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Consequences of space-time invariances in quantum mechanics and direct interaction theories Kalyniak, Patricia Ann 1978

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c.l CONSEQUENCES OF SPACE-TIME INVARIANCES IN QUANTUM MECHANICS AND DIRECT INTERACTION THEORIES by PATRICIA ANN KALYNIAK B . S c , U n i v e r s i t y of C a l g a r y , 1976 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Phys ics ) We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA October , 1978 © P a t r i c i a Ann K a l y n i a k , 1978 In presenting th i s thes is in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree l y ava i lab le for reference and study. I fur ther a g r e e t h a t permission for extensive copying of th i s thes is for scho lar l y purposes may be granted by the Head of my Department or by h is representat ives . It is understood that copying or pub l i ca t ion of th is thes is fo r f i n a n c i a l gain sha l l not be allowed without my wr i t ten permission. Department of Physics The Univers i ty of B r i t i s h Columbia 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 September 29. T978 A b s t r a c t The problem of d e s c r i b i n g a quantum mechanica l system i s c o n s i d e r e d . For a system which i s i n v a r i a n t under the P o i n c a r e (inhomogeneous Lorentz ) t r a n s f o r m a t i o n s t h i s d e s c r i p t i o n i s p rov ided by the generators of those t r a n s f o r m a t i o n s , which s a t i s f y the u s u a l commutation r e l a t i o n s . Genera l exp ress ions f o r the cen t re of mass p o s i t i o n and i n t e r n a l angular momentum o p e r a t o r s , i n terms of these g e n e r a t o r s , a re o b t a i n e d . The generators of the P o i n c a r e t r a n s f o r m a t i o n s are w r i t t e n i n terms of the fundamental dynamical v a r i a b l e s f o r s e v e r a l systems. The systems cons idered a re those c o n s i s t i n g of a s i n g l e f r e e s p i n l e s s p a r t i c l e , a s i n g l e f r e e p a r t i c l e w i t h s p i n , a s i n g l e f r e e D i r a c p a r t i c l e , TL n o n i n t e r a c t i n g p a r t i c l e s , and ft i n t e r a c t i n g p a r t i c l e s . In each c a s e , the c e n t r e of mass p o s i t i o n and i n t e r n a l angular momentum are g i v e n i n terms of the fundamental dynamical v a r i a b l e s of the system. For the f i r s t two systems l i s t e d above, these two o p e r a t o r s are found to be equal to the C a r t e s i a n c o o r d i n a t e s and s p i n of the p a r t i c l e , r e s p e c t i v e l y . In the case of the D i r a c p a r t i c l e , these opera to rs a re seen to be r e l a t e d to the C a r t e s i a n c o o r d i n a t e s and s p i n of the p a r t i c l e v i a the Pryce -Fo ldy -Wouthuysen t r a n s f o r m a t i o n . F o l l o w i n g Bakamjian and Thomas, i n t e r a c t i o n i s in t roduced to the n p a r t i c l e system v i a a s i n g l e operator which depends o n l y on i n t e r n a l v a r i a b l e s . The c o n d i t i o n of "asymptot i c c o v a r i a n c e " of the s c a t t e r i n g operator i s d i s c u s s e d f o r two p a r t i c l e s c a t t e r i n g . The s c a t t e r i n g operator f o r a two p a r t i c l e system w i t h no bound s t a t e s and w i t h Po incar£ generators of the Bakamjian-Thomas form i s seen to be a s y m p t o t i c a l l y c o v a r i a n t . CONTENTS A b s t r a c t i i Acknowledgement v Chapter 1 I n t r o d u c t i o n 1 Chapter 2 Consequences of Space - t ime I n v a r i a n c e s i n Quantum Mechanics 5 2 . 1 Lorentz I n v a r i a n c e of a P h y s i c a l System 7 2 .2 Commutation R e l a t i o n s f o r the Generators of the Poincare" Group 11 2 .3 V a r i o u s Transformed Operators 24 2.4 Poincare" I n v a r i a n t s M1" and U" 33 2 .5 F o u r - v e c t o r N o t a t i o n 36 2 .6 The Space I n v e r s i o n Operator 6° 39 2.7 The Time R e v e r s a l Operator'T 43 2 .8 The Centre of Mass P o s i t i o n %., I n t e r n a l Angular Momentum i , and Centre of Mass V e l o c i t y V Operators 47 2.9 T rans fo rmat ion P r o p e r t i e s of the Observables and V 51 2.10 The Operators T and % (T) 58 Chapter 3 D e s c r i p t i o n s of Some Lorentz I n v a r i a n t Systems 62 3 . 1 S i n g l e S p i n l e s s Free P a r t i c l e 63 3 .2 S i n g l e Free P a r t i c l e w i t h Sp in 68 3 . 3 S i n g l e Free D i r a c P a r t i c l e 73 3 .4 n. N o n i n t e r a c t i n g P a r t i c l e s w i t h Sp in 76 3 . 5 71 P a r t i c l e s i n I n t e r a c t i o n 78 Chapter 4 Two P a r t i c l e S c a t t e r i n g 84 4 . 1 Lorentz I n v a r i a n c e f o r Two P a r t i c l e S c a t t e r i n g Systems 84 Footnotes 93 i i i B i b l i o g r a p h y 95 Appendices A Proof tha t Equat ion ( 2 . 1 . 8 ) I m p l i e s Equat ion ( 2 . 1 . 9 ) 97 B D e r i v a t i o n of Equat ion ( 2 . 8 . 3 ) f o r the Generator K 100 C D e r i v a t i o n of Equat ions ( 2 . 8 . 5 ) and ( 2 . 8 . 6 ) f o r %, and 4 103 D I n t e g r a l Forms of the Space - t ime T ransformat ion Operators 105 E Centre of Mass P o s i t i o n %. and I n t e r n a l Angular Momentum S Operators f o r a Free D i r a c P a r t i c l e 109 F D e r i v a t i o n of Equat ions ( 3 . 3 . 1 3 ) through ( 3 . 3 . 1 8 ) f o r a Free D i r a c P a r t i c l e 112 G D e r i v a t i o n of H B , and S „ 115 H V e r i f i c a t i o n tha t P „ , X „ ^ , and a re I n t e r n a l Operators and D e r i v a t i o n of the Commutation R e l a t i o n s ( 3 . 5 . 3 1 ) 120 I The G a l i l e a n Group 124 i v Acknowledgement I w i s h to thank Malcolm M c M i l l a n f o r a s s i s t a n c e i n the r e s e a r c h and w r i t i n g of t h i s work and the N a t i o n a l Research C o u n c i l and the U n i v e r s i t y of B r i t i s h Columbia f o r f i n a n c i a l a s s i s t a n c e . v 1 Chapter 1 I n t r o d u c t i o n We a re concerned here w i t h the c o n s t r u c t i o n of a d e s c r i p t i o n of a p h y s i c a l quantum mechanica l system. By r e q u i r i n g that a system be i n v a r i a n t under c e r t a i n s p a c e - t i m e t r a n s f o r m a t i o n s , we a re l e d to a set of o p e r a t o r s which p rov ide a d e s c r i p t i o n of tha t system. F i r s t , however, the meaning of " i n v a r i a n c e under a s p a c e - t i m e t r a n s f o r m a t i o n " must be c l a r i f i e d . S ince any i n f o r m a t i o n about a quantum mechanica l system i s the r e s u l t of the measurement of o b s e r v a b l e s , represented by H e r m i t i a n opera to rs i n the quantum t h e o r y , our b a s i c p o s t u l a t e i s a c o n d i t i o n of the r e s u l t s of such measurements. Consider a system which has been prepared by some apparatus i n a s t a t e liy> and an observab le A which i s to be measured by another a p p a r a t u s . . Now i f both the p r e p a r a t i o n and measuring apparatus undergo the same t r a n s f o r m a t i o n i n space and t i m e , then the s t a t e IS prepared and the observab le A i s measured. The r e s u l t of the measurement of the observab le A i n the s t a t e ll//) must be e q u a l , on the average , to the r e s u l t of the measurement of the observab le A i n the s t a t e I ty'). S y m b o l i c a l l y , we w r i t e <T|/iAi-uy) = onAMT) Th is p o s t u l a t e must ho ld f o r a l l s t a t e s and a l l observab les A . The system i s then s a i d to be i n v a r i a n t under the s p a c e - t i m e t r a n s f o r m a t i o n . The t r a n s f o r m a t i o n s of concern here a r e , f o r the most p a r t , the P o i n c a r e (inhomogeneous Lorentz ) t r a n s f o r m a t i o n s . These i n c l u d e t ime t r a n s l a t i o n s , space d i s p l a c e m e n t s , r o t a t i o n s , and Lorentz b o o s t s . They are g i ven e x p l i c i t l y i n S e c t i o n 2.1. We r e q u i r e tha t any r e a l i s t i c system be i n v a r i a n t under the Poincare ' t r a n s f o r m a t i o n s . T h i s requirement i m p l i e s , as we show, tha t to each one-parameter t r a n s f o r m a t i o n there corresponds a 2 u n i t a r y l i n e a r operator 7J.^(T) which may be expressed as T i s the parameter of the t r a n s f o r m a t i o n . These u n i t a r y opera to rs p r o v i d e the r e l a t i o n s between the s t a t e s and and between the observab les A and A'. The H e r m i t i a n opera to rs C„, a re c a l l e d the generators of the t r a n s f o r m a t i o n and p rov ide the c h a r a c t e r i z a t i o n of the system which we seek . (The l a b e l s <* i d e n t i f y the one-parameter t r a n s f o r m a t i o n . ) There a re ten such opera to rs cor responding to the P o i n c a r e t r a n s f o r m a t i o n s . A system may a l s o be i n v a r i a n t under two t r a n s f o r m a t i o n s which are not P o i n c a r e t r a n s f o r m a t i o n s . These are the o p e r a t i o n s of space i n v e r s i o n and t ime r e v e r s a l which we i n t r o d u c e i n S e c t i o n s 2 .6 and 2 . 7 , r e s p e c t i v e l y . As f o r the Poincare' t r a n s f o r m a t i o n s , there e x i s t s a u n i t a r y l i n e a r operator cor respond ing to the o p e r a t i o n of space i n v e r s i o n . An a n t i u n i t a r y a n t i l i n e a r opera to r corresponds to the t ime r e v e r s a l o p e r a t i o n . The ten generators of the P o i n c a r e t r a n s f o r m a t i o n s obey a set of commutation r e l a t i o n s , which we d e r i v e i n S e c t i o n 2 . 2 . Thus, to c o n s t r u c t a d e s c r i p t i o n of a p h y s i c a l system which i s i n v a r i a n t under the P o i n c a r e t r a n s f o r m a t i o n s , one must f i n d express ions f o r the ten g e n e r a t o r s , i n terms of the fundamental dynamical v a r i a b l e s of the system, which s a t i s f y the proper commutation r e l a t i o n s . S i x of the generators may a l s o be expressed , i n a g e n e r a l fo rm, i n terms of two o p e r a t o r s , )N and ^ , which we i n t r o d u c e i n S e c t i o n 2 . 8 . t i s the cen t re of mass p o s i t i o n operator and _> i s the t o t a l i n t e r n a l angu lar momentum o p e r a t o r . They obey a set of commutation r e l a t i o n s which ensures tha t the generators s a t i s f y the proper set of commutation r e l a t i o n s . We w r i t e the s i x : g e n e r a t o r s i n terms of these opera to rs and, i n v e r s e l y , g i v e 3 g e n e r a l e x p r e s s i o n s f o r ^ and 4 i n terms of a l l the generators of the Po incar£ t r a n s f o r m a t i o n s . In the t h i r d c h a p t e r , we c o n s i d e r s e v e r a l quantum mechanica l systems. T h e i r fundamental dynamical v a r i a b l e s are in t roduced and the generators of the Poincare' t r a n s f o r m a t i o n s are g i ven i n terms of these v a r i a b l e s . The f i r s t two systems cons idered are those of a s i n g l e f r e e s p i n l e s s p a r t i c l e and a s i n g l e f r e e p a r t i c l e w i t h s p i n , i n S e c t i o n s 3 . 1 and 3 . 2 , r e s p e c -t i v e l y . The opera to rs _\ and are expressed p a r t i c u l a r l y s imply i n terms of the fundamental dynamical v a r i a b l e s f o r these systems. A l s o , the u n i t a r y opera to rs cor responding to the P o i n c a r e t r a n s f o r m a t i o n s and to space i n v e r s i o n as w e l l as the a n t i u n i t a r y t ime r e v e r s a l operator are expressed i n i n t e g r a l form f o r these systems. We a l s o g i v e the generators of the P o i n c a r e t r a n s f o r m a t i o n s f o r a s i n g l e D i r a c p a r t i c l e i n S e c t i o n 3 . 3 and o b t a i n e x p r e s s i o n s f o r _v and 3> which are i n agreement w i t h s e v e r a l a u t h o r s . In S e c t i o n 3 . 4 , f o r a system of a f i x e d number of n o n i n t e r a c t i n g p a r t i c l e s , we g i v e the generators i n terms of the s i n g l e p a r t i c l e g e n e r a t o r s . As a f i n a l example, we cons ide r a f i x e d number of i n t e r a c t i n g p a r t i c l e s i n S e c t i o n 3 . 5 . The i n t e r a c t i o n i s comple te l y s p e c i f i e d by an operator hi , the mass o p e r a t o r , when the H a m i l t o n i a n takes the Bakamjian-Thomas f o r m . 1 C e r t a i n r e s t r i c t i o n s are imposed on M and , i n order to s a t i s f y these r e s t r i c t i o n s , i n t e r n a l v a r i a b l e s a re d e f i n e d . We do not cons ide r the q u e s t i o n of the s e p a r a b i l i t y of the n p a r t i c l e H a m i l t o n i a n . In Appendix I, we cons ide r a system of 71 i n t e r a c t i n g n o n r e l a t i v i s t i c p a r t i c l e s wherein the i n t e r a c t i o n i s aga in s p e c i f i e d by a s i n g l e o p e r a t o r . We f i n d tha t the generators of the Poincare' t r a n s f o r m a t i o n s f o r t h i s system may be obta ined by t a k i n g the l i m i t c—*•«> i n the express ions f o r the r e l a t i v i s t i c system. 4 In a c o n s i d e r a t i o n of two p a r t i c l e s c a t t e r i n g i n Chapter 4 , we f i n d tha t s e v e r a l c o n d i t i o n s are imposed on the M o l l e r operators -0 .+and the s c a t t e r i n g operator S as a r e s u l t of the requirement of Poincare" i n v a r i a n c e . The f u r t h e r c o n d i t i o n of "asymptot ic c o v a r i a n c e " of the s c a t t e r i n g operator i s in t roduced and d i s c u s s e d . E q u i v a l e n t c o n d i t i o n s are w r i t t e n i n terms of the generators of the P o i n c a r e t r a n s f o r m a t i o n s and the Bakamjian-Thomas express ions g i v e n i n S e c t i o n 3 . 5 f o r the generators f o r a system of two i n t e r a c t i n g p a r t i c l e s are seen to s a t i s f y these c o n d i t i o n s . 5 Chapter 2 Consequences of Space - t ime I n v a r i a n c e s i n Quantum Mechanics T h i s chapter p r o v i d e s a g e n e r a l d i s c u s s i o n of the d e s c r i p t i o n of a quantum mechanica l system i n terms of the generators of the t r a n s f o r m a t i o n s under which the system i s i n v a r i a n t . In S e c t i o n 2 . 1 , the P o i n c a r e t r a n s f o r m a t i o n s are in t roduced and the concept of Lo rentz i n v a r i a n c e of a p h y s i c a l system i s d i s c u s s e d i n some d e t a i l . The ten generators of the P o i n c a r e t r a n s f o r m a t i o n s , H, P'^ , 7'^ , K'J> ( j = 1 ) , a re i d e n t i f i e d and, i n S e c t i o n 2 . 