UBC Theses and Dissertations

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UBC Theses and Dissertations

Cross-sections for the gravitational scattering of massless particles Peet, Frederick Gordon 1970

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CROSS-SECTIONS FOR THE GRAVITATIONAL SCATTERING OF MASSLESS PARTICLES b y FREDERICK GORDON PEET B,A., U n i v e r s i t y o f Saskatchewan, 1963 M.A., Columbia U n i v e r s i t y , 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN THE DEPARTMENT OF PHYSICS We a c c e p t t h i s t h e s i s as conforming t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA October, 1970 , In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r equ i r emen t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I ag ree tha t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree tha p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y pu rposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d tha t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l owed w i t hou t my w r i t t e n p e r m i s s i o n . Department o f P h y s i c s The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouve r 8. Canada Date October 16, 1970 ABSTRACT The purpose of t h i s t h e s i s i s t o examine the g r a v i -t a t i o n a l s c a t t e r i n g o f m a s s l e s s s c a l a r p a r t i c l e s , photons, f o u r component n e u t r i n o s , and two component n e u t r i n o s "by one another. A m o d i f i c a t i o n o f the quantum t h e o r y o f the weak g r a v i t a t i o n a l f i e l d d eveloped by Gupta i s used as a b a s i s f o r the c o n s i d e r a t i o n s . C r o s s - s e c t i o n s are g i v e n f o r the g r a v i t a t i o n a l s c a t t e r i n g o f s c a l a r p a r t i c l e s by: s c a l a r p a r t i c l e s , photons, f o u r component n e u t r i n o s , and two component n e u t r i n o s ; o f photons by: photons, f o u r component n e u t r i n o s , and two component n e u t r i n o s ; o f f o u r component n e u t r i n o s by f o u r component n e u t r i n o s , and o f two component n e u t r i n o s by two component n e u t r i n o s . The c r o s s - s e c t i o n g i v e n by Bar k e r et a l and by B o c c a l e t t i e t a l f o r the s c a t t e r i n g o f photons by photons i s c o n f i r m e d . The c r o s s - s e c t i o n f o r the s c a t t e r i n g o f massive s c a l a r p a r t i c l e s by massive s c a l a r p a r t i c l e s quoted by DeV/itt and t h e c r o s s - s e c t i o n f o r the s c a t t e r i n g o f photons by massive s c a l a r p a r t i c l e s g i v e n by B o c c a l e t t i e t a l are found t o be i n e r r o r and are c o r r e c t e d . i i i CONTENTS ABSTRACT i i LIST OF FIGURES v i ACKNOWLEDGMENTS v i i INTRODUCTION 1 1. The C l a s s i c a l and Quantum D e s c r i p t i o n s of the Weak G r a v i t a t i o n a l F i e l d 5 a. C l a s s i c a l Theory 5 b. Quantum Theory 9 '2. The D e s c r i p t i o n o f the S c a l a r F i e l d and I t s I n t e r a c t i o n w i t h the G r a v i t a t i o n a l F i e l d 23 3. The D e s c r i p t i o n of the Photon F i e l d and I t s I n t e r a c t i o n w i t h the G r a v i t a t i o n a l F i e l d 25 4 . The D e s c r i p t i o n o f t h e Four Component N e u t r i n o F i e l d and I t s I n t e r a c t i o n w i t h the G r a v i t a t i o n a l F i e l d 27 5. The D e s c r i p t i o n of the Two Component N e u t r i n o F i e l d and I t s I n t e r a c t i o n w i t h the G r a v i t a t i o n a l F i e l d 37 6 . The M a t r i x Elements f o r the C o l l i s i o n s 4-3 a. The M a t r i x Elements f o r S c a l a r - S c a l a r S c a t t e r i n g 4-6 b. The M a t r i x Elements f o r S c a l a r - P h o t o n S c a t t e r i n g 4-8 c. The M a t r i x Element f o r S c a l a r - F o u r Component N e u t r i n o S c a t t e r i n g 51 d. The M a t r i x Element f o r Scalar-Two Component N e u t r i n o S c a t t e r i n g 53 e. The M a t r i x Element f o r Photon-Four Component N e u t r i n o S c a t t e r i n g ' 54-f . The M a t r i x Element f o r Photon-Two Component N e u t r i n o S c a t t e r i n g 56 g. The M a t r i x Elements f o r Four Component N e u t r i n o -Four Component N e u t r i n o S c a t t e r i n g 58 h. The M a t r i x Elements f o r Two Component N e u t r i n o -Two Component N e u t r i n o S c a t t e r i n g 60 7. The C r o s s - s e c t i o n s f o r the C o l l i s i o n s 62 a. The C r o s s - s e c t i o n f o r S c a l a r - S c a l a r S c a t t e r i n g 65 b . The C r o s s - s e c t i o n f o r S c a l a r - P h o t o n S c a t t e r i n g 64 c. The C r o s s - s e c t i o n f o r S c a l a r - F o u r Component N e u t r i n o S c a t t e r i n g 65 d. The C r o s s - s e c t i o n f o r Scalar-Two Component N e u t r i n o S c a t t e r i n g 66 e. The C r o s s - s e c t i o n f o r Photon-Four Component N e u t r i n o S c a t t e r i n g 67 f . The C r o s s - s e c t i o n f o r Photon-Two Component N e u t r i n o S c a t t e r i n g 69 g. The C r o s s - s e c t i o n f o r Four Component N e u t r i n o -Four Component N e u t r i n o S c a t t e r i n g 71 h. The C r o s s - s e c t i o n f o r Two Component N e u t r i n o -Two Component N e u t r i n o S c a t t e r i n g 72 8. C o n c l u s i o n s and D i s c u s s i o n ' 75 BIBLIOGRAPHY 85 APPENDIX A. The R e d u c t i o n o f the Four Component N e u t r i n o L a g r a n g i a n 87 APPENDIX B. The E v a l u a t i o n of a T r a c e f o r Photon-Four Component N e u t r i n o S c a t t e r i n g 88 APPENDIX C. The Photon P o l a r i z a t i o n Sums f o r Photon-Four Component N e u t r i n o S c a t t e r i n g 91 APPENDIX D. T r a c e C a l c u l a t i o n s f o r Four Component N e u t r i n o - F o u r Component N e u t r i n o S c a t t e r i n g 94 APPENDIX E. T r a c e C a l c u l a t i o n s f o r Two Component Neutrino-Two Component N e u t r i n o S c a t t e r i n g 97 APPENDIX P. The C r o s s - s e c t i o n f o r the S c a t t e r i n g of Txro I d e n t i c a l M a s s i v e S c a l a r P a r t i c l e s 100 APPENDIX G. Notes on N e u t r i n o s 108 LIST OF FIGURES 1. C r o s s - s e c t i o n f o r s c a l a r - s c a l a r s c a t t e r i n g 82 2. C r o s s - s e c t i o n f o r s c a l a r - p h o t o n s c a t t e r i n g 82 3. C r o s s - s e c t i o n f o r s c a l a r - n e u t r i n o s c a t t e r i n g ' 83 4-. C r o s s - s e c t i o n f o r p h o t o n - n e u t r i n o s c a t t e r i n g 83 5. C r o s s - s e c t i o n f o r f o u r component n e u t r i n o -f o u r component n e u t r i n o s c a t t e r i n g 84-6. C r o s s - s e c t i o n f o r two component n e u t r i n o -two component n e u t r i n o s c a t t e r i n g 84 v i i ACKNOWLEDGMENTS T h i s t h e s i s t o p i c was suggested hy P r o f e s s o r P. A. Kaempffer. I t would have been d i f f i c u l t t o f i n d a more s u i t a b l e t o p i c f o r me. F o r t h i s , I thank him. I n a d d i t i o n , h i s comments and c r i t i c i s m s g u i d e d the t h e s i s t o i t s c o m p l e t i o n . The remarks o f Dr. L. de S o b r i n o were v e r y much a p p r e c i a t e d . I thank B. G. K e e f e r f o r r e a d i n g the t h e s i s i n d r a f t form. The N a t i o n a l R e s e a r c h C o u n c i l of Canada p r o v i d e d generous f i n a n c i a l s u p p ort i n the form o f a Po s t g r a d u a t e S c h o l a r s h i p . 1 INTRODUCTION D u r i n g the p a s t decade s e v e r a l a r t i c l e s , f o r which r e f e r e n c e s will be g i v e n s h o r t l y , have appeared d e a l i n g w i t h the quantum m e c h a n i c a l treatment of the g r a v i t a t i o n a l i n t e r a c t i o n between p a r t i c l e s . I n most of t h e s e the i n t e r a c t i n g p a r t i c l e s have had a non-zero r e s t mass. Because of the a l l p e r v a s i v e n a t u r e of the g r a v i t a t i o n a l f i e l d , p a r t i c l e s of zero r e s t mass a l s o i n t e r a c t g r a v i t a -t i o n a l l y . I n t h i s t h e s i s the g r a v i t a t i o n a l i n t e r a c t i o n of p a r t i c l e s of zero r e s t m a s s — p h o t o n s , n e u t r i n o s , and mass-l e s s s c a l a r p a r t i c l e s , i s s t u d i e d , u s i n g a m o d i f i c a t i o n of the quantum t h e o r y of the weak g r a v i t a t i o n a l f i e l d developed by G u p t a ( l 9 5 2 ) . T h i s l a t t e r t h e o r y a l l o w s one t o t r e a t the g r a v i t a -t i o n a l i n t e r a c t i o n i n much the same manner as the e l e c t r o -dynamic i n t e r a c t i o n i s t r e a t e d i n quantum e l e c t r o d y n a m i c s . The p a r t i c l e c o r r e s p o n d i n g to the photon i s the g r a v i t o n . Ten t y p e s of g r a v i t o n s are p o s s i b l e . I n a f r e e g r a v i t a -t i o n a l f i e l d e i g h t o f t h e s e can be e l i m i n a t e d by means of a s u b s i d i a r y c o n d i t i o n j u s t as l o n g i t u d i n a l and t i m e - l i k e photons are e l i m i n a t e d from the f r e e e l e c t r o m a g n e t i c f i e l d by the L o r e n t z c o n d i t i o n . When so u r c e s a r e p r e s e n t the e i g h t g r a v i t o n s cannot be e l i m i n a t e d , but i n s t e a d mediate the i n t e r a c t i o n . The i n t e r a c t i o n L a g r a n g i a n f o r a g i v e n f i e l d and t h e g r a v i t a t i o n a l f i e l d may be found i n the f o l l o w i n g manner,. E s s e n t i a l l y , one s t a r t s from the L o r e n t z c o v a r i a n t L a g r a n g i a n f o r the f i e l d and r e p l a c e s a l l p a r t i a l d e r i v a t i v e s w i t h c o v a r i a n t d e r i v a t i v e s . The c o v a r i a n t 2 d e r i v a t i v e of a s p i n o r has been g i v e n by Fock(l929). A l t e r n a t i v e l y , one may use the method of the compensating f i e l d ( U t i y a m a 1956) which y i e l d s the same r e s u l t . The L a g r a n g i a n thus o b t a i n e d t r a n s f o r m s as a s c a l a r d e n s i t y under g e n e r a l c o o r d i n a t e t r a n s f o r m a t i o n s . One expands t h i s L a g r a n g i a n i n terms of the g r a v i t a t i o n a l c o u p l i n g c o n s t a n t and e x t r a c t s the i n t e r a c t i o n L a g r a n g i a n . One may now a p p l y the S-matrix f o r m a l i s m ( B o g o l i u b o v and S h i r k o v 1959) and work out m a t r i x elements and c r o s s -s e c t i o n s j u s t as i s done i n quantum e l e c t r o d y n a m i c s . T h i s i s the procedure which i s f o l l o w e d h e r e . Though t h e r e are no e x p e r i m e n t a l o b s e r v a t i o n s t o e i t h e r c o n f i r m or r e f u t e the r e s u l t s of the c a l c u l a t i o n s performed t o d a t e , t h e r e are some problems i n the quantum r e g i o n i n which the g r a v i t a t i o n a l f i e l d does not p l a y an i n s i g n i f i c a n t r o l e . F o r example, the g r a v i t a t i o n a l con-t r i b u t i o n t o the c r o s s - s e c t i o n f o r the s c a t t e r i n g of photons by photons dominates the e l e c t r o d y n a m i c c o n t r i b u t i o n , f o r v e r y low and v e r y h i g h f r e q u e n c i e s . T h i s r e s u l t f o l l o w s upon comparison of the g r a v i t a t i o n a l c r o s s - s e c t i o n t o be g i v e n here and the e l e c t r o d y n a m i c c r o s s - s e c t i o n g i v e n by A k h i e z e r and B e r e s t e t s k i i ( l 9 6 5 ) . Another example i s the f o l l o w i n g . I f one s e t s the mass to zero i n the D i r a c e q u a t i o n one o b t a i n s the e q u a t i o n f o r a f o u r component n e u t r i n o ( M u i r h e a d 1 9 6 5 , L u r i e 1 9 6 8 ) . A l t o g e t h e r t h e r e are f o u r t y p e s of p a r t i c l e s — t w o n e u t r i n o s and two a n t i n e u t r i n o s . These can be put i n t o correspondence w i t h the e l e c t r o n and muon n e u t r i n o s and t h e i r a n t i n e u t r i n o s ( A p p e n d i x G). I f one 3 wants t o d i s c u s s j u s t the e l e c t r o n n e u t r i n o , t h e n one imposes a s u b s i d i a r y c o n d i t i o n . T h i s has the' e f f e c t of r e d u c i n g t h e D i r a c f o u r component s p i n o r t o a two component s p i n o r . These m a t h e m a t i c a l e n t i t i e s are t r e a t e d i n the books by Corson ( l 9 5 3 ), Roman ( l960) , and Aharoni ( l 9 6 5 ). B e f o r e the d i s c o v e r y of the muon n e u t r i n o , Kobzarev and Okun( l963) p o i n t e d out t h a t the g r a v i t a t i o n a l i n t e r a c t i o n c o u l d be used t o d e t e c t t h e t h e n s o - c a l l e d anomalous(muon) n e u t r i n o s . Here, c r o s s - s e c t i o n s are g i v e n f o r c o l l i s i o n s i n v o l v i n g two component n e u t r i n o s and f o r c o l l i s i o n s i n v o l v i n g un-p o l a r i z e d beams of f o u r component n e u t r i n o s . The h i s t o r y o f the s u b j e c t runs as follo\\rs. The weak g r a v i t a t i o n a l f i e l d was q u a n t i z e d , as mentioned e a r l i e r , by Gupta i n 1952. The f o r m a l i s m was used by him i n the same y e a r i n c o n s i d e r a t i o n s on the g r a v i t a t i o n a l s e l f - e n e r g i e s of the photon and the e l e c t r o n . C o r i n a l d e s i ( 1 9 5 6 ) , u s i n g Gupta's f o r m a l i s m , c o n s i d e r e d the two-body problem. V ladimirov ( l 9 6 4 ) t r e a t e d the g r a v i t a t i o n a l a n n i h i l a t i o n of f e r m i o n s . B a r k e r e t al (1966) worked out the m a t r i x elements f o r the s c a t t e r i n g of massive p a r t i c l e s of v a r i o u s s p i n s and i n 1967 gave the g r a v i t a t i o n a l c r o s s - s e c t i o n f o r photon-photon s c a t t e r i n g . B o c c a l e t t i et al ( l 9 6 9 ) c o n s i d e r e d t h i s problem a g a i n and a r r i v e d a t the r e s u l t of B a r k e r e t a l . Kuchowicz ( l 9 6 9 ) reviewed n e u t r i n o dynamics i n quantum and non-quantum t h e o r i e s of g r a v i t a t i o n . In the p r e s e n t work c r o s s - s e c t i o n s are g i v e n f o r c o l l i s i o n s ( d u e to the g r a v i t a -t i o n a l i n t e r a c t i o n ) between s c a l a r p a r t i c l e s and: s c a l a r s , photons, f o u r component n e u t r i n o s and two component n e u t r i n o s between photons and: f o u r component n e u t r i n o s and two component n e u t r i n o s ; between f o u r component n e u t r i n o s and f o u r component n e u t r i n o s ; and, between two component n e u t r i n o s and two component n e u t r i n o s . I n Chapter 1 the r e l e v a n t n o t i o n s of the c l a s s i c a l ( E i n s t e i n 1918, 1956) and quantum(Gupta 1952) t h e o r i e s o f the weak g r a v i t a t i o n a l f i e l d a r e g i v e n . The s c a l a r f i e l d , photon f i e l d , f o u r component n e u t r i n o f i e l d and two component n e u t r i n o f i e l d a r e d e s c r i b e d and the g r a v i t a t i o n a l i n t e r a c t i o n L a g r a n g i a n f o r each i s g i v e n i n Chapters 2, 3, 4, and 5» r e s p e c t i v e l y . The m a t r i x elements f o r the v a r i o u s c o l l i s i o n s a re g i v e n i n Chapter 6 and i n Chapter 7 the c r o s s -s e c t i o n s are c a l c u l a t e d . The t h e s i s i s c o n c l u d e d i n Chapter 8 w i t h a d i s c u s s i o n of the r e s u l t s . N a t u r a l u n i t s (#--c=l) and an im a g i n a r y time c o o r d i n a t e are used. A l l Greek i n d i c e s r u n from 1-4 and are summed i f r e p e a t e d , u n l e s s o t h e r w i s e s t a t e d . The L a t i n i n d i c e s i , j , k , l,m, and n r u n from 1-3 and are summed i f r e p e a t e d , u n l e s s o t h e r w i s e s t a t e d . The L a t i n i n d i c e s p , q , r , s , and t r u n from 1-4 and are summed i f r e p e a t e d . The L a t i n i n d i c e s b and c r u n from 1-4, u n l e s s o t h e r w i s e s t a t e d , and are never summed i f r e p e a t e d . The symbols |, *, t , and T s i g n i f y r e s p e c t i v e l y , p a r t i a l d i f f e r e n t i a t i o n , complex c o n j u g a t i o n , H e r m i t i a n c o n j u g a t i o n and t r a n s p o s i t i o n . The s c a l a r p r o d u c t of 4 - v e c t o r s i s denoted by (V,W) or j u s t Vw", and of 3 - v e c t o r s by p-q. 5 1. The C l a s s i c a l and Quantum D e s c r i p t i o n s of the Weak G r a v i t a t i o n a l F i e l d a) C l a s s i c a l Theory A c c o r d i n g t o Einstein ( l 9 5 6 ) the g r a v i t a t i o n a l f i e l d v a r i a b l e s , g , are the c o e f f i c i e n t s which appear i n the d e f i n i t i o n o f the s c a l a r p r o d u c t , (V,W), of two v e c t o r s i n space-time, V and V/, whose components are and wv: (1.1) (V,W) = g ^ v v ^ wv . The s c a l a r p r o d u c t i s assumed t o be symmetric so t h a t (1.2) - g ^ . I f one can f i n d a c o o r d i n a t e system such t h a t (1.3) R =& + xh , « 1 , one says t h a t the g r a v i t a t i o n a l f i e l d i s weak. In (1.3) 6 i s the Kro n e c k e r d e l t a and K i s a c o n s t a n t whose v a l u e w i l l be g i v e n s h o r t l y . The h ^ v c o u l d be used as f i e l d v a r i a b l e s , b ut the f i e l d e q u a t i o n s have a s i m p l e r form i f one i n t r o d u c e s new v a r i a b l e s y d e f i n e d by (1.4) Y ^ v = - ihb^)/2 , h = T r a c e ( h ^ ) . A c c o r d i n g t o Einstein ( l 9 1 8 ) one may impose on the y the s u b s i d i a r y c o n d i t i o n The L a g r a n g i a n f o r a weak g r a v i t a t i o n a l f i e l d i s ( G u p t a 1952) (1.6) 4 = -(V4 ) [ V| . T V | T - ( Y | T ? | T > / 2 ] ' Y = T r a c e ( Y f x v ) . The L a g r a n g i a n d e n s i t y which d e s c r i b e s the i n t e r a c t i o n between the g r a v i t a t i o n a l f i e l d and any one of t h e f i e l d s t o be c o n s i d e r e d , whose energy-momentum t e n s o r i s T , i s , as w i l l be shown i n Chapters 2, 3, 4, and 5» ( 1 . 