n l e s s one demands <|s = I. (e) R e f l e c t i o n R e f l e c t i o n i s d e f i n e d as an o p e r a t i o n under w h i c h a p a r t i c l e i n a l e f t handed frame I s t r a n s f o r m e d i n t o an a n t i p a r t i c l e i n a r i g h t handed frame, i . e . I t i s the p r o d u c t o f space i n v e r s i o n and p a r t i c l e c o n j u g a t i o n . Hence one has: I I . 5 . 3 0 \/MoCWroO I t i s c l e a r t h a t a r e p r e s e n t a t i o n f o r A w i l l d i f f e r from o n l y 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 i i . 5 . 3 2 t = B)^ N(M^O~) ^ t h e r e f o r e 11 , 5 * 3 3 A c \u20ac *\"r [iS ^L^ +CWm)-hlCfk\u00a3-A l s o , w r i t i n g g-^ - { ^ 11.5 . 3 k A = &f{ I^2, ^ (i+ ^ . ) NCt*w s go o v e r i n t o b+\/S and v i c e v e r s a , so t h a t the o r d e r o f o p e r a t o r s i s i n t e r c h a n g e d . The above o b s e r v a t i o n s u g g e s t s two ways to remedy the s i t u a t i o n . (A) To p e r f o r m i n a d d i t i o n to the 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 i o n o f complex c o n j u g a t i o n , so t h a t the R.H.8. o f ( b ) commutators, b e i n g pure i m a g i n a r y , remain unchanged under the combined o p e r a t i o n . Such an o p e r a t i o n i m p l i e s t h a t exp(iKX) i s t r a n s f o r m e d i n t o ar t ) -26-I I . 1 . 2 2 A T - VlL , Where U. i s u n i t a r y and L_ I s complex c o n j u g a t i o n . T h i s i s the d e f i n i t i o n used by S i g n e r , Luders and o t h e r s . (B) The'above i s the s o l u t i o n from the p o i n t o f view o f (k ) commutators. The r e v e r s a l o f the o r d e r o f o p e r a t o r s i n the ( a ) commutators suggests a n o t h e r method (Schwinger 1951) \u2022 Here one r e p l a c e s the time r e v e r s e d f i e l d f u n c t i o n by i t s complex c o n j u g a t e t o g e t h e r w i t h t r a n s p o s i t i o n o f a l l the o p e r a t o r s . Thus; (a) e+cxtje\"*1 * % 4>+t*,-f), I I . 2.7 (b) w.iere 0* U k , Y ~ t r a n s p o s i t i o n - - ' I t f o l l o w s t h a t , i i . 2. s e^ +j^ c+Je\"1 = u ^ a ' a ^ a m \" 1 ~ 4>t-t) and the transformation formulae are the same as f o r the symmetry operator r\\ \u2022 Thus IV.6.5 T l - l ' l U - n v c t i . ^ - ^ t s ^ - ^ C a v t W . r i ^^(-4,^ )5 where the prime on \"T* indicates the product f o r a l l V x > fe<, One e a s i l y checks that W~ =\u2022 \\ \u201e Hence, using IV. 6.2, one obtains: iv.6.6 H (n1jcsir)-'n^f)*^r1l?bts\u00bbr^ Ci'r)3). (2) V tV . Then -<\\ so that IV.6.4 (A) and (B) d i f f e r i n sign. In order to restore the sign one can use the operator Also, Let IV.6.7 Then - ^ -^k ts,r) -TL^rr)-*. x T i ^ l i . r ) ^ . ^ is, f) iv.6.8 ^ ^ T ' ^ n Cs,r) One then checks that IV.6.9 n^T^fc-MLTV5'^ such that iv.6.io VC~ C v ^ t S i r ) ^ \" 4 - - - i s \u00bb r ) ( i i ) P a r t i c l e conjugation ( \\~ ) Here one obtains IV.6.11 tS) =\u2022 l-0 where _ J 1 ^ L S ) - _ z A-\u00bbc (.j) and VV\\ - I Hence IV.6.12 r - I I ( \\ - ^ C S , i ^ 1 \\ K t s ^ ) ^ C ^ \u00a3 $ i 0 + ^ C t e C t i ^ one then checks that IV.6.13 ^ \\ > so t h a t T\"1\"\" \\, - 8 8 -( i i i ) R eflection ( A ) I f one writes i v . 6 . i i * . ^ _ \\ f o r n r one obtains from IV.I4..15 iv . 6 . 1 5 ^--I ,Jl^^,v^\u00ab.o +\u00bbJ\u00ab 1.J .1m f t l l t 4 , l ^ t \u201e * $ (iv) Strong Reflection ( S ) Prom IV.i|..20 one has (v) Weak Reflection ( X )* being antiunitary,may be written as, IV.6.17 T ^ 3\" \\_ where 3 i s the unitary f a c t o r . To f i n d T one can use the b i l i n e a r s & K <-S|T) of IV.2? one gets, iv . 6 . 1 8 - H - - L t^ j tk t t ^ - G u c \\ , 0 ) ^ (vi) Inversion ( 7X) \u00b0 Since inversion i s an antiunitary operator, i t may be written as: IV.6.19 X ~ X ^- where X i s the unitary f a c t o r . .89-The c h a r a c t e r i s t i c b i l i n e a r for constructing \"X_ was found to be l S i T ) . i n this case one gets two representations, One f o r S~\\ , the other f o r S - 2- , because f o r the same T ; tSiO and ts',r3 have both the same phase fact o r whereas i n forming an -O- , i f <-S,Y-} has co-e f f i c i e n t \"\\ then that of T>k is'.v) w i l l be ' \" l * \" , so that the transformation property f o r O.^ isS-r) w i l l be wrong. One obtains f o r S \u2014 \\ , IV.6.19 H \\ C ^ C i A ) - K ^ ' 0 ) ^ C b u ^ O - ^ v ^ L ^ BIBLIOGHAPKY 1\u00ab Blatt, 3.n* and tfeiaskopf F.S. Theoretical Muelear Physics* Jofaa Wiley and Sons, Kew York* 1952. ,.:2. 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