2 , t h e i r commutation r e l a t i o n s a re d e r i v e d . In S e c t i o n 2 . 3 , the opera to rs lA^^JMp a re d e r i v e d , where ^ i s a u n i t a r y operator cor responding to a one-parameter P o i n c a r e t r a n s f o r m a t i o n and i s the generator of a one-parameter Poincare" t r a n s f o r m a t i o n . I f i s measured by some appara tus , then ZL/gC/aTX^ i s the observab le measured by tha t apparatus a f t e r i t has undergone the t r a n s f o r m a t i o n to which Xl^ co r responds , w h i l e Zz/C^Z/g i s the observab le measured when the p r e p a r a t i o n apparatus has undergone the t r a n s f o r m a t i o n . Two Poincare" i n v a r i a n t s H and IT*'axe d e f i n e d i n S e c t i o n 2 . 4 ; such opera to rs a re i n v a r i a n t under a l l the Poincare" t r a n s f o r m a t i o n s and , t h u s , as we show, commute w i t h a l l the generators of the Poincare" t r a n s f o r m a t i o n s . Some of the r e s u l t s of the f i r s t th ree s e c t i o n s are put i n t o f o u r - v e c t o r n o t a t i o n i n S e c t i o n 2 . 5 . The o p e r a t i o n s of space i n v e r s i o n and t ime r e v e r s a l , to which correspond operators.(P and 1 " , a re in t roduced i n S e c t i o n s 2 .6 and 2 . 7 , r e s p e c t i v e l y . The i r e f f e c t , i n the sense of S e c t i o n 2 . 3 , on the generators of the P o i n c a r e t r a n s f o r m a t i o n s i s i n v e s t i g a t e d . That i s , we o b t a i n a n d f C ^ T . We i n t r o d u c e the c e n t r e of mass p o s i t i o n % , the t o t a l i n t e r n a l angular momentum sS , and t h e . c e n t r e of mass v e l o c i t y V i n S e c t i o n 2.8 and w r i t e ^ J and K J\^= i n terms of % and $> . The commutation r e l a t i o n s obeyed by 6 % and 4 a r e g i v e n and g e n e r a l e x p r e s s i o n s f o r these o p e r a t o r s , i n terms of the generators of the P o i n c a r ^ t r a n s f o r m a t i o n s , a re o b t a i n e d . The i r p r o p -e r t i e s under the Po incar£ t r a n s f o r m a t i o n s , i n the sense of S e c t i o n 2 . 3 , a r e g i v e n i n S e c t i o n 2 . 9 . In that s e c t i o n , we see tha t ^>t ransforms i n a compl i ca ted manner under L o r e n t z b o o s t s . T h e r e f o r e , we i n t r o d u c e , i n S e c t i o n 2 , 1 0 , opera to rs T and ^ . ( T ) which t rans form more s i m p l y . 2.1 L o r e n t z I n v a r i a n c e of a P h y s i c a l System An apparatus used to prepare a s t a t e ll^} of some p h y s i c a l system i s p laced i n a f i x e d space - t ime r e f e r e n c e frame i n a l a b o r a t o r y . (We s h a l l except when noted always use S c h r o d i n g e r ' s p i c t u r e of quantum mechanics . ) The placement of the apparatus i n the l a b o r a t o r y i s s p e c i f i e d by ten r e a l numbers which may be chosen to be the t ime i t was switched on (one number), the p o s i t i o n of a f i x e d p o i n t on the apparatus ( three numbers), the ang les axes f i x e d on the apparatus make w i t h the f i x e d l a b o r a t o r y frame ( three numbers) , and the v e l o c i t y of i t s c e n t r e of mass i n the l a b o r a t o r y ( three numbers) . The placement of the apparatus i n the l a b o r a t o r y i s now changed wi thout changing i t s i n t r i n s i c s t r u c t u r e . The changed apparatus prepares the s t a t e lljj}. The change i n v o l v e s changing one or more of the above ten numbers and can be c h a r a c t e r i z e d by the equat ions where the x^=(ct ,x ) a re the space - t ime c o o r d i n a t e s of a p o i n t on the apparatus be fo re the change (t i s the t ime the apparatus was switched o n ) , x^* a re the c o o r d i n a t e s of the p o i n t on the apparatus a f t e r the change, and -ATv and a'^'are q u a n t i t i e s independent of x '^and x ( / ^ which c h a r a c t e r i z e the change i n p lacement . More s p e c i f i c a l l y , l ) i f the apparatus i s switched on e. seconds e a r l i e r : < x , t ) - - ( V ,t') = ( x , t - e ) Li.) i f i t i s d i s p l a c e d a long the 1 - a x i s by a d i s t a n c e a : ( x , t ) - * ( x ' , t , ) = ( x < ° + a , x a ' , , x m , t ) U.I. 3) • L L L ) i f i t i s r o t a t e d about the 1 - a x i s by an a n g l e © : 8 ( x , t ) (_',_' ) = (x < , ) ,x l x l cose - x < 3 , s i n 9 , x t t „ n O + x' J ,cos 8 , t ) U.l.^ iv ) i f i t i s boosted a long the 1 - a x i s by speed v : ( _ , t ) — ^ ( _ ' , t ' )=( K (x'n+ v t ) .x'nx'n * <t + v/& x w ) ) a.i.«-^ or (x' o ),x) »(xL<",V ) = (x r o , coshTt+ x " , s inhu ,x < "coshu + x""sinhu,x"-\x < i V) ( X. iSb) where c i s the speed of l i g h t , and t a n h u = v/c (XA.l) S i m i l a r c o n s i d e r a t i o n s can be g i v e n to a second apparatus which measures the v a l u e of some observab le of the system. We denote by A the observab le measured i n the f i r s t placement and by A the observab le measured i n the second p lacement . A p h y s i c a l system i s s a i d to be Lorentz i n v a r i a n t i f f o r every Po incar£ t r a n s f o r m a t i o n , i . e . , every orthochronous inhomogeneous Lo rentz t r a n s -f o r m a t i o n ( 2 . 1 . 1 ) one o b t a i n s the same number on the average when one t ransforms both the p r e p a r a t i o n and the measuring a p p a r a t u s . That i s , <VIA'IV//> = (WAty) (L.i.i) Moreover , equat ion ( 2 . 1 . 8 ) must ho ld f o r every p r e p a r a t i o n and measuring appara tus ; tha t i s , i t must be t r u e f o r every s t a t e I'ty'} and every observab le A . We show i n Appendix A that equat ion ( 2 . 1 . 8 ) i m p l i e s tha t cor respond ing p r o b a b i l i t i e s are e q u a l . That i s , 9 Wigner has shown that equat ion ( 2 . 1 . 9 ) i m p l i e s tha t \1P') = UIW) (L.I. 10) where Zl i s e i t h e r a l i n e a r u n i t a r y operator or an a n t i l i n e a r a n t i u n i t a r y o p e r a t o r . 2 Equat ion ( 2 . 1 . 8 ) then f o l l o w s f o r a l l and A p rov ided A ' = 1/ A (t.l.ll) where ll. i s the a d j o i n t of Zl . F u r t h e r , i t f o l l o w s from the group p r o p e r t i e s of the P o i n c a r e t r a n s -fo rmat ions tha t IC can be w r i t t e n i n terms of p roducts of one or more of ten one-parameter o p e r a t o r s <*• = 1 , 2 , . . . , 10, which have the p r o p e r t i e s U«(o)--\ ¥ oc (z. UZ) S i n c e the square of a u n i t a r y l i n e a r operator or of an a n t i u n i t a r y a n t i l i n e a r operator i s u n i t a r y , i t f o l l o w s from equat ion ( 2 . 1 . 1 3 ) that the 2/»are l i n e a r u n i t a r y o p e r a t o r s . The u n i t a r y o p e r a t o r s ZL^Ct) can be s p e c i f i e d f u r t h e r . I t f o l l o w s from S t o n e ' s Theorem that one can w r i t e where (CK i s a H e r m i t i a n o p e r a t o r . 3 The opera to rs C „ a re c a l l e d the generators of the Poincare" group. I t i s convenient to l a b e l the ten generators as f o l l o w s : H generates t ime t r a n s l a t i o n s ; P '^ P ™ P' 3 > generate space t r a n s l a t i o n s a long the 1 - , 2 - , 3 - a x i s , r e s p e c t i v e l y ; T ' * T ' " ' J ' ^ g e n e r a t e 10 r o t a t i o n s about the l - , 2 - , 3 - a x e s , r e s p e c t i v e l y ; K°] K'** generate L o r e n t z boosts a long the l - , 2 - , 3 - a x e s , r e s p e c t i v e l y . The cor responding u n i t a r y opera to rs w i l l be w r i t t e n as U (t)- e"1- : t ime t r a n s l a t i o n or e v o l u t i o n operator fx.I.Iff) D'CecV5 t,L^ rd isplacement operator a long j - a x i s (l>.I.lfc) 6{U%\- e" L^ e ^ : r o t a t i o n operato r about j - a x i s (%..l.n) L J (\A)= e" L C ^ rLorentz boost operator a long j - a x i s (l..t.\*) Thus U (t) corresponds to t r a n s f o r m a t i o n ( 2 . 1 . 2 ) ; D"VcO corresponds to t r a n s f o r m a t i o n ( 2 . 1 . 3 ) ; (R'(e) corresponds to t r a n s f o r m a t i o n ( 2 . 1 . 4 ) ; L ("^ corresponds to t r a n s f o r m a t i o n ( 2 . 1 . 5 ) . ^ i s c a l l e d the H a m i l t o n i a n of the system; r , r , r a re c a l l e d the components of the t o t a l l i n e a r momentum of the system; T"*T'" J < J > are the components of the t o t a l angular momentum of the system. 11 2 .2 Commutation R e l a t i o n s f o r the Generators of the P o i n c a r e Group The generators of the P o i n c a r e group obey a set of commutation r e l a t i o n s which we now d e r i v e . Two i n f i n i t e s m a l t r a n s f o r m a t i o n s generated by C „ and w i t h parameters t and cr, r e s p e c t i v e l y , a re a p p l i e d to the p r e p a r a t i o n apparatus of the system. ( « ,p = 1 , 2 , . . . , 10) They a re succeeded by t h e i r i n v e r s e s . The cor responding u n i t a r y opera to rs can be expanded i n a power s e r i e s as f o l l o w s : where the commutator The r e s u l t of the same set of four s u c c e s s i v e i n f i n i t e s m a l t r a n s f o r -mat ions can be e x h i b i t e d upon a p p l i c a t i o n to a s p a c e - t i m e p o i n t ( x , t ) on the p r e p a r a t i o n appara tus . Th is r e s u l t i s some net t r a n s f o r m a t i o n w i t h the cor responding u n i t a r y operator Comparison of the terms which are f i r s t order i n To- i n equat ions ( 2 . 2 . 1 ) and ( 2 . 2 . 3 ) y i e l d s the commutator ( 2 . 2 . 2 ) . However, the u n i t a r y opera to rs ( 2 . 2 . 1 ) and ( 2 . 2 . 3 ) may d i f f e r by a phase f a c t o r . That i s , we may w r i t e eio-C / 9 £ L x C « ^-IcCo _ ^sa-Zap L-c<r( k w X +• •£.««) = e p ^ where b,. i s a r e a l number and I i s the i d e n t i t y o p e r a t o r . Thus i t f o l l o w s 12 tha t The m u l t i p l e s of the i d e n t i t y are inc luded f o r now because of the p o s s i b i l i t y of a phase f a c t o r . We w i l l show l a t e r how they are e l i m i n a t e d from the commutation r e l a t i o n s of the P o i n c a r e group. Case 1 The p r e p a r a t i o n apparatus of the system i s d i s p l a c e d a long the 1 - a x i s by a , then a long the 2 - a x i s by b. These are f o l l o w e d by the i n v e r s e d i s p l a c e m e n t s . c i P , u b A ftiP«£/R £-LP"Vfc e-.P'V* _ , + ^ L p < o p«»2 + ... • t f -( x '° ,x m , x "U ) — (xm+ a.x^x ^ t ) _ » (x'"+ a , x U H - b,x<3U) _* (x"V a-a,x t l '+ b,x°U) •—•(x ,x + b - b , x , t ) Thus ( x < r t x ( W ' , x ( i > / , t ) = (x<",x<*xt31, t ) The i n i t i a l and f i n a l placements of the p r e p a r a t i o n apparatus are the same; t h e r e f o r e we can take I P'\ Pml = i * a,.I 13 S i m i l a r l y , Case 2 The p r e p a r a t i o n apparatus i s now d i s p l a c e d a long the 1 - a x i s by a and then a t ime t r a n s l a t i o n by tr i s per formed. The i n v e r s e t r a n s f o r m a t i o n s succeed t h e s e . e - t H W * £ I P ^ / * e - i . H « r / * e - i T » u V « S . , + ^ r_p«» V4") + . . . (xw,x'",x"\t) (x"V a,x"»,x"U) (x°v a , x a ; x < 3 ; t - < r ) _ * (xc,>+ a-a,x,M,x<rt,t-«-) — * (x^x'^x^ t-cr+ O = (x w,x , xix a ) , t ) A g a i n , the i n i t i a l and f i n a l placements of the p r e p a r a t i o n apparatus a re the same, so we can take r_p"\ H ! = t * c i S i m i l a r l y , Case 3 Consider next the s u c c e s s i v e t r a n s f o r m a t i o n s c o n s i s t i n g of a t r a n s l a t i o n through a a long the 1 - a x i s , a r o t a t i o n through 9 about the 2 - a x i s , and t h e i r i n v e r s e s . 14 e e e e = t + o.e L P . o J ( x ^ x ^ x ^ t ) — • (x0>+ a ,x w , x ( 3 U) ((x0 >+ a)cos ©'+ x 0 , s i n 9 , x ' V ( x " + a ) s i n 9 + x "cos 9 , t ) _ » ((x c 0+ a ) c o s 9 + x " s i n e -a , x < lJ - ( x ' r t + a ) s i n 6 + x ( 5 ) cose , t ) —•((x">+ a ) c o s 1e+ x ' , s i n e c o s e - a c o s e + (x f , , +a)sin*-© - / c o s Q s i n © , x™, (x'V a ) c o s 9 s i n 6 + x ' 3 s i n * 9 - a s i n © - ( x ° V a) s i n 8 cos6 + x' c o s * 6 , t ) = (x f , )+ a -acos 0 j X ^ x ' ^ - a s i n S , t ) In the l i m i t of a , 6 i n f i n i t e s m a l , (x , x ,x , t ) —*• (x ,x ,x - a 9 , t ) Th is corresponds to a space t r a n s l a t i o n through -a© a long the 3 - a x i s . Thus, r_p<« jc«3= ^(p<*\eJ^ S i m i l a r l y , Case 4 Cons ider a r o t a t i o n through 0 about the x " - a x i s f o l l o w e d by a t ime t r a n s l a t i o n of <r. The i r i n v e r s e s succeed them. (x , x ,x , t ) — » (x ,x cos© - x s m © ,x s x n © + x cos 6 , t ) —*• Cx ,x cos© - x s in© ,x s m o + x cosQ , t^cj) — C x , x cos e - x s m o c o s y +x s m ©+ x cos 9 s in© , - x ( 1cos8 s in© + x ' 3 s i n i § + x ^ s i n ^ cos© + x < 3 c o s X B , t - c r ) Taking the i n f i n i t e s m a l l i m i t , = (x < r t,x a ,,x w,t) The i n i t i a l and f i n a l placements of the p r e p a r a t i o n apparatus a re unchanged so we can take I T ' * H ] = l * f , I S i m i l a r l y , Case 5 Under r o t a t i o n s , the space c o o r d i n a t e s of a p o i n t on the p r e p a r a t i o n apparatus t rans fo rm as x —* x ' =7/lx where x i s the column m a t r i x / land Tn i s a three—by—three m a t r i x which \ w may be w r i t t e n i n terms of p roducts of the u n i t a r y m a t r i c e s : 16 cos 9 O \ - sin 6 O o m 6 \ 0 C O S © / / c o s e sin 8 o i T l / e l , 771l(e), and 7?Z_fG) d e s c r i b e r o t a t i o n s about the l - , 2 - , and 3 - a x i s , r e s p e c t i v e l y . S ince •for r e o J ©..Oi. and 171,(0), i S t o n e ' s Theorem, can be used to w r i t e where the ("V'are H e r m i t i a n o p e r a t o r s . 3 Now Te 6=0 so i t f o l l o w s from equat ion (2..2..^ ) that rT° = /o o 0 \ 0 O - i . \ 0 I 0 j M = /o -i o' l O 0 O O 0/ These m a t r i c e s s a t i s f y the f o l l o w i n g commutation r e l a t i o n . Consider a r o t a t i o n of the p r e p a r a t i o n apparatus of a system about the 1 - a x i s through tp, f o l l o w e d by a r o t a t i o n through © about the 2 - a x i s . These are f o l l o w e d by the i n v e r s e r o t a t i o n s . Jn.J-e)mi(-<p)mi(B)7ril(<p) - » . + < p » [ M t r t 4 M < M ] + = i -+• i. <p e |*\ (31 Th is corresponds to a r o t a t i o n about the 3 - a x i s through an angle of -<$>8. T h e r e f o r e , the operator V B J W / * ^ j " > A e l 6 J < "^ e T ^ 3 i n = i - _ [ J"\ <- • • • •V must equal Consequent ly , we can w r i t e and s i m i l a r l y , 18 Case 6 A space d isp lacement through a a long the 1 - a x i s i s f o l l o w e d by a L o r e n t z boost through v , a l s o a long the 1 - a x i s . These t r a n s f o r m a t i o n s are f o l l o w e d by t h e i r i n v e r s e s . e e. e. e- = i + a -u -c L r K J + - • • ( x l " ) x w , x l , , ( t ) - ^ (xc"+ a,x*",x'",t) = (x<,,,,x<M/,x<w',t/ ) ( (x,,)'+.vt/),x,"/,xt"'> V (t' + v/c?x w')) = (^'".x^.x"'".