7 ) 4 n t 1 8 ~ ( H / 2 > Y - (y& )/2 The f i e l d e q u a t i o n s which f o l l o w from the L a g r a n g i a n d e n s i t y (i.s) x. 4 + 4 n t a r e t h o s e g i v e n by E i n s t e i n f o r a weak g r a v i t a t i o n a l f i e l d w i t h a source whose energy-momentum t e n s o r i s T ' An e x a m i n a t i o n o f the s o l u t i o n s o f t h e s e e q u a t i o n s f o r two c h o i c e s o f T , p r o v i d e s the l i n k w i t h Newton's law of g r a v i t a t i o n and p r o v i d e s an i n t r o d u c t i o n t o the quantum tr e a t m e n t . I f one assumes a s t a t i c d i s t r i b u t i o n o f mass w i t h d e n s i t y u., t h e o n l y non-zero component of T i s (1.10) T ^ = u . I f t h e y v a r e assumed to be time independent and 0 a t i n f i n i t y , t h e n i t f o l l o w s from (1.9) t h a t a l l the y are 0 e x c e p t . f o r Yz^* which i s , (1.11) = - ( K / 4 7 t j / ( ^ / r ) dx . The e q u a t i o n of motion of a t e s t p a r t i c l e i s ( E i n s t e i n 1956) 7 (1.12) d 2 r / d t 2 = -(H/4) g r a d C y ^ ) . T h i s agrees w i t h Newton's law of motion i f one takes ( 1 . 1 3 ) H 2 = 16TCG . I n c.g.s. u n i t s the r e l a t i o n s h i p i s ( 1 . 1 4 ) H 2 = 1 6 7 I G / C 4 . I f one s e t s T^^ t o zero i n ( 1 . 9 ) , t h e wave e q u a t i o n s i f o r a f r e e g r a v i t a t i o n a l f i e l d are o b t a i n e d : C1-15) tfv = °-S o l u t i o n s o f ( 1 . 1 5 ) are (1.16) Y = (V/V) S ( l / / 2 S k ) ( a (k) e i k x + a (k) e " i k x ) . The a ^ y s a t i s f y t he f o l l o w i n g r e a l i t y p r o p e r t i e s : (1.17) a ± j = a ± ^ , a ± 4 = - a ± 4 , a ^ - a ^ . I f i t i s assumed t h a t the wave propagates a l o n g the x a x i s , the s u b s i d i a r y c o n d i t i o n (1.5) y i e l d s (1.18) a ^ + i a ^ = 0 , u -1,4 ( 1 . 1 9 ) + i a ^ 4 = 0 , u=l,4 . The H a m i l t o n i a n d e n s i t y i n such a wave i s ( 1 . 2 0 ) # = ( 1 / 2 ) Y ^ v Y ^ v - (1/4) Y Y - / . The H a m i l t o n i a n , .H, i s ( 1 . 2 1 ) H = y ^ d 3 x = 2 co k( | ( a i ; L - a 2 2 ) / 2 | 2 + | a 1 2 | 2 ) 8 where the c o n d i t i o n s (1.18) and (1.19) have been used. I f one makes the s u b s t i t u t i o n s i i ( 1 . 2 2 ) a i ; L = ( a i : L - a 2 2 ) / 2 , a 2 2 = (an + a 2 2 ^ 2 ' H becomes ( 1 . 2 3 ) H = S w k ( a l l 2 I I 2 > + | a 1 2 | ). Thus, i n a f r e e g r a v i t a t i o n a l wave the energy depends o n l y on two independent modes. The s u b s i d i a r y c o n d i t i o n ( 1 . 5 ) p r e v e n t s the o t h e r modes from c o n t r i b u t i n g t o the energy. On the o t h e r hand, p a r t i c u l a r s o l u t i o n s o f the inhomogeneous e q u a t i o n s ( 1 . 9 ) a u t o m a t i c a l l y s a t i s f y ( 1 . 5 ) as a consequence of the v a n i s h i n g d i v e r g e n c e o f b) Quantum Theory The momenta n , c o n j u g a t e to the v , which f o l l o w [J* V Ll V from the L a g r a n g i a n (1.6) are (1.24a) = V / S y p = y^ , u/v (1.24b) 7 i b b = '>>Vb%h = Y b b / 2 - Y A . The Y ^ v and are d e f i n e d to be o p e r a t o r s which s a t i s f y the commutation r e l a t i o n s ( a t e q u a l t i m e s ) (1.25a) i ( V 6v(3 + 5u.p 6 v a ) 6 ( f " ? , ) » ^ V > (1.25b) b V = i 6 ( r - r ' ) . A l l o t h e r b a s i c commutators are z e r o . From (1.24b), (1.25b), and the c o n d i t i o n (1.26) ^bb' 7 1 cc = o, bye , i t f o l l o w s t h a t (1.27) [ r b b , i b b ] - i&C?-?' ) and t h a t (1.28) r T b b ( r ) , Y ^ ( r ' ) | = - i 6 ( r - r « ) , b/c. cc One may stu d y the spectrum of the o p e r a t o r s Y ^ y "by decomposing t h e y a c c o r d i n g t o (1.16). The r e l a t i o n s (1.25a), (1.27), and (1.28) l e a d t o (1.29a) au.v'acc|3 10 (1.29b) [ a b b , a - b b ] = 1 (1.29c) [ a b b > a c c ] = -1, b/c. A l l o t h e r b a s i c commutators are z e r o . The r e a l i t y c o n d i t i o n s (1.17) now r e a d (1.30) a ^ = a±.. , a i 4 T = - a i 4 , a ^ 1 = a ^ . One would l i k e t o , i n the customary f a s h i o n , use a r e p r e s e n t a t i o n i n which the o p e r a t o r s (1.31) N = a a (no summation i m p l i e d ) IX v | X V LX t/ are d i a g o n a l . T h i s i s not p o s s i b l e because, by (l.29c), N,, and N do not commute. T h i s i s a d i r e c t consequence bb cc of t h e appearance of y i n the L a g r a n g i a n (1.6). I n the tr e a t m e n t g i v e n by Gupta(l952) t h i s problem does not appear e x p l i c i t l y . There, Gupta t r e a t e d y as an independent v a r i a b l e and imposed a s u b s i d i a r y c o n d i t i o n (1.32) Y = T r a c e ( y ) . Here, a d i f f e r e n t method o f c i r c u m v e n t i n g the problem i s f o l l o w e d . One f i r s t o f a l l w r i t e s the L a g r a n g i a n (1.6) i n the form (1.33) / G = -(lAX/^ +/2) where (1.34-) / x = y ^ Y ^ | T and 11 4 (1.35) / 2 = S Y b b j T Y b b | T - (1/2)Y| T Y| T • The "bar under u.v i n ^ s i g n i f i e s summation over the o f f -d i a g o n a l terms o n l y . I f one now i n t r o d u c e s nexv v a r i a b l e s Y^ y d e f i n e d by (1.36a) Y ' V = - Y , tfv (1.36b) = (1//2) Y i i ^22 Y33 V ' <44 then ^ reduces to (1.37) / 2 = 2( s Y b b | T T B B ) T 1//2" -1//2 0 0 0 1//2 1/2 1/2 -1/2 1/2 1/2 1/2 Y b b I T ~ y 4 4 | T T 4 4 0 -1/2 Y l l ' Y22 Y33 Y 4 4 -). I n (1.37) the q u a d r a t i c form o/^ has been reduced t o a sum and d i f f e r e n c e o f s q u a r e s , i n c o n t r a d i s t i n c t i o n t o (1.35) which c o n t a i n s the term Y j T Y | T» A commutation r e l a t i o n l i k e (1.29c) will not a r i s e now. The c o n j u g a t e momenta are (1.38a) = Y ^ v , (l ^ v ) / ( 4 4 ) (1.38b) = - Y ^ . The commutation r e l a t i o n s f o r t h e Y 1 send ix' , which are 1 [IV (J.V 7 e q u i v a l e n t t o the r e l a t i o n s (1.25a,b) f o r the Y v a n ( l a re [.t V [-IV (1.39a) (1.39b) i ( 6 6 ' + 6 D 6 ) 6 ( r - r ' ) , v [icx v[3 u-(3 va ' u/v, a/p, Y b b ( ? ) ' H b ( ? , ) ] = " ( r - r 1 ) • The Y ^ v a r e expanded as 12 (1.40) y' = (1//V) E - ( l / J ^ O (c (k) U.V > ± l 0 C + H l A V ( k ) e - i k x ) . The commutation r e l a t i o n s ( l . 3 9 a - b ) t o g e t h e r w i t h ( l . 3 3 a - b ) imply (1.41a) [ c ^ v , S a p ] = + ^ p ^ ) . H / v , a/B, (1.41b) [ c b b , 3 b b ] = 1 , b/4 (1.41c) [ c ^ c ^ = -1. A l l o t h e r b a s i c commutators are z e r o . A commutation r e l a t i o n l i k e ( l . 2 9 c ) does not appear now. From the expansions (1.16) and (1.40) and the t r a n s -f o r m a t i o n (1.36b) i t f o l l o w s t h a t the c ^ y and c ^ v are r e l a t e d t o the a ^ v and a ^ v by (1.42a) c (1.42b) '11 ;22 ; 3 3 ;44 l / f 5 - 1 / J 2 " 0 0 0 0 1/12 - 1 / J 2 1/2 1/2 - 1 / 2 - 1 / 2 1/2 1/2 1/2 1/2 a l l " a 2 ? a 3 3 a 4 4 and t h e above two e q u a t i o n s w i t h c ^ v and a ^ v r e p l a c e d w i t h c and a . The symbols c and c were used i n _ i (1.40) i n s t e a d of t h e symbols a ^ y and a ^ y , which are more n a t u r a l i n view of ( l . 3 6 a , b ) , i n o r d e r t o a b b r e v i a t e the n o t a t i o n . I t f o l l o w s from the c o n j u g a t i o n p r o p e r t i e s (1.30) f o r the a and t h e t r a n s f o r m a t i o n ( l . 4 2 a , b ) t h a t the c o n j u g a t i o n p r o p e r t i e s of the c are (1.43) c 10 13 The o p e r a t o r s N (k) d e f i n e d by mn (1.44a) N. .(k) = c. .(k)c• .(k) (no summation i m p l i e d ) (1.44b) N i Q ( k ) = c i 0 ( k ) c i 0 ( k ) (1.44c) N^Ck) = c ^ G ^ c ^ k ) where (1.45) c i 0 = c o i = - i c i 4 , c i 0 = c o i = - i c i 4 form a complete s e t of commuting o p e r a t o r s . From the commutation r e l a t i o n s and c o n j u g a t i o n p r o p e r t i e s of the c and c i t f o l l o w s t h a t the e i g e n v a l u e s , n (k), of the [iv ° 7 mn N „(k) are the non - n e g a t i v e i n t e g e r s . F o r g i v e n k and mn * . * g i v e n (mn) t h e e i g e n v e c t o r s , | n m n ( ^ ) / > °£ ^mn^) a r e J u s ^ the e i g e n v e c t o r s o f the harmonic o s c i l l a t o r . I f E (k) denotes the v e c t o r space spanned by t h e s e e i g e n v e c t o r s , the space E of a l l p o s s i b l e d y n a m i c a l s t a t e s can be w r i t t e n U-V (1.46) E = where ® denotes the t e n s o r p r o d u c t w i t h the p r o v i s o t h a t o n l y one of the p a i r E and E i s t o be i n c l u d e d s i n c e N _ ^ x mn nm mn i s symmetric i n m and n. An a r b i t r a r y v e c t o r | J> i n E can be w r i t t e n as a l i n e a r c o m b i n a t i o n o f t e n s o r p r o d u c t s of v e c t o r s i n the ^ m n ( k ) . The o p e r a t o r s ., C ^ Q , and t h e i r c o n j u g a t e s a c t i n the f o l l o w i n g maimer where, f o r — i b r e v i t y , the argument k has been s u p p r e s s e d , and o n l y the r e l e v a n t n _ has been w r i t t e n i n I > : mn (1.47) c ^ l ^ > . / ^ j | n 1 ; 1 - l > 14 S i o l n i o > = f ^ i o l n i o - i ; > c i o | n i O > = F i C ^ I V ^ One says t h a t c ; | / | , the C^Q, and the c^.. d e s t r o y g r a v i t o n s of momentum k and p o l a r i z a t i o n s (4-4-), ( i O ) and ( i j ) r e s p e c t i v e l y . S i m i l a r l y , c ^ , the C^ Q and the c\^ c r e a t e g r a v i t o n s o f momentum k and p o l a r i z a t i o n s ( 4 4 ) , ( i O ) and ( At t h i s p o i n t however, a new problem a r i s e s . The quantum m e c h a n i c a l form o f the s u b s i d i a r y c o n d i t i o n (1.5) i s ( G u p t a 1952) where y~ i s t h e n e g a t i v e f r e q u e n c y p a r t of y . T h i s c o n d i t i o n , t o g e t h e r w i t h the i n v e r s e of (l.4-2b), y i e l d s (1.49a) ( c 1 5 + i c 1 4 ) | /> = 0 (1.49b) ( c 2 3 + i c 2 4 ) | > = 0 (1.49c) ( c 2 2 - ( c 3 5 - cm)//2 + i c $ 4 ) | > = 0 (1.49d) ( i c 4 3 + c 2 2 + ( c 3 5 - c^)//2)\ > = 0. The l a s t two im p l y (1.50a) ( c 2 2 + i c 3 4 ) | > = 0 15 (1.50b) ( c 3 3 - c ^ ) ! > = 0. That p a r t o f ^ a n a r b i t r a r y v e c t o r which d e s c r i b e s the (33) and (4-4-) p o l a r i z a t i o n s can be w r i t t e n as (1.51) |33,4-4> = 2 2 A ( n $ 5 , n 4 Z ) _ ) | n 3 3 , n ^ ^ I t f o l l o w s from (1.50b) t h a t (1.52) 2 2 A(n35,n^)|jliJ3 | n 3 5 - l , n 4 4 > - | n J 3 ^ n ^ + l ^ O . I t can be shown(Akhiezer and B e r e s t e t s k i i , 1 9 6 5 , p a g e 168) t h a t a r e l a t i o n o f t h i s form i m p l i e s t h a t |33,44> cannot be n o r m a l i z e d . S i m i l a r c o n c l u s i o n s can be drawn from (1.49a), (1.49b) and (1.50a). A s o l u t i o n t o t h i s problem i s t o i n t r o d u c e the i n d e f i n i t e m e t r i c f o r m a l i s m . One assumes t h a t the o p e r a t o r s Y]_4.J Y 2 v a n d which i s d e f i n e d by (1.53) i m = i Y ^ , a r e H e r m i t i a n . Then c ^ , c3/p a n d a r e ^ e H e r m i t i a n c o n j u g a t e s of c-^,, c*2^} ^5^' a n d w ^ - e r e $i\>\ and ^ / | y [ a r e d e f i n e d by (1.54) ~ ^-c44 » ^44 = ^-c>\i\ * The commutation r u l e f o r tf, ^ and j ? y [ / | i s , from ( l , 4 1 c ) , (1.55) [ ^ , ? 4 4 ] = I-The e i g e n v a l u e s , n , of t h e o p e r a t o r s (J. V (1.56a) N^Ck) = c^(X)c^(X) , ( i i v ) / ( 4 4 ) 16 and (1.56b) N ^ t k ) = ^ ( k ) j z ^ ( k ) are the n o n - n e g a t i v e i n t e g e r s because of the commutation r u l e s (1.41a), (1.41b) and (1 . 5 5 ) . The e i g e n v e c t o r s of each N (k) are j u s t t he harmonic o s c i l l a t o r e i g e n v e c t o r s . The space E of dynamical s t a t e s i s the space spanned by the t e n s o r p r o d u c t s of the e i g e n v e c t o r s o f t h e N ( k ) . One now d e f i n e s a u n i t a r y and H e r m i t i a n o p e r a t o r r\ by i t s m a t r i x elements ( 1 . 5 7 ) < n ' h | n > = ( - l ) n W + n 2 ^ + n ^ 6 m , where |n> and |n'> are b a s i s v e c t o r s o f E formed from the t e n s o r p r o d u c t o f e i g e n v e c t o r s o f the N ( k ) . A g e n e r a l i z e d s c a l a r p r o d u c t and a g e n e r a l i z e d e x p e c t a t i o n v a l u e are d e f i n e d by (1 . 58 ) <¥|4)> G =<Y|rj|(|>> and (1 . 5 9 ) <A> Q = < |nA| > One now i n t e r p r e t s a l l e x p e c t a t i o n v a l u e s as g e n e r a l i z e d e x p e c t a t i o n v a l u e s . I f an H e r m i t i a n o p e r a t o r commutes w i t h r\ i t has a r e a l g e n e r a l i z e d e x p e c t a t i o n v a l u e and i f i t anti-commutes w i t h n i t has a pure i m a g i n a r y g e n e r a l i z e d e x p e c t a t i o n v a l u e . From (1 . 57 ) i t f o l l o w s t h a t c-^, c 2 / p c-,^, and t o g e t h e r w i t h t h e i r c o n j u g a t e s anti-commute w i t h n; hence, Y^/p y 2 4 7 y J 4 a n < ^ ^ 44 ^ • A V E P U R E i m a g i n a r y g e n e r a l i z e d e x p e c t a t i o n v a l u e s . A l l t h e o t h e r commute w i t h n and hence a l l the o t h e r y* have r e a l g e n e r a l i z e d 17 e x p e c t a t i o n v a l u e s . T h e r e f o r e , the c o r r e c t r e a l i t y p r o p e r t i e s f o r the<Y ,j l v>Q a r © o b t a i n e d . The o p e r a t o r s c t i a c t on the e i g e n v e c t o r s of N (k) i n ^ (IV ^ JJ.V ' the f o l l o w i n g f a s h i o n : (1.60) c ^ n ^ / ^ . > . . - ! > c i 4 | n i Z f > = - i / n ~ 4 | n i 4 - l > ^ z r 4 i n 4 4 > = " i f ^ 4 - l n 4 4 " i ; > c 1 4 - l n i 4 > = i J n i 4 + 1 l n i 4 + 1 > The s u b s i d i a r y c o n d i t i o n (l.50b) now reads (1.61) ( c 3 5 + i ^ ) | > = 0. I f one e x p r e s s e s , as i n (1.51)> t h a t p a r t of an a r b i t r a r y v e c t o r which d e s c r i b e s t h e (53) and (44) p o l a r i z a t i o n s , one o b t a i n s from (1.61) (1.62) s S A( 1 1 ^ , 1 1 ^ ) [~n^3 \n^-±,n^y + J n ^ |n-,-, . n ^ - l / ? ] = 0. I t can be shown(Akhiezer and B e r e s t e t s k i i , 1 9 6 5 ) t h a t t h i s c o n d i t i o n i m p l i e s t h a t |33,44> i s n o r m a l i z a b l e w i t h non-n e g a t i v e norm. ^ i m i l a r c o n c l u s i o n s f o l l o w from the s u b s i d -i a r y c o n d i t i o n s (l.49a,b) and (l.50a). Thus t h e problem of i n f i n i t e norms has been removed. The c o n d i t i o n (1.61) t o g e t h e r with, i t s c o n j u g a t e (1.63) < | ( c 5 5 - i ^ ) = < | T J ( C 5 5 + i ? ^ ) = 0 18 l e a d s t o (1.64a) < N „ > G + < V | > G - = °-The s u b s i d i a r y c o n d i t i o n s ( 1 . 