^ ) (x">"-a,x'0",xrt,",t" ) = (x t , ,",x w"V w"',t" /) ( 8 (x<»'" - v t ) ,x ' l > ' " , x ^ " ' , tf (t'" - v/c 1 x"1'" ) ) = (x< 0 ' \x^ ' v ,x° , w,t w ) x " " v = (xt0+ a + v O ^ - a * - ( v t + v l/c l (x' l ,+ a ) ) * 1 x w , y= x<* x w , v= xt5> t ' \ = (t + v/tf- (x<0+ a ) ) )$ 1 - v/c l (x'n+ a) 8 V / c l t X + v a / c l In the i n f i n i t e s m a l l i m i t , (x ' " ,x l " ,x" , , t ) (x t r t,x"W + v a / c 1 ) Th is corresponds to a t ime t r a n s l a t i o n through - v a / c 1 - s o we w r i t e C P ' " , k u , J = L* (H/c- • k,I) 19 S i m i l a r l y , j-K<j» p < K , j = _ ^ ( 8 . K H / C * + h.D (%.*,. If) Case 7 A Lorentz boost of v e l o c i t y v a long the 1 - a x i s i s f o l l o w e d by a t ime t r a n s l a t i o n of or and then by t h e i r i n v e r s e s . e c H a / * e i c K « ' W * e . i H * / * e . o i K " w + l = , + ^ r _ ^ H ] + • • • (x'^x-nx'^t) — • ( t (x0 ,+ vt),x , u,x" l X ( t + v/clx")) = (x w,x B",x" ,' )t' ) ( x . ^ V x ' ^ ' . X ^ ' . t ' - C T ) = (x<rt" ,x"-,<f,xCM" , t " ) - * ( U ( x ">" - v t " ),x' 1 >",x 'V", * ( t " - v/c^x '*" ) ) = (x""",xw"' )x<, ,'" )t"') ( x ' ° ' " ,x ,"",x < » > " ' , t ' " + <0 = («fc(x,0+ v t ) - v ( t + v/c lx< r t)lf+ vcrY , x ( 1 \ x ° ; ( t +v/c* x*") X1- oT5 - v / c x (x"V v t ) V a1) In the i n f i n i t e s m a l l i m i t , ( x " W , t ) — • (x'V v c r , x < i ; x ^ , t ) Th is corresponds to a t r a n s l a t i o n through v c a l o n g the 1 - a x i s . Thus, and s i m i l a r l y , 20 Case 8 A r o t a t i o n through 0 about the 1 - a x i s i s f o l l o w e d by a Lo rentz boost through v a long the 2 - a x i s and then by the i n v e r s e t r a n s f o r m a t i o n s . ( x " \ x , x ; x i v , t ) — * ( x m , x , l , c o s 9 - x " s i n 6 ,x(°sine + x C i cosQ , t ) = ( x < » ' , x ~ V ' , t ' ) _ * (x < r t # , * (x'"'+ v t ' ) , x w ; V ( t ' + v / c l x < l V ) ) , ( x " w ' , x < " " , x ' " " , t " ) - * ( x « r t " , x , M " c o s 6 + x " v ' s i n 6 , - x t l W ' s i n © + x , J , " c o s 6 , t " ) = ( x < r t " / , x " - , , " , x < M " ' , t ' " ) <x < r t /", X ( x w ' % t " ' ) , x ' " " ' , U (t '" - v / c l x 1 1 " " ) ) = ( x t 0 , v , x t " w , x t r t , v , t , v ) In the l i m i t of v,Q i n f i n i t e s m a l , ( x w , x t t , , x '* ; t ) — ( x < n , x < M , x ™ - e v - t , t - v & /c* x r t 1) Th is corresponds to an i n f i n i t e s m a l Lo rentz boost through -v6 a long the 3 - a x i s so we w r i t e S i m i l a r l y , 21 Case 9 A L o r e n t z boost a long the 1 - a x i s of v e l o c i t y v i s f o l l o w e d by a second L o r e n t z b o o s t , a l s o of v e l o c i t y v ( fo r s i m p l i c i t y ) , a long the 3 - a x i s . These are succeeded by t h e i r i n v e r s e s . ; d c ' J , u 7 K L c l O ' V / * , - c c K ' ^ V / V , - U k ! * ' V / * , , r . . ^ . , o > i e e e. e = \ + Lis , is I K , , ( x , , , , x ' u , x , » t) *( if (x'°+ vt),x a i )x' 5 , ,rt+lSv/c 1x" 1) x (xw ,,x t M ',x">',t') • ( x 4 r t ' , x < « ' , If (x ( 5 )'+ vt ' ) , J ( t '+ v / c 1 ^ ' ) ) = (x , r t" , x l " " , x ' ^ " , t " ) •( J (x^" - v t " ) , x , M " , x , M " , X ( t " - v / c ^ x ' 0 " ) ) = (x < ' ," , ,x <^"',x ( 3 V",t"' ) -*(x" v",x'-"' , X (x^"'-vt"' ) , « (t'" -v/<?-x»>"')) = ( x < n , v , x i u i v , x " v v , t ' v ) x' r t W = f(x'"+ vt) - 5 3 (vt + v V c l O - ^ ( v V ^ x ' * ) X • t v l t f = U x"V V v ( t + v/c 1 -^) - . H v(t + v/c l x^) - * V / c * x a , + ftV/c*- (xl,,+vt) t w = ( t + v/c l x w ) + f v / c - x 3 1 - tfv/c1 (x"+ v t ) - y \ / c l x ' - ] i v / c v (vt••.^(•v/i0') In the l i m i t of v i n f i n i t e s m a l , (x ,x ,x , t ) — * (x - x V /C ,X ,X + X V /c , t ) Th is corresponds to a r o t a t i o n through an i n f i n i t e s m a l ang le -vVc1" about the 2 - a x i s . T h e r e f o r e ? we can w r i t e S i m i l a r l y , We now show how the m u l t i p l e s of the i d e n t i t y a re e l i m i n a t e d from the commutation r e l a t i o n s . Us ing the J a c o b i i d e n t i t y and equat ion (Z.Z.5), we have = [ [ P t , , , P , M ] , T , n ] + [ t J , , \ P ' n 3 1 P , M ] = 0 so o-3l= 0 and, s i m i l a r l y , a^= 0 , J ^ K = I J X ) 3 . Us ing the J a c o b i i d e n t i t y w i t h equat ion (2 . .2 .U) , we f i n d SO Here , we e l i m i n a t e the m u l t i p l e of the i d e n t i t y by adding a term to the H a m i l t o n i a n H. That i s , 23 Then, A l l the m u l t i p l e s of the i d e n t i t y are e l i m i n a t e d from the commutation r e l a t i o n s e i t h e r by u s i n g the J a c o b i i d e n t i t y or by adding r e a l numbers to the g e n e r a t o r s . In summary, the generators of the P o i n c a r e t r a n s f o r m a t i o n s s a t i s f y the f o l l o w i n g commutation r e l a t i o n s : L P ^ P < K , ] = o [ P ^ H 3 = o a - - i . Thus, to d e s c r i b e a system which i s L o r e n t z i n v a r i a n t , one must c o n -s t r u c t from the fundamental dynamical v a r i a b l e s of the system ten H e r m i t i a n o p e r a t o r s s a t i s f y i n g the above commutation r e l a t i o n s . 24 2.3 Various Transformed Operators Recall that an observable A measured by a Poincare transformed measuring apparatus i s related to A measured by the o r i g i n a l apparatus i n thi s manner. A'=UA& + a . a.O where li i s the unitary operator which corresponds to the Poincare trans-formation. Further,!/, may be written i n terms of products of one or more of the ten one-parameter operators lL^(x). We define A("C,j) as The parameter T and a x i s , j , along or about which the transformation takes place specify the label« . That i s , they specify the transformation. Thus, i n t h i s section we use the notation ALt^J) to id e n t i f y the observable measured by the transformed apparatus. Then Equation (2.3.4), along with the i n i t i a l condition A ( 0 > i V - A can be solved for Ad,]") when the commutator tCKlA} i s known. Some results follow. 25 UL(t) H U+(t)= H (%.3.5^ i u o p u + m = p (i.3. 6CMP"<5TW P"1 (u.ioa) A C M P ' U T'W- P < 0 cos* * P°V.«« (X3.100 ( S T U p ^ R " ^ = P W c o S * - P'" m « a.3.io^ L"s(u')P',,,Lm\u>i- P'cosViu.-r UsinKu. (x,.3.a<x) r y P ' - L 0 ' ^ * P'" (x.3.ab) L " U P t " L " , t ( u V P < " a . 3 . . x c ^ U(t) K 'UHi. ) = K+ Pt (i.3.\*rt DiVJtfoVi^ -arxP (x.s.is) 26 ATM J , r t(R i n t(^ --T"1 (i^.n«> R , r t W K w r t M = K'rt (1.3. \ M L u , U U ( l ) L . < n H u V - J ' ^ D s U u K ^ m W rz,.3.iqb") where D M = D " \ a « " ) D ' " ( V " ) D ^ Y a ' ^ = e 1 * * ' * 0 , 3 ^ 0 and <R(e^ < R , s , ( « ) ( R w ( ^ ( R , v ( n e . - 1 * 4 ' * (i.i.xti where 8= 0n and n i s a u n i t v e c t o r a long the a x i s of r o t a t i o n . Before d e r i v i n g any of the above e q u a t i o n s , we i n t r o d u c e the space - t ime d isp lacement o p e r a t o r , , where 27 £fu) = D<*> act) Taking L (uV-- L'Yu^ on the displacement operator and using the transfor-mat ion p r o p e r t i e s of P and H from above, = e = D(x / ) lUt ' ) (L.3.Z4-) where i s the point on a Lorentz boosted preparation apparatus given by equation ( 2 . 1 . 5 ) . Thus the measurement of an observable by a measuring apparatus which has undergone a space-time displacement through x' yields the same result as the measurement of that observable by an apparatus which has undergone a Lorentz boost through U, a space-time displacement through x, and, f i n a l l y , a Lorentz boost through -U. That i s , We now derive some of the equations ( 2 . 3 . 5 ) through ( 2 . 3 . 2 0 ) . Equations ( 2 . 3 . 5 ) through ( 2 . 3 . 9 ) and equation ( 2 . 3 . 1 3 ) follow immediately since the corresponding commutators i n equation ( 2 . 3 . 4 ) vanish. Equations ( 2 . 3 . 1 0 ) are now derived. For a rotation through « about the 1-axis, define £fW)A &\*) - L u , t(u\ #(V) L"V) AL'nV> # +ML"\u> 28 Then and P ' j \ 0 ) , ) x P ^ Since , we have drwplrtvTVi= p c 0 Forj = x andj=3, we obtain the following coupled equations. d o t D i f f e r e n t i a t i n g with respect to *. , Also, These equations have equations (2.3.10b) and (2.3.10c) as their respective solutions. Thus (P , P , ? 0transform under rotations l i k e the components of a three-vector so we write P= (P l,P'"P ). Equations (2.3.17) and (2.3.18) are derived i n exactly the same way. Thus, we write T= (J4* J ' " 3"°*) and K = (K K,'" K'*). We now derive equations (2.3.11) and (2 .3.12). Under a Lorentz boost through u along the 1-axis, define d l P ( " ( V U . P < ! , ( , , ^ <A«V = 1 . X.3 P'^o,^ P^ O b v i o u s l y , f o r ^-x, and 3 =3 we have the r e s u l t s s i n c e the commutators v a n i s h . We a l s o d e f i n e Then ifc a_B(u,A - L U X^ CcK°\ H"\|_u*(u> - ^ V U ^ c P ' ^ L ' ^ M and M(o,^  = H The coupled equat ions have as t h e i r s o l u t i o n s equat ions ( 2 . 3 . 1 1 ) and ( 2 . 3 . 1 2 a ) , r e s p e c t i v e l y . We now d e r i v e equat ion ( 2 . 3 . 1 4 ) above. For a t ime t r a n s l a t i o n through t , d e f i n e 30 Then cW\<(V) = U It") L H , K l (t) dt -- U(i^ (it P^U+(tN) = P and K(6)= K T h e r e f o r e , we have equat ion (2.3.14) as the s o l u t i o n . To d e r i v e equat ion (2.3.15), cons ide r one component of J , T J , d i s p l a c e d by a a long the K - a x i s . D e f i n e Then A J T ' J V " , K) = D ' ^ C a " " ) I P U ^ , T^3 D W D ' K V - M ^ f e i K 1 P m > ) D < K , t ( a ^ = - L-K £ j K j . P A l s o , J*(o,tf = J 1 * T h e r e f o r e , we have s o , f o r a g e n e r a l d isp lacement through g.r a'"e, + a.<t)e, + c'"e , we have Thus equat ion (2.3.15) f o l l o w s . Equat ion (2.3.16) i s d e r i v e d s imply as (2.3.14) was. Equat ions (2.3.19) 31 and ( 2 . 3 . 2 0 ) are d e r i v e d below. Under Lorentz boosts through u. a long the 1 - a x i s , d e f i n e Then \*h dT^(u,0> = C\,A L C K ^ T ^ J L ^ M du and J ^ f l . ^ T ^ In the s i m p l e s t c a s e , j=> and du so tha t equat ion (2 .3 .15a ) f o l l o w s . For j = ^ , we have du The cor responding equat ion f o r xs du - ^ J ^ ( u C ' A l s o , K"'(o,\) = K ' * Now we have a system of two coupled equat ions . du d u Th is system has as i t s s o l u t i o n equat ions (2 .3 .19b) and ( 2 . 3 . 2 0 c ) S i m i l a r l y , we have du du w i t h equat ions ( 2 . 3 . 1 9 c ) and (2 .3 .20b) as s o l u t i o n s . A l s o , 33 2.4 P o i n c a r e I n v a r i a n t s H and U There are c e r t a i n combinat ions of opera to rs which a re i n v a r i a n t under a l l P o i n c a r e t r a n s f o r m a t i o n s . E q u i v a l e n t l y , such i n v a r i a n t s commute w i t h a l l the generators of the P o i n c a r e group. L e t the operator M*" be d e f i n e d by From the commutation r e l a t i o n s of H and P i t i s c l e a r t h a t t h i s ob jec t commutes w i t h a l l the generators of the P o i n c a r e group. M i s t h e r e f o r e an i n v a r i a n t under a l l P o i n c a r e t r a n s f o r m a t i o n s . The nonnegat ive q u a n t i t y (M 1-) 7 1" = r\ i s c a l l e d the t o t a l mass of the system. We a l s o d e f i n e the P a u l i - L u b a n s k i f o u r - v e c t o r ( ^ W ) where # = J - P a . 4 ^ W= c ( K*P) + i HT U.H-.^  i s c a l l e d the t o t a l h e l i c i t y of the system. A l s o , we note tha t •kftH- W - P = o CL.HAV) H a n d W t rans fo rm under P o i n c a r e t r a n s f o r m a t i o n s as f o l l o w s : 34 ft'"(.W'Tw = W ° c o s « - W " e i n « U.H.\0O L " \ v > W L t r t + ( ^ » * f o o s h u - W " U W (X.4 .U ) r w w ^ r ^ ^ y ^ - U . 4 . H 0 ) The above t r a n s f o r m a t i o n equat ions a re a l l ob ta ined from equat ions ( Z . ^ . Z ) and (Z..H.3) and from 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 of the generators H,P, T, and K of the p rev ious s e c t i o n . For example, We now d e f i n e an operator ZJ~ i n terms of the opera to rs "H a n d W . i s i n v a r i a n t under the P o i n c a r e t r a n s f o r m a t i o n s . That i s , UL(tMJ'7-Uf(t) = V " x fai.f.i^ 35 (R(9)l/%6?(e)-- 2^" ci.*. it) These i n v a r i a n c e r e l a t i o n s are a r r i v e d a t e a s i l y . For example, under a Lo rentz boost a long the 1 - a x i s , t ransforms as cos/) It - W s /'n A U « /V1- w-w 36 2 .5 F o u r - v e c t o r N o t a t i o n Equat ions ( 2 . 3 . 6 ) , ( 2 . 3 . 1 3 ) , and ( 2 . 3 . 1 4 ) show that P ,5 , and K t rans fo rm l i k e t h r e e - v e c t o r s under r o t a t i o n s . In t h i s s e c t i o n we show that combinat ions of the generators t rans form under Lo rentz boosts l i k e f o u r - v e c t o r s and l i k e a rank two t e n s o r . Equat ions ( 2 . 3 . 7 ) and ( 2 . 3 . 8 ) may be w r i t t e n as (i.s-.O where I c o aV\ i»-•6ir\Ku 0 0 O 0 and P^--(^H (x.S\ 3") Thus P^  t ransforms under Lo rentz boosts l i k e a c o n t r a v a r i a n t v e c t o r . We take the m e t r i c tensor ^/*v to be such that Thus, Pj^ = J ^ ^ P ^ ( s H s - P ^ . One can thus w r i t e equat ion ( 2 . 4 . 1 ) as (x.S'.'V) Equat ions ( 2 . 4 . 4 ) . , ( 2 . 4 . 1 1 ) , ( 2 . 4 . 1 2 ) , and ( 2 . 4 . 1 3 ) may be w r i t t e n as P^W^=0 (x.s-.s") (x.s-.O (xs-.i) 37 where W ) Fur thermore , the equat ions ( 2 . 3 . 1 5 ) and ( 2 . 3 . 1 6 ) may be w r i t t e n as where ( r V \ fT^ = 1= ( J ^ ^ ' l and ( r r , n°\ n M)= C K = ( C K u : t r t n AISO, r r = - r r ^ Thus, the t rans fo rm under Lo rentz boosts as a rank two c o n t r a -v a r i a n t t e n s o r . The commutation r e l a t i o n s f o r the generators may be w r i t t e n more compactly i n terms of and YV* as f o l l o w s . 1 * L P ^ P * ] --0 U.S. si) Equat ion ( 2 . 5 . 1 1 ) corresponds to L P ' ^ P ^ ^ O c,nd [ P ^ H K O , equat ion ( 2 . 5 . 1 2 ) to and equat ion ( 2 . 5 . 1 3 ) to 38 Equat ions ( 2 . 5 . 1 1 ) , ( 2 . 5 . 1 2 ) , and ( 2 . 5 . 1 3 ) can be r e w r i t t e n i n terms of the c o v a r i a n t q u a n t i t i e s P ^ and as 39 2 .6 The Space I n v e r s i o n Operator (P We now extend the above c o n s i d e r a t i o n s to i n c l u d e a l s o space i n v e r s i o n . That i s , we cons ide r the t r a n s f o r m a t i o n ( x , t ) - * ( x ' , t ' ) = ( - x , t ) (x.fe.O Th is i s not a P o i n c a r e t r a n s f o r m a t i o n ( 2 . 1 . 1 ) and i t i s not c h a r a c t e r i z e d by a cont inuous parameter as a re the P o i n c a r e t r a n s f o r m a t i o n s . The t r a n s f o r m a t i o n ( 2 . 6 . 1 ) corresponds to v iew ing the p r e p a r a t i o n or measuring process i n a m i r r o r . We assume t h a t equat ions ( 2 . 1 , 8 ) and ( 2 . 1 . 9 ) ho ld f o r a l l s t a t e s and observab les prepared and measured by apparatuses r e l a t e d by equat ion (2.6.1). Thus, by Wigner ' s Theorem,2 one has ly> = (Pl^> (Ub.Vi A ' = (P A (P+ where (P i s e i t h e r a l i n e a r u n i t a r y operator or an a n t i l i n e a r a n t i u n i t a r y operator which corresponds to the t r a n s f o r m a t i o n ( 2 . 6 . 1 ) . Moreover , s i n c e the product of two i n v e r s i o n s l e a v e s the apparatus unchanged one can w r i t e so We now i n v e s t i g a t e v a r i o u s combinat ions of P o i n c a r e t r a n s f o r m a t i o n s and space i n v e r s i o n s . 40 O b v i o u s l y , a space i n v e r s i o n f o l l o w e d by a t ime t r a n s l a t i o n and then by a second space i n v e r s i o n i s e q u i v a l e n t to a t ime t r a n s l a t i o n a l o n e . ( x , t ) — * ( - x , t ) — * ( - x , t - € ) ( x , t - e ) T h e r e f o r e , A space i n v e r s i o n f o l l o w e d by a space d isp lacement and then by another space i n v e r s i o n y i e l d s the t r a n s f o r m a t i o n ( x , t ) — » ( - x , t ) — f r ( -x + a , t ) — f r ( x - a , t ) which corresponds to the i n v e r s e space d isp lacement so tha t (? D^(P += D(-^ {T-.b.D For the case of an i n v e r s i o n f o l l o w e d by a r o t a t i o n , then another i n v e r s i o n , we have ( x , t ) — f r ( - x , t ) — • ( - x R , t ) — f r ( x t t , t ) where x. R i s the p o i n t on the r o t a t e d a p p a r a t u s . T h e r e f o r e , For an i n v e r s i o n f o l l o w e d by a Lorentz boost through a v e l o c i t y v a long the 1 - a x i s , then another i n v e r s i o n , ( x , t ) — f r ( - x , t ) _ f r (* (-x'°+ vt ) , - x ' 1 > , - x C 3 > , ( t - v/c -x< 1 , )X ) — ( 5 (x" ' -vt) , x u , , x in ( t - v / c - x"1) ) 41 Th is corresponds to a Lorentz boost through - v a long the 1 - a x i s so In summary, we have (?U(t)C? + = U(t) ix.b.til CPD(a){?+-- D(-a> (x.b.-rt ( P & f e ^ - &(§) a.e.O Since U (•(:)=e~iH we have from equat ion ( 2 . 6 . 6 ) tha t 9 H = H i f (P i s l i n e a r or <5>H(P+=-H i f (P i s a n t i l i n e a r However, H and (P H (P+have the same spectrum s o , i f H and -H have d i f f e r e n t s p e c t r a ( fo r example, i f H i s bounded below but not above) , then (P must be l i n e a r . I t f o l l o w s that the a c t i o n of (P on the generators of the P o i n c a r e group i s (P H (P^  = H (7,.b.\& (pp(p + - , p (x.fc.lrt ®Z (P+ = J a.fe.Krt 42 ( ? K ( P f = - K (X.fe.\3) These a re c o n s i s t e n t w i t h the commutation r e l a t i o n s f o r the g e n e r a t o r s . For example, I P L P * * K ' 1 0 ] S>+= [ I P P ^ I P * ( P \ 0 " < P + J -- L P ^ K 1 ^ - - W i U ^ H / c ^ ( P + = ^ S J K H I* as r e q u i r e d . 43 2.7 The Time R e v e r s a l Operator T We now extend the above c o n s i d e r a t i o n s to i n c l u d e t ime r e v e r s a l as w e l l . That i s , we cons ide r the t r a n s f o r m a t i o n ( x , t ) — * ( x ' : , t ' ) = ( x , - t ) (x,.7.r> T h i s i s not a P o i n c a r e t r a n s f o r m a t i o n ( 2 . 1 . 1 ) and i t i s not c h a r a c t e r i z e d by a cont inuous parameter as are the P o i n c a r e t r a n s f o r m a t i o n s . The t r a n s f o r m a t i o n ( 2 . 7 . 1 ) corresponds to v i e w i n g i n r e v e r s e a mot ion p i c t u r e of the preparat ion-measurement p r o c e s s . We assume that equat ions ( 2 . 1 . 8 ) and ( 2 . 1 . 9 ) ho ld f o r a l l s t a t e s and observab les prepared and measured by apparatuses r e l a t e d by equat ion ( 2 . 7 . 1 ) . Then, one has lT|/'> = T ITJO (v i i ) A' = T" A T ^ a.-?.3\ whereT* i s e i t h e r a l i n e a r u n i t a r y operator o r . a n a n t i l i n e a r a n t i u n i t a r y operator which corresponds to the t r a n s f o r m a t i o n ( 2 . 7 . 1 ) . Moreover , s i n c e the product of two t ime r e v e r s a l s l eaves the apparatus unchanged one can w r i t e so nr + = T ( - L . 7 . 5 ) We now i n v e s t i g a t e v a r i o u s combinat ions of P o i n c a r e t r a n s f o r m a t i o n s and time r e v e r s a l s . 44 A t ime r e v e r s a l f o l l o w e d by a t ime t r a n s l a t i o n and then by a second t ime r e v e r s a l y i e l d s the t r a n s f o r m a t i o n ( x , t ) — * ( x , - t ) — • ( x , - t - e ) _ f r ( x , t + & ) Th is i s e q u i v a l e n t to t ime t r a n s l a t i o n through - € . In terms of the u n i t a r y o p e r a t o r s , O b v i o u s l y , i f the t ime t r a n s l a t i o n i s r e p l a c e d by a space d isp lacement i n the sequence of t r a n s f o r m a t i o n s above the net t r a n s f o r m a t i o n i s j u s t the space d i s p l a c e m e n t . That i s , The same i s t r u e f o r a t ime r e v e r s a l f o l l o w e d by a r o t a t i o n , then by a t ime r e v e r s a l s i n c e the r o t a t i o n o n l y t ransforms: the space c o o r d i n a t e s and the t ime r e v e r s a l s ac t on o n l y the t ime c o o r d i n a t e . T h e r e f o r e , For a t ime r e v e r s a l f o l l o w e d by a Lorentz boost through v a long the 1 - a x i s , f o l l o w e d by another t ime r e v e r s a l , ( x , t ) — * ( x , - t ) — » ( (x<°-vt) , x m , x i n S ( - t + v/tf-x" 1)) ( * (x'^-vt) , x i nx i n ( t - v/c" x < r t)) Th is corresponds to a Lo rentz boost through - v a long the 1 - a x i s so 45 In summary, we have ro(c i N ir + = < x.i.i) The a c t i o n of f on the generators of the Poincare - group w i l l be i n v e s t i g a t e d . From equat ion ( 2 . 7 . 6 ) above, we see that t H T ^ . - H i f f i s l i n e a r THT + = H if T is a n t i l i n e a r The s p e c t r a of THT^and H are i d e n t i c a l . I f H and -H have d i f f e r e n t s p e c t r a , then f must be a n t i l i n e a r . The a c t i o n o f f on the generators of the P o i n c a r e group i s such that T H T ^ = H (%.i.\o) T P T + = -P (VT.III f 2 f + - - I cx.7.\n T KT+= K (X.7.1-3) These are c o n s i s t e n t w i t h the commutation r e l a t i o n s f o r the g e n e r a t o r s . For example, 46 TTP,J: \<<W!T^ -LP^K< K ,J = r , r ^ s J K T ( H / c ^ / r + = . - ^ s . K H / ^ so t h a t as r e q u i r e d . 47 2.8 The Centre of Mass P o s i t i o n ^ , I n t e r n a l Angular Momentum 4, and Centre of Mass V e l o c i t y V Operators We now i n t r o d u c e new operato rs % and $ which we c a l l the p o s i t i o n of the c e n t r e of mass of the system and the t o t a l i n t e r n a l angular momentum of the system, r e s p e c t i v e l y . ( I t i s shown, i n the next s e c t i o n tha t % and $ t rans form l i k e t h r e e - v e c t o r s under r o t a t i o n s . ) In terms of these o p e r a t o r s , we express the t o t a l angular momentum of the system as where P i s the t o t a l momentum of the system. To r e t a i n the commutation r e l a t i o n s ( 2 . 2 . 1 5 ) of the generators c o n d i t i o n s must be imposed on % and $ . In p a r t i c u l a r , the commutation r e l a t i o n s (2 .2 . 15c ) , (2.2.15J), and (2.2.)5e) f o l l o w when (x.^.x/) Moreover , as we show i n Appendix B, the remain ing commutation r e l a t i o n s i n v o l v i n g J and K f o l l o w w i thout f u r t h e r c o n d i t i o n s on % and & when wherein M t 1 - - ( H 1 - c ^ P x V / x (X.X.M-') and ^ = £ H" where F_ - + -/H1" A l s o shown i n Appendix B are the i n v e r s e r e l a t i o n s 48 M X = -ft M c x ( V T ' K -h K H~") - f x J -f P x ( P x K ) (x.S.S) /$>M$ = K x P +_H_-2 -Equat ion (2.8.5) i s the same as the t o t a l p o s i t i o n operator g i ven by Osborn f o r a system of two n o n i n t e r a c t i n g r e l a t i v i s t i c p a r t i c l e s and equat ion (2.8.6) agrees w i t h Schwinger 's e x p r e s s i o n f o r the t o t a l angular momentum . 5 Thus i f one has determined the ten generators H , P , J , and K i n terms of the fundamental dynamical v a r i a b l e s of the system then % and $ a re g i ven by equat ions (2.8.5) and (2.8.6). A l t e r n a t i v e l y , i f one has determined ten opera to rs H, P , % , $ which s a t i s f y LP' J; ?'"]--o C P y \ H ] = o <»..«.7) i n a d d i t i o n to the commutation r e l a t i o n s (2.8.2), then I and K a re g i v e n by equat ions (2.8.1) and (2.8.3). That i s , equat ions (2.8.2) and (2.8.7) are s u f f i c i e n t c o n d i t i o n s f o r a p h y s i c a l system to be Lo rentz i n v a r i a n t . I t i s c l e a r from equat ions (2.8.1) and (2.8.3) tha t the operator §>*~- % commutes w i t h J and IS, as w e l l as w i t h P and H . I t i s t h e r e f o r e a L o r e n t z i n v a r i a n t . In f a c t , T h i s r e l a t i o n i s d e r i v e d as f o l l o w s . S u b s t i t u t i n g equat ions (2.8.1) and (2.8.3) f o r T and K i n t o equat ions (2.4.2) and (2.4.3) f o r fa"and W, 49 W = i H ( X * P ) + £ H $ + I c P x ( | H + H ^ ) - _ c P x ( $ * B ) (i.%.\6) Us ing the i d e n t i t y P * (-1 x P ) = P " $ - ( £ • & ) P c^.S. l l ) and the commutator we r e w r i t e as = c ^ n $ + _ c ( P - ^ P S u b s t i t u t i n g equat ions ( 2 . 8 . 9 ) and ( 2 . 8 . 1 3 ) above i n t o equat ion ( 2 . 4 . 1 3 ) f o r If* I t i s convenient to d e f i n e a H e r m i t i a n operator V by LS 5H] = M cM.m Then the cor responding Heisenberg opera to rs 50 V(t) = U r(0 V U(t) a..*.is) and a re r e l a t e d v i a equat ion ( 2 . 8 . 1 5 ) by V(0= d%-(t) (vB.U) i t Thus V i s the v e l o c i t y of the c e n t r e of mass of the system. I d e n t i f y i n g equat ion ( 2 . 8 . 1 2 ) and equat ion ( 2 . 8 . 1 4 ) , we have V = P H" ( Z . S . I S ) F i n a l l y , i n a d d i t i o n to the commutation r e l a t i o n s ( 2 . 8 . 2 ) we can w r i t e [K«\ r ° ] - i-n {-^(%°"\J'^W) + vM(v\>$r\<>)\ (V\+finery1"} h.%.ix» LK v, v<K,l = -i-n(sjK+i>v''VKV) u.s.W) 51 2 .9 T rans fo rmat ion P r o p e r t i e s of the Observables % , ^ , and V An observab le A measured by a t ransformed apparatus i s r e l a t e d to A measured by the o r i g i n a l apparatus by K - - U k l L i cz.q.O where li i s the u n i t a r y or a n t i u n i t a r y operator which corresponds to the t r a n s f o r m a t i o n - The observab les X , S ,V t rans form as f o l l o w s under the Poincare - t r a n s f o r m a t i o n s , space i n v e r s i o n , and t ime r e v e r s a l . iLMXUiHt) = X- V t U<\.TA D(±)% D + ( ^ = t - a U*-^ r w r L ^ u ^ r ^ r r v" + v" r i a.q.s-^ L,,(uU<°L,,+w= ( r v<o + v" r i 1 ^ , 3 U ( t ^ u + i t w 4 a ^ 52 0 (O4 D+(<x> = & cx.<*.<0 CO-=,Y| -t- ( l - CO&T^ N - sir\^ ( N X-S^ (x.<*. '0 where M - - P u , U . S . U ) and X 7 tan" /eu(H-ce,-P) t M c ^ - Un7 U - c P ^ ^  \ (X.<U^ and e.i i s a unit vector along the 1-axis. ( p | ( p + = 4 (-L<\.M) T 4 T + - - & (x.<uer) U U W l A + ( t ^ V (z. A.I fa) m" ,WV" ,r tW = V'" (x . i .r^ 1- V"V/c-53 r y r t = -v d.<\.ao) A l s o , s i n c e /$ commutes w i t h a l l the g e n e r a t o r s , < R ' J W < R , I } V > = ^ (2.1.13) L^M/frL^M- i r , ^ , 3 (z.q.m) T h u s , ^ i s a L o r e n t z i n v a r i a n t . Most of the above equat ions a re d e r i v e d as i n S e c t i o n 2 . 3 . We d e r i v e the except ions below. Equat ions ( 2 , 9 . 5 ) a re found as f o l l o w s . 6 L e t us c o n s i d e r the commutator where we have used equat ions ( 2 . 3 . 1 2 a ) and ( 2 . 8 . 2 ) . Taking l_"u). . . L°^(u) on both s i d e s of the equat ion above y i e l d s L i ^ u Y t V t a P a r U ( c o 5 U t U " V W " , L " , t ( u U , „ U S l 1 - VV/o1- / 54 or [L ' " [u)rL M t W, - _ i h _ _ UA.is) X(\-V"V/c>-) S i m i l a r l y , L r w V" LMfiu), H i = Equat ions ( 2 . 9 . 2 5 ) and ( 2 . 9 . 2 6 ) a re s a t i s f i e d when i s g i ven by equat ion ( 2 . 9 . 5 a ) . S i m i l a r l y , LL 4 r t(uU L"V , +(^^\ lP* , ,] - ^ V " V d . q .n ) and [ r ^ r r b . H i - (JL.,.^ Equat ions ( 2 . 9 . 2 7 ) and ( 2 . 9 . 2 8 ) are s a t i s f i e d when L" X l l < > i l " Y u ) i s g i ven by equat ion ( 2 . 9 . 5 b ) . Equat ions ( 2 . 9 . 6 ) and ( 2 . 9 . 7 ) f o l l o w from That i s , so (PX(P + , - 1 S i m i l a r l y , so r x r ^ x • 55 Equat ions ( 2 . 9 . 1 4 ) and ( 2 . 9 . 1 5 ) f o l l o w s i m i l a r l y from and equat ions ( 2 . 9 . 1 9 ) and ( 2 . 9 . 2 0 ) f o l l o w from We now d e r i v e equat ion ( 2 . 9 . 1 1 ) . 7 For a Lorentz boost of v e l o c i t y v a long the 1 - a x i s , we d e f i n e Then, du The commutator can be w r i t t e n as where e.( i s a u n i t v e c t o r a long the 1 - a x i s . Thus, i_<$(-u,»W c (r\(u,^^ + ^ ^c-V , (U,x P U ^ x l l u , ^ du where H(u,rt i s g i ven by equat ion ( 2 . 3 . 1 1 ) . A l s o , &, x. £ (u,rt = t , x P Now, we l e t N - j,*P a n d dji - c PI ijio)- 0 Then, The s o l u t i o n of the above equat ion i s g i ven by equat ion ( 2 . 9 . 1 1 ) . Equat ion ( 2 . 9 . 2 9 ) i n t e g r a t e s to g i v e equat ion ( 2 . 9 . 1 3 ) . We now d e r i v e equat ions ( 2 . 9 . 1 8 ) . They f o l l o w from equat ion ( 2 . 8 . 1 8 ) . That i s , S i n c e i t f o l l o w s t h a t so C W H - ' L ^ l u ^ ( C'^ Urt L"nWT' T h e r e f o r e , Thus, s C - P " ' H " - c U L u Equat ion (2 .9 .18b) f o l l o w s s i m i l a r l y . 58 2.10 The Operators T and %(T) We have seen tha t the cen t re of mass v e l o c i t y V t ransforms i n the f a m i l i a r manner under Lo rentz b o o s t s . That i s , equat ions ( 2 . 9 . 1 8 ) rep resent the E i n s t e i n a d d i t i o n of v e l o c i t i e s . However, as i s seen from equat ions ( 2 . 9 . 5 ) , the c e n t r e of mass p o s i t i o n operator does not t rans fo rm i n a f a m i l i a r or even s imple way under Lorentz b o o s t s . We have not yet g i ven an operator equat ion analogous to the f a m i l i a r s p a c e - t i m e t r a n s f o r -mat ion ( 2 . 1 . 6 ) . We now r e c t i f y t h i s m a t t e r . In so do ing we c o n s t r u c t opera to rs T and whose p h y s i c a l i n t e r p r e t a t i o n r e q u i r e s some comment.6 Th is i s taken up at the end of t h i s s e c t i o n . Le t the H e r m i t i a n operator T be d e f i n e d by L ^ W T L ^ V ^ t c o s h V L - cT X"\t) s ink u. ( L . I O . l ) where )|\ (t) i s the component of the Heisenberg p o s i t i o n operator a long the 1 - a x i s . That i s , t'\t)= u+(t} f u n ) - r ' + v ' t (1.1 0.1,) F u r t h e r , we d e f i n e the H e r m i t i a n operator %. (T)by X (TV£ + ±(\/T + T\n (h.\o.i) Upon t a k i n g C CuY-.L^ui on both s i d e s of equat ion ( 2 . 1 0 . 3 ) and u s i n g equat ions ( 2 . 9 . 5 ) , ( 2 . 9 . 1 8 ) , and ( 2 . 1 0 . 1 ) , we f i n d tosku. - ct sinhu U . 1 0 . H - ) (1.10.5"") 59 Along w i t h equat ion ( 2 . 1 0 . 1 ) , these equat ions are the d e s i r e d operator analogs to the space - t ime t r a n s f o r m a t i o n ( 2 . 1 . 