4 9 a ) , (1.49b) and ( l . 5 0 a ) and t h e i r c o n j u g a t e s l e a d to (1.64b) < N 1 5 > G + <N 1 / +> G 0 (1.64c) < N 2 3> G + <N 2 4> 0 ( l . 6 4 d ) <N 2 2> & + <N 34> G 0. E q u a t i o n s ( l . 6 4 a - d ) a r e u s e f u l i n the computation of the d y n a m i c a l v a r i a b l e s o f t h e f r e e f i e l d . F o r example, the H a m i l t o n i a n which f o l l o w s from the L a g r a n g i a n ( 1 . 3 3 ) i s T h i s i s j u s t t h e quantum m e c h a n i c a l form of ( 1 . 2 3 ) i f one ta k e s i n t o c o n s i d e r a t i o n ( l . 4 2 a , b ) . The s u b s i d i a r y c o n d i t i o n a l l o w s o n l y t he ( l l ) and ( 1 2 ) g r a v i t o n s t o c o n t r i b u t e t o the H a m i l t o n i a n . From (1.60) i t f o l l o w s t h a t t e n t y p e s of g r a v i t o n s are p o s s i b l e , though i n the f r e e f i e l d , o n l y t he ( l l ) and ( 1 2 ) g r a v i t o n s c o n t r i b u t e t o the dy n a m i c a l v a r i a b l e s as e x e m p l i f i e d i n ( 1 . 6 6 ) . I n Gupta's tr e a t m e n t e l e v e n t y p e s o f g r a v i t o n s a re p o s s i b l e . T h i s i s because he t r e a t s y as an independent v a r i a b l e . C o n t r i b u t i o n s from y t o the f r e e f i e l d v a r i a b l e s a re removed by i m p o s i t i o n of the a d d i t i o n a l ( 1 . 6 5 ) H = ( 1 / 2 ) 2 co, fN ,(k) + 2 2 N, , ( k ) . Hence, because of ( l . 6 4 a - d ) ( 1 . 6 6 ) 19 s u b s i d i a r y c o n d i t i o n (1.32). The c o v a r i a n t commutation r e l a t i o n s f o r the f i e l d s Y 1 a r e , from ( l . 4 1 a , b ) and (1.55)» (1.67a) [ Y' v(x),Yip(y)] = K V 6 v - p + ^ p 6 v a ) D ( x - y ) , u ^ , a / B (1.67b) [ Y b b U ) , Y b b ( 7 ) ] = i D ( x - y ) , b/4-(1.67c) [ Y i 4 ( x ) , Y i 4 ( y ) ] = - [ ^ U ) , / ^ ) ] = - i D ( x - y ) where (1.68) D(x-y) = (1/(2TI) 5) J(1/a^) s i n ( k ( x - y ) ) d 3 k I f one uses the commutation r e l a t i o n s (1.4-la-c) f o r the c and the t r a n s f o r m a t i o n (1.42a,b) r e l a t i n g t he c and the a , one o b t a i n s f o r the commutation r e l a t i o n s f o r the [XV ' a the r e l a t i o n s ( l . 2 9 a - c ) . The c o v a r i a n t commutation r e l a t i o n s f o r the f i e l d s Y ^ y which f o l l o w from ( l . 2 9 a - c ) a re (1.69a) [ Y ^ ( x ) , Y a p ( y ) ] = i ^ B v + 6 ^ 6 y a ) D ( x - y ) ,u^v,a/B (1.69b) [ Y b b U ) , Y b b ( y ) ] = i D ( x - y ) (1.69c) [ Y b b ( x ) , Y c c ( y ) ] = - i D ( x - y ) A l s o u s e f u l a r e (1.70a) [ Y b b ( x ) , Y ( y ) ] = - 2 i D ( x - y ) (1.70b) [Y(x),Y(y)] = - S i D ( x - y ) The c o v a r i a n t r e l a t i o n s f o r the h are L i V (1.71) [ V x ) > h a P ( y ) ] - [ Vv U ) - 6 ^ ( x ) / 2 - W ^ ' V i z l " 20 i ( 6 6 D + 6 D 6 - 6 5 o ) D ( x - y ) , v (j.a vp u.|3 va (iv a P ' v an expression.--which i s good f o r a l l (pv) and a l l ( a p ) . The vacuum s t a t e , |0)>, can "be d e f i n e d as the s t a t e f o r which (1.72) c^|o> = 0, (u-v)/(44) ; ^|o> = ivJo>= c^|0> = 0. The vacuum e x p e c t a t i o n v a l u e of P ( h ^ y ( x ) n a p ( y ) ) , where P i s the time o r d e r i n g o p e r a t o r , i s r e q u i r e d f o r l a t e r use. By f o l l o w i n g t h e same procedure as was f o l l o w e d f o r the commutators, t h a t i s , t r a n s f o r m i n g back from the y ^ v to the y t o the h , one o b t a i n s \iv [XV (1.73) <0|qP(h^ v(x) h a p ( y ) ) | 0 > = <0|P(h^ v(x) h a p ( y ) ) | 0 > = ~ i ( V 6 v p + 6 ^ p 6 v a - 6 ( x v 6 a P ) D P ( x - y ) where (1.74) D f f(x-y) = l i m ( l / ( 2 T u ) 4 ) / ( l / ( k 2 - i e ) ) e i k ( x - y ) d \ . e-»0 J T h i s r e s u l t i s good f o r a l l (pv) and a l l ( a p ) . I t agrees w i t h the e x p r e s s i o n g i v e n by Gupta(l952). As an example of the f o r m a l i s m c o n s t r u c t e d , the change i n energy of two n o n - q u a n t i z e d mass p o i n t s , s i t u a t e d a t and x 2 , due t o t h e i r g r a v i t a t i o n a l i n t e r a c t i o n i s computed. A s u i t a b l e i n t e r a c t i o n H a m i l t o n i a n d e n s i t y i s , s i n c e i t l e a d s t o the c o r r e c t f i e l d e q u a t i o n s (1.9), (1.75) # ± N T = (H/2) T ^ h 4 4 = (H/2) LX h ^ 21 where (1.76) p. = 2 m 6(x-x_) n= ?l n n and (1.77) h ^ = Y44 - Y/2 - ~ Y 2 2 ~ ^ 3 3 / ^ + ^ 4 4 / ^ - Y/2 S i n c e the H a m i l t o n i a n (1.75) a f f e c t s o n l y the ( 2 2 ) , (33) and (44-) g r a v i t o n s i t i s s u f f i c i e n t t o l a b e l a s t a t e by | n 2 2 , n 3 3 , n ^ > . ^ e i n i t i a l and f i n a l s t a t e s c o n t a i n no g r a v i t o n s . The non-zero m a t r i x elements are (1.78) . <000|H|lOO>= <OOl|H|000>* = A <000|H|010> = <010|H|000>* = A//2~ <000|H|001> = -<100|H|000> = A//2~ , where (1 . 7 9 ) H =jM±nt a V and 2 ik-x (1.80) A = -(H/2 ) ( n / / v)(l//2w k) 2 m n e " " n n=l The f i r s t o r d e r c o r r e c t i o n t o the energy i s ze r o s i n c e t h e r e are no non-zero m a t r i x elements of the form <^ 0 |H|0>. The second o r d e r c o r r e c t i o n i s (1.81) U = 2 < 0 0 0 | H | n 2 2 ( k ) n 5 3 ( k ) n ^ ( k ) > < n 2 2 ( k ) n 5 5 ( k ) n ^ ( k ) | lc H|000>/(-cu k) 0 ik<x -x ,; P = ( - H " / 8 V ) S S 2 ni m . e n n (1+1/2^1/2)/(co,^) k n n 1 n n K n/n« = _ ( H V t - V)m nm 0 2 e ^ /(co v ) 1 ^ k K 2 —** —* = - H m^m 2/(16Tcr), r= jx-^-x^ . From (1.13) one has (1.82) U = - G m ^ / r which i s j u s t the Newtonian p o t e n t i a l energy. A t t e n t i o n i s now t u r n e d t o a more d e t a i l e d t reatment of i n t e r a c t i o n s . 2. The D e s c r i p t i o n of the S c a l a r F i e l d and I t s I n t e r a c t i o n w i t h the G r a v i t a t i o n a l F i e l d In a f l a t space-time the L a g r a n g i a n d e n s i t y f o r the ma s s l e s s and H e r m i t i a n s c a l a r f i e l d 0 % when r e f e r r e d t o r e c t i l i n e a r c o o r d i n a t e s , i s ( C o r s o n 1953) (2.1) / » - ( 6 ^ i V ^ 0 | v ) / 2 . The f i e l d e q u a t i o n which f o l l o w s from the L a g r a n g i a n (2.1) i s (2.2) Q2tf = 0 f o r which a s o l u t i o n i s (2.3) ft = (1//V) 2 ( l / / 2 ^ ) ( a « ) e i p x + a t ( p ) e " i P x ) . P P When one makes t h e t r a n s i t i o n t o quantum f i e l d t h e o r y a(p) i s i n t e r p r e t e d as an o p e r a t o r which d e s t r o y s a s c a l a r p a r t i c l e of momentum p and a (p) i s i n t e r p r e t e d as an o p e r a t o r which c r e a t e s a s c a l a r p a r t i c l e of momentum p. The c a n o n i c a l energy-momentum t e n s o r which f o l l o w s from the L a g r a n g i a n (2.1) i s (2.4) T = 0, + 6 ( c * l a 0. )/2 , Gfl a=g a P I n g e n e r a l r e l a t i v i t y one seeks, f o r a t e n s o r f i e l d , " a L a g r a n g i a n d e n s i t y which 1) t r a n s f o r m s as a s c a l a r d e n s i t y under g e n e r a l c o o r d i n a t e t r a n s f o r m a t i o n s , and 2) reduces t o the known f l a t space-time L a g r a n g i a n when no g r a v i t a t i o n a l f i e l d i s p r e s e n t and r e c t i l i n e a r c o o r d i n a t e s are used. Such a L a g r a n g i a n d e n s i t y can be o b t a i n e d from the f l a t 24 space-time form by 1) r e p l a c i n g the p a r t i a l d e r i v a t i v e by the c o v a r i a n t d e r i v a t i v e , 2) r e p l a c i n g the m e t r i c 6^ v by the m e t r i c g^y, and 3) i n t r o d u c i n g a f a c t o r /g where g i s the de t e r m i n a n t of the m e t r i c t e n s o r . F o r a s c a l a r f i e l d t he c o v a r i a n t d e r i v a t i v e i s the same as the p a r t i a l d e r i v a t i v e . Hence, the a p p r o p r i a t e g e n e r a l i z a t i o n of (2.1) i s (2.5) /'= -/g g ^ $ ^ tf|v/2. When the g r a v i t a t i o n a l f i e l d i s weak (2.6) g ^ V = 6^ V - H h ^ and (2.7) / g = 1 + H 6 a p h a p / 2 so t h a t (2.8) / ' = - ( 1 + K 6 a p h a g / 2 ) ( 6 l i U - K h ^ ) 0, (*|v/2 where T a ^ i s t h e energy-momentum t e n s o r d e f i n e d by ( 2 . 4 ) . T h e r e f o r e , the d e s i r e d q u a n t i t y , the i n t e r a c t i o n L a g r a n g i a n , i s (2.9) ^ = - ( n / 2 ) h Q T Q where the d i s t i n c t i o n between upper and lower i n d i c e s has been dropped s i n c e 6 i s t h e Kron e c k e r d e l t a . 3. The D e s c r i p t i o n o f • t h e Photon P i e l d and I t s I n t e r a c t i o n w i t h the G r a v i t a t i o n a l F i e l d In a f l a t space-time the L a g r a n g i a n d e n s i t y f o r the photon f i e l d A , when r e f e r r e d t o r e c t i l i n e a r c o o r d i n a t e s , i s ( C o r s o n 1953) (3.1) / = -(1/4)6°^ 6 P v F R F 0 0 7 <x$ [xv where The f i e l d e q u a t i o n s which f o l l o w from the L a g r a n g i a n (3.1) and the r e l a t i o n s (3.2) are (3.3) a\ = 0 f o r which s o l u t i o n s are ( 3 . 4 - ) A u = (1//V) S ( l / ^ ) ( a T ( p ) e T a e 1 ^ + a t T ( p ) e T * e " 3 * * ) . r p P u p u H-where the e a r e p o l a r i z a t i o n v e c t o r s . When one makes the t r a n s i t i o n t o quantum t h e o r y , a T ( p ) d e s t r o y s a photon of momentum p and p o l a r i z a t i o n e and a T ( p ) c r e a t e s a photon of momentum p and p o l a r i z a t i o n e . The energy-momentum t e n s o r which f o l l o w s from the L a g r a n g i a n (3.1) i s , a f t e r s y m m e t r i z a t i o n , (3.5) T = -F a F + (1/4)6 F D F a ( 3 . •/J \xv [x va s ' J [xv a(3 I n o r d e r t o f i n d a L a g r a n g i a n f o r the photon f i e l d i n the r e a l m of g e n e r a l r e l a t i v i t y , one a p p l i e s the r u l e s l a i d down i n Chapter 2. The r e s u l t i s (3.6) / ' = - ( l / 4 ) / g g 0 ^ g ^ F a p F ^ 26 where The s e m i c o l o n denotes c o v a r i a n t d i f f e r e n t i a t i o n . When the g r a v i t a t i o n a l f i e l d i s weak one o b t a i n s u s i n g (2.6) and ( 2 . 7 ) (3.8) -(1/40(1, + H h _ 6 ^ T / 2 ) ( 6 a ' 1 - H h a ^ ) ( 6 p V - H h P v ) x F D F ap u-v = -(l/4-)6°^ 6 P v F P F - ( l / 2 ) n h D T a P aB where T p i s the energy-momentum t e n s o r d e f i n e d by ( 3 . 5 ) . T h e r e f o r e , the d e s i r e d q u a n t i t y , the i n t e r a c t i o n L a g r a n g i a n , i s vrtiere the d i s t i n c t i o n between upper and lower i n d i c e s has been dropped s i n c e i s the Kronecker d e l t a . 2 7 4. The D e s c r i p t i o n o f the .Four Component N e u t r i n o F i e l d and I t s I n t e r a c t i o n w i t h the G r a v i t a t i o n a l F i e l d I n a f l a t space-time the L a g r a n g i a n f o r the ma s s l e s s f o u r - s p i n o r f i e l d cj)-the f o u r component n e u t r i n o f i e l d , when r e f e r r e d to r e c t i l i n e a r c o o r d i n a t e s , i s ( C o r s o n 1955) ( 4 . 1 ) / = ( 1 / 2 ) ( $ ( p Yp,ci) - $ Y P <D"jp) where (4.2) $ = cD1" and where the y m a t r i c e s a re 4x4 m a t r i c e s which s a t i s f y (4.3) Y P Y q + Y V = 2&P^ . C a l c u l a t i o n s are e a s i e r i f one uses the r e p r e s e n t a t i o n (4.4a) y P = 0 - i a p p 0 I O Y I 0 0 - I where. (4.4b) a 1 = " o l " ' ^ v. b - i 1 o " 1 o " 1 0_ i q 0 -1 0 1 The o n l y a d d i t i o n a l p r o p e r t i e s o f the Y m a t r i c e s which a re r e q u i r e d a re t h e f o l l o w i n g t r a c e theorems which f o l l o w from the e q u a t i o n (4.3) ( 4 . 5 ) T r ( T P Y q ) = 4 & p q T r ( Y P Y q Y r Y S ) = 4 ( 6 p q 6 r S - 6 p r & q s + 6 p S 6 q r ) . Under an i n f i n i t e s i m a l L o r e n t z t r a n s f o r m a t i o n w i t h 28 parameters co , cj) and if; t r a n s f o r m as f o l l o w s (4.6a) = ( 1 + ( l / 2 ) o j p q M p q ) ( J ) (4.6b) = $ ( 1 + (l/2)co H p ( 1) i r where (4.6c) I1 p q = ( l / 2 ) Y P Y q (4.6d) H p q = ( l / 2 ) Y q Y P . These e x p r e s s i o n s are r e q u i r e d l a t e r when g r a v i t a t i o n i s i n t r o d u c e d . The f i e l d e q u a t i o n s which f o l l o w from the L a g r a n g i a n ( 4 . 1 ) are ( 4 . 7 ) Y P c | ) . p = 0 , $ Y P = 0 f o r which s o l u t i o n s are (4.8) d>+ = u ( p ) e i ( p - x - ! E l t : ) , = v ( p ) e i ( p * x + l E l t } where u and v s a t i s f y ( 4 . 9 ) CY-P + i l E h V = 0 , ( Y - P - i | E | Y 4 ) v = 0 S o l u t i o n s f o r u and v are ( 4 . 1 0 ) u,(n) = 1 1 & 1 & *1 ^ 1 ' ^ 1 , u 2 ( n ) = 1 J2 , v2(n) 1 /2 ^2 r)2 where (4.11) n = p/[E| 29 and where (4.12) n t = ( l / / 2 ( l + n z ) ) MI = i f + r)2 w 2 = < T)" = (l//2(l-Kn z|)) 1 + n n + m x y n - m x y 1 + In. ( l / / 5 ( l + n z ) ) r," = ( l / / 2 ( l + | a z p ) -n + m x y 1 + 'n_ 1 + |ns i f n >y 0 2 i f n z < 0 i f n z>,0 i f n z < 0 The components and r ) 2 s a t i s f y (4.13) a.nn^- = T J 1 , a.nr) 2 = - r ) 2 and have t h e n o r m a l i z a t i o n t t (4.14) T ) 1 T ) 1 = n 2 r| 2 = 1. With t h i s n o r m a l i z a t i o n f o r rf-^ and r ) 2 , u r and v r have the n o r m a l i z a t i o n (4.15) u j ' u = v T v = 1 r r r r The s o l u t i o n s u and v can be c h a r a c t e r i z e d a c c o r d i n g t o r r the f o l l o w i n g scheme(Lurie 1968) (4.16a) hu^ = U-JL , h u 2 = - u 2 , hv-^ = v ^ , h v 2 = - v 2 (4.16b) y u x = - u 1 , Y u 2 = u 2 , y'v1 = v x , Y' v 2 = - v 2 30 5 where h, the h e l i c i t y o p e r a t o r , and y are (4.16c) h cf.n • 0 0 o «n Y5 - 0 - I - I 0 The s o l u t i o n s u r and v r have the f o l l o w i n g a d d i t i o n a l p r o p e r t i e s which are r e q u i r e d l a t e r (4.17a) v 1 ( - n ) = u 2 ( n ) (4.17b) v 2 ( - n ) = U l ( n ) (4 . 1 7 c ) 2 u (n) i W n ) = 2 v (-n) v (-n) = - i ( y n ) / 2 r = l r r r = l r r where (4.18) n = ( n , i ) . An a r b i t r a r y s o l u t i o n o f the f i e l d e q u a t i o n ( 4 . 7 ) can be expanded as (4 . 1 9 ) <1> = (1//V) 2 2 ( a r ( p ) u r ( p ) e I P X + b ' ^ p O u ^ p ) e " 1 ^ ) . P r When one makes the t r a n s i t i o n t o quantum f i e l d t h e o r y the a r ' ^ r ' a r> ^ r a r e i n t e r p r e t e d as o p e r a t o r s such t h a t the a r d e s t r o y n e u t r i n o s , the b d e s t r o y a n t i n e u t r i n o s , t h e a' c r e a t e n e u t r i n o s and the b 1 c r e a t e a n t i n e u t r i n o s . The symmetrized energy-momentum t e n s o r i s (4.20) T n = (1/4) ((]), y t|) - $ Y <|>.„ + <5, Y„<1> - cpy (|>, ) pq w w | q ' p Y v r p ^ | q v | p ' q ^'q^|p The system d e s c r i b e d by. the L a g r a n g i a n ( 4 . 1 ) i s i n v a r i a n t under the charge c o n j u g a t i o n o p e r a t i o n and under the space i n v e r s i o n o p e r a t i o n i n c o n t r a s t t o the system t o be d e s c r i b e d i n Chapter 5« 31 The g e n e r a l i z a t i o n of the L a g r a n g i a n (4-.1) t o g e n e r a l I" r e l a t i v i t y i s not as simple as i t was i n the p r e v i o u s two cases where t e n s o r f i e l d s were c o n s i d e r e d . T h i s i s because cl; t r a n s f o r m s a c c o r d i n g to a r e p r e s e n t a t i o n of the L o r e n t z group which cannot be extended t o the group of a l l l i n e a r t r a n s f o r m a t i o n s , which g e n e r a l c o o r d i n a t e t r a n s f o r m a t i o n s are when c o n s i d e r e d as changes of the n a t u r a l b a s i s ( t o be d e f i n e d s h o r t l y ) a t each p o i n t of the space-time m a n i f o l d (Weyl 1929). Thus a d e f i n i t i o n of a c o v a r i a n t d e r i v a t i v e of (J) r e q u i r e s s p e c i a l a t t e n t i o n . Fock(1929) a r r i v e d a t a s u i t a b l e d e f i n i t i o n by c o n s i d e r i n g 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 v e c t o r determined by cb. H i s method w i l l be a p p l i e d to the two component s p i n o r i n Chapter 5. I n t h i s c h a p t e r the method of Utiyama(l956) w i l l be used which i s s i m i l a r t o t h a t o f V/eyl(l93l). No m atter which method i s used, the v i e r b e i n f o r m a l i s m i s r e q u i r e d and so t h i s i s now d e s c r i b e d . At each p o i n t of the m a n i f o l d the s e t of v e c t o r s j e ^ which are tangent t o the c o o r d i n a t e l i n e s a t t h e g i v e n p o i n t span a v e c t o r space c a l l e d the tangent space. The s e t of v e c t o r s je ( i s c a l l e d the n a t u r a l b a s i s . I t i s not i n g e n e r a l o r t h o n o r m a l a t every p o i n t but i f i t i s , the space i s f l a t . C o n t r a v a r i a n t v e c t o r s ana more g e n e r a l l y t e n s o r s can be d e f i n e d v/ith r e s p e c t t o t h i s b a s i s and i t s t e n s o r p r o d u c t s . By the Gram-Schmidt p r o c e s s one can i n t r o d u c e an orthonormal b a s i s {e j a t each p o i n t which i s v a r i o u s l y c a l l e d a v i e r b e i n b a s i s , a t e t r a d or an o r t h o g o n a l quad-r u p l e . Thus, one can w r i t e ' 52 (4.21a) 5 p . f a p e a i and (4.21b) e = F p e s a a p where (4- 22) f a F p „ =6 a f a F q = 6 q From the d e f i n i t i o n ( I . l ) of the m e t r i c t e n s o r one o b t a i n s (4.25) g a p - ?VVp4 , 6 p q - f a p f P q B a p G i v e n a c o n t r a v a r i a n t v e c t o r A, i t s components m the n a t u r a l b a s i s are r e l a t e d t o i t s components [ S p j i n the v i e r b e i n b a s i s by (4.24) S p = F P a A a . For a c o v a r i a n t v e c t o r one has ( 4 . 2 5 ) S = f a A v p p a and i n p a r t i c u l a r i f (4.26) A a = V 6 x a one may c a l l ( 4 . 2 7 ) d = f a A x p p a the v i e r b e i n p a r t i a l d e r i v a t i v e . F o r a cu r v e d m a n i f o l d t h e r e does not e x i s t a s e t o f c o o r d i n a t e s y p which would a l l o w one t o .write (4.28) d = d/fcyP . 33 I f a t each p o i n t of the m a n i f o l d a v i e r b e i n frame i s d e f i n e d , so t h a t one has a s e t of v i e r b e i n frames, one can o b t a i n a new s e t by a p p l y i n g a L o r e n t z t r a n s f o r m a t i o n t o the o r i g i n a l s e t . The new s e t of v i e r b e i n frames i s j u s t as good a s e t of b a s i s frames as t h e o l d s e t , even when the L o r e n t z t r a n s f o r m a t i o n a t each p o i n t of the m a n i f o l d depends on the p o i n t . Under such a r e o r i e n t a t i o n o f frames the components of a v e c t o r t r a n s f o r m a c c o r d i n g t o a L o r e n t z t r a n s f o r m a t i o n . However, the q u a n t i t y (4 . 2 9 ) d 5 q = f a b l q /ax a which may be c a l l e d t he v i e r b e i n p a r t i a l d e r i v a t i v e of the v e c t o r A does n o t t r a n s f o r m as a mixed t e n s o r o f rank two under c o o r d i n a t e dependent L o r e n t z t r a n s f o r m a t i o n s . One can, however d e f i n e a new d e r i v a t i v e D such t h a t D S q » P P does t r a n s f o r m as a second rank t e n s o r under c o o r d i n a t e dependent L o r e n t z t r a n s f o r m a t i o n s . T h i s i s done by adding the term(Utiyama 1956) (4 . 3 0 ) - f a B r S M / 2 p a r s ' t o d t o o b t a i n D . The M „ are the m a t r i c e s which r e p -p p r s * r e s e n t the i n f i n i t e s i m a l L o r e n t z t r a n s f o r m a t i o n s f o r the v e c t o r A, and the B a r s are g i v e n by (4 . 3 1 ) B r s = 6 S p fU ( F r , - F r R P P ) s ' a p ^ u j a 3 ua ' = - 6 S p 6 r q F* Y (Y =6 ' f a F s fP ^  0 0 * a Ypqt> ^ p q t b q s 1 p;B * a 1 t ; where the"!" 1^ a re t h e C h r i s t o f f e l symbols and the Ypq^ a r e t h e R i c c i c o e f f i c i e n t s o f r o t a t i o n . The new q u a n t i t y 34 (4.32) D / = f°V ( c 3 l V o x a - ( 1 / 2 ) B / S ( K J j I*) i s r e l a t e d t o t h e u s u a l c o v a r i a n t d e r i v a t i v e by (4-.33) A a . p = f a Q I * ( D l * ) . ) P 4. P P Hence, D can be c a l l e d the v i e r b e i n c o v a r i a n t d e r i v a t i v e . I n summary, V^lfi- i s a q u a n t i t y which t r a n s f o r m s as a second rank mixed t e n s o r under c o o r d i n a t e dependent L o r e n t z t r a n s f o r m a t i o n s , and i s an i n v a r i a n t under g e n e r a l c o o r -d i n a t e t r a n s f o r m a t i o n s . Utiyama has shown t h a t i f <\> i s a s p i n o r , t h e n under c o o r d i n a t e dependent L o r e n t z t r a n s f o r m a t i o n s t h e q u a n t i t y Dp(J) d e f i n e d by (4.34-) Dpcl) = f a p(dci>/6x a - ( l / 2 ) B a r s M r s cl)) , where the M a r e t h e m a t r i c e s which r e p r e s e n t the i n f i n i t e s i m a l L o r e n t z t r a n s f o r m a t i o n s of ( J ) , t r a n s f o r m s as a c o v a r i a n t v e c t o r and as a s p i n o r s i m u l t a n e o u s l y . The passage t o g e n e r a l r e l a t i v i t y can now be made f o r a s p i n o r f i e l d . One seeks a L a g r a n g i a n which 1) t r a n s f o r m s as a s c a l a r d e n s i t y under g e n e r a l c o o r d i n a t e t r a n s f o r m a t i o n s , 2) i s i n v a r i a n t under c o o r d i n a t e dependent r e o r i e n t a t i o n s of t he v i e r b e i n frames, and 3) reduces t o the f l a t space-time L a g r a n g i a n when no g r a v i t a t i o n a l f i e l d i s p r e s e n t and r e c t i l i n e a r c o o r d i n a t e s are used. Such a L a g r a n g i a n d e n s i t y can be o b t a i n e d from t h e f l a t space-time form by 35 1) r e p l a c i n g the p a r t i a l d e r i v a t i v e by the v i e r b e i n c o v a r i a n t d e r i v a t i v e D cb, ,^ P * 2) r e p l a c i n g the m e t r i c 6 by the m e t r i c g , and 3) i n t r o d u c i n g a f a c t o r /g where g i s the d eterminant of the m e t r i c t e n s o r . When the above r u l e s are a p p l i e d t o the L a g r a n g i a n (4.1) one o b t a i n s ( 4 . 3 5 ) / ' = y g ( ( d p $ ) Y P c b - 45Y P(d pcb ) )/2 -•§<P 1 % B a r S ( H r s Y P - Y P M r s)cb/2 . When the g r a v i t a t i o n a l f i e l d i s weak one w r i t e s ( 4 . 3 6 ) F p = 6 p + n p , f a = 6 a + . a a ' a ' p p ^ p b u t the o r t h o g o n a l i t y c o n d i t i o n s (4 . 2 2 ) imply (4 . 3 7 ) ^ p = - 1 1 % and the c o n d i t i o n s (4 . 2 3 ) imply (4 . 3 8 ) n p q + n q p = x h p q . The system i s i n v a r i a n t under change of v i e r b e i n frame so one may choose (4 .39) n p q - H h p q / 2 . The term B r s becomes a (4.40) . H6PP 6 ^ ( h a a | p - h a p | a ) • I t i s shown i n Appendix A t h a t when (4.40) and (4 . 6 c,d) are s u b s t i t u t e d i n t o the L a g r a n g i a n (4 . 3 5 ) the second term on the r i g h t i n e q u a t i o n (4-. 3 5 ) v a n i s h e s as p o i n t e d out by V l a d i m i r o v ( 1 9 6 4 ) c T h e r e f o r e , (4.41) • / ' = ( 1 / 2 ) ( 1 + K h a p & a p / 2 ) { ( 6 a p - nap)!P|a Y P 4 -* Y P ( 6 - p - ^ , 0 ; ] = (l/2)(jP|p y P <\> - $ Y P 4>|p) -( l / 2 ) n a p ( $ | a Y P (|> - <P Y P <D|a) + ( n / 4 ) h a p 6 a P (jP|p y P t|) - ip Y P *| p) The t h i r d term v a n i s h e s when the f i e l d e q u a t i o n s are s a t i s f i e d . T h e r e f o r e , one o b t a i n s < 4 - 4 2 ) / i n t = -ClA)xh a p($ | a Y P t - l Y f *, a) where the symmetry of h a ^ has been used t o w r i t e the l a s t l i n e and where the d i s t i n c t i o n between upper and lower i n d i c e s has been dropped s i n c e 6 i s the Kron e c k e r d e l t a . 5 . The D e s c r i p t i o n of the Two Component N e u t r i n o F i e l d and I t s I n t e r a c t i o n w i t h the G r a v i t a t i o n a l F i e l d I n a f l a t space-time the L a g r a n g i a n d e n s i t y f o r the m a s s l e s s two component s p i n o r f i e l d ¥ - t h e two component n e u t r i n o f i e l d , when r e f e r r e d t o r e c t i l i n e a r c o o r d i n a t e s , i s ( C o r s o n 1 9 5 3 ) ( 5 . 1 ) / = ( i / 2 ) 0 F + a p Y| p - | p a p Y) where the a p are the fundamental s p i n - t e n s o r s . A s u i t a b l e r e p r e s e n t a t i o n i s (5.2) a1 = ( o 1 A B ) = b l 0 - i , a 3 . " i o" 1 0 i 0 0 - 1 These s p i n - t e n s o r s have c o n t r a v a r i a n t s p i n o r i n d i c e s . The c o r r e s p o n d i n g s p i n - t e n s o r s w i t h c o v a r i a n t s p i n o r i n d i c e s are 1CD< ( 5 . 5 ) d1 = ( a 1 ^ ) = ( e AC eBD ° ') 0 1 0 1 0 - 1 - 1 0 1 0 1 o. 0 -1 -1 0 -2 a = b - i , a 5 - - 1 6 , -s = i o " i 0 0 I 0 i The a and o s a t i s f y (5.4a) 5 p T o q + a q T a p = - 2 6 p q (5.4b) a p a q T + a q a p T = - 2 6 p q from which f o l l o v / the t r a c e theorems 38 (5.5a) T r ( a p T a q ) = T r ( a p a q T ) = - 2 6 p q (5.5b) T r ( a p T a q a p T a s ) = 2 ( 6 p q 6 r s " I 6 p r 6 q s + 5 p s 6 q r ) Under an i n f i n i t e s i m a l L o r e n t z t r a n s f o r m a t i o n w i t h parameters eo p q (5.6a) = ( I + ( l / 2 ) c o p q n p q ) ^ (5.6b) •ft-*^«t=>?i'(i + ( l / 2 ) a ) p q H p q ) where (5.6c) MPQ = - ( l / 2 ) a p T a q (5.6d) H p q = - ( l / 2 ) a q a p T . The f i e l d e q u a t i o n s which f o l l o w from the L a g r a n g i a n (5.1) are (5.7) a p = 0 , Y f | p a p = 0 f o r which s o l u t i o n s a re (5.8) Y + = u ( p ) e i ( p * x - l E l t } , Y_ = v ( p ) e i ( p , x + I 3 3!*) where u and v s a t i s f y (5.9) (a.p - |E|)u = 0 , (a.p + |E| )v =0 . S o l u t i o n s f o r u and v a r e (5.10) u(n) = •r) 1 , v ( n ) = r)2 where "n i s g i v e n by (4-. 11) and where and are g i v e n by (4-. 12). They are e i g e n v e c t o r s of the h e l i c i t y o p e r a t o r a-n 3 9 (5.11) a.nr) 1 = , a-nr) 2 = - T ) 2 . The s o l u t i o n s u and v have the f o l l o w i n g a d d i t i o n a l p r o p e r t i e s which are r e q u i r e d l a t e r (5.12a) u(n)u 1'(n) = -(na T ) / 2 ( 5 . 1 2 b ) v ( n ) v ' ( n ) = - ( n a ) / 2 (5.15) v(-n) = u(n) where n i s d e f i n e d b y , ( 4 . 1 8 ) . An a r b i t r a r y s o l u t i o n of the f i e l d e q u a t i o n ( 5 . 7 ) can be expanded as (5.14) ¥ = (1//V) S ( a ( p ) u ( p ) e i P x + b t ( p ) u ( p ) e " i P x ) . P When the t r a n s i t i o n t o quantum f i e l d t h e o r y i s made, a(p) d e s t r o y s a n t i n e u t r i n o s , a'(p) c r e a t e s a n t i n e u t r i n o s , b (p) d e s t r o y s n e u t r i n o s and b (p) c r e a t e s n e u t r i n o s . Here, the u s u a l c o n v e n t i o n I s f o l l o w e d — n e u t r i n o s have n e g a t i v e h e l i c i t y and a n t i n e u t r i n o s have p o s i t i v e h e l i c i t y . The symmetrized energy-momentum t e n s o r i s (5.15) T = (i/4)(¥ Ya ¥. + ¥ T o ¥, - a ? ! a The system d e s c r i b e d by the L a g r a n g i a n (5.1) i s not i n v a r i a n t under the space r e f l e c t i o n and charge c o n j u g a t i o n o p e r a t i o n s b u t I t i s i n v a r i a n t under t h e combined space r e f l e c t i o n and charge c o n j u g a t i o n o p e r a t i o n s . The L a g r a n g i a n f o r the two component n e u t r i n o i n g e n e r a l r e l a t i v i t y , which s a t i s f i e s t he c r i t e r i a l a i d down i n Chapter 4, can be found by u s i n g Utiyama's 4-0 p r e s c r i p t i o n where the m a t r i c e s r e p r e s e n t i n g the i n f i n i t -e s i m a l L o r e n t z t r a n s f o r m a t i o n s are g i v e n by ( 5.6c,d). However, i n s t e a d o f u s i n g t h i s method, Fock's method(1929) of g e n e r a l i z i n g the D i r a c e q u a t i o n w i l l be used. Fock's method runs as f o l l o w s . From V f , a p and Y one can form a c o n t r a v a r i a n t (under L o r e n t z t r a n s f o r m a t i o n s ) v e c t o r A w i t h v i e r b e i n components (5.16) A p = ^']'o^)/2 . T h i s v e c t o r i s d e f i n e d a t each p o i n t o f the space-time m a n i f o l d . I f one d i s p l a c e s the v e c t o r A an amount dx^ by p a r a l l e l d i s p l a c e m e n t , the change i n t h e v i e r b e i n components i s (5.17) 6 ( A P ) = - 6 P Q Y q r s A r F s ^ dx^ . F o l l o w i n g Fock, one w r i t e s , from (5.16) (5.18) 6 ( A p ) = (1/2) ( 6 ^ ) 0 % + ¥ T o P ( 6 ¥ ) and assumes t h a t under p a r a l l e l d i s p l a c e m e n t (5.19) '6f =-C Y F s dx^ , = -Y T Gl F S m dx^ where the C are t o be determined. By s u b s t i t u t i n g (5.19) s i n t o (5.18) and e q u a t i n g the r e s u l t t o ( 5 .17 ) , one o b t a i n s (5.20) Y t ( G t a p + a pG )Y F s dx^ = 6 P Q y Y + a rY F s dx^ or (5.2D 0 * aP + aP 0s - 6P1 Y a r 4 - 1 f o r which s o l u t i o n s are ( 5.22a) C s = - ( i / 4 ) < o P T a q ) Y (5.22b) C* = - ( 1 / 4 ) ( o q o P T ) Y pqs pqs One now d e f i n e s t h e v i e r b e i n c o v a r i a n t d e r i v a t i v e of t h e s p i n o r Y t o be T h i s i s a s a t i s f a c t o r y d e f i n i t i o n because under c o o r d i n a t e t r a n s f o r m s as a v e c t o r and a s p i n o r s i m u l t a n e o u s l y . The d e f i n i t i o n ( 5 . 2 5 ) i s the same as t h a t o b t a i n e d from Utiyama's p r o c e d u r e . I n g e n e r a l r e l a t i v i t y , a L a g r a n g i a n which i s s u i t a b l e f o r d e s c r i b i n g the two component n e u t r i n o can be o b t a i n e d from the L a g r a n g i a n ( 5 - 1 ) by a p p l y i n g the r u l e s l a i d down i n Chapter 4 . Thus one r e p l a c e s Y. and V , v/ith D ¥ and P P P t and D ¥ t o o b t a i n I n t h e weak f i e l d a p p r o x i m a t i o n the second term can be shown t o v a n i s h by e x a c t l y the same procedure as g i v e n i n Appendix A f o r the f o u r component n e u t r i n o . The f i r s t term becomes, w i t h t h e h e l p of ( 4 . 3 6 ) and ( 4 . 3 9 ) » ( 5 . 2 3 ) D T = f f I - C ¥ . w P P I cx p L o r e n t z t r a n s f o r m a t i o n s i t can be shown t h a t D ¥ P P ( 5 . 2 5 ) / ' = / g ( i / 2 ) { V a p ( D p Y ) - ( D p Y f ) a p ^} = a p ( d p ¥ ) - ( d p Y T ) a p Yj y g ( i / 2 ) Y T ( C ^ a p - a p C p ) ¥ 42 (5.26) / ' = ( i / 2 ) ( T + a p ¥| p - Y| p a p Y) + ( i A ) H h a a ( 1 ? t c p ¥ | p - ¥Jp o p ¥ ) -( i / 4 ) H h a p ( ¥ T a p - Y| a a p . The. second term v a n i s h e s when the f i e l d e q u a t i o n s are s a t i s f i e d so t h a t where T R i s the energy-momentum t e n s o r d e f i n e d by (5.15). The symmetry of h a ^ has been used t o o b t a i n (5.27) from (5.26) and the d i s t i n c t i o n between upper and lower i n d i c e s has been dropped s i n c e 6 i s the Kronecker d e l t a . 43 6. The M a t r i x Elements f o r the C o l l i s i o n s I n the i n t e r a c t i o n p i c t u r e (6.1) i l _ | p / ( t ) > = H T ( t ) |0-(t)> . bt One l e t s (6.2) |i> = |0(-«>)> and l o o k s f o r a s o l u t i o n of (6.1) such t h a t (6.3) |0( + «>)> = S|i> Such a s o l u t i o n i s ( M u i r h e a d 1965) (6.4) S = J Q ( - i ) n / n l / - - / d x n P[ ^ ( x ^ . - . Z n t( oo = S S n=0 n where P i s the c h r o n o l o g i c a l p r o d u c t . One i s i n t e r e s t e d i n t h e m a t r i x element < f | S | i > where |f)> i s some p a r t i c u l a r f i n a l s t a t e . The c o l l i s i o n s under c o n s i d e r a t i o n a r e t h o s e which take p l a c e t h r o u g h t h e c r e a t i o n and a n n i h i l a t i o n o f a s i n g l e g r a v i t o n . The lowest o r d e r c o n t r i b u t i o n t o such events i s c o n t a i n e d i n S 2 which i s Z (6.5) S 2 = ( - 1 / 2 ^ dx dy P[ « { n t ( x ) *d i n t(y) The q u a n t i t y P j can be decomposed by means of Wick's Theorem. The term i n the d e c o m p o s i t i o n which c o n t r i b u t e s t o the afo r e m e n t i o n e d c o l l i s i o n s i s (6.6) I = (H 2/4) h (X) h p ( y ) N L V x ) V y ) . 