6 ) . One a l s o has the rear ranged equat ions which f o l l o w . X"\t)= L ' V i r t n r V i coaku + c L ' V i T L ^ a i n h u U . \ 0 . a X < K ,(^= L W ( u U ' M ( T ) LM\<A * = Z , 3 ( X . I 0.7) We now d e r i v e equat ion ( 2 . 1 0 . 4 ) . From equat ion ( 2 . 1 0 . 3 ) , we have r°m= r +i(v"T,Tvn u.»o.^  Taking .U'WV.. LV\-u.) on both s i d e s of t h i s equat ion y i e l d s w h i c h , on u s i n g equat ions ( 2 . 9 . 5 ) , ( 2 . 9 . 1 8 ) , and ( 2 . 1 0 . 1 ) , may be r e w r i t t e n as l - V " V l t l l - W c 1 - u-v%/^ + ( i tosku- c (% "'+- V""-t) a'. nUu ) _V^iy_ ~| I- V '^ / t 1 " J A f t e r some a l g e b r a , t h i s reduces to equat ion ( 2 . 1 0 . 4 ) as r e q u i r e d . Equat ion ( 2 . 1 0 . 5 ) f o l l o w s s i m i l a r l y . I t must be s t r e s s e d tha t equat ions ( 2 . 1 0 . 1 ) , ( 2 . 1 0 . 4 ) , and ( 2 . 1 0 . 5 ) a re on ly analogous to the t r a n s f o r m a t i o n ( 2 . 1 . 6 ) . They do i n c o r p o r a t e the c o r r e c t quantum mechanica l asymmetry between p o s i t i o n and t i m e ; tha t i s , 60 on the r i g h t hand s i d e of each e q u a t i o n , p o s i t i o n appears as a H e r m i t i a n operato r w h i l e t ime i s a parameter of the t h e o r y . The equat ions l o o k a p p r o p r i a t e i n form but c o n t a i n the o b j e c t s I and 1 ( f ) . These opera to rs do not have an immediate p h y s i c a l i n t e r p r e t a t i o n but we w i l l now o b t a i n from the above r e s u l t s equat ions which depend o n l y on o b j e c t s which are p h y s i c a l l y m e a n i n g f u l . Assume that l ip) i s p r a c t i c a l l y an e igenvec to r of LTMY LU'YU). (a l though t h i s operator has no r e a l e igenvec to rs ) Taking on both s i d e s of equat ion ( 2 . 1 0 . 3 ) , L'^XmirW- Yl'XMl^U) + t L l " ( a U \ / T + T V K u , + ( 0 A l s o , T h e r e f o r e , where i ' = <l|j \ U " W> T YJ '^ULMI^ Since the Heisenberg p o s i t i o n operator i s i t f o l l o w s tha t r w l L ^ l u ^ L l \ ) I ( t ' ) L w t W - C H u W L ' H ^ - t ' S u b s t i t u t i n g the above equat ion i n t o equat ion ( 2 . 1 0 . 1 1 ) , we f i n d 61 Thus, when averaged over the " e i g e n v e c t o r s " \^)) of L ' ' U W C & , equat ions ( 2 . 1 0 . 1 ) , ( 2 . 1 0 . 4 ) , ( 2 . 1 0 . 5 ) , ( 2 . 1 0 . 6 ) , ( 2 . 1 0 . 7 ) , and ( 2 . 1 0 . 8 ) become, r e s p e c t i v e l y , t ' ^ tcosWu- b V\V> s \ A u t l . 1 0 . l t i The equat ions above do not c o n t a i n the opera to rs T and X(T"). R a t h e r , they depend on the parameter t and the average v a l u e of the Heisenberg p o s i t i o n o p e r a t o r . 62 Chapter 3 D e s c r i p t i o n s of Some Lorentz I n v a r i a n t Systems In t h i s c h a p t e r , a set of ten generators s a t i s f y i n g the commutation r e l a t i o n s ( 2 . 2 . 1 5 ) i s c o n s t r u c t e d f o r each of s e v e r a l systems. F i r s t , i n S e c t i o n 3 . 1 , we c o n s t r u c t from the fundamental dynamical v a r i a b l e s the generators f o r a system c o n s i s t i n g of a s i n g l e f r e e s p i n l e s s p a r t i c l e . We then cons ider a system c o n s i s t i n g of a s i n g l e f r e e p a r t i c l e w i t h s p i n i n S e c t i o n 3 . 2 . In each of these s e c t i o n s we f i n d that the c e n t r e of mass p o s i t i o n % i s equal to the C a r t e s i a n c o o r d i n a t e s of the p a r t i c l e and t h a t the t o t a l i n t e r n a l angular momentum 4> i s equal to the s p i n ^ of the p a r t i c l e . A l s o , the u n i t a r y o p e r a t o r s cor responding to the P o i n c a r e t r a n s f o r m a t i o n s and to space i n v e r s i o n a long w i t h the a n t i u n i t a r y operator cor responding to t ime r e v e r s a l a re expressed i n i n t e g r a l _ form i n these s e c t i o n s . The generators of the P o i n c a r e group are cons t ruc ted i n S e c t i o n 3 . 3 f o r a system c o n s i s t i n g of a s i n g l e f r e e D i r a c p a r t i c l e i n terms of the fundamental dynamical v a r i a b l e s of the system. Express ions f o r % and i n terms of the fundamental dynamical v a r i a b l e s are o b t a i n e d . F i n a l l y , the opera to rs % and $ are seen to be r e l a t e d to the fundamental dynamical v a r i a b l e s X and S v i a the w e l l known Pryce -Fo ldy -Wouthuysen t r a n s f o r m a t i o n . 8 In S e c t i o n 3 . 4 , a system of P- n o n i n t e r a c t i n g p a r t i c l e s i s cons idered where in the generators of the P o i n c a r e group are seen to be sums of s i n g l e p a r t i c l e g e n e r a t o r s . I n t e r a c t i o n i s in t roduced to the system i n S e c t i o n 3 . 5 . In s p e c i f y i n g the i n t e r a c t i o n , i t becomes necessary to d e f i n e i n t e r n a l opera to rs and express ions are g i ven f o r some of these a long w i t h v a r i o u s commutation r e l a t i o n s which they s a t i s f y . The r e s u l t s a re compared w i t h those f o r a system of Ti i n t e r a c t i n g n o n r e l a t i v i s t i c p a r t i c l e s i n Appendix I. 63 3.1 Single Spinless Free P a r t i c l e Let us consider a single spinless free p a r t i c l e of massWt. We take the fundamental dynamical variables of th i s system to be the Cartesian , A , A and the momenta of the p a r t i c l e which s a t i s f y the fundamental quantum conditions. [P 9 ,P ' K , ] - - o tXu\ P l w ] - - i U i K j4k=»,x,3 (3.1.0 y a l y<-W D<,i _tn We assume that the operators h , A, A and r , r , r each form a complete set of compatible observables. We denote their eigenkets by lx"> = x<3,> and 1 = I p'* p1*'p<5V> . The eigenkets may be used as bases for the Hilbert space. X < j ' = J d \ lx> <&l ( 3 1 ^ I = j d ^ l*>^*\ = $d3p l p > < ^ 4x1 4') = &(4-X.O (3.1.5) <£\£'> = & Cp-o.0 (3.1.6) Also i t follows from the commutation relations (3.1.1) that 64 We now seek a c o n s t r u c t i o n of the generators of the Poincare* group i n terms of the fundamental dynamical var i a b l e s X and V ( k = l , 2 , 3 ) . A l l of the commutation r e l a t i o n s ( 2 . 2 . 1 5 ) ho ld when J = X x P <3.l.8) K - ( XH + H>0 O.i.q) H= ( P V + Tn^cV'*- = j a 3 P l ^ e ( P ) ( p l (3.1.10) where e(p") = (p^c1- + m c^1*) ''^ ( 3 . \ i ^ Thus the p a r t i c l e i s s a i d to be a Lo rentz i n v a r i a n t system when T, K , and H are g i ven by the above e q u a t i o n s . Fur thermore , i t f o l l o w s from equat ion (3.1 .10) tha t the v e l o c i t y V of the p a r t i c l e i s Y= c 1 P H" = L X , H ] / i * (3.\.ixl Thus X s a t i s f i e s the commutation r e l a t i o n s ( 2 . 8 . 2 ) and indeed s u b s t i t u t i o n of equat ions ( 3 . 1 . 8 ) , ( 3 . 1 . 9 ) , and ( 3 . 1 . 1 0 ) i n t o equat ions ( 2 . 8 . 4 ) , ( 2 . 8 . 5 ) and ( 2 . 8 . 6 ) shows that % ^ X 0 ( 3 . U 3 ) F i n a l l y , the cor responding u n i t a r y opera to rs f o r the P o i n c a r e t r a n s f o r - ; m a t i o n s , the space i n v e r s i o n o p e r a t o r , and the a n t i l i n e a r t ime r e v e r s a l opera to r may be w r i t t e n s imply i n terms of space and momentum i n t e g r a l s . 6 We show i n Appendix D that where l x „> = hx"\ x m tos« - x ' 3 ) a ' i r \ « , x.<v> cos<=t-»-x<,-',sin«>> and l p R > ^ I p"\ p t x , c o s ^ - p < 3 , s i n « , p C 3^o&a. + p ' " s i n a > where &= ( p" c >m 1 - t ^ V ' 7 " & L = e. c-oslvu. •+• c p^VrnVvu. l^L>= l p , r t c o s K u + £ e s i n W u , p " \ p , 3 > > (P = ^ <i\ J-X^ U \ r i T V > = ^ 4 3x ix><i\)U> - JVP i - ^ x i y u > ¥ Thus D (CK) \ x"> = U + f t > 6Mx> = \-x> 66 (Plp>= (3.\.30) The v a r i o u s i n t e g r a l forms of the space - t ime opera to rs have a c e r t a i n i n t u i t i v e appeal i n that they make the correspondence w i t h equat ions ( 2 . 1 . 3 ) , ( 2 . 1 . 4 ) , ( 2 . 1 . 5 ) , and ( 2 . 6 . 1 ) more m a n i f e s t . In a d d i t i o n they may be used d i r e c t l y to g i v e the v a r i o u s t r a n s f o r m a t i o n equat ions d e r i v e d i n S e c t i o n 2 . 3 . For example, we d e r i v e L0(OA) H L'"^'^. where S'Cp1!- eCp") coshu. -*- c p,,',c>\r^ln'UL e . ' ( (J^= e. ( c p CQ&V- I T J. -v- t (| ( r t sinlru. Thus, L'"wHL W = Jd'pd*f <i\ ( ) \ £'> 8( fc(p~i S(p- %Hnt-,) < '^ I n t e g r a t i n g over ^ , I n t e g r a t i n g over p , Now a change of v a r i a b l e s i s made. Put p"* = p^cosKu. + t £ ( p ) sinhw. Since d J = dj> , and £<-p"> = e'Cpl , C ' ^ H L ^ W = J J 3 p i$r> eCp^ < £\ But e (p^ - fe(p) cosViU. - c p'* sinhu. so L -^u.") H l_ \ - u ) =• HcosKu.- C P'°.si nk "ix Th is i s the r e s u l t obta ined i n S e c t i o n 2 . 3 . The other t r a n s f o r m a t i o n equat ions of that s e c t i o n are obta ined s i m i l a r l y . 68 3 . 2 S i n g l e F ree P a r t i c l e w i t h Sp in For a s i n g l e f r e e p a r t i c l e of mass m and s p i n s the fundamental dynamica l v a r i a b l e s i n c l u d e the s p i n opera to rs S I S", S " as w e l l as the and the momenta r of the p a r t i c l e . In a d d i t i o n to the commutation r e l a t i o n s (3.1.1), these, v a r i a b l e s s a t i s f y the f o l l o w i n g r e l a t i o n s . We assume that the operato rs each form a complete set of compat ib le o b s e r v a b l e s . We denote t h e i r e igenkets by I x ^ " O and I f . s . m ^ . The e igenkets may be used as bases f o r the H i l b e r t space . m l S - « J P= £ U3P ,^s,n>.>p <p,s,"0 S r = fc (d 3 x U s , m 5 > s ( s +0 ^ < x . , s , m s \ ^.1.4^1 = £ LJp I p . S . O sCs+rttf ( ? ) S , m J m r - s J 69 <x.,S> tn5 \ X ' 3 S , ms'> = V(x - & ') S ^ ^ ' ( 3 . X . T ) < p , s , r y 1 J ^ ' > 5 ) m ^ = " f ^ 6 m , « ; (3 . X . . S ) A l s o , The commutation r e l a t i o n s ( 2 . 2 . 1 5 ) a re a l l s a t i s f i e d when the generators a re expressed i n terms of the fundamental dynamical v a r i a b l e s as J= X x P + S <3.JU<ft K= - rz- (XH* R)0 + (M + mc^V l ( S x P ) ( 3 . x . lO H - C P 1 ^ + r ^ c 4 ) V x (3.X.1^ where e ( r ^ ( i * v > c * ) , / v O.Z.\3) A l s o , i t f o l l o w s from the commutation r e l a t i o n s ( 3 . 1 . 1 ) and equat ion ( 3 . 2 . 1 2 ) f o r H that so the v e l o c i t y of the p a r t i c l e , V > i s V= iV HX,HJ= c l P H " o . v i s - - ) Thus X s a t i s f i e s the commutation r e l a t i o n s ( 2 . 8 . 2 ) and s u b s t i t u t i o n of equat ions ( 3 . 2 . 1 0 ) , ( 3 . 2 . 1 1 ) , and ( 3 . 2 . 1 2 ) i n t o equat ions ( 2 . 8 . 4 ) , ( 2 . 8 . 5 ) , 70 and ( 2 . 8 . 6 ) y i e l d s $ = & (3.X.^) The u n i t a r y opera to rs cor responding to the space - t ime t r a n s f o r m a t i o n s under c o n s i d e r a t i o n and the a n t i u n i t a r y operator T* can be w r i t t e n i n i n t e g r a l form as f o l l o w s . D(o.) = ^ d3< U + x , S , m s > < K ^ s , V Y \ s l (3.X.X0) (R(«^= E D*. ^ , ^ | ( l 5 p l p 8 > s m s ' H ^ 5 , ^ (a.x.xO where 6*,^^ a re the E u l e r ang les cor responding to £ . The m a t r i x i s the (2s + 1 ) - d i m e n s i o n a l i r r e d u c i b l e r e p r e s e n t a t i o n of the r o t a t i o n group. I t ensures the proper t r a n s f o r m a t i o n of the s p i n under r o t a t i o n s . To v e r i f y that the above equat ion i s the c o r r e c t form f o r 9,(9) we separate the s p i n and momentum p a r t s of the e x p r e s s i o n . From S e c t i o n 3 . 1 , we have and i t i s w e l l known that 71 SO as r e q u i r e d , where (*u,^ u.O a re the E u l e r ang les cor respond ing to the Wigner r o t a t i o n of the s p i n ,S. 9 R e c a l l the t r a n s f o r m a t i o n of & under Lo rentz boosts g i v e n by equat ion ( 2 . 9 . 1 1 ) . Here , the operator |j i s equal to the s p i n , S . The s p i n undergoes a r o t a t i o n under Lo rentz b o o s t s . That i s , t a k i n g m a t r i x elements of the operator equat ion ( 2 . 9 . 1 1 ) would y i e l d the Wigner r o t a t i o n . Thus, the Lo rentz boost operator must take the form above to ensure the proper t r a n s f o r m a t i o n of the s p i n . m s = - » J ' Thus, s s 73 3.3 S i n g l e Free D i r a c P a r t i c l e For a system c o n s i s t i n g of a s i n g l e f r e e D i r a c p a r t i c l e of mass TO., we take the C a r t e s i a n c o o r d i n a t e s X , the momenta P , and four D i r a c o p e r a t o r s si and /& as the fundamental dynamical v a r i a b l e s . These opera to rs s a t i s f y the f o l l o w i n g c o n d i t i o n s . [ x * X I K V ] --o iP'j', r 1 ] = o cx t j : pn - i * siK We now seek a c o n s t r u c t i o n of the generators of the P o i n c a r e group i n terms of the fundamental dynamical v a r i a b l e s . F o l l o w i n g D i rac , 1 0 we w r i t e the H a m i l t o n i a n as H =• c «• P + /6 , m,c i ( 3 . 3 . ^ The c o n d i t i o n s (3.3.1) on °i and /S ensure tha t H 1 = P ^ C > + TY^C* (3.3.3) Thus ri d e f i n e d i n equat ion (2.8.4) i s equal t o r n , the mass of the p a r t i c l e . F i n a l l y , i t i s s t r a i g h t f o r w a r d to check that the commutation r e l a t i o n s (2.2.15) a re s a t i s f i e d when H i s g i ven by equat ion (3.3.2) above and P= P K = -£c* (XH + H>0 (*.i.s) 74 J = X + S (3.3. b) where S - X *h 5 (3.3.71) and S" = ( et X. (3.3,^ We note that H * EL where E i s g i ven by equat ion ( B . 1 6 ) . T h e r e f o r e , /£ * \ Th is i s , of c o u r s e , expected s i n c e the D i r a c H a m i l t o n i a n ( 3 . 3 . 2 ) has both p o s i t i v e and n e g a t i v e spectrum. A l s o , s i n c e and [^1 -0= i . - n c«*P (3.3. lo) the o p e r a t o r s X and S a re not equal to the cent re of mass p o s i t i o n operator % and the i n t e r n a l angular momentum operator S. Indeed, as i s shown i n Appendix E, s u b s t i t u t i o n of the above express ions f o r the generators i n t o the d e f i n i t i o n s ( 2 . 8 . 5 ) and ( 2 . 8 . 6 ) of X and S y i e l d s %= X - cT- 6 x P + [ij8~ - c c " / 3 P ( F « s ) " l (3.3.11) Z, EL E (E t->vvc^  (3.3 .iz\ We r e c o g n i z e tha t these opera to rs are those g i v e n by P r y c e , Fo ldy and Wouthuysen, Jordan and Mukunda, and de Groot and S u t t o r p . 1 1 We show i n Appendix F that F ^ F ' U X (3.z.\3) F $ F + = S C3.3.W) p p p t _ p (3.3.\-5) F H F T = j3 £ (3.-5. n ) F / ^ F + = / d ? < 3 - 3 1 ^ where ( 2 . E ( E + 7 v 1 c - ' ) V / l ' The u n i t a r y operator F i s the Pryce -Fo ldy -Wouthuysen t r a n s f o r m a t i o n 1 9 o p e r a t o r . 