44 where h (x) h R ( y ) i s the g r a v i t o n p r o p a g a t o r , g i v e n by (1 .73)> N i s t h e normal p r o d u c t and T i s the energy-momentum t e n s o r f o r the two f i e l d s which d e s c r i b e the p a r t i c l e s b e i n g s c a t t e r e d — i f t he s c a t t e r i n g o f a s c a l a r p a r t i c l e and a photon i s under c o n s i d e r a t i o n t h e n T i s the sum of (2.4) and (3.!?)• The q u a n t i t y I , g i v e n by ( 6 . 6 ) , w i l l be r e f e r r e d t o as the i n t e g r a n d o f the s c a t t e r i n g m a t r i x . One i s now i n a p o s i t i o n t o c a l c u l a t e the m a t r i x elements f o r t he s c a t t e r i n g problems l i s t e d i n t h e I n t r o d u c t i o n . Only the h i g h l i g h t s of the c a l c u l a t i o n s a re g i v e n i n t h i s Chapter. T h i s i s because any one c a l c x i l a t i o n can s e r v e as an example, and i n Appendix F the d e t a i l e d c a l c u l a t i o n f o r the s c a t t e r i n g o f two massive s c a l a r p a r t i c l e s i s g i v e n . The l a t t e r c a l c u l a t i o n can ser v e as the example f o r the r e s u l t s p r e s e n t e d i n t h i s Chapter. B r i e f l y , however, t o work out a m a t r i x element-one s u b s t i t u t e s the a p p r o p r i a t e T i n t o t he i n t e g r a n d I and expands the p r o d u c t T (x)T R ( y ) , t h u s o b t a i n i n g an expa n s i o n o f the i n t e g r a n d I . The Feynman graphs then f o l l o w from the exp a n s i o n o f I . The m a t r i x element, s, i s computed by p e r f o r m i n g the i n t e g r a t i o n s i n d i c a t e d i n (6.5) and e v a l u a t i n g the s c a l a r p r o d u c t <T|S 2|i>. In the f o l l o w i n g pages the i n d i v i d u a l s c a t t e r i n g problems are c o n s i d e r e d . F o r each, the r e l e v a n t ' energy-momentum t e n s o r , the ex p a n s i o n of the i n t e g r a n d I , the Feynman graphs and the m a t r i x elements a r e g i v e n . The p o s i t i v e and n e g a t i v e f r e q u e n c y p a r t s o f the f i e l d s , 4-5 t h a t i s , t h o s e p a r t s which c o n t a i n e and e r e s p e c t i v e l y , are denoted by a f f i x i n g a + or - t o the symbols f o r t h e f i e l d s . The i n i t i a l momenta of the two i n t e r a c t i n g p a r t i c l e s a re denoted by p and q and the f i n a l momenta are denoted by p' and q'. The center-of-momentum frame i s used so t h a t (6.7a) p = -q , p' = -q' . The p a r t i c l e e n e r g i e s a re a l l e q u a l and t h i s common energy ^ 2 _i.P i s denoted by p . The q u a n t i t i e s k and s are d e f i n e d by (6.7b) k 2 = |t>-p'| 2 = 4 p o 2 s i n 2 ( 9 / 2 ) -n2 !-»•-» i 1 2 .. 2 2 s = |p+p'| = 4- p o cos where 0 i s the s c a t t e r i n g a n g l e : P e 'A ' I n the Feynman graphs a dashed l i n e r e p r e s e n t s a g r a v i t o n and a s o l i d l i n e r e p r e s e n t s a s c a l a r p a r t i c l e , photon or n e u t r i n o . Where n e c e s s a r y , s c a l a r , photon, and n e u t r i n o l i n e s are d i s t i n g u i s h e d by j u x t a p o s i n g an n, y or v r e s p e c t i v e l y . F i n a l l y , : : denotes normal o r d e r i n g . 4 6 a) The M a t r i x Elements f o r S c a l a r - S c a l a r S c a t t e r i n g The energy-momentum t e n s o r f o r two m a s s l e s s , H e r m i t i a n s c a l a r f i e l d s $ and % i s (s.e) - \jX J | v - 6^v 0 | T / 2 ) : . When t h i s i s s u b s t i t u t e d i n t o I, d e f i n e d by ( 6 . 6 ) , one o b t a i n s , a f t e r a l g e b r a i c m a n i p u l a t i o n ( 6 . 9 ) i = ( * 2 A ) t ^ W i f c p C y ) N • <2*2> f a p ( y ) * V x ) V y ) . (*y*> *V*> Vy) Vy) - ( W 2 ) *V*> M y ) ^ V y ) "< V 2 5 ^ | T ( X ) *V x ) ^Vy> V |p (7 ) + < V V 0 ^ x ( x ) '*Vx) V y ) * i l C y )  +^V (X) *l« (7 ) ^ l P ( y ) -<6a{/2) ^V(X) ^V y ) ^"«(X) "(^v/ 2) a^<y> V X ) r ip ( y ) + < V 6ap/4) ^V y ) ^ i n C x ) ^ ( y ) + V X ) V y ) *|v (x) -<6ap/2> *'V(X) ^> ( y > ^ "^X; ' " < V / 2 ) ^ a ( y ) ^ |P ( y ) ^ W + < V W 4 ) i^n(x) ^ ( y ) M y ) • 47 Upon i n s p e c t i o n o f ( 6 . 9 ) one sees t h a t : l ) the f i f t h t o n i n t h terms c o n t r i b u t e t o a p r o c e s s whose graph i s 2) the l a s t f o u r terms c o n t r i b u t e t o a p r o c e s s whose graph i s 3) the f i r s t f o u r terms c o n t r i b u t e t o a p r o c e s s whose graph i s The c o r r e s p o n d i n g m a t r i x elements f o l l o w from ( 6 . 9 ) . They a r e , i n the center-of-momentum frame, (6.10a) s 1 = i ( H 2 / V 2 ) ( 2 7 t ) 4 6 ( p + q - p ' - q , ) [ c o s 2 ( 6 / 2 ) / s i n 2 ( 0 / 2 ) j / 2 (6.10b) s 2 = i ( H 2 / V 2 ) ( 2 T x ) 4 6 ( p + q - p ' - q ' ) { s i n 2 ( 9 / 2 ) / c o s 2 ( 0 / 2 ) j / 2 (6.10c) s 3 = i ( H 2/V 2 ) ( 2 7 i ; ) / 4 " 6(p+q-p»-q'){sin 2oJ/8 where p and q denote t h e i n i t i a l momenta o f the two p a r t i c l e s and p' and q 1 denote the f i n a l momenta. 48 b) The M a t r i x Elements f o r S c a l a r - P h o t o n S c a t t e r i n g The energy-momentum t e n s o r f o r the combined s c a l a r f i e l d 0 and photon f i e l d A i s r (6.11) T = :-ef. d, + 6 <zf. cfi /2 - F F + 6 F D F D/4 v u , i i y [xv w k r |v [xv ^  I T 7 (xa va [xv aB aB' When t h i s i s s u b s t i t u t e d i n t o the i n t e g r a n d I , one o b t a i n s (6.12) I = (HVO h^Cx) h a p ( y ) N T (x) T n ( y ) [xv ' a p w _ Y HLX A y | a * ] v A Y | B * A Y | a ^|v A i | -^"|[x A a | Y * ] v A Y | B + <rjn A a | Y A P | Y ^ 6 a B / 2 ^ | [ x A Y > + ^aP/2)^V ^ l T ^ ^ - c^ v/2)^ T A ; | O C tT\x A ^ j p + ( 6 ^ v / 2 ) / | T A ; | A ^ T A - | Y ^ 6 [ X V / 2 ^ | T A a | Y ^ A Y | P ~ ( 6 u v / 2 ^ + | x A a | y * ] T A p | Y I n (-6.12) ci i s a f u n c t i o n of x and A i s a f u n c t i o n of y. r A l l t h e terms i n (6.12) c o n t r i b u t e t o a p r o c e s s whose graph i s < The c o r r e s p o n d i n g m a t r i x element i s , i n the c e n t e r - o f -momentum frame and i n the Coulomb gauge, (6.13) s = i ( * 2 / V 2 ) ( 2 T 0 4 6(p+q-p ,-q') ( l / k 2 ) * | e . e ' * ( p o 2 + p.p*') - ( p . e ' * ) ( e - p ' ) j . 4 9 where p and p' denote the i n i t i a l and f i n a l momenta of the s c a l a r p a r t i c l e , q and q' denote the i n i t i a l and f i n a l momenta of the photon, and e and e 1 denote the i n i t i a l and f i n a l p o l a r i z a t i o n v e c t o r s o f the photon. The e q u a t i o n (6.13) has a s i m p l e r form i f expressed i n a d i f f e r e n t b a s i s . I f "e^ i s a u n i t v e c t o r p e r p e n d i c u l a r to t he p l a n e determined by "q and~q', t h e n (6.14) "e 2 = q ^ e 1 / p o i s a second u n i t v e c t o r which t o g e t h e r w i t h "e-^  can be used as. a b a s i s f o r the space spanned by the p o l a r i z a t i o n v e c t o r s . T Thus, e can be ex p r e s s e d as (6.15) e = a-,e In a b a s i s e e d e f i n e d by (6.16) e + (e-1 + iep)//"2 , e = (e, - ie*p)//*2 one has (6.17) e + a e where (6.18) a + i a 2 ) / / 2 ~ . S i m i l a r l y one can w r i t e , f o r the s c a t t e r e d photon, (6.19) e' = 5 0 where ( 6 . 2 0 ) e.J = e 1 , e 2 = ^' xei/P 0 • I f one introduces ( 6 . 2 1 ) = (ej[ + i e 2 ) / / 2 , e^ = (ej_ - i e 2 ) / / 2 = (a{ - i a 2 ) / / 2 s a_^  = (a^ + i a 2 ) / / 2 then e' becomes ( 6 . 2 2 ) e' = a j e j + a _ ^ . In the basis |e +, e , e|, the matrix element ( 6 . 1 3 ) i s ( 6 . 2 3 ) s = i ( H 2 / V 2 ) ( 2 n - ) 4 6(p+q-p'-q') (a +a.J* + a_a^*)x { c o s 2 ( 9 / 2 ) / s i n 2 ( e / 2 ) j / 2 . • 5 1 c) The M a t r i x Element f o r S c a l a r - F o u r Component N e u t r i n o S c a t t e r i n g The energy-momentum t e n s o r f o r the combined s c a l a r f i e l d 0 and f o u r component n e u t r i n o f i e l d c | ) i s (6 . 2 1 ) T ^ ( x ) =:-^,/ l v *vtv/2 + Y v * -When t h i s i s s u b s t i t u t e d i n t o the i n t e g r a n d I , one o b t a i n s (6 . 2 2 ) I - ( H 2 A ) h ^ W j i p ( , ) H [ V x ) V y ) ] - (K 2/2) h ^ O O j a p ( y ) X \fy * + Y a 1 > ^"jv - *\ * ] a ^ p * " f\v In (6 . 2 2 ) 0 i s a f u n c t i o n o f x and i s a f u n c t i o n o f y. A l l t he terms i n ( 6 . 2 2 ) c o n t r i b u t e t o a p r o c e s s whose graph i s The c o r r e s p o n d i n g m a t r i x element i s , i n t h e c e n t e r - o f -momentum frame, (6 . 2 3 ) ,s = i(H 2/V 2)(27T;) 4 6(p+q-p' -q« ) ( 1 / 2 ) ( l / k 2 ) ( p • + . 5 p 0 ) n r , ( p ' ) Y4. u r(p) where q and q 1 denote the i n i t i a l and f i n a l momenta of the s c a l a r p a r t i c l e , p and p' denote the i n i t i a l and f i n a l 52 momenta of the n e u t r i n o , and u and u , are the i n i t i a l and ' r r f i n a l h e l i c i t y s t a t e s of the n e u t r i n o . 53 d) The M a t r i x Element f o r Scalar-Two Component N e u t r i n o S c a t t e r i n g The energy-momentum t e n s o r f o r the combined s c a l a r f i e l d $ and two component n e u t r i n o f i e l d ¥ i s (6.24) T(iv(x) =:-^/w + V ' V l * / 2 + iC'1,, av * -When t h i s i s s u b s t i t u t e d i n t o the i n t e g r a n d I , one o b t a i n s • (x) T R ( (6.25) I = (H 2/4) h ^ ( x ) h a p ( y ) N T M l i ( x ) „ R ( y ) N " i ( H 2 / 2 ) ^ v ( x ) h a p ( y ) x V | , * * " *a Y |P - ^ Y l a ° P ¥ + I n (6.25) $ i s a f u n c t i o n o f x and ¥ i s a f u n c t i o n of y. A l l t he terms i n (6.25) c o n t r i b u t e t o a p r o c e s s whose graph i s The c o r r e s p o n d i n g m a t r i x element I s , i n t h e center-of-momentum frame, (6.26) s = -i(H 2/V 2)(2n) 4 a t p + q - p ' - q O ^ X l A ^ P - p ' + 3p02) u f ( p ) u(p') where q and q 1 denote the i n i t i a l and f i n a l momenta of the s c a l a r p a r t i c l e , and p and p' denote the i n i t i a l and f i n a l momenta of the n e u t r i n o . 5 4 e) The M a t r i x -Element f o r Photon-Four Component N e u t r i n o S c a t t e r i n g The energy-momentum t e n s o r f o r the combined photon f i e l d A and f o u r component n e u t r i n o f i e l d cb i s r (6.2?) T (x) =:-F F + 6 F Q F D + (cp > y,. $ - T> Y> <!> i )/2l When t h i s i s s u b s t i t u t e d i n t o I , one o b t a i n s (6.28) I = (H 2/4) l j ^ v ( x ) h a ( 3 ( y ) N T (x) T D ( y ) [iv J a p w _ - ( * 2 / 2 > V W V a p ( ^ x N * T(i *|v A x | a A x j p " * V *'|v A x | a A P | T . - i + Y^ cD"jv A ^ T A ; 1 R + t Y ^ ] v A + j T A-JT -C6 a p/2 ) I P + Y ( a 0 ] v < h + ( 6 a p / 2 ) $ + ^ ^ A ^ A~ | T Y v *" A T j a A x | p + V*~ A ^ ) a A"|T V * ~ A ; 1 T A ; 1 P - ^ Y V * - A ; J T A-J T + ^ 6 a P / 2 H V Y v *" A x | n A x | n " ( 6 a p / 2 ) ^ V Y v *" < | r) A n j x j I n (6.28) cb i s a f u n c t i o n of x and A i s a f u n c t i o n o f y. r A l l t h e terms i n (6.28) c o n t r i b u t e t o 'a; p r o c e s s .whose graph i s The c o r r e s p o n d i n g m a t r i x element i s , i n the c e n t e r - o f -momentum frame and i n t h e Coulomb gauge, 5 5 (6 . 2 9 ) s = - ( H 2 / V 2 ) ( 2 7 r ) ' l 6 ( p + q - p ' - q ' ) ( l / 8 p 0 ) ( l / k 2 ) e.. e!_*x ^ , ( p ' ) (Cyq') + ( Y q ) ) [ 6 i d ( - p . p ' - 3 p 0 2 ) + q.q'. where q and q 1 denote the i n i t i a l and f i n a l momenta of the photon, e and e' denote the i n i t i a l and f i n a l p o l a r i z a t i o n v e c t o r s of the photon, p and p' denote the i n i t i a l and f i n a l momenta of t h e n e u t r i n o , and u and u , denote the ' r r ' i n i t i a l and f i n a l h e l i c i t y s t a t e s o f the n e u t r i n o . 56 f ) The M a t r i x Element f o r Photon-Two Component N e u t r i n o S c a t t e r i n g The energy-momentum t e n s o r f o r the combined photon f i e l d A and two component n e u t r i n o f i e l d \' i s (6.50) T (x) = ;-F P +6 F DF D + i ^ t a Y - T f a Y . )/2 I When t h i s i s s u b s t i t u t e d i n t o I , one o b t a i n s (6.31) I = ( H 2 A ) h ^ ( x ) h a p ( y ) Nfl^Cx) T a p ( y ) - i(H 2 /2 ) h ( X) h R ( y ) x N j Y T ~ a At, A ( u [ i |v x j a T | p " ¥' a u A x | a A p l x <V *|v Aa|x A " | P + * * " % *}v Aa|r A i | x -(6 A R/2)Y F- o^ Y+Y A ^ A ; h + (6 A R /2)V i- YJ V A " ^ -Y 1" a Y + A + |n % 1 *r|a Axjp + ^  av YA T | a P |T +YI~ a Y + A + v •a|T A T | p ~ ^ | JJL a v ^ A a | T A P|T + (6^ R/2)Y T- o Y"1" A r + » + + , + 'ap'*' 1 |u. u v 1 Ax|n * T | T ] - ( 5 a p / 2 ) T V ^  ¥" ^  h A ^ 1 T I n (6.31) Y i s a f u n c t i o n o f x and A i s a f u n c t i o n o f y. A l l t he terms i n (6.31) c o n t r i b u t e t o a p r o c e s s whose graph i s \ The c o r r e s p o n d i n g m a t r i x element i s , i n the c e n t e r - o f -momentum frame and i n the Coulomb gauge, 5 7 ( 6 . 3 2 ) s = -i(H 2/V 2)(27i) 4 6(p+q-p'-q')(l/8p 0)(lA 2) e!_* o i d(-p-p' - 3P Q ) + u'(p){((aq') + (aq)) q i q j ] + ( a d q i + V ^ ) ( 4 - p 0 2 ) ] u ( P ' ) where q and q 1 denote the i n i t i a l and f i n a l momenta of the photon, "e and ~e' denote the i n i t i a l and f i n a l polarization vectors of the photon and p.and p 1 denote the i n i t i a l and f i n a l momenta of the neutrino. 58 g) The M a t r i x Elements f o r Four Component N e u t r i n o - F o u r Component N e u t r i n o S c a t t e r i n g The energy-momentum t e n s o r f o r two, f o u r component n e u t r i n o f i e l d s , cj> and i s (6.53) T ^ x ) - (1/2) I | ;(|. Y y I - | Y y l{v) I 1 X X 1 When t h i s i s s u b s t i t u t e d i n t o I , one o b t a i n s (6.54) I = U 2 / 4 ) b t i u ( x ) h a p(y ) N V x ) V y ) . Ijcp ( x ) Y ) i <l>"j v (x) (p ( y ) y a <l>"jp(y) ( x ) Y < K „ ( x ) $ , R ( y ) Y „ < l ~ ( y ) - c p j v ( x ) Y ^ <|T(x) t|> ( y ) Y A <l> | p ( y ) +cp, ( x ) Y ^ <|)"(x) cp j p ( y ) Y A t|)"(y) y + - - t -+cp ( x ) Y ^ <J>~[v(x) STCy) Y A «i>"jp(y) -<j> ( x ) Y ^ *"jv(x) cp |p(y) Y A <\> (y) -$"| v(x) Y ^ <T(x) <T(y) Y A <l>"jp(y) Upon i n s p e c t i o n o f (6.34) one sees t h a t the f i r s t f o u r terms c o n t r i b u t e t o a p r o c e s s whose graph i s (0 and t h a t t he l a s t f o u r terms c o n t r i b u t e t o a p r o c e s s 59 whose graph i s The c o r r e s p o n d i n g m a t r i x elements a r e , i n the c e n t e r - o f -momentum frame, (6.35a) s 1 (6.35b) s 2 where p, p 1 and u , u r , denote the i n i t i a l and f i n a l momenta and h e l i c i t y s t a t e s o f one n e u t r i n o and q, q 1 and u o > u o i denote t h e c o r r e s p o n d i n g q u a n t i t i e s f o r the second n e u t r i n o . i ( H 2 / V 2 ) ( 2 T i ) 4 6(p +q-p'-q»)(l/8k 2 ) 1 (p.p' + 5 p 0 2 ) u r , ( p « ) u r ( p ) u s , ( q « ) Y ^ n s ( q ) + ( 8 p 0 2 ) u r , ( p ' ) Y 4 u r ( p ) u s , ( q ' ) Y 4 u g ( q ) ] = - i ( H 2 / V 2 ) ( 2 7 t ) 4 6 ( p + q - p ' - q , ) ( l / 8 s 2 ) { ( - p ^ ' + 5 p o 2 ) S r , ( p , ) Yu- u s ( q ) S s , ( q , ) Yu. u r ( p ) + ( 8 p o 2 ) 5 r , ( p ' ) Y 4 u s ( q ) u s , ( q ' ) Y ^ C p ) ] 60 h) The M a t r i x Elements f o r Two Component Neutrino-Two Component N e u t r i n o S c a t t e r i n g The energy-momentum t e n s o r f o r two, two component n e u t r i n o f i e l d s ¥ and ¥, i s 2 2 i . j i . j (6.36) T_,(x) = ( i / 2 ) .2 2 :(¥,,, o\, ¥ - Y T o\, ¥ , , , ) : i = l j = When t h i s i s s u b s t i t u t e d i n t o I , one o b t a i n s ( 6 . 3 7 ) I - (K 2/4) h ^ ( x ) h a p ( y ) H J ^ C x ) T a p ( y ) • - ( " 2 / 2 > y « fo<y> N -¥ T-x) o u ^ ( x ) f + - ( y ) a p l | a ( y ) x) o v ^ ( x ) l ] ; ( y ) a p l + ( y ) ; t --¥, :x) ¥ + ( -+¥, :x) a v ¥+ ( : :x) °v w :x) a v :x) a v ¥+ ( :x) a v ¥+ ( r p ° p y + ( x ) ¥"|-(y) a p ¥ + ( y ) } Upon i n s p e c t i o n o f (6.37) one sees t h a t the f i r s t f o u r terms c o n t r i b u t e t o a p r o c e s s whose graph i s and t h a t the l a s t f o u r terms c o n t r i b u t e t o a p r o c e s s 61 whose graph i s 2/ (2) The c o r r e s p o n d i n g m a t r i x elements a r e , i n the c e n t e r - o f -momentum frame, (6.38a) s± = - i ( H 2 / V 2 ) ( 2 7 i ) 4 6(p+q-p »-q • ) ( l / 8 k 2 ) | (p • p » + 3 p Q )u (p) u(p') u'(q) u(q') -( 8 p 0 2 ) u + ( p ) u(p') u f ( q ) u(q')} (6.38b) s 2 = i ( H 2 / V 2 ) ( 2 K ) Z | - 5 ( p + q - p » - q ' ) ( l / 8 s 2 ) j ( - p . p ' + 2 t i* 3p 0 )u'(q) u(p') u (p) 0 u(q') -( 8 p 0 2 ) u t ( q ) u(p') u ^ p ) u ( q ' ) | where p and p 1. denote the i n i t i a l and f i n a l momenta of one n e u t r i n o and q and q 1 denote the c o r r e s p o n d i n g q u a n t i t i e s f o r the o t h e r n e u t r i n o . 62 7. The C r o s s - s e c t i o n s f o r the C o l l i s i o n s For each m a t r i x element g i v e n i n Chapter 6 a q u a n t i t y M i s d e f i n e d by (7.1) s = i ( 2 r c ) 4 6(p+q-p'-q') M . The t r a n s i t i o n p r o b a b i l i t y per u n i t of space-time i s then (7.2) co = ( 2 T I ) 4 6(p+q-p'-q') |M| 2 and the c r o s s - s e c t i o n i s (7.3) da = (co/J) dN where J i s the incoming f l u x d e n s i t y and dN i s the number of f i n a l s t a t e s ( M u i r h e a d 1965). F o l l o w i n g Muirhead(l965) one o b t a i n s f o r the c r o s s -s e c t i o n f o r u n p o l a r i z e d p a r t i c l e s i n the c e n t e r - o f -momentum frame (7.4) d a / d i l = ( V 4 / ( 2 r t ) 2 ) ( p 2/4) s E|M|2 0 i f where E i n d i c a t e s an average over i n i t i a l s p i n s t a t e s and i E i n d i c a t e s a sum over f i n a l s p i n s t a t e s , f With the h e l p of the r e s u l t s o b tained i n the p r e v i o u s Chapters, the c r o s s - s e c t i o n s f o r the c o l l i s i o n s l i s t e d i n the I n t r o d u c t i o n can now be worked out. a) The C r o s s - s e c t i o n f o r S c a l a r - S c a l a r S c a t t e r i n g The m a t r i x element i s the sum of (6.10a), (6.10b) and ( 6 . 1 0 c ) . T h e r e f o r e , from ( 7 . 1 ) , (7.5) M = ( K 2 / 2 V 2 ) c o s 2 ( 9 / 2 ) / s i n 2 ( e / 2 ) + s i n 2 ( 9 / 2 ) / c o s 2 ( 9 / 2 ) + ( s i n ^ 6 ) / 4 S i n c e s c a l a r p a r t i c l e s have s p i n 0, t h e r e are no p o l a r i z a t i o n sums. Hence, from (7.4), (7.6) da/dn = (HV(2T02)(PO2/16) c o s 2 ( e / 2 ) / s i n 2 ( 0 / 2 ) + s i n 2 ( G / 2 ) / c o s 2 ( 0 / 2 ) + ( s i n 2 e ) / 4 J 64 b) The C r o s s - s e c t i o n f o r S c a l a r - P h o t o n S c a t t e r i n g From ( 6 . 2 3 ) and (7.1) one o b t a i n s (7.7) M = ( K 2 / 2 V 2 ) ( a + a ; * + a_a^*) c o s 2 ( 9 / 2 ) / s i n 2 ( 9 / 2 ) . For a photon t h e r e are two p o s s i b l e i n i t i a l s t a t e s c o r r e s p o n d i n g t o (7.8) |a+| 2 = 1 and | a _ | 2 = 1 and two f i n a l s t a t e s c o r r e s p o n d i n g t o (7.9) |a|| 2 = 1 a n d | a ^ | 2 = 1 . T h e r e f o r e , from (7.4), (7.10) da/dJL= ( H 4 / ( 2 T I ) 2 ) ( p Q 2 / 1 6 ) c o s 4 ( 0 / 2 ) / s i n 4 ( 9 / 2 ) . i 65 c) The C r o s s - s e c t i o n f o r S c a l a r - F o u r Component N e u t r i n o S c a t t e r i n g From (6.23) and (7.1) one o b t a i n s (7.11) M = ( H 2 / V 2 ) ( l / 2 k 2 ) ( p . p » + 3p Q 2) u r , ( p ' ) Y 4 u r ( p ) From (7.4) one has, t h e r e f o r e , (7.12) d o / d i i = ( H 4 / ( 2 T i ) 2 ) ( p 0 2 / 1 6 k 2 ) ( p . p ' + 3 p Q 2 ) 2 x E E r r 1 ^ . ( p 1 ) Yz. u f p ) . F o r the f o u r component n e u t r i n o t h e r e are two i n i t i a l and two f i n a l s t a t e s . Hence, 2 (7.13) 2 E r r ' ur i ( P ! ) Y 4 u p ( p ) = (1/2)E £ u r , ( p ' ) Y 4 u (p) u r ( p ) y 4 ^ . ( p 1 ) r r 1 =(1/2)S ( u r , ( p ' ) y 4 [ - i ( Y P ) / 2 p 0 ] Y 4 u r , ( p « ) = ( - l / 8 p o 2 ) T r a c e [ ( Y p ' ) Y 4 ( Y P ) Y 4 ] = cos 2 ( e/2) where the H e r m i t i a n p r o p e r t y o f Y / j . i n "the r e p r e s e n t a t i o n ( 4 . 4 a ) , the p r o p e r t y (4.17c) of the s o l u t i o n s u r ( p ) , and the t r a c e theorems f o r the y m a t r i c e s ( M u i r h e a d 1965) have been used. The s u b s t i t u t i o n o f (7.13) back i n t o (7.12) and the use o f (6.7b) y i e l d s (7.14) d a / d i l = (H 4 / ( 2K ) 2 ) ( P 0 2 / 6 4 ) [ c o s 4 ( 9 / 2 ) / s i n 4 ( e / 2 ) ] x (1 + c o s 2 ( 0 / 2 ) ) 2 66 d) The C r o s s - s e c t i o n f o r Scalar-Two Component N e u t r i n o S c a t t e r i n g From (6.26) and (7.1) one o b t a i n s (7.15) M = - ( K 2 / V 2 ) ( l / 2 k 2 ) ( 5 . S ' + 3p 0 2) u + ( p ) u(p') . There i s no p o l a r i z a t i o n sum f o r a two component n e u t r i n o s i n c e t h e r e i s o n l y one i n i t i a l and one f i n a l s t a t e . T h e r e f o r e , from (7.4), (7.16) da/dJl= (H 4/(27i) 2)(p 0 2 / 1 6 t 2)(p.p' + 3p Q 2) | u t ( p ) u ( p ' ) | Now, (7.17) |u +'(p) u ( p ' ) | 2 = u + ( p ) u(p«) u + ( p ' ) u(p) 2 = - u ^ p ^ C p ' c )/2p Q] u(p) = ( l / 4 p o 2 ) T r a c e [ ( p a T ) ( p ' a T ) ] = - ( l / 4 p o 2 ) T r a c e = cos2(0/2) ( p a T ) ( p ' a T ) oL where the p r o p e r t i e s o f i n the r e p r e s e n t a t i o n ( 5 . 2 ) , the t r a c e theorem ( 5 . 5 b ) and t h e p r o p e r t y (5.12a) o f t h e s o l u t i o n u(p) have been used. The s u b s t i t u t i o n o f (7.17) i n t o (7.16) and the use o f (6.7b) y i e l d (7.18) d a / d i l = ( H V(27i) 2)(p o 2/64 ) r c o s 4(9/2)/sin 4(9/2)] x (1 + c o s 2 ( 0 / 2 ) ) 2 . 67 e) The C r o s s - s e c t i o n f o r Photon-Four Component N e u t r i n o S c a t t e r i n g With the h e l p o f (7.19) u r , ( p « ) [ ( Y l 1 ) + (Y<l)]u r(p) = i 4 y 4 p Q u r l ( p » ) u p ( p ) one o b t a i n s , from (6.29) and (7.1), (7.20) M = i ( H 2 / Y 2 ) ( l / 8 p o k 2 ) e!_* e.. u r , ( p ' ) X±. u r ( p ) where (7.21) X = ( 4 i Y ^ P 0 ) L 6 i d(-p.p' " 3 P 0 ) + q i + Y i q i ) ( Z l - p o } T h e r e f o r e , from (7.4) one has (7.22) d o / d J l (H 4 / ( 2 T t ) 2 ) (p0 2A) ( l / 6 4 p o 2 k 2 ) 2 2 i f i ed X i . i Now, (7.23) 2 2 i f = ( - l / 8 p 0 2 ) S S ( e ! * e . e' e*)x Tr a c e (YP') X (yp) ^ where (4.17c) has been used and where (7.24) X. . . ^ X? . Y 4 . The t r a c e i s e v a l u a t e d i n Appendix B and t h e photon p o l a r i z a t i o n sum i s worked out i n Appendix C. One o b t a i n s (7.25) 2 2 i f I2 - ^ o 6 c o s2 ( 0 / 2 ) - cos^Ce/2) + 4 c o s 6 ( 6 / 2 ) 68 • The s u b s t i t u t i o n o f (7.25) i n t o (7.22) and the use of (6.7b) y i e l d (7.26) da/dJl= ( H Z t " / ( 2 7 i ) 2 ) ( p o 2 / 6 4 ) ( l / s i n 4 ( 6 / 2 ) ) x [ c o s 2 ( 0 / 2 ) - c o s 4 ( 6 / 2 ) + 4 c o s 6 ( 9 / 2 ) 69 f ) The C r o s s - s e c t i o n f o r Photon-Two Component N e u t r i n o S c a t t e r i n g With'the h e l p of (7.27) u t ( p)[(aq') + (aq)]u(p«) = -^u^pMp ' ) one o b t a i n s , from (6.32) and (7.1), (7.28) M = - ( K 2 / V 2 ) ( l / 8 p ^ k 2 ) e j _ * e u^p) X u(p') where (7.29) X = - 4 P q 6 (--p.p' - 3P 0 2) + q ± q i 0 J T h e r e f o r e , from (7.4) one has (7.30) do/dJl= ( H 4 / ( 2 T t ) 2 ) ( p o 2 / 4 ) ( l / 6 4 p o 2 k 2 ) x Nov/, 2 2 i f e! e d u T ( p ) X±. u(p') (7.31) 2 2 i f I 2 = ( l / 4 p 2 ) 2 2 ( e ! * e . e' e* )X I ° e e' T r a c e (pa") X (p'aT) X ^ where the p r o p e r t y (5.12a) o f the s o l u t i o n u(p) and t h e H e r m i t i a n p r o p e r t y o f X. . have been used. The t r a c e can be e v a l u a t e d w i t h the h e l p of the t r a c e theorems (5.5a,b). One o b t a i n s f o r (7.31) the r e s u l t g i v e n i n Appendix B f o r t h e p h o t o n - f o u r component n e u t r i n o case. Thus, the photon p o l a r i z a t i o n sum i s the same as t h a t g i v e n i n Appendix C f o r the p h o t o n - f o u r component n e u t r i n o case. Hence, one o b t a i n s 70 (7.32) S 2 i f = 6 4 p c ( 0 0 3 ^ ( 0 / 2 ) - c o s 4 ( 6 / 2 ) + 4 c o s D ( 9 / 2 ) ) 16,, o The s u b s t i t u t i o n o f (7.32) i n t o (7.30) and t h e use of (6.7b) y i e l d (7.33) d o / d i l = ( K V ( 2 T i ) 2 ) ( p o 2 / 6 4 ) ( l / s i n 4 ( © / 2 ) ) x [ c o s 2 ( 0 / 2 ) - c o s 4 ( e / 2 ) + 4 c o s 6 ( 0 / 2 ) 71 g) The C r o s s - s e c t i o n f o r Four Component N e u t r i n o - F o u r Component N e u t r i n o S c a t t e r i n g The m a t r i x element i s the sum o f (6.35a) and (6.35b) From (7.1) one then o b t a i n s (7.34) M = ( H 2 / 8 V 2 ) A u r , ( p ' ) u r ( p ) u s , ( q ' ) u g ( q ) + B u r , ( p ' ) Y Z F u r ( p ) u s , ( q ' ) Y 4 u g ( q ) - C u r I ( p ! ) Y ^ u g ( q ) u s , ( q ' ) Y ^ u p ( p ) - Du r,(p') Y4 u s ( q ) ^ . ( q 1 ) Y 4 U r ( p ) where (7.35) A = (p.p* + 3 p 0 2 ) / k 2 P -*2 B = 8 P < ) V k^ C = ( - p ^ ' + 3 p Q 2 ) / - s 2 2 -^ 2 D = 8 p c V s d I t i s n e c e s s a r y t o compute £ E M . T h i s i s done i n i f Appendix D. The s u b s t i t u t i o n of the r e s u l t o b t a i n e d t h e r e i n t o (7.4) y i e l d s (7.36) da/dJl= ( K 4 / ( 2 K ) 2 ) ( P o 2 / 5 1 2 ) X | [ l + 6 c o s 2 ( e / 2 ) + 1 8 c o s 4 ( e / 2 ) + 6 c o s 6 ( 0 / 2 ) + c o s 8 ( e / 2 ) ] /sin 4(0/2) + [ l + 6 s i n 2 ( e / 2 ) + 1 8 s i n 4 ( e / 2 ) + 6 s i n 6 ( e / 2 ) + s i n 8 ( 6 / 2 ) ] / c o s 4 ( e / 2 ) + 2 [ 4 + 9 s i n 2 ( 0 / 2 ) c o s 2 ( e / 2 ) ] / [ s i n 2 ( e / 2 ) c o s 2 ( e / 2 ) ] j 72 h) The C r o s s - s e c t i o n f o r Two Component Neutrino-Two Component N e u t r i n o S c a t t e r i n g The m a t r i x element i s the sum of (6.38a) and (6.38b), From (7.1) one t h e n o b t a i n s (7.37) M = - ( x 2 / 8 V 2 ) ^ A u + ( p ) G [ i u(p') u +(q) u(q') - Bu (p) u(p') u (q) u ( q f ) - C u f ( q ) a u(p') u T ( p ) o" u(q') + D u f ( q ) u(p') u f ( p ) u(q') where A,B,C, and D a r e the same as i n (7.35). I t i s n e c e s s a r y t o compute [ M | . T h i s i s done i n Appendix E. The s u b s t i t u t i o n o f the r e s u l t o b t a i n e d t h e r e i n t o (7.4) y i e l d s (7.38) da/d_0.= (HV(2TC)2)(P02/512)X | [ l + 6 c o s 2 ( e / 2 ) + 1 8 c o s 4 ( e / 2 ) + 6 c o s 6 ( G / 2 ) + c o s 8 ( 0 / 2 ) ] / s i n Z , ' ( € / 2 ) + [ l + 6 s i n 2 ( e / 2 ) + 1 8 s i n 4 ( e / 2 ) + 6 s i n 6 ( e / 2 ) + s i n 8 ( e / 2 ) ] / c o s Z j ' ( e / 2 ) + 4 [ 4 + 9 s i n 2 ( e / 2 ) c o s 2 ( G / 2 ) ] / | s i n 2 ( e / 2 ) c o s 2 ( e / 2 ) ] j 73 8. C o n c l u s i o n s and D i s c u s s i o n I n t h i s Chapter the r e s u l t s of the p r e v i o u s Chapters are r eviewed, summarized, and compared, where p o s s i b l e , w i t h p r e v i o u s l y p u b l i s h e d r e s u l t s ; a comparison w i t h e l e c t r o -dynamics i s made; a simple s e l f - e n e r g y c a l c u l a t i o n i s g i v e n ; and f i n a l l y , a recent ( l 9 7 0 ) approach, i n v o l v i n g g r a v i t a t i o n , to the s e l f - e n e r g y problem i n e l e c t r o d y n a m i c s i s mentioned. In Chapter 1, P a r t a, the g r a v i t a t i o n a l f i e l d v a r i a b l e s are d e f i n e d . The L a g r a n g i a n f o r a weak g r a v i t a t i o n a l f i e l d i s w r i t t e n down and two s o l u t i o n s ( f o r a s t a t i c s ource and f o r a f r e e f i e l d ) o f the f i e l d e q u a t i o n s are mentioned. I n P a r t b o f Chapter 1, a quantum t h e o r y o f the weak g r a v i t a t i o n a l f i e l d i s developed. Ten ty p e s of g r a v i t o n s a r i s e i n c o n t r a s t t o the Gupta f o r m a l i s m In which t h e r e a re e l e v e n . In Chapter 2 the i n t e r a c t i o n L a g r a n g i a n f o r a ma s s l e s s and H e r m i t i a n s c a l a r f i e l d i s e x t r a c t e d from, the L a g r a n g i a n which i s p o s t u l a t e d t o d e s c r i b e the s c a l a r f i e l d i n the presence o f g r a v i t a t i o n . I n Chapter 3 the i n t e r a c t i o n L a g r a n g i a n f o r the photon f i e l d i s e x t r a c t e d from the L a g r a n g i a n which i s p o s t u l a t e d t o d e s c r i b e t h e photon f i e l d i n the pre s e n c e o f g r a v i t a t i o n . I n Chapter 4 the r e q u i r e d p r o p e r t i e s o f t h e f o u r component n e u t r i n o are g i v e n . To'pass t o the re a l m of g e n e r a l r e l a t i v i t y one imposes the requirement t h a t the L a g r a n g i a n be i n v a r i a n t under a c o o r d i n a t e dependent r e o r i e n t a t i o n o f v i e r b e i n frames. Utiyama's p r e s c r i p t i o n i s used t o o b t a i n a L a g r a n g i a n which s a t i s f i e s the r e q u i r e -ment. The i n t e r a c t i o n L a g r a n g i a n i s then e x t r a c t e d i n the 74-weak f i e l d a p p r o x i m a t i o n . I n Chapter 5 t h e r e q u i r e d p r o p e r t i e s of the two component n e u t r i n o are g i v e n . The method o f Fock i s used t o extend the f l a t space-time L a g r a n g i a n t o a form which s a t i s f i e s t he above re q u i r e m e n t . The i n t e r a c t i o n L a g r a n g i a n i s t h e n e x t r a c t e d i n the weak f i e l d a p p r o x i m a t i o n . I n Chapter 6 the m a t r i x elements f o r the c o l l i s i o n s are g i v e n . Some of thes e can be compared v / i t h p r e v i o u s l y -p u b l i s h e d r e s u l t s . B a r k e r e t a l ( l 9 6 6 ) worked out the m a t r i x elements f o r the g r a v i t a t i o n a l i n t e r a c t i o n between photons, s c a l a r p a r t i c l e s w i t h non-zero r e s t mass, and s p i n 1/2 p a r t i c l e s w i t h non-zero r e s t mass. By s e t t i n g t h e masses t o zero i n t h e i r r e s u l t s , one may o b t a i n some of the m a t r i x e l -ements g i v e n here. I f one s e t s the masses t o zero i n the s c a l a r - p h o t o n , s c a l a r - s p i n 1/2, and p h o t o n - s p i n 1/2 m a t r i x elements o b t a i n e d by the above a u t h o r s one o b t a i n s the m a t r i x elements g i v e n h e r e f o r s c a l a r - p h o t o n , s c a l a r - f o u r component n e u t r i n o , and p h o t o n - f o u r component n e u t r i n o s c a t t e r i n g . When c o n s i d e r i n g p a r t i c l e s ,of the same s p i n , the above authors assume t h a t the masses are d i f f e r e n t . T h e r e f o r e , i f one s e t s the masses t o zero i n t h e i r r e s u l t s f o r p a r t i c l e s of the same s p i n , o n l y p a r t o f the m a t r i x element f o r the s c a t t e r i n g of i d e n t i c a l m a s s l e s s p a r t i c l e s i s o b t a i n e d . A d d i t i o n a l terms a r i s e which are due to the i d e n t i t y of the p a r t i c l e s . F or s c a l a r - s c a l a r s c a t t e r i n g , the zero mass v a l u e of t h e i r m a t r i x element i s the m a t r i x element (6.10a) g i v e n here and c o r r e s -ponds t o graph (1) on page 4-7. The o t h e r m a t r i x elements g i v e n here, (6.10b) and (6 . 1 0 c ) , c o r r e s p o n d i n g t o graphs (2) 75 and (3) on page 47, are due to the i d e n t i t y of the p a r t i c l e s . The zero mass v a l u e o f the m a t r i x element f o r the s c a t t e r i n g of two massive s p i n 1/2 p a r t i c l e s , g i v e n by the above a u t h o r s , i s j u s t the m a t r i x element (6.35a) g i v e n here and corresponds t o graph ( l ) on page 58* The o t h e r m a t r i x element g i v e n on page 58, (6.35b), i s due to the i d e n t i t y of the n e u t r i n o s . I n Chapter 7 the c r o s s - s e c t i o n s are worked out f o r the v a r i o u s c o l l i s i o n s . These are now d i s c u s s e d i n the o r d e r g i v e n t h e r e . The c r o s s - s e c t i o n (7.6) g i v e n here f o r t h e s c a t t e r i n g of two ma s s l e s s s c a l a r p a r t i c l e s does not agree w i t h t h e extreme r e l a t i v i s t i c l i m i t o f the c r o s s - s e c t i o n f o r the s c a t t e r i n g o f two massive s c a l a r p a r t i c l e s quoted by DeWitt(l967). The r e a s o n i s t h a t the term (8.1) (3 - v 2 ) ( l + v 2 ) i n h i s e q u a t i o n (3.10) s h o u l d r e a d (8.2) (3 + v 2 ) ( l - v 2 ) . The p r o o f of t h i s statement i s g i v e n i n Appendix F. As mentioned on page 44, the c a l c u l a t i o n i n Appendix F can ser v e as the example f o r the c a l c u l a t i o n s o f the m a t r i x elements g i v e n i n Chapter 6. The c r o s s - s e c t i o n (7.10) g i v e n here f o r s c a l a r - p h o t o n s c a t t e r i n g does not agree w i t h t h e extreme r e l a t i v i s t i c l i m i t o f the r e s u l t g i v e n by B o c c a l e t t i et al(l969a) f o r the s c a t t e r i n g o f a photon by a massive s c a l a r p a r t i c l e . The r e a s o n i s t h a t the f a c t o r , 76 (8.3) vV(W-K)2 i n t h e i r equation' (12) , s h o u l d r e a d (8.4) W2 . The extreme r e l a t i v i s t i c l i m i t o f t h e i r r e s u l t i s then gust (7.10) g i v e n h e r e . Upon comparison of (7.14) and (7.18) one sees t h a t the c r o s s - s e c t i o n s f o r s c a l a r - f o u r component n e u t r i n o s c a t t e r i n g and s c a l a r - t w o component n e u t r i n o s c a t t e r i n g are i d e n t i c a l ( t o t h i s o r d e r of p e r t u r b a t i o n t h e o r y ) . Upon comparison o f (7.26) and (7.33) one sees t h a t the c r o s s - s e c t i o n s f o r ph o t o n - f o u r component n e u t r i n o s c a t t e r i n g and photon-two component n e u t r i n o s c a t t e r i n g are i d e n t i c a l ( t o t h i s o r d e r o f p e r t u r b a t i o n t h e o r y ) . By comparing (7.36) and (7.38) one sees t h a t the c r o s s -s e c t i o n f o r f o u r component n e u t r i n o - f o u r component n e u t r i n o s c a t t e r i n g i s d i f f e r e n t from the two component n e u t r i n o - t w o component n e u t r i n o c r o s s - s e c t i o n . The quantum m e c h a n i c a l exchange t e r m ( t h e term which comprises t h e l a s t l i n e o f (7.36)) f o r the f o u r component case i s s m a l l e r , by a f a c t o r of one-h a l f , than the quantum m e c h a n i c a l exchange term(the term which comprises the l a s t l i n e o f (7 .38)) f o r the two component n e u t r i n o case. The re a s o n f o r t h i s i s the f o l l o w i n g . The quantum m e c h a n i c a l exchange term f o r the f o u r component case c o n t a i n s terms l i k e 1. In the meantime an Erratum has a p p e a r e d ( B o c c a l e t t i et a l 1969b) thus making i t u n n e c e s s a r y t o w r i t e down the p r o o f h e r e . 