76 3.4 7i N o n i n t e r a c t i n g P a r t i c l e s w i t h Sp in L e t us c o n s i d e r a system of a f i x e d number ,n, of f r e e p a r t i c l e s of masses fn,,^, ... ,7nn and s p i n s s, , S l 9 ••- , s^. The fundamental dynamical v a r i a b l e s of t h i s system are the components of the C a r t e s i a n c o o r d i n a t e s , of the momentum, and of the s p i n of the i n d i v i d u a l p a r t i c l e s , ^ ^ F , , ^ ^ . . - , ^ ^ ^ These o p e r a t o r s s a t i s f y the f o l l o w i n g commutation r e l a t i o n s . Here the Greek s u b s c r i p t s are the p a r t i c l e l a b e l s w h i l e the Roman s u p e r s c r i p t s l a b e l the component of the v a r i a b l e . S i n c e the p a r t i c l e s are n o n i n t e r a c t i n g each generator must be a sum of s i n g l e f r e e p a r t i c l e g e n e r a t o r s . That i s , H0= £ H„ <*.<+. o,-1 P0= E P. TL T -- £ I „ (3.4-H) where (3.4-.^ We have appended a s u b s c r i p t zero to these generators to d i s t i n g u i s h them from t h e . g e n e r a t o r s f o r an i n t e r a c t i n g system, wh ich we cons ide r i n the next s e c t i o n . These generators s a t i s f y the commutation r e l a t i o n s ( 2 . 2 . 1 5 ) . I t i s convenient to h e n c e f o r t h denote the t o t a l i n t e r n a l angular momentum by the symbol [ T . That i s , i n t e r n a l angular momentum 3 0 a re g i ven by equat ions ( 2 . 8 . 4 ) , ( 2 . 8 . 5 ) , and ( 2 . 8 . 6 ) . We w r i t e these equat ions here e x p l i c i t l y f o r l a t e r convenience . The t o t a l mass M 0 , the cen t re of mass p o s i t i o n and the t o t a l (3.M-.I0') n0%^ -n, c-( H : K 0 + K „ V O - P„ + _ c i I S O (5.4-.\\) M ^ 0 = K n * P 0 + Ho C T . - P ^ P . (4= ') 78 3.5 TI P a r t i c l e s i n I n t e r a c t i o n Le t us c o n s i d e r a system of n i n t e r a c t i n g p a r t i c l e s of masses m , , . . , ^ ^ and s p i n s s , , . . . , s n . We assume that no p a r t i c l e c r e a t i o n or a n n i h i l a t i o n takes p l a c e . That i s , the number of p a r t i c l e s , 7 t , i s f i x e d . The re fo re , we may take the fundamental dynamical v a r i a b l e s to be the i n d i v i d u a l p a r t i c l e p o s i t i o n , momentum, and s p i n opera to rs X „ , P 1 ( . . . tP n S n . These opera to rs s a t i s f y the commutation r e l a t i o n s (3.4.1). We seek a c o n s t r u c t i o n of the generators of the P o i n c a r e group f o r t h i s i n t e r a c t i n g system. F o l l o w i n g Bakamjian and Thomas,1 we see that the commutation r e l a t i o n s (2.2.15) a re s a t i s f i e d when P =. P " la.s.ti — — o j""=T C3.5-. 3") — •— O K - - - ^ C l 0 H + r \ X 0 ~ ) + ( H ^ f n ^ V ( 5 . * ^ U.^f-H) where , *X„ , , and T 0 a re g i ven by equat ions (3.4.3), (3.4.4), (3.4.10), and (3.4.11), r e s p e c t i v e l y , and where the mass operato r M s a t i s f i e s Lrn,p]-- [ ^ , £ . " 1 = [ M . t l = o (3 .5 .5) The i n t e r a c t i o n i n the system i s , t h u s , s p e c i f i e d complete ly by the mass operator M . If. M = n a , g i ven by equat ion (3.4.10), the p a r t i c l e s are f r e e whereas, i f M * H,,, the p a r t i c l e s i n t e r a c t amongst themselves . F i n a l l y , we may d e f i n e the p o t e n t i a l V of the system by 79 One should n o t i c e that the p o t e n t i a l f o r a P o i n c a r e i n v a r i a n t T L p a r t i c l e system w i l l i n g e n e r a l be a f u n c t i o n of the t o t a l momentum of the system. To complete the d e s c r i p t i o n of the system, we must show how to c o n s t r u c t an operator M s a t i s f y i n g equat ion ( 3 . 5 . 5 ) . Th is i s a n o n t r i v i a l problem. One can e a s i l y f i n d many opera to rs which commute w i t h P (eg. P 2S«rXs,^-X„, ) but i t i s more d i f f i c u l t to f i n d o p e r a t o r s which a l s o commute w i t h the compl icated o p e r a t o r s % B and J c . F o l l o w i n g Osborn, we s o l v e t h i s problem u s i n g a t r a n s f o r m a t i o n analogous to tha t in t roduced by Gartenhaus and Schwartz . We f i r s t d e f i n e a u n i t a r y operator £KX) by where X i s a r e a l number and B = - i ( X 0 - P + P - X / ) o.tr.fi Using equat ion (3.4.10) f o r one can a l s o w r i t e B i n the form B « T ( K 0 - V O + y.-JO ( 3 . ^ ) where Thus, ^(X^ reminds one of a Lo rentz boost o p e r a t o r . I t f o l l o w s on u s i n g the methods of S e c t i o n 2 .3 that 80 J8(\) J 0 M » = I „ ( 3 . 5 . so tha t f o r any operator A , LE, uooAtfiti] = i/(A)L^+(A)P£(A\A]Mv> = exi/(x^LP,MA\XS (3.S.IH) and LX 0 ^(xU^ + (A^ = & -^ (x\L% 0 i Al JfHti ( 3 . 5 . i ^ Thus, i f [P A] = 0 ^.S.lfe) and i f \im (ICV M L X . Al^ 1(A^ ex-.sis (s-s.n) x-*« then where The above g i v e s the c o n d i t i o n s f o r an operator to commute w i t h both P. and % o . As we s h a l l see , some w e l l chosen opera to rs which s a t i s f y these c o n d i t i o n s w i l l a l l o w us to f i n d opera to rs commuting w i t h J D as w e l l . Osborn s t a t e s tha t the c o n d i t i o n f o r the e x i s t e n c e of A i s tha t A commute w i t h P . l t + H e g i v e s as an example the operator %_o which does not 81 commute with P and which does indeed diverge under J2/(AY in the limit A—**> s i n c e However, cons ide r the operator P which commutes w i t h P . By the methods of S e c t i o n 2 . 3 , i t f o l l o w s that #(V> P-Zib\£\ = e> P " 1 <3.S\xrt Thus P x d i v e r g e s . A l s o , cons ide r the operator ^v0 which does not commute w i t h P . We f i n d that so tha t X* e x i s t s . C l e a r l y , Osborn 's c o n d i t i o n does not ensure the e x i s t e n c e of A . That i s , C[| 0 >Al,Pi-o does not guarantee tha t c o n d i t i o n ( 3 . 5 . 1 7 ) h o l d . I t f o l l o w s from equat ion ( 3 . 5 . 1 1 ) t h a t In v iew of t h i s e q u a t i o n , we c a l l A f o r H e r m i t i a n A " t h e observab le A i n A the zero-momentum p i c t u r e " . I f equat ion ( 3 . 5 . 1 6 ) h o l d s , we c a l l A a "d isp lacement i n v a r i a n t zero-momentum p i c t u r e o p e r a t o r " . I f , i n a d d i t i o n , c o n d i t i o n ( 3 . 5 . 1 7 ) h o l d s , we c a l l A an " i n t e r n a l o p e r a t o r " . Of p a r t i c u l a r i n t e r e s t are the f o l l o w i n g d isp lacement i n v a r i a n t z e r o -momentum p i c t u r e o p e r a t o r s . 82 U . S . X H l (One might suppose that the operator should be i n c l u d e d i n t h i s l i s t . However, t h i s operator i s not an i n t e r n a l o p e r a t o r . ) Thus, f o r example, P^ i s the momentum of p a r t i c l e <* i n the zero-momentum p i c t u r e . I t f o l l o w s from the commutation r e l a t i o n s (3.4.1) t h a t the f o l l o w i n g commutation r e l a t i o n s are s a t i s f i e d . l^ «yfl 1 J = (Sa-t,- fe^T,^ * > j t r p cji ~ m n £ (S^  We show i n Appendix G tha t P US".xi) from which (3.5.xV) and 1 P.= o ( 3 . 5 . 83 as r e q u i r e d . Bakamjian and Thomas g i v e opera to rs of t h i s f o r m . 1 5 We show i n Appendix H tha t Pa, X«a, and SA a re indeed i n t e r n a l o p e r a t o r s . (This i s t r u e p r o v i d i n g tha t and 6^ e x i s t . Osborn has g i v e n e x p l i c i t exp ress ions f o r X ( i and f o r the two-body p r o b l e m .J Thus, Moreover we show that Thus, f o r example, the f o l l o w i n g i n t e r n a l opera to rs commute w i t h I 0 . Th is completes the problem of c o n s t r u c t i n g the mass operator M . That i s , M may be a f u n c t i o n of the opera to rs ( 3 . 5 . 3 2 ) above. In Appendix I, we cons ide r a system of n i n t e r a c t i n g p a r t i c l e s i n v a r i a n t under the G a l i l e a n t r a n s f o r m a t i o n s . 84 Chapter 4 Two P a r t i c l e S c a t t e r i n g 4 . 1 Lo rentz I n v a r i a n c e f o r Two P a r t i c l e S c a t t e r i n g Systems We now cons ide r a system of two s p i n l e s s p a r t i c l e s of masses m, and mL s c a t t e r i n g o f f each o t h e r . We assume that no p a r t i c l e c r e a t i o n or a n n i h i l a t i o n takes p l a c e . The fundamental dynamical v a r i a b l e s of t h i s system can be taken to be the C a r t e s i a n c o o r d i n a t e s of the p a r t i c l e s , X . , , X , , and the i n d i v i d u a l p a r t i c l e momenta, P, , P t , which obey the commutation r e l a t i o n s ( 3 . 4 . 1 ) . S ince the fundamental dynamical v a r i a b l e s of d i f f e r e n t p a r t i c l e s commute, the H i l b e r t space ^ of the system may be expressed as the d i r e c t product of the i n d i v i d u a l p a r t i c l e H i l b e r t spaces , fr, and That i s , We a l s o d e f i n e c e n t r e of mass p o s i t i o n and momentum, % 0 and P , and i n t e r n a l ( r e l a t i v e ) v a r i a b l e s X^ and 2, , as i n S e c t i o n 3 . 5 : where P = P . ^ P r and % 0 and Pw a r e g i ven by equat ions ( 3 . 4 . 1 1 ) and ( 3 . 5 . 2 7 ) , r e s p e c t i v e l y . These opera to rs obey the commutation r e l a t i o n s ( 2 . 8 . 2 ) , ( 3 . 5 . 2 5 ) , and ( 3 . 5 . 2 0 ) . That i s , (V. /. /) (.H.I. Vl 85 i x ^ x ^ ] - [ x ; j ;p u < > ] - L P ^ M = [ p ^ p ^ i - o i | K . , , * . * (4.1.M) The cen t re of mass v a r i a b l e s commute w i t h the i n t e r n a l ( r e l a t i v e ) v a r i a b l e s . Thus, the H i l b e r t space may be expressed as the d i r e c t product of the cent re of mass and r e l a t i v e H i l b e r t spaces , fe^m and }4 t }7 That i s , ftL • K„ ty • K We now d e f i n e the important o p e r a t o r s of s c a t t e r i n g t h e o r y . The M o l l e r opera to rs JQ+ on the two body H i l b e r t space are fl, = l ;m U+(OU0(P) (H.U) where U m = e - l H i / * LH.il) U 0(t)-e- L H o t /^ (4.1.3) wherein H i s the H a m i l t o n i a n f o r the system and H 0 i s the H a m i l t o n i a n f o r two f r e e p a r t i c l e s . That i s , H6= t . ( P ^ m i c - V 7 1 (H.i.9) The M o l l e r opera to rs are i s o m e t r i c } 8 That i s , Xll fl+ -- 1 (H.I.IO) runX-- i- © (v././/) 86 where (& i s the p r o j e c t i o n operator onto the bound s t a t e s of H . Of c o u r s e , i f H has no bound s t a t e s , the M o l l e r opera to rs a re u n i t a r y . Fur thermore, the M o l l e r o p e r a t o r s s a t i s f y the " i n t e r t w i n i n g r e l a t i o n " : 1 9 In the above, we have, of c o u r s e , assumed that the M o l l e r opera to rs e x i s t . T h i s requirement (" the asymptot ic c o n d i t i o n " ) p l a c e s r e s t r i c t i o n s on the H a m i l t o n i a n H which we do not i n v e s t i g a t e i n d e t a i l h e r e . 2 0 At the very l e a s t , the p o t e n t i a l V g i v e n by H-H.+V (H.I.I3) must v a n i s h s u f f i c i e n t l y r a p i d l y as the i n t e r p a r t i c l e d i s t a n c e becomes l a r g e . We have a l s o assumed "asymptot i c completeness" ; that i s , the ranges of and -Q_ a re i d e n t i c a l . 2 1 The s c a t t e r i n g operator S i s d e f i n e d as s ^ l n , (H.I.IH) and i t f o l l o w s from equat ions ( 4 . 1 . 1 0 ) and ( 4 . 1 . 1 1 ) that S i s u n i t a r y . That i s , S S + - S 1 S = I ( 4 . U S ) Moreover , i t f o l l o w s from equat ions ( 4 . 1 . 1 2 ) and ( 4 . 1 . 1 4 ) that lL(t)Su!(t)=5 (*A.\b) and thus , LH0,S] = o (4.1.17) 87 As s t r e s s e d by T a y l o r ? t h i s equat ion corresponds to the statement t h a t the s c a t t e r i n g process conserves e n e r g y . 2 2 T h a t equat ion ( 4 . 1 . 1 7 ) i n v o l v e s the f r e e H a m i l t o n i a n H0 corresponds to the f a c t that the s c a t t e r i n g operator i s a mapping of the asymptot ic f r e e o r b i t s which l a b e l the p a r t i c l e s ' s t a t e on ly when they are f a r apar t and do not f e e l the p o t e n t i a l , when t h e i r energy i s g i ven by Ha. The i n v a r i a n c e of the system under the Poincare* t r a n s f o r m a t i o n s , i n the sense of S e c t i o n 2 . 1 , imposes c o n d i t i o n s on A £ and S . We a l s o assume the system to be i n v a r i a n t under space i n v e r s i o n and time r e v e r s a l . We g i v e the r e s u l t i n g c o n d i t i o n s on -0.+ and S be low. D(^ru D+(3>-Q-v D(OSD+(ftVS <R(fiULt&V)=rLt (R(8)Stft+(gWS The c o n d i t i o n s ( 4 . 1 . 1 8 ) above f o l l o w from equat ions ( 2 . 3 . 7 ) , ( 2 . 3 . 9 ) , ( 2 . 6 . 6 ) , and ( 2 . 7 . 6 ) . We note i n p a r t i c u l a r tha t i t f o l l o w s from the c o n d i t i o n s ( 4 . 1 . 1 8 ) t h a t the s c a t t e r i n g operato r i s i n v a r i a n t under space d isp lacements and r o t a t i o n s and , t h u s , tha t C P , S ] = [3 Si = 0 (HI.19) T h e r e f o r e , S conserves t o t a l momentum and t o t a l angular momentum. I t w i l l be n o t i c e d that equat ions ( 4 . 1 . 1 8 ) do not i n v o l v e Lorentz b o o s t s . F o l l o w i n g Fong and Sucher , we now assume that 88 where L 0 i s the Lo rentz boost operato r f o r the f r e e p a r t i c l e s . 2 3 That i s , where in K „ i s g i v e n by equat ion ( 3 . 4 . 5 ) . The s c a t t e r i n g operator i s s a i d to be " a s y m p t o t i c a l l y c o v a r i a n t " when equat ion ( 4 . 1 . 2 0 ) h o l d s . In analogy w i t h equat ion ( 4 . 1 . 1 6 ) , equat ion ( 4 . 1 . 2 0 ) i n v o l v e s the f r e e Lorentz boost opera to rs L 0 s i n c e the s c a t t e r i n g operator i s a mapping of asymptot ic f r e e o r b i t s . (We note tha t i t f o l l o w s from the obvious two f r e e p a r t i c l e equat ion which g e n e r a l i z e s the s i n g l e f r e e p a r t i c l e equat ion ( 3 . 1 . 3 0 ) t h a t , i n the momentum r e p r e s e n t a t i o n , equat ion ( 4 . 1 . 2 0 ) i s which i s the f a m i l i a r equat ion s p e c i f y i n g " t h e Lorentz i n v a r i a n c e of the s c a t t e r i n g m a t r i x " . ^ E q u a t i o n ( 4 . 1 . 2 0 ) guarantees tha t one measures the same v a l u e s f o r observab les on the average i n a s c a t t e r i n g experiment when one Lo rentz boosts both the p r e p a r a t i o n and measuring appara tuses . As 2 5 p o i n t e d out by Fong and Sucher , one needs to i n t r o d u c e equat ion ( 4 . 1 . 2 0 ) as an e x t r a assumption a t t h i s stage because a s c a t t e r i n g experiment i n v o l v e s a concept not i n c o r p o r a t e d i n the d i s c u s s i o n of Chapter 2 which l e d to the commutation r e l a t i o n s of the P o i n c a r e group; namely, a s c a t t e r i n g experiment i n v o l v e s a comparison between i n t e r a c t i n g and f r e e systems. C o n d i t i o n ( 4 . 1 . 2 0 ) can be expressed i n another way as f o l l o w s . Rearranging equat ion ( 4 . 1 . 2 0 ) , U S = S L e 89 M u l t i p l y i n g by i"L_ from the l e f t and .TL+ from the r i g h t , B u t , s i n c e we f i n d whence Th is c o n d i t i o n i s e q u i v a l e n t to c o n d i t i o n ( 4 . 