77 ( 8 . 5 ) X = ( V 4 - ) Trace"" where ( y p 1 ) ( Y P ) Y * v (yq') Y ^ (yq) Y ^ 2 . g i v e n on page ( 8 . 6 ) y v = Y 4 Y* Y 4 ( t h i s f o l l o w s from the e x p r e s s i o n f o r £ E hi i f 96 and, f o r example, the e x p r e s s i o n f o r £ £ Z-, g i v e n on i f 0 page 95) • The quantum m e c h a n i c a l exchange term f o r the two component case c o n t a i n s terms l i k e ( 8 . 7 ) X« = n T r a c e [ ( p o T ) ( p ' o T ) o y (qcT) ( q ' o T ) o y ( t h i s f o l l o w s from the e x p r e s s i o n f o r *~ g i v e n on page 99 and, f o r example, the e x p r e s s i o n f o r g i v e n on page 9 8 ) . r] i s a number which i s the same i n b o t h cases and the 1/4 i n ( 8 . 5 ) a r i s e s from the a v e r a g i n g over i n i t i a l s t a t e s . One can show t h a t the t r a c e i n ( 8 . 5 ) i s t w i c e the t r a c e i n ( 8 . 7 ) . Then, because o f the 1/4 i n ( 8 . 5 ) , one has ( 8 . 8 ) X = X'/2 which shows t h a t the quantum m e c h a n i c a l exchange term f o r t h e f o u r component n e u t r i n o case i s o n e - h a l f the quantum m e c h a n i c a l exchange term f o r the two:..component case. I n the c o u r s e of p e r f o r m i n g the o r i g i n a l c a l c u l a t i o n s f o r the r e s u l t s g i v e n h e r e , the photon-photon s c a t t e r i n g c r o s s - s e c t i o n was c a l c u l a t e d . The r e s u l t of Barker e t a l ( 1.967) and B o c c a l e t t i e t a l ( l 9 6 9 a ) was c o n f i r m e d . I t i s ( 8 . 9 ) d o / d i l = (HV(2TC)2)(P02/2) 1 + c o s 1 6 ( 0 / 2 ) + s i n 1 6 ( 0 / 2 ) J / s i n Z , e . 7 8 T h i s c r o s s - s e c t i o n can dominate the e l e c t r o d y n a m i c c r o s s - s e c t i o n f o r photon-photon s c a t t e r i n g . The e l e c t r o -dynamic c r o s s - s e c t i o n i s (8.10) da/dii * I O " 1 5 5 co6 f o r v e r y low f r e q u e n c i e s (#io«m c ), and (8.11) da/dii lo 1 2/co 2 f o r v e r y h i g h f r e q u e n c i e s (#ao»m c ) ( A k h i e z e r and B e r e s t e t s k i i 1965). The g r a v i t a t i o n a l c r o s s - s e c t i o n f o r photon-photon s c a t t e r i n g i s , from ( 8 . 9 ) , (8.12) d a / d i i * 1 0 " 1 5 1 co2 Thus f o r v e r y low f r e q u e n c i e s (co<10 sec""*") and f o r v e r y h i g h f r e q u e n c i e s (co>10 4^ sec"^") the g r a v i t a t i o n a l i n t e r a c t i o n dom-i n a t e s i n the c r o s s - s e c t i o n f o r photon-photon s c a t t e r i n g . F o r v e r y s m a l l s c a t t e r i n g a n g l e s a l l the c r o s s - s e c t i o n s have the form (8.13) d o / d / L = ( H 4 / ( 2 n ) 2 ) ( p o 2 / e 4 ) . I n c.g.s. u n i t s the r e l a t i o n s h i p i s (8.14) da/da = ( 8 G ^ / c 4 ) 2 w 2 / e 4 = 4 . 4 x 1 0 " 1 5 1 O J 2 / 0 4 cm 2 . The graphs f o r the v a r i o u s c r o s s - s e c t i o n s are drawn on pages 82 to 84. I n quantum e l e c t r o d y n a m i c s t h e r e are p r o c e s s e s c o n t a i n e d i n S 2 ( t he second o r d e r term i n the S -matrix expansion) which l e a d to d i v e r g e n t i n t e g r a l s . The graphs f o r these 7 9 p r o c e s s e s a re where a s o l i d l i n e r e p r e s e n t s an e l e c t r o n or p o s i t r o n and a dashed l i n e r e p r e s e n t s a photon. The graph on the l e f t i s c a l l e d the s e l f - e n e r g y graph of the e l e c t r o n and the one on the r i g h t i s c a l l e d the s e l f - e n e r g y graph of the photon. The same d i v e r g e n t i n t e g r a l s which appear i n the m a t r i x elements f o r thes e simple p r o c e s s e s a l s o appear i n h i g h e r o r d e r p r o c e s s e s such as S i m i l a r l y , i n the d e c o m p o s i t i o n of f o r t h e g r a v i t a -t i o n a l i n t e r a c t i o n , s e l f - e n e r g y p r o c e s s e s appear. The q u e s t i o n a r i s e s as t o whether d i v e r g e n t i n t e g r a l s appear. F o r s i m p l i c i t y the i n t e r a c t i o n between the g r a v i t a t i o n a l f i e l d and the s c a l a r f i e l d d e s c r i b e d i n Chapter 2 i s c o n s i d e r e d . The graph f o r the p r o c e s s under c o n s i d e r a t i o n i s where a s o l i d l i n e r e p r e s e n t s a s c a l a r p a r t i c l e and a dashed l i n e r e p r e s e n t s - a g r a v i t o n . The energy-momentum t e n s o r i s , from ( 2 . 4 ) , (8 . 1 5 ) T = -0 tf\r ^ l T / 2 . y j (IV ^ (1 r V JJ.V r T ^ T' 80 One t h e n o b t a i n s f o r I , the i n t e g r a n d of the s c a t t e r i n g m a t r i x , d e f i n e d by ( 6 . 6 ) , (8.16) I . 2,<2 h^U) h a p(y)x -^ 6ap/ 2 )f f |n ( y ) >*VX) ^ ]l(y) " ( 5 a p / 2 ) f | v ( x ) f | n ( y ) * V X ) ^ 6 a p V / 4 ) f | T ( x ) f l n ^ <r„C7)} where (8.17) 0 , ( x ) 0 (y) = l i m ( - i / ( 2 7 c ) 4 ) / a a e l i ^ ? d \ • I f one denotes the i n i t i a l momentum b,v p and the f i n a l momentum by p', one o b t a i n s f o r the m a t r i x element between the i n i t i a l and f i n a l f r e e p a r t i c l e s t a t e s (8.18) s =<p' | ( - l / 2 j ^ I d x dy |p> = (H2/2Vpo) 6(p-p')(pp ,)y (1A 2) d \ = 0 , 2 s i n c e p =0. The s e l f - e n e r g y m a t r i x element f o r a s c a l a r p a r t i c l e i s z e r o , i n c o n t r a d i s t i n c t i o n t o t h e m a t r i x elements f o r t he photon and e l e c t r o n s e l f - e n e r g y graphs i n e l e c t r o -dynamics. I n o r d e r t o c o n s i d e r h i g h e r o r d e r p r o c e s s e s one cannot j u s t proceed as i n e l e c t r o d y n a m i c s and c o n s i d e r . h i g h e r o r d e r terms i n the S-matrix expansion. I t i s f i r s t n e c e s s a r y t o c a r r y t h e ex p a n s i o n of g ^ v and ,/g t o h i g h e r o r d e r s i n H. The 81 c o m p l e x i t y of the problem thus i n c r e a s e s r a p i d l y . Salam and S t rathdee(1970) have s t a t e d t h a t one s h o u l d t r e a t problems i n g r a v i t a t i o n n o n - p e r t u r b a t i v e l y . They have g i v e n an example to i n d i c a t e how g r a v i t a t i o n might be t r e a t e d n o n - p e r t u r b a t i v e l y t o suppress d i v e r g e n t i n t e g r a l s i n quantum e l e c t r o d y n a m i c s . -82 50° 60° 90° /80° /tro° F i g . 1. C r o s s - s e c t i o n f o r s c a l a r - s c a l a r s c a t t e r i n g 3©° 6°° 90° F i g . 2. C r o s s - s e c t i o n f o r s c a l a r - p h o t o n s c a t t e r i n g 83 30" 60° po" /eo° /SO" F i g . 3. C r o s s - s e c t i o n f o r s c a l a r - n e u t r i n o s c a t t e r i n g ig» 4. C r o s s - s e c t i o n f o r p h o t o n - n e u t r i n o s c a t t e r i n g 84 3°° 60° 90° U0° /CO" F i g . 5 . C r o s s - s e c t i o n f o r f o u r component n e u t r i n o -f o u r component n e u t r i n o s c a t t e r i n g F i g . 6 . 30° ^O 3 ^To5 ISO" C r o s s - s e c t i o n f o r two component n e u t r i n o -two component n e u t r i n o s c a t t e r i n g 85 BIBLIOGRAPHY A l i a r o n i , J . 1965.' The S p e c i a l Theory of R e l a t i v i t y . ( O x f o rd U n i v e r s i t y P r e s s , London). A k h i e z e r , A . I . and B e r e s t e t s k i i , V . B . 1965. Quantum E l e c t r o d y n a m i c s . ( I n t e r s c i e n c e P u b l i s h e r s , New Y o r k ) . Barker,B.M., Gupta,S.N., and Haracz,R.D. 1966. Phys. Rev., 14-9, 1027. Barker,B.M., Bhatia,M.S., and Gupta,S.N. 1967. Phys. Rev., 162, 1750. Bludman,S.A. 1963. Nuovo Cimento, 2_2, 75L B o c c a l e t t i , D . , de Sabbata,V., G u a l d i , C , , and F o r t i n i , P . 1969a. Nuovo C i m e n t o , 60B, 320. 1969b. Nuovo Cimento, 64B, 419. Bogoliubov,N.N. and Shirkov,D.V. 1959. I n t r o d u c t i o n t o the Theory of Q u a n t i z e d F i e l d s . ( I n t e r s c i e n c e P u b l i s h e r s , New Y o r k ) . C o r i n a l d e s i , E . 1956. P r o c . Phys. Soc.(London), A69, 189. Corson,E. 1953. I n t r o , t o T e n s o r s , S p i n o r s and R e l a t i v i s t i c Wave E q u a t i o n s . ( B l a c k i e and Son, London). DeWitt,B.S. 1967. Phys. Rev., 162, 1239. E i n s t e i n , A . 1918. S i t z u n g s b e r i c h t e d e r K o n i g l i c h P r e u s s i s -chen Akademie der W i s s e n s c h a f t e n , 8, 153. E i n s t e i n , A . 1956. The Meaning of R e l a t i v i t y . ( P r i n c e t o n U n i v e r s i t y P r e s s , P r i n c e t o n ) . Fock,V. 1929. Z e i t s . f . Phys., £ Z , 261. Gupta,S.N. 1952. P r o c . Phys. Soc.(London), A6£, 161, 608. Kerimov,B.K. and Romanov,Yu.I. 1965. S o v i e t P h y s i c s JETP, 86 Kobzarev,!.Yu. and Okun,L.B. 1963. S o v i e t P h y s i c s JETP, 16, 134-3. Kuchowicz,B. 1969. F o r t s c h r i t t e der P h y s i k , 17, 517. Lederman,L.M. 1967. N e u t r i n o P h y s i c s , i n H i g h Energy P h y s i c s ( E d i t o r - E . Burhop), V o l . 2. (Academic P r e s s , New Y o r k ) . Lee,T.D. and Vu,C.S. 1965. Weak I n t e r a c t i o n s , i n Annual Review of N u c l e a r S c i e n c e , _Tp_, 381. L u r i e , D . 1968. P a r t i c l e s and F i e l d s . ( I n t e r s c i e n c e P u b l i s h e r s , New Y o r k ) . Marshak,R.E., R i a z u d d i n , R y a n , C P . 1969. Theory o f Weak I n t e r a c t i o n s . ( W i l e y - I n t e r s c i e n c e , New Y o r k ) . Muirhead,H. 1965. The P h y s i c s o f Elementary P a r t i c l e s . (Pergamon P r e s s , London). Roman,P. I960. The Theory of Elementary P a r t i c l e s . ( N o r t h - H o l l a n d , Amsterdam). Salam,A. and S t r a t h d e e , J . 1970. Quantum G r a v i t y and I n f i n i t i e s i n Quantum E l e c t r o d y n a m i c s . ( I n t e r n a t i o n a l C e n t r e f o r T h e o r e t i c a l P h y s i c s , T r i e s t e ) , ( p r e p r i n t ) . Schweber,S. 1961. An I n t r o d u c t i o n t o R e l a t i v i s t i c Quantum F i e l d Theory. (Harper and Row, New Y o r k ) . Utiyama,R. 1956. Phys. Rev., 101, 1597. Vl a d i m i r o v , Y u . S . 1964. S o v i e t P h y s i c s JETP, 18, 176. Weyl,H. 1929. P r o c . N a t i o n a l Acad, o f S c i e n c e , 15_, 323. 1931. The Theory o f Groups and Quantum. Mechanics. (Dover P u b l i c a t i o n s , I n c . , New Y o r k ) . 87 Appendix A. The R e d u c t i o n of the Four Component N e u t r i n o L a g r a n g i a n I n t h i s Appendix the term f 0 ^ B r s (H y p - y p M 0 ) which appears i n e q u a t i o n (4.35) i s shown t o v a n i s h . / ; ^ / Y A L r - * ' n J - f SUM v - r m t ' m , z 2 88 Appendix B. The E v a l u a t i o n of a T r a c e f o r Photon-Four Component N e u t r i n o S c a t t e r i n g I n t h i s Appendix the t r a c e i n e q u a t i o n (7.23) i s e v a l u a t e d . One has, from (7.23), ("fey) I I (e '* e. e'„ e*£ ) Tr«ce[ H/Jfy f/fJ where 7rtfcc [Qff ') /fyj Y+ & 3 Tr«ce [ (f/UZfl * fifr (Ytp *£fe')KJ The t r a c e i n (7.23) has now been b r o k e n down i n t o f o u r t r a c e s which can be e v a l u a t e d d i r e c t l y . 89 •. B* /> 7 [(e*-e)(-2A 2 c ^ z 9 / z - y -f- (f. t*)(f '.e)J* C = U4/>/)[(e'-e*) (-2/>,l<^zeA-zAz) + Cf.e')(f-e*)]* e'i eje',c€*e Ti [CY/)(lfi j-f;fj)iYf)Yf(Y(f/<+Y<f*)i-] = - (f-e*)(f. e'J ZT[fr/J ( t e ) ( t y ) (te*)] - (f-e'*)(f'e*) r,[(Y/>')(Ye)aF)(^')3 90 - ( f e ) ( f - e ' ) z r ( r / ) ( t f e ' * ) ( y f ) a e V l - ( f - ) K r ( i f / ) ( r e ' V ( f y ) ( f e ' ) J -fff>e'V(f'e')[(^e)(f.e')~ -f(?-e)(f 'e')t-(f>r')(e'%*) + (p'-e*)( + f (}'• e) (j'. e*)[ CPF0 (e'- e 'V J The t r a c e i n (7.23) has now been e v a l u a t e d . I n • c o n c l u s i o n , t h e r e f o r e , ZIIe'*e u„,(r<))(^(F)lZ -(fo^)ZI (/1+B+C + D) where Z denotes an average over the i n i t i a l photon s t a t e s and 2i denotes a sum over t h e f i n a l photon s t a t e s , e' 9 1 Appendix C. The Photon P o l a r i z a t i o n Sums f o r Photon-Pour Component N e u t r i n o S c a t t e r i n g I n t h i s Appendix the average over i n i t i a l p o l a r i z a t i o n s t a t e s and the sum over f i n a l p o l a r i z a t i o n s t a t e s of the photon i s performed f o r p h o t o n - f o u r component n e u t r i n o s c a t t e r i n g . I t i s c o n v e n i e n t to t r a n s f o r m t o the b a s i s (6.16) which r e p r e s e n t s s t a t e s of c i r c u l a r p o l a r i z a t i o n . One t h e n has, + (a- a'/)^ lo/z + {a- a'J)AX* %&A (e • e J = (a+ a+)t<v &/z + (4+ ct-J***? 0/1 7r (e j') = i/><> (as - a~ ) IT (f.e')-~ -Afo ^ & (0+ - ai ) ( p • e J = ^<9(a^ - a„ J ft (f'-ey) - -A pa a^e (a* - a*) rr 92 (p'- e) - ~JJ& /x^ e (a+ - a~) r? ( p • e ' ) - -+fj> - aL ) f t The problem i s t o compute I (A e e' where A,B,C and D are g i v e n i n Appendix B. S u b s t i t u t i o n of t h e r e s u l t s g i v e n immediately above i n t o A,B,C and D as d e f i n e d i n Appendix B y i e l d s e e' e e e e From t h e s e r e s u l t s one can show t h a t ( 1Sp0z)II ( A + P) = / ^ g 94-Appendix D. T r a c e C a l c u l a t i o n s f o r Four Component N e u t r i n o -Four Component N e u t r i n o S c a t t e r i n g I n t h i s Appendix Z ZlM/Z i s e v a l u a t e d f o r f o u r component n e u t r i n o - f o u r component n e u t r i n o s c a t t e r i n g . M i s g i v e n by (7.34-). I t i s c o n v e n i e n t t o s e t Wi = ">>(/)tfs,(f'K Mf, Wi - fa<(/)Y4 tf,(/ fa>(/) fa fa4 ^ r uA,(f)Y» fafy)fa,fy) to* = £. (/Xt tt (f) fa, (fX+ fa ty) . Then 5 from (7.34-), one has II/nl1-- (P/i*<v*)2l(£z) i / i * A- ' where AzW,Wif Iz'-A3WlWzt 2^-Acw.w/ ?7 = -3c Wi w/ Iff = -^iPWi. w/ 71 - w, f ~£io - ~CBWjW^ i „ = c z W j W s f 2/z = CD i4/j w + f 2,z - D A W * W, : -pa W i " Z j * = DC H 4 V 7 2,t-- DlU4li// 95 One can show t h a t (where - Y4Yy K- ) ll2, = yV'W) K[(rt)tlZ[(tf'J U$)I]- *JY/ III,-- (A3/fa/if)r,[(0Y,aF)Y,jz[()ff)t(r7)^j=^£(^&A 11 lz-- kAc//^J)7^[(Yj>n,(^)t(^rMaf)lJ ^ SAC Z"J^- ( : - / ? z > / / ^ ^ ^ r / " ^ i ; r ^ - ; ^ ( % ; ^ 7 - 4 A D ^ Z Z / Z 12 h s A *)T«t ty) Y+(ty)tJT* RY/J Yf (fy K1493 (#» 111,= tBV/tfiVzr(p)Y* ty)&]Z[YY/JY? (fykJ = 13z^4e/a II ? 