1 . 2 0 ) . Fong and Sucher have shown the Lo rentz boost operator f o r an i n t e r a c t i n g system w i t h an a s y m p t o t i c a l l y c o v a r i a n t s c a t t e r i n g operato r to be u n i q u e l y g i v e n 2 5 i n the absence of bound s t a t e s ( i n which case £1^ are u n i t a r y ) , by so tha t !< = n _ K o n S - f l + K 0 < T U ( f 90 Thus, from equat ions (4.1.16), (4.1.17), and (4.1.19), i t f o l l o w s that the generators of the P o i n c a r e group f o r an i n t e r a c t i n g system w i t h an a s y m p t o t i c a l l y c o v a r i a n t s c a t t e r i n g operator and wi thout bound s t a t e s a re K = f l ± K 0 n.-^ (4.1.1"?) The set of generators (4.1.29) obeys the commutation r e l a t i o n s (2.2.15). We now c o n s i d e r a two p a r t i c l e system w i t h the P o i n c a r e group generators of the Bakamjian-Thomas form and we show that equat ions (4.1.29) ho ld when the system has no bound s t a t e s . That i s , we show that the s c a t t e r i n g operator of the system i s a s y m p t o t i c a l l y c o v a r i a n t . I t f o l l o w s from equat ions (3.5.23) and (4.1.9) that we can w r i t e H0= (P^+- r C y / 2 (4./.SO) where ! I x i H0= T> ( P « V + 7ntt HV / l (4.1.3 1) Moreover , the Bakamjian-Thomas form f o r the H a m i l t o n i a n i s g i v e n by equat ion (3.5.1). Thus, where 91 where V - V(2 P) (H.I.3H) and t V . I . l - O ( f 1.35) We now define operators XL t by These operators w i l l e x i s t i f V i s sui t a b l y r e s t r i c t e d . Moreover, they are unitary i f H has no bound states and they s a t i s f y the intertwining r e l a t i o n : ^ /V ^ . Furthermore, since f l t are r o t a t i o n a l l y invariant functions of i n t e r n a l v a r i a b l e s , i t follows that j Kato has proven the remarkable r e s u l t ("invariance of the wave operators") i that Iim e. e. =• l i m e. e. (4-.l.3ci) for a wide c l a s s of functions 0 . 2 7 In p a r t i c u l a r , for H 0 and H given by equations (4.1.30) and (4.1.32), r e s p e c t i v e l y , i t follows that 92 vs Thus when V\ has no bound states, i t follows from equations (3.4.5), (3.5.4), (4.1.12), (4.1.30), (4.1.32), (4.1.37), and (4.1.38) that H --si t w6 nX P . - - n t P0 - ^ t where H , K , V\ 0 , and K 0 are given by equations (3.5.1), (3.5.4), (3.4.2), and (3.4.5), respectively. Thus the scattering operator for a two p a r t i c l e system with no bound states and with Poincare* group generators of the Bakamjian-Thomas form.is asymptotically covariant. We wish to emphasize the importance of the role of Kato's theorem and of the resul t (4.1.40). This res u l t allows one to determine the scattering operator solely by solving the scattering problem for H„ + V • on Frre| . This i s exactly analogous to the procedure one follows i n the n o n r e l a t i v i s t i c case. There, however, the P /ww. term i n H& and Hoc cancels out i n the expression for the Moller operators and one does not need to appeal to Kato's theorem. F i n a l l y , we note that Fong and Sucher and also Heller, Bohannon, and Tabakin assume the result (4.1.40) without discussion or proof. Fong and Sucher have generalized the above to include a discussion of bound s t a t e s . 2 9 93 Footnotes 1. B. Bakamjian & L.H. Thomas, " R e l a t i v i s t i c P a r t i c l e Dynamics I I , " Phys. Rev. 92, 1301 (1953). 2. Eugene P. Wigner, Group Theory and i t s Application to the Quantum  Mechanics of Atomic Spectra (New York: Academic Press, 1959), pp.233-236, 326-328. 3. Thomas F. Jordan, Linear Operators for Quantum Mechanics (New York: Wiley, 1969), p.52. 4. Silvan S. Schweber, Introduction to R e l a t i v i s t i c Quantum F i e l d  Theory (Evanston: Row, Peterson, & Co., 1961), p.45. 5. Hugh Osborn, " R e l a t i v i s t i c Center-of-Mass Variables for Two-P a r t i c l e Systems with Spin," Phys. Rev. 176, 1516 (1968); J u l i a n Schwinger, P a r t i c l e s , Sources, and Fields (Reading: Addison-Wesley Co., 1970), p.19. 6. Malcolm McMillan, private communication. 7. Osborn, Phys. Rev. 176, 1517-1518 (1968). 8. M.H.L. Pryce, "The Mass-centre i n the Restricted Theory of R e l a t i v i t y and i t s Connexion with the Quantum Theory of Elementary P a r t i c l e s , " Proc. Roy. Soc. (London) A195, 71 (1948); L e s l i e L. Foldy & Siegfried A. Wouthuysen, "On the Dirac Theory of Spin ^ P a r t i c l e s and i t s Non-Relativistic Limit," Phys. Rev. 78, 31 (1950). 9. Stephen Gasiorowicz, Elementary P a r t i c l e Physics (New York: Wiley, 1966), pp.73-75. 10. Paul Adrien Maurice Dirac, The Pr i n c i p l e s of Quantum Mechanics (Oxford: Clarendon Press, 1958), p.261. 11. Pryce, Proc. Roy. Soc. (London) A195, 70-71 (1948) ; Foldy & Wouthuysen, Phys. Rev. 78, 32 (1950); Thomas F. Jordan & N. Mukunda, "Lorentz-Covariant Position Operators for Spinning P a r t i c l e s , " Phys. Rev. 132, 1846 (1963); S.R. de Groot & L.G. Suttorp, Foundations of Electro- dynamics (Amsterdam: North-Holland Co., 1972), p.460. 12. S.R. de Groot & L.G. Suttorp, Electrodynamics, p.418. 13. Osborn, Phys. Rev. 176, 1514-1522 (1968); S. Gartenhaus & C. Schwartz, "Center-of-Mass Motion i n Many-Particle Systems." Phys. Rev. 108, 482-484 (1957). 14. Osborn, Phys. Rev. 176, 1515-1517 (1968) . 15. B. Bakamjian and Thomas, Phys. Rev. 92,1301 (1953) , Eq. ( 3 . 4 ) , ( 3 . 5 ) , ( 3 . 6 ) . 16. Osborn, Phys. Rev. 176, 1518-1519 (1968), Eq. (3.13) & (3.23). 17. John R. Taylor, Scattering Theory (New York: Wiley, 1972), p.58 18. Taylor, Scattering Theory, pp. 14-16. 19. Taylor, Scattering Theory, pp.39-40. 20. Taylor, Scattering Theory, pp.28-31. 21. Taylor, Scattering Theory, p.31. 22. Taylor, Scattering Theory, p.39. 23. R. Fong & J. Sucher, " R e l a t i v i s t i c P a r t i c l e Dynamics and the S Matrix," Jour. Math. Phys. _5, 462 (1963), Eq. (5-13). 24. Marvin L. Goldberger & Kenneth M. Watson, C o l l i s i o n Theory (New York: Wiley, 1967), p.85, Eq. (110). 25. Fong & Sucher, Jour. Math. Phys. j>, 461-462 (1963). 26. Fong & Sucher, Jour. Math. Phys. J5, 462 (1963). 27. Tosio Kato, "Wave Operators and Unitary Equivalence," P a c i f i c  Journal of Mathematics 15, 172-174 (1965). 28. Fong & Sucher, Jour. Math. Phys..5, 456-470 (1963); Leon Heller G.E. Bohannon, & F. Tabakin, Phys. Rev. C 13, 742-748 (1976). 29. Fong & Sucher, Jour. Math. Phys. j>, 466-467 (1963). 30. Gartenhaus & Schwartz, Phys. Rev. 108, 482 (1957). 95 Bibliography Bakamjian, B. & L.H. Thomas. " R e l a t i v i s t i c P a r t i c l e Dynamics I I , " Physical  Review, 92 , 1300-1310 (1953) . Berg, R.A. "Position and I n t r i n s i c Spin Operators i n Quantum Theory," Journal of Mathematical Physics, 6, 34-39 (1965) . 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Quantum Mechanics Volume I: Fundamentals. Reading, Mass.: W.A. Benjamin, Inc., 1966. Groot, S.R. de & L.G. Suttorp. Foundations of Electrodynamics. Amsterdam: North-Holland Publishing Co., 1972. H e l l e r , Leon. "Lectures on the Nucleon-Nucleon Interaction." Vancouver Physics Summer School, 1973. Heller, Leon, G.E. Bohannon, & F. Tabakin. "Dependence of the Off-energy Shell T Matrix on the Total Three-momentum," Physical Review C, 13 , 742-748 (1976) . Jordan, Thomas F. Linear Operators for Quantum Mechanics. New York: Wiley, 1969. Jordan, Thomas F. & N. Mukunda. "Lorentz Covariant Position Operators for Spinning P a r t i c l e s , " Physical Review, 132, 1842-1848 (1963) . Kato, Tosio. Perturbation Theory for Linear Operators. B e r l i n : Springer-Verlag,1976. 96 K a t o , T. "Wave Operators and U n i t a r y E q u i v a l e n c e , " P a c i f i c J o u r n a l of  Mathemat ics , 15, 171 (1965) . M c M i l l a n , Ma lco lm. Unpubl ished l e c t u r e s on quantum mechanics . M o l l e r , C. The Theory of R e l a t i v i t y . Ox fo rd : Clarendon P r e s s , 1962. Osborn, Hugh. " R e l a t i v i s t i c C e n t e r - o f - M a s s V a r i a b l e s f o r T w o - P a r t i c l e Systems w i t h S p i n , " P h y s i c a l Review, 176, 1514-1522 (1968) . P r y c e , M .H .L . "The M a s s - c e n t r e i n the R e s t r i c t e d Theory of R e l a t i v i t y and i t s Connexion w i t h the Quantum Theory of Elementary P a r t i c l e s , " Proceedings of the Roya l S o c i e t y , London, A195, 62-81 (1948) . Schweber, S i l v a n S . I n t r o d u c t i o n to R e l a t i v i s t i c Quantum F i e l d Theory. Evanston , I l l i n o i s : Row, P e t e r s o n , & C o . , 1961. Schwinger , J u l i a n . P a r t i c l e s , Sources , and F i e l d s . Read ing , M a s s . : Addison-Wesley C o . , 1970. T a y l o r , John R. S c a t t e r i n g Theory: The Quantum Theory on N o n r e l a t i v i s t i c  C o l l i s i o n s . New Y o r k : W i l e y , 1972. Weinberg, S teven . "Feynman Rules f o r Any S p i n , " P h y s i c a l Review B, 133, 1318-1332 (1964) . Wigner , Eugene P. Group Theory and i t s A p p l i c a t i o n to the Quantum Mechanics  of Atomic S p e c t r a . New Y o r k : Academic P r e s s , 1959. 97 Appendix A Proof tha t Equat ion ( 2 . 1 . 8 ) Imp l ies Equat ion ( 2 . 1 . 9 ) Consider a set of apparatuses which prepares a set of or thonormal b a s i s v e c t o r s •flOl. Let{la;Yl be a set of or thonormal b a s i s v e c t o r s prepared by the apparatuses a f t e r they have undergone a change i n p lacement . Then, and a r b i t r a r y s t a t e s prepared by the apparatuses i n t h e i r two placements may be w r i t t e n as ii//>= H o n \ o From equat ion ( 2 . 1 . 8 ) i t f o l l o w s that <?|/IAW> = Z UM„ A c J O - <mA'W'> = I I < a j A ' c ^ 1 O R e w r i t i n g , Now equat ion ( 2 . 1 . 8 ) must h o l d f o r a l l observab les and s t a t e s so we may choose some p a r t i c u l a r o b s e r v a b l e , A , say , i n order to o b t a i n r e s t r i c t i o n s on the c o e f f i c i e n t s cl. . For i n s t a n c e , l e t Then i t f o l l o w s on s u b s t i t u t i n g A i n t o equat ion ( A . l ) tha t We o b t a i n a second c o n d i t i o n on the c o e f f i c i e n t s by choos ing a p a r t i c u l a r s t a t e I w h i l e the observab le A i s l e f t a r b i t r a r y . For i n s t a n c e , l e t 1(D) = c , l O + c j a ^ C.;IOL;> + C'„\OL'„> Since the o v e r a l l phase of i s a r b i t r a r y , we choose c, = c v . Then - -<f lA ' l<p%W\a; lAV^c;\^ Thus c*cn<a,IAlcO cXaJAlO^c^c^c^A' la '^ + c;*c, <a'B\A'U;> Since A r e p r e s e n t s an o b s e r v a b l e , i t i s H e r m i t i a n so M u l t i p l y i n g by , c f ( c ; x ) 1 - + c , \ c ; \ i - c ; ( o f c „ v c * c , \ - 0 and s o l v i n g f o r , = c n or (c, /cf) c*n Choosing the phase of I <p) so that c, i s r e a l , we have Now, l e t /<p,) and I<f0 be two a r b i t r a r y s t a t e s . so 99 Then 100 Appendix B D e r i v a t i o n of Equat ion ( 2 . 8 . 3 ) f o r the Generator K I t i s c l e a r that the generator K w i l l s a t i s f y the commutation r e l a t i o n s (2.2.\5$ and (2.2. I5, ) i f i t takes the form K = -&(%H+H&) + a ( ^ P ) ( B . O where a*-a.(H) i s an operator which may depend on H. (as w e l l as upon the i n v a r i a n t s and $ .) The operator a(H) i s determined by r e q u i r i n g that the commutation r e l a t i o n (2.2.is;) h o l d s . That i s , by r e q u i r i n g tha t where Z i s g i ven by equat ion ( 2 . 8 . 1 ) . The q u a n t i t y K x K i n v o l v e s th ree terms which may be s i m p l i f i e d as f o l l o w s . One can use equat ions ( 2 . 8 . 2 a ) , ( 2 . 8 . 2 b ) , and ( 2 . 8 . 2 c ) to w r i t e CXH + HS,1 x (XH + HX>) = xP) A l s o one can w r i t e the r i g h t hand s i d e of which may be s i m p l i f i e d u s i n g C&,<0 = LX ,H]^ = iU l PH- ' (^ H ) (B.O 101 to Fur thermore , one has Thus K x K = { X x P + ( X a H t P " c - ( j S ) ) | - C M P •4^^ + ^ P ' ] (B.10) Hence equat ion (B.2) i s s a t i s f i e d when and That i s , i f PVo, 1 - + 1 = 0 ( B . l ^ S o l v i n g t h i s q u a d r a t i c operator equat ion f o r a(W) y i e l d s where ^ = E H " 1 and E = 102 Choosing the nons ingu la r s o l u t i o n g i v e s equat ion ( 2 . 8 . 3 ) . F i n a l l y , i t i s s t r a i g h t f o r w a r d to show that the commutation r e l a t i o n (2.2.I5K) i s s a t i s f i e d when 3" and K a re g i ven by equat ions ( 2 . 8 . 1 ) and ( 2 . 8 . 3 ) , r e s p e c t i v e l y . 103 Appendix C D e r i v a t i o n of Equat ions ( 2 . 8 . 5 ) and ( 2 . 8 . 6 ) f o r % and $ We determine our e x p r e s s i o n f o r % by m a n i p u l a t i n g equat ions ( 2 . 8 . 1 ) and ( 2 . 8 . 3 ) f o r J and K . Th is l a t t e r equat ion may be r e w r i t t e n u s i n g equat ion ( 2 . 8 . 2 c ) as K = - ^ H X - r T ' P - ( H t ^ h c 1 ) " ' ( P x ^ (C.I) The q u a n t i t y ( P x & ) may be e l i m i n a t e d from t h i s equat ion u s i n g P x 3 > P x ( X x P ) + P x & = P 1 * . - P ( P - X W P x | > (C.Z) and the q u a n t i t y B % may be e l i m i n a t e d from t h i s equat ion u s i n g P . K = Vi P-% - '4 H " P 1 - (C .3) Thus s o l v i n g equat ion ( C . l ) f o r % y i e l d s / f r H X = - K + _ c L _ _ P ( P - K V j P x T - l U M c ^ P (C .4-) Th is l a t t e r equat ion may be w r i t t e n as equat ion ( 2 . 8 . 5 ) u s i n g E (P-K} = P . * C p x K ) + P x K (c.er) and LiS P H " 1 = L K , H M 1 ( C . ^ F i n a l l y , ^ may be determined on w r i t i n g * * I - * * P (C.7) and u s i n g equat ion (C.4) f o r X. Our e x p r e s s i o n ( 2 . 8 . 6 ) f o l l o w s when one uses P ( P . K ) n P = 0 104 (C. 2) ( P x l ^ * P - - C X - P ^ P + I P 5 (c.<rt 105 Appendix D I n t e g r a l Forms of the Spacer-time T rans fo rmat ion Operators The u n i t a r y opera to rs cor responding to the P o i n c a r e t r a n s f o r m a t i o n s are determined by the equat ion dx t-0 and the c o n d i t i o n Uj0) = \ <D.2> We now show that equat ions ( 3 . 1 . 1 4 ) , ( 3 . 1 . 1 5 ) , and ( 3 . 1 . 1 8 ) f o r D(a\ & ("\ and L' ("U.1), r e s p e c t i v e l y , s a t i s f y these c o n d i t i o n s . In the case of Lorentz boosts we must show that d_ <p\L<0("u.)\P'>l = -i_c<|>\K<rt\£'> (0.3^ and | _ < r t ( < r t = l <D-^  When U= 0 , we have so tha t equat ion ( 3 . 1 . 1 8 ) becomes as r e q u i r e d . Thus, the second c o n d i t i o n i s f u l f i l l e d . A l s o , 106 D i f f e r e n t i a t i n g , where Us ing and the equat ions ( 3 . 1 . 2 ) and ( 3 . 1 . 1 0 ) , we have < P I X ^ H - r l X H o ' ) = L* 6 ( ? - j » ' U * p " " e and <^\X'nHi^'>= ^ 6 , " ' ( f - ^ £ so t h a t 2/nC T h e r e f o r e , equat ion ( 3 . 1 . 1 8 ) does indeed s a t i s f y the c o n d i t i o n s (D.3) and ( D . 4 ) . S i m i l a r l y , we have, f o r the cases of space d isp lacements and r o t a t i o n s , d o . ' a = 0 * 107 f o r D (o-^  and g i ven by equat ions (3.1.14) and (3.1.15), r e s p e c t i v e l y . Equat ion (3.1.14) can be obta ined more d i r e c t l y from the e x p o n e n t i a l form of the space d isp lacement o p e r a t o r . That i s , «T l p-* /*' = [ d > l ^ e * " * ' * < f l To o b t a i n the i n t e g r a l form of the space i n v e r s i o n o p e r a t o r , we seek a l i n e a r operator (P such that That i s , d*x \(fV> * < (Px\= -^aV llf> 1 <*\ The above equat ion i s s a t i s f i e d i f P\x> = \-%> M u l t i p l y i n g each s i d e by x^l and i n t e g r a t i n g , S i m i l a r l y , f o r the t ime r e v e r s a l o p e r a t o r , we seek an a n t i l i n e a r operator T such that 108 That i s , Th is ho lds i f We a l s o r e q u i r e rp r + = - P That i s , jd> |T>> f <T f I = -pfp lf>> p Col Th is ho lds i f r I (O.\0) An a n t i l i n e a r form f o r ^ which s a t i s f i e s equat ions (D.9) and (D.10) above i s T Ill>> = \^>0^\tsi (O.i i) In terms of momentum k e t s we r e w r i t e equat ion ( D . l l ) as 109 Appendix E Centre of Mass P o s i t i o n \ and I n t e r n a l Angular Momentum S Operators f o r a Free D i r a c P a r t i c l e % i s g i v e n i n terms of the generators by equat ion ( 2 . 8 . 5 ) . In to t h i s e x p r e s s i o n , we s u b s t i t u t e and equat ion ( 3 . 3 , 6 ) f o r T to o b t a i n &=X +_k±L_ « - i f r c £<P-sV_J P x S - i J ^ - P ( E . X l where we have made use of the i d e n t i t y p *(x%F) - P X X - p( P X " ) (E.3) Now, s i n c e = E / = VA/£ and ( « • P ) ( P - o . N ) = P X ( E . ^ then )K can be r e w r i t t e n |4 + A L ^ - i u ^ _ P ( P « K j L ( f £ ) * - J £xS-ii>_£ (ES") We now use equat ions ( 3 . 3 . 7 ) and ( 3 . 3 . 8 ) to w r i t e S i n terms of « and use the i d e n t i t i e s to w r i t e the l a s t th ree terms i n equat ion (E.5) as 110 Ifr (g-fte - J PxS -iVE _ iW (a x»)*P = - c " ^ P (E.S) We thus/obtain equation (3.3.11) f o r X . 4 i s given i n terms of the generators by equation (2.8.6). Sub-s t i t u t i n g equations ( E . l ) and (3.3.6) i n t o equation (2.8.6), we o b t a i n $=_H_ ("•CE-y&HXxP + ' A c « x P ) + E. (XxP+ £*)-_! (s-P) P (E.S) Em wt1 Y A (E+ T n c 1 - ' ) where we have used the d e f i n i t i o n of $ . From equation (3.3.2) f o r H and the i d e n t i t y ( S - P ^ P = p^S - P x(SxP) (E.\o) we have 4= - i . - k c^gxP) + S - ifr (g-P) ( a x P U i ( P H ( | K P ) ) (E.H) We now use equations (3.3.7) and (3.3.8) to w r i t e S i n terms of « and use equations (E.4), (E.6), and (E.7) to o b t a i n Px'(sxP) = -i£ ((e^(px«v (Px«)(p.«^ (E.\ti From equation (E.7), we have ( P « U « K P ) = (EM) so that I l l 5- L * C/3(Q.X.P) - c " ( P K(5 KP^)) We thus o b t a i n equat ion ( 3 . 3 . 1 2 ) f o r 4. Appendix F 112 D e r i v a t i o n of Equat ions ( 3 . 3 . 1 3 ) through ( 3 . 3 . 1 8 ) f o r a Free D i r a c P a r t i c l e The operator %. i s g i ven by equat ion ( 3 . 3 . 1 1 ) . We now show that i t can be w r i t t e n as equat ion ( 3 . 3 . 1 3 ) . One has F + X F = F + ( F X - l F , ^ = X - F ^ F \ X l ( R O where F i s g i ven by equat ion ( 3 . 3 . 1 9 ) . Us ing the commutators LE''1- xi = -«£ C-P E - i / J - ( R X ) [ ( E + ™ c O " V " X l = ^ C - P E - ^ E + T n t - V ^ ( £ . 3 ^ we o b t a i n S ince we can w r i t e (RM-) F f = E t m ^ - c ^ a . P (F.5"i (xECE-^oV*"*-• + LF,X> [ (E - v v n c > - c>dg.EMc^-P E"»c/3a) - e E ( E - ' ( E m r i A-Q] 1 Z E ( E + -mc>) J = -L* ( - c'ygfa-E^E t c^((gP)g -gH (F. fe> (• r e 1 El(E-i-wvO) ZE<E+-»ne) j 113 Us ing the i d e n t i t y (o,.p)<±-P = Z l S x P (F.-|) i n equat ion (F .6) and s u b s t i t u t i n g tha t equat ion and equat ion ( 3 . 3 . 1 1 ) i n t o equat ion ( F . l ) , we o b t a i n equat ion ( 3 . 3 . 1 3 ) f o r % . We now d e r i v e equat ion ( 3 . 3 . 1 4 ) for>$ from equat ion ( 3 . 3 . 1 2 ) . One has F + 5 F - F + ( F S - L F , S ? ) - 5 - F + t F . s l (r.ft and L F , S > £ (•tV.cftxP>) (F °i) ( l £ ( E + m e > ) ) V l s u b s t i t u t i o n of which i n t o equat ion (F .8) y i e l d s F t S F = S - f E - h T v i c > - c ^ t t - £ \ ' jQ ( A c ( « . P ) ) V ( l E ( E + m ^ ) " 1 / ( 2 . E ( E + THC^V/X 5 - ^ c.ggyP + tr\e ( « g ) ( P ^ (F. IO) On comparison w i t h equat ion ( E . 1 4 ) , we see that equat ion ( 3 . 3 . 1 4 ) f o l l o w s . S i m i l a r l y , s i n c e L F } P ] = [ F , T 1 = O ( F . 10 we o b t a i n equat ions ( 3 . 3 . 1 5 ) and ( 3 . 3 . 1 6 ) . We now d e r i v e equat ion ( 3 . 3 . 1 7 ) . One has F H F t = F ( F + H - t F + , H]) = H - F [ F + , H j (F.IZ.) and 114 (lE(E+™e)V- (%ECEV-mc?-y)Vl-s u b s t i t u t i o n of which i n t o equat ion (F .12) y i e l d s E ( E + TrvC^) E (E -t--rv\c>) = /?E where we have used F i n a l l y , F+ySF = H/(F^EF)= H / E =- ^ o= i5-) 115 Appendix G D e r i v a t i o n of PA , H„, and Q„ We use the method of S e c t i o n 2 .3 to determine .P, . F i r s t , we d e f i n e A(X^) by A(X^ = #(X} A # +CX} (o.O where A i s some a r b i t r a r y o p e r a t o r . I t f o l l o w s that L*<LPa(XW i / U U B ^ P j # + ( X ) dX H„(X^ Us ing equat ion ( 3 . 5 . 1 1 ) f o r P(\) , we f i n d i P„M = - e A >jU>L>P <iX H„(X) S i m i l a r l y , dX = P(XV P M (X) which we r e w r i t e as A = - c l e ^ P . ( X V P H 0 (X) Now, l e t «±B - « t " X P c / H 0 ( V > N U P / P (Qr.l ' i d. \ 116 Then L£«m* -± r\„LX)U (G-.3) and i,U„aU-c Pa,<\VN (Cr.4) These equations are l i k e those for Lorentz boosts. Thus a solution i s P„(X\= P. + (coshS-rKP.-NON ~ E -UmK^^H«N ( 0 . 5 s ) Since we have where x= Pet*' (b.ti) and also 6,(o^ - 0 Therefore, \ n 0 c - / \ r v 1 / Thus 117 SO s i n and co^S,U) = H„ /noc>-T h e r e f o r e , H«=lim Ha(x)= (H«H0-c-P«.PV/nec-- (o io) P. - 1 , ^ P.M = P. + fx , 1 (P„. PVP - fe. vA. P = P-+U1-£.-P -H From equat ion ( G . 1 0 ) , SO By the method of S e c t i o n 2 . 3 , we have dA where P- = P«- I P « ^ « ( P (QrAtf 118 Let N = P / P and r e c a l l that dA H.M Then i 6 . W . _ c L(h!x P.,(xVW&..UY\ From equat ion (G.5), we see that N * P « ( X ^ = N * P t t Now l e t Then A. = ciT[ N x S^Xl A s o l u t i o n of t h i s equat ion i s We have 119 T h i s i n t e g r a t e s to = t*.*(e*( H » - c ( P , - N V U m.c^ ) - tan'/ H « - c ( P , - N ) + n y . c A We leave the c a l c u l a t i o n a t t h i s p o i n t as i t becomes more compl icated than u s e f u l . 120 Appendix H V e r i f i c a t i o n tha t P « , XK / 3 , and a re I n t e r n a l Operators and D e r i v a t i o n of the Commutation R e l a t i o n s ( 3 . 5 . 3 1 ) We have shown that i f then L&.,A] = 0 (W.TJ\ To check tha t equat ion ( H . l ) i s s a t i s f i e d f o r A= P„, X.,- X^,, 5 w we compute the commutators of % w i t h these o p e r a t o r s . Thus, we have v,™ flea P " ! ^ - A H * & J K <H.*) \ — » t o We have g i ven H» e x p l i c i t l y ( i . e . H* e x i s t s . ) so A = B» s a t i s f i e s c o n d i t i o n (H.X). S i m i l a r l y , 121 Thus, we have We have g i ven H^, P „ , and e x p l i c i t l y ( S K i s g i ven i n Appendix G.) so A = S K s a t i s f i e s c o n d i t i o n ( H . 2 ) . F i n a l l y , we cons ide r A = X , * - ) ^ . (ft) + terms which c o n t a i n a f a c t o r of P (and hence v a n i s h under $ ) + terms which c o n t a i n a f a c t o r of P a ) From equat ions ( 2 . 8 . 2 0 ) , ( 2 . 8 . 1 2 ) , and ( 3 . 4 . 5 ) , + -A (o pr W: - ^  p;w o H ; ( H X j 1 . x ; ) + terms which c o n t a i n f a c t o r s of P'*^  S i n c e we f i n d W OL-X-Y""] = -^ fc-£^  ( x M - x^]-^f^prH« ( x . - x „ Y I R -122 + terms which contain a factor of r IH.T) Thus, u tott*(xw-x,YK,]^(xw£ U ^ Y " 1 ] ^ ( p « w A g ^ " j L ) ^ We assume that % ^ exists so A= X - satisfies equation (H..2) Therefore, We now compute the commutators of J o with the operators PA , S a, and (X^)Q. Therefore, Similarly, Therefore, Finally, 123 [ ^ i v ^ r i - ? n £ > , ( V X , Y ^ + t r* < X * - X;H ^ + terms which c o n t a i n a f a c t o r of P*"1 We s i m p l i f y the f i r s t te rm. From £ M . V , = M v t f - P V X . - ^ l - ^ ( V o l - V ^ H o we o b t a i n so that S ince t h i s term c o n t a i n s a f a c t o r of P , i t van ishes under % . The re fo re , + terms c o n t a i n i n g f a c t o r s of P ^ (H.w\ so I,™ i/u^j.* (X--X„YKW(^ = L3r6li\ %™1 - x £ (H.K) The commutation r e l a t i o n s ( 3 . 5 . 3 1 ) f o l l o w a t once from equat ions ( H . l l ) , ( H . 1 3 ) , and (H .15 ) . 124 Appendix I The G a l i l e a n Group The G a l i l e a n group c o n s i s t s of the t r a n s f o r m a t i o n s ( 2 . 1 . 2 ) , ( 2 . 1 . 3 ) , ( 2 . 1 . 4 ) , and the G a l i l e a n b o o s t s : ( x , t ) * . ( x ' , t ' ) = ( x + v t . t ) ( l . O The u n i t a r y operator cor responding to a G a l i l e a n boost a long the j - a x i s of speed v w i l l be w r i t t e n as where K & i s the generator of the b o o s t . Us ing the methods of S e c t i o n 2 . 2 , one f i n d s tha t the ten generators of the G a l i l e a n group.obey the f o l l o w i n g set of commutation r e l a t i o n s . LC p r i - A ^ 6 I K [c.Hj-iftp;P i ^ r u ^ c wio-o where W& i s a parameter c h a r a c t e r i z i n g the system (which cannot be e l i m i n a t e d u s i n g the J a c o b i i d e n t i t y as i n S e c t i o n 2.2) and where we have appended a s u b s c r i p t & to the generators of the t r a n s f o r m a t i o n s ( 2 . 1 . 2 ) , ( 2 . 1 . 3 ) , and ( 2 . 1 . 4 ) as a reminder tha t we now cons ide r the G a l i l e a n group. The on ly G a l i l e a n group commutation r e l a t i o n s which d i f f e r i n form from those f o r the P o i n c a r e group a r e , not s u r p r i s i n g l y , the f i r s t and l a s t ones i n the l a s t l i n e of equat ions ( 1 . 3 ) . ( In the P o i n c a r e c a s e , these commutators c o n t a i n f a c t o r s of Ycl.) We now i n t r o d u c e the cent re of mass p o s i t i o n operator % & and the t o t a l 125 i n t e r n a l angular momentum 3*& by r e q u i r i n g that Then the commutation r e l a t i o n s (1.3) f o l l o w when K f r = - ™ & * & ( 1 5 ) p r o v i d i n g tha t te, - i*. i K tr t ^ : p u:\ H a- ti:: c i - ° Equat ions (1.4) and (1.5) may be so lved immediately f o r % b and J & to y i e l d I ^ I ^ ^ K ^ x P , (I.S) which equat ions a re the d e f i n i t i o n s of X^ . and 3^ i n terms of the generators of the G a l i l e a n group. I t a l s o f o l l o w s that if.) [ K ^ f n - o a.^ A system i s s a i d to be G a l i l e a n i n v a r i a n t i f one can c o n s t r u c t opera to rs 126 s a t i s f y i n g equat ions (1 .3) i n terms of the fundamental dynamical v a r i a b l e s of the system. For a system of TI p a r t i c l e s of masses m, and s p i n s S i , . . . , 5 x w i t h fundamental dynamical v a r i a b l e s the C a r t e s i a n c o o r d i n a t e s , K i , . . •, & n , momenta, P , P , , . , and s p i n s , S , , . . . , § n , s a t i s f y i n g fundamental commutation r e l a t i o n s ( 3 . 4 . 1 ) , the commutation r e l a t i o n s (1 .3) are s a t i s -f i e d when f v - £ P-provided the operator V& satisfies C f l . V j - - L X ^ . V j - L f & , V j = o ( I . I H ) The operator i s c a l l e d the p o t e n t i a l of the system and s p e c i f i e s the i n t e r a c t i o n s i n the system. A l s o , from L K cr\ P f r - - i.^ rn. & S j K we have that -- m.„ (IAS) The cen t re of mass p o s i t i o n %. b and the t o t a l i n t e r n a l angular momentum 5 ^ can be w r i t t e n e x p l i c i t l y i n terms of the fundamental dynamical v a r i a b l e s of the system. In p a r t i c u l a r , i t f o l l o w s from equat ions (1 .7) and (1.15) 127 that To complete the problem, one must find r o t a t i o n a l l y invariant "internal variables" for the system, operators commuting with P t,% b, and T &. This can be done at length using the method of Section 3.5. Thus, one defines the unitary operator where (This i s precisely the Gartenhaus-Schwartz transformation. f° I t follows that ^ & ^ P & J # C + ( A ^ ^ - x P c a.II) #&+(^ = e X X f r (1.10) so that i f [ f & , A l = o (I.*.l) and i f l.« *>/e.M£&«r,M»8s.M ex\*U ( l . Z ^ then L P & , A ] - £ } L , X l (1.2.3) where 128 C o n d i t i o n s (1 .21) and (1.22) are s a t i s f i e d by the opera to rs P K, X Ks X^ - X &, X^X-X^ , 5^. P roceed ing as i n S e c t i o n 3 . 5 , one f i n d s that X a / 8- X J O l-X / S= X ^ d. i-i) The above opera to rs do indeed commute w i t h .Pfr andX f r. Moreover , L^c/, X a 1 = l-fc fej^j X« r =t © « « - i _ * (M R o t a t i o n a l l y i n v a r i a n t i n t e r n a l v a r i a b l e s (operators commuting a l s o w i t h 7j) may thus be found by t a k i n g s c a l a r products of P« , X I , X ^ , and F i n a l l y , the i n t e r n a l angular momentum may be w r i t t e n as W - l The G a l i l e a n case i s c l e a r l y s i m p l e r than the Poincare ' c a s e . Th is i s due to the f a c t that the G a l i l e a n boost (1 .1) does not i n v o l v e a change i n the t ime c o o r d i n a t e whereas the Lo rentz boost ( 2 . 1 . 5 ) does . Th is l e a d s to l e s s c o u p l i n g i n the commutation r e l a t i o n s (1 .3) f o r the generators of the 129 G a l i l e a n group than i n the commutation r e l a t i o n s ( 2 . 2 . 1 5 ) f o r the P o i n c a r e g e n e r a t o r s . The G a l i l e a n . g e n e r a t o r s ( 1 . 1 0 ) - ( 1 . 1 3 ) f o r the p a r t i c l e system are c o r r e s p o n d i n g l y l e s s compl icated than the P o i n c a r e generators ( 3 . 5 . 1 ) - ( 3 . 5 . 4 ) , as are the c e n t r e of mass p o s i t i o n , i n t e r n a l angular momentum, and the i n t e r n a l v a r i a b l e s . In both c a s e s , the i n t e r a c t i o n s i n the system are s p e c i f i e d by a s i n g l e o p e r a t o r . In the G a l i l e a n case on ly the H a m i l t o n i a n i s changed by the i n t e r a c t i o n , whereas, i n the P o i n c a r e c a s e , both the H a m i l t o n i a n and the generator of Lo rentz boosts are a l t e r e d by an i n t e r a c t i o n . One can a l s o a r r i v e at the above p a r t i c l e r e s u l t s f o r the G a l i l e a n case by t a k i n g the l i m i t C - -*" 0 of the a p p r o p r i a t e equat ions f o r the P o i n c a r e c a s e . Thus, one has (1.30 (I.31> (I. 4^ (T.-J4) C — » t o (1.35) i f M e x i s t s Moreover , i f lim V= Vfr (1.36) then 1 1 vn 130 H = H & (1.37) 1 Gr where we have omit ted the ( i n f i n i t e ) term 7ft & c x on the r i g h t hand s i d e of the above e q u a t i o n . 

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