7 -• (c^//6/)K[(Y/)Yi (Y/jY+(Yf)YrJ-- c^e/z II I, = C'^/u/jZffY/)^ (Yf')Yt (ty*,] - 2AD^le>/i ^ 1 e/z (c*c//tf)T*[(/)r«(Yj)l(Y/Y»(YF)il-- *AC 112,0-- (-^/Uf/)T^[(Y/IY* (Yf)Y+ (ty)Y„ (Y?)Y+J = +0C ^  z£>/2 l l I, Cc x/uf) 7*[(//)YM (Yf)I 7 7*t(ty)L ft)?,]- <?c Y/+*S*A) II l,t = (CP/K/J%[(Yf>')YM (tyY+J T« [ (*<?')YM (Yf)Y+J:4CD Y*~> le/z 111/3 -- V I^Yf'J Y+ (Y9) % W)^ (Y?)Y* J = A A 96 11 In = Q>lM/) ~£f {Y/Jb(ff)Yf JT«I(ty)Y* (Yf)Y4J* 4Dl^#/z One a l s o has /tfp/i s^'tyl Cco+e/z eaS^/l. 97 Appendix E. T r a c e C a l c u l a t i o n s f o r Two Component N e u t r i n o -Two Component N e u t r i n o S c a t t e r i n g I n t h i s Appendix /M/Z i s e v a l u a t e d f o r two component n e u t r i n o - t w o component n e u t r i n o s c a t t e r i n g . M i s g i v e n by (7.37). I t i s c o n v e n i e n t t o s e t V v l - - uf(/>)u(f>)uf(f) ufy'J Then, from (7.37), one has IMP* WIN*) ti; where ?+= APV.W+ I 7 = .BCW1W3 2S =-BAW.WLF t l i t - D B W + W z 1 Z3 = -AcW,W3 2JZ - C D W 3 W J £ / R = - PC \A4 W3 98 One can show t h a t 1ir; (cZ//t/t')T*.[(iirT)cZ (/>'rr)^J7k[CptrT)K C<J'^)CTZJ--2CY/^46>/Z) 99 One a l s o has AK- (/-t^fa J#^J_ (/+c#»le/z) B * _ £ JsuSe/i yttSe/z C = _J_ ( / f ^ e / i f CD-J_ (/+^1&A) £)= 4&>*0/2 Ofr+e/z AC- __/ (/-ts**le>/z)0+&ol'6/i) AP - _Z (/+&>*'L0/z) B C - . _ L . ' ( / S P - _±_ ~ ( X f a l / * ) J (1 + 6 u&L9A + /fc&e/z / V/ +£&**A)//H»? *g>/2 100 Appendix F. The G r o s s - s e c t i o n f o r the S c a t t e r i n g of Two I d e n t i c a l M a s s i v e S c a l a r P a r t i c l e s . I n t h i s Appendix the c r o s s - s e c t i o n f o r the s c a t t e r i n g o f two i d e n t i c a l , m assive, n e u t r a l s c a l a r p a r t i c l e s i s worked out i n d e t a i l . The i n i t i a l momenta of the p a r t i c l e s a re denoted by p and q and the f i n a l momenta by p' and q 1 . One must compute \ where ift) Si -- {-*l/e)JJJ*Jy<ojpc/,/,,(*)/><£(y))le?/Y[T^/*) Zsfy)] Here, W M -- -Zr/i t - 4»£ },„ & - k' j One decomposes f i n t o p o s i t i v e and n e g a t i v e f r e q u e n c y p a r t s (fi) fa- -rfa Then (fs) fa fa Jfa + f a fa -if"£)( fa fa + i fa fa + fa fa) where use has been made of the f a c t s t h a t e v e n t u a l l y t he normal o r d e r i n g w i l l be t a k e n and t h a t T^ y i s m u l t i p l i e d b y <o/P/^y/z) h/3(y))l^> which i s symmetric i n u- and v. A 101 s i m i l a r e x p r e s s i o n h o l d s f o r T o ( j ) t I t i s seen t h a t i n the e x p r e s s i o n s f o r T and T n t h e terms [LV aB are of the g e n e r a l form, n e g l e c t i n g d i f f e r e n c e s i n s u b s c r i p t s , (f7) f f , , / V " I n the normal p r o d u c t , /[//"Z^ (*) ~~&/3(y)J , terms o f the g e n e r a l form do not c o n t r i b u t e to the p r o c e s s under c o n s i d e r a t i o n . ^he r e m a i n i n g terms. i n the normal p r o d u c t are of the f o l l o w i n g forms or can be put i n one of the f o l l o w i n g forms ( n e g l e c t i n g s u b s c r i p t s ) ( f 9 ) fr(*)Jf(») frfy) $-(/) J fa) *f<y) 'ft/) . The terms i n the normal p r o d u c t of the form g i v e n i n ( F 9 )» c o n t r i b u t e t o a p r o c e s s whose graph i s That p a r t of the i n t e g r a n d of the s c a t t e r i n g m a t r i x 102 which, c o n t r i b u t e s to t h i s p r o c e s s ( t h a t i s , t h a t p a r t of the i n t e g r a n d c o n t a i n i n g terms of the form (F10)) i s ^ i* 5- }-+ & ^ fa fa ; / ; ^ _ ^ » * f a f a far + ^ z ftU fa ft>~^~ ' £f ^ l f t * ^ ' f y ' f t fa 4 7 1 IJ j 4- / ' - (2Xz)^olp(^)^4(Y)loy where 'fa fa fa j 3--- 4- f l / , //V f j 'fm h- - &y fa far fa 4 - fa ?,= <S~ mlfit *ffv j h -C C Z ' 2 j ^i-faj 1, - &„(/3 /y>* d> f- 7 7 ' ^ -I n ( F l l ) and (F12) the p o s i t i v e f r e q u e n c y p a r t s are f u n c t i o n s of x and the n e g a t i v e f r e q u e n c y p a r t s are f u n c t i o n s of y. 103 One l e t s f fv P fivr ' (V 1 l Z ! j t 7 / v p T t / y (V 1 S i n c e e v e n t u a l l y one forms the s c a l a r p r o d u c t ^p'f11 one can use f o r purposes of computation 7 (v (zUF v T? fT^ Each d e r i v a t i v e i n ?/~?? i n t r o d u c e s the c o r r e s p o n d i n g component of />, /> or j . These components;, can be summed w i t h v which appears i n < olP(hM/(*) >K<A (y))M. One has - f(fWf) + (/fXff'J - (/f'J(/f)J + 104 ?6A - Z^Crf') f'frfrfi-l l A -- - m2 A/fJ fifaft~ 7,A - ft'fififi J/ + 2j+ >/~ h-Then where In the center-of-momentum frame (W) (/>f)= Cf -AV-AY^^J ^r°fa' A V 2 i j 'A " z<^°&) •tyf'J- p 1 ^ 9 "fa - 78 z6- t/l6&&) 105 • where (Flo) if-- ///> . (F18) becomes, w i t h the h e l p of ( F 1 9 ) , The m a t r i x element i s t h e n The terms i n the normal p r o d u c t of the form g i v e n i n ( F 9 ) , c o n r i b u t e t o c e s s e s whose: g aphs are 106 0) \ if (a) The matrix element, s^, f o r process ( l ) can be obtained i n the same manner as s^. Or, one can obtain s^ from the matrix element given by Corinaldesi(1956) or Barker et al(l966) f o r the i n t e r a c t i o n of two sc a l a r p a r t i c l e s with d i f f e r e n t masses. One obtains •F t / v ) c 0 3 . l&/2 } The matrix element, s 2 , f o r process (2) can be obtained from that f o r process ( l ) by interchanging p and q. One obtains •F ^ Y>v i/ /W? } The complete matrix element i s (FU) S = j ; y j i + f3 and M, defined by ( 7 . 1 ) , i s 107 The c r o s s - s e c t i o n i s (Flfi) (J*fa si) - (X 4k rr) ////1 Thus i t i s seen t h a t the term g i v e n by DeWitt(l967) i n h i s e q u a t i o n (3.10) s h o u l d r e a d (f3o) (ifv^ci-vy . 108 Appendix G. Notes on N e u t r i n o s T h i s Appendix complements Chapters 4 and 5 i n which f o u r and two component n e u t r i n o s are d e s c r i b e d . I n p a r t i c u l a r , a l l the dyn a m i c a l v a r i a b l e s and c o n s t a n t s of the motion due t o i n t e r n a l symmetries are worked out; the r e d u c t i o n of the f o u r component n e u t r i n o e q u a t i o n t o two, two component e q u a t i o n s i s g i v e n e x p l i c i t l y ; and, the l a t t e r two e q u a t i o n s are o b t a i n e d from f i r s t p r i n c i p l e s . The L a g r a n g i a n which d e s c r i b e s the f o u r component n e u t r i n o i s by d e f i n i t i o n t he e l e c t r o n - p o s i t r o n L a g r a n g i a n w i t h t h e mass s e t t o z e r o . Thus, (c7i) / - - -f IP where ($1) f.sf'.f (S3) (C74) ^ , 10 -A. i 0 <r3- /, o o -I I-- I I 0 \ 6 I T h i s L a g r a n g i a n d i f f e r s from (4.1) i n t h a t i t i s not r e a l . T h i s does not make any d i f f e r e n c e t o t h e f o l l o w i n g remarks, The f i e l d e q u a t i o n i s (cnS) Vft^O There are two s o l u t i o n s c o r r e s p o n d i n g t o E=iJpj. 109 They are (GG) ft - fa (p)e ft Kffje where (67; fa, / M , \ fa: fa I^ W ^ ( f. w i t h p and fjL s a t i s f y i n g n i s g i v e n by ( 4 . l l ) . I t i s (Gi9) n = P//£/ E x p l i c i t forms o f /J, and fjL are g i v e n by (4.12). They have the n o r m a l i z a t i o n and o r t h o g o n a l i t y p r o p e r t i e s (CIO) |£ ^ = &s The s o l u t i o n s UlX and l£ s a t i s f y (6Jl) ' = <L J = ^ fan O (612) Tffa-f)- a*(P) > = # ^ (<?/-*) - ^ = -fa / i i i . -irz (Ctf) Y*a, - -fa, Xsfa =- i/i Ysv, if, r s n - K where h, the h e l i c i t y o p e r a t o r , and J r , t h e c h i r a l i t y 110 o p e r a t o r , a re (Cy/S) h « f(T.ti O \ V ~- I O -I-An a r b i t r a r y s o l u t i o n of the f i e l d e q u a t i o n can be expanded as • (C7/6) j ^ - J . II Qft(?)UJp)e +£Jr)K(r)e • where (Cn/7) p = (f,UE/) . I f one now d e f i n e s 4- (/>) by im) k (?) far) , A (?) = fa-?), and uses (G12), one o b t a i n s f o r the expansion (G16) (Cnl?) fa^ITaA (?) £C ( ? ) e ^  + fa?) 4. (?) e ^ One can w r i t e where (CnZl) fR= j.(i-r)<f> , # = J.(, + r* so t h a t The l e t t e r s R. and L s t a n d f o r r i g h t and l e f t . T h i s I l l (V F n o t a t i o n i s used because (Gi3) Jfe = fa 9 i f i --</i . One then has (V F These forms are u s e f u l when the two component t h e o r y i s c o n s i d e r e d . I f one now makes the t r a n s i t i o n t o quantum f i e l d t h e o r y t h e tf/i, 4. 5 O-t•» and <4 are i n t e r p r e t e d as o p e r a t o r s such t h a t the Zn. d e s t r o y n e u t r i n o s , t h e 4i d e s t r o y a n t i n e u t r i n o s , the <3i c r e a t e n e u t r i n o s , and t h e 4 c r e a t e a n t i n e u t r i n o s . The anticommutation r e l a t i o n s a re (<S2S) la*,a*fl = {k^l}= X* a 1 1 o t h e r s a r e z e r o -The d y n a m i c a l v a r i a b l e s and c o n s t a n t s o f the motion due t o i n t e r n a l symmetries are now worked out i n or d e r t o c h a r a c t e r i z e the v a r i o u s n e u t r i n o s . The energy d e n s i t y i s (tee) - : f r r i + ' V 1 and the energy i s (CZi) //=J&c/Jx - 1 1 aJp{ aj (?) fat?) + hUr)k(r)] t h * The k component of t h e momentum d e n s i t y i s 112 t h and the k component of momentum i s (CZ9) ft =J(?J* = J I f r UXf) aJp) * I'ft)i(f)l The component of the s p i n d e n s i t y i n the d i r e c t i o n n i s too; Jf * i\f&--t<?: where / (631) <r; = / <r; o and the component of the s p i n i n the d i r e c t i o n ~n i s (C-?3Z) Sn = ft*?* = '/jff*ir«> + ^ k l The L a g r a n g i a n ( G l ) i s i n v a r i a n t under the t r a n s f o r m a t i o n (63S) (/-> e ^ f The i n v a r i a n c e l e a d s t o a c u r r e n t (634) f = -\U (A f)<f\ The q u a n t i t y which i s conserved i n time i s (C3S) Q-.-JjY** -'.Jfl% Vs)fd3* ; p which i s the same as (G32). The L a g r a n g i a n ( G l ) i s a l s o i n v a r i a n t under the t r a n s f o r m a t i o n 113 The i n v a r i a n c e l e a d s t o a c u r r e n t X The conserved q u a n t i t y i s (0,38) Z = :Jtf l$*yi*x\ ^(-A ) l [ a h + 4*i - l > t h i T h i s s t a t e s t h a t t he t o t a l number of n e u t r i n o s minus the t o t a l number of a n t i n e u t r i n o s i s a c o n s t a n t . T h i s i s j u s t c o n s e r v a t i o n o f l e p t o n number. A s u i t a b l e assignment of l e p t o n numbers i s L=+l f o r n e u t r i n o s and L = - l f o r a n t i -n e u t r i n o s . T h i s completes the s p e c i f i c a t i o n of the v a r i o u s n e u t r i n o s . The p r o p e r t i e s a re summarized i n the f o l l o w i n g t a b l e . Part/cYe ft y, Creation Operator Desi ritct/on . Operator a? a, r b,f k Energy IE/ IEI in Momentum p P P ^-F SP)>n J Project ten Yz -Yt Fe/ic/fy 1 -1 -I / Leplon M> I 1 -1 -/ Afaiwriitk y* The n e u t r i n o s 14. , , FM , and which appear i n the r e a c t i o n s + e + y„ J T ~ - P fa< yes * e**- +fa 114 may be. i d e n t i f i e d w i t h • P{ , and ^  as f o l l o w s : Upon t a k i n g l i n e a r combinations o f ( G 3 5 ) and (G38), one f i n d s t h a t (639*) I, I (a/a, -ML r (6391) L - j ( f a - i / k ) are conserved. I n c o n j u n c t i o n w i t h the above assignment of l e p t o n number, the charged l e p t o n s e~ and \i+ have l e p t o n number +1, and the charged l e p t o n s e + and u"~ have l e p t o n number - 1 . The f o r m a l i s m d e s c r i b e d here f o r the ^  and ^  n e u t r i n o s i s d i s c u s s e d by Bludman ( l963). Kerimov and Romanov(l965) g i v e o t h e r r e f e r e n c e s . I n t h i s f o r m a l i s m t h e r e are two gauge t r a n s f o r m a t i o n s , (G33) and (G36), of a s i n g l e f o u r component f i e l d , which l e a d t o two independent c o n s e r v a t i o n laws (G35) and (G38). An a l t e r n a t i v e xvay o f d e s c r i b i n g the i/@ and ^  n e u t r i n o s i s t o d e f i n e two independent f o u r component f i e l d s (/? and /z . The f i e l d , l//c , which d e s c r i b e s t h e 4> n e u t r i n o , and the f i e l d , /, , which d e s c r i b e s t h e ^  n e u t r i n o , are The e q u a t i o n s f o r the two ty p e s of n e u t r i n o s a re then i d e n t i c a l . There i s one c o n s e r v a t i o n law f o r each f i e l d (/^ and /i/M . T h i s a l l o w s one t o d e f i n e an e l e c t r o n number 115 L which, i s c o n s e r v e d i n a l l r e a c t i o n s , and a muon number L which i s a l s o c o n served i n a l l r e a c t i o n s . T h i s i s the f o r m a l i s m used by Lee and Wu(l965) and by Lederman(l967). A d i s c u s s i o n o f the ti^o f o r m a l i s m s i s g i v e n by Ilarshak, R i a z u d d i n , and Ryan(l968). I n t h i s c o n n e c t i o n , the u.+ i n t h e i r T a b l e 3.6 s h o u l d be a u~. I t i s now. shown t h a t and (/{ d e f i n e d by (C-21) can be reduced t o a form i n which they have o n l y two non-zero components. I f one s e t s where (W) S = _L { 1 I ) T l i -1/ one o b t a i n s 116 and T h i s shows t h a t and ^ have o n l y two non-zero components. The <f r e p r e s e n t a t i o n used above i s c a l l e d the Kramers r e p r e s e n t a t i o n of the D i r a c m a t r i c e s . The L a g r a n g i a n ( G l ) i n the Kramers r e p r e s e n t a t i o n i s T h i s L a g r a n g i a n i s i n v a r i a n t under the t r a n s f o r m a t i o n Mi) % -> > & -> r/- eMr^ T h i s y i e l d s a c u r r e n t i^e) f = -M ~ rtx)rt<r>yt 0 d 'g. jp where The q u a n t i t y c o n s e r v e d i n time i s (ClSo) L', - . J t i ' K = Z ( « ' < > . - W A ^ ) which i s j u s t (GJ9a). The L a g r a n g i a n (G4-6) i s a l s o i n v a r i a n t under the t r a n s f o r m a t i o n 117 which y i e l d s the conserved q u a n t i t y ($si) / / = I(faat - / / / J p which i s gust (G39b;. The sum of and I ^ t h e n y i e l d s (G38), and the d i f f e r e n c e of and L 2 y i e l d s (GJ5). The f i e l d e q u a t i o n s f o r ^ and ^ f o l l o w e i t h e r from the L a g r a n g i a n (G46) or from the a p p l i c a t i o n of the t r a n s f o r m a t i o n (G4-2) t o the f i e l d e q u a t i o n s (G5). They are (C7S3) I O i c k z + i i 0 ){%J - 0 or 1 = 0 cr O There are two s o l u t i o n s f o r each e q u a t i o n . They are where ( C S 7 ) U--ft } A r b i t r a r y s o l u t i o n s of (G54-) and (G55) can be expanded as 1 1 8 (GSS) > i ' ^ J « <f)«(?)<* ** ^ t/fi e ' % --XZa, tt) *<f) e'f* + I (?) *ft e r " M ) I f one s e t s Us 9) l.(p= £(-?) J AJP)--A('P) and uses the f a c t t h a t ( G 6 0 ) v ( - p ) one o b t a i n s (66/) YA*^J 0fafj#f/)e*x s J / f t s f l e ^ * which agree w i t h (G4-4-) , as they must. The e q u a t i o n s f o r 9*. and ?£ can be o b t a i n e d from f i r s t p r i n c i p l e s . There a r e two, two d i m e n s i o n a l r e p r e s e n t a t i o n s of the L o r e n t z group which y i e l d e q u a t i o n s f o r m a s s l e s s p a r t i c l e s o f s p i n l/2(Schweber 1961). The two f i e l d e q u a t i o n s f o r the f i e l d s f and j" which t r a n s f o r m a c c o r d i n g t o t h e s e two r e p r e s e n t a t i o n s a re where 1 1 9 ^ = - W V>^  The f i e l d e q u a t i o n s become, upon s u b s t i t u t i o n of (G63) i n t o (G62), and 0 ' i ^ 4- • . i i + / 2 i ~ - i AxJ 0 -f ° 1 ) -i I 0/ o \b + ( l o U _ //1 o U which a r e , r e s p e c t i v e l y , the same as the e q u a t i o n s (G54-) and (G55) f o r and . Thus, (&et) Y, - I , % -. J The i n v a r i a n t s a re IP where 'o I 2 O -A and where 120 dp. and H-L are j u s t the L a g r a n g i a n d e n s i t i e s which appear i n (G46). One f i n a l word i s n e c e s s a r y . I n Chapter 5 the two component n e u t r i n o f i e l d ^ i s i s expanded as 1 Fv r S i n c e n e u t r i n o s , i . e . l e f t - h a n d e d p a r t i c l e s , a r e under c o n s i d e r a t i o n , h ana b' are used as a n n i h i l a t i o n and c r e a t i o n o p e r a t o r s . These a n n i h i l a t e and c r e a t e l e f t -handed p a r t i c l e s . T h i s i s the approach f o l l o w e d by Roman(l960). An a l t e r n a t i v e procedure i s t o choose"^" as Y . Then ^ i s expanded as f I n t h i s case, a and a are a n n i h i l a t i o n and c r e a t i o n o p e r a t o r s f o r n e u t r i n o s , i . e . l e f t - h a n d e d p a r t i c l e s , f o r t h e y a n n i h i l a t e and c r e a t e l e f t - h a n d e d p a r t i c l e s . The m a t r i x elements f o r the p r o c e s s e s i n v o l v i n g two component n e u t r i n o s are independent of the f o r m a l i s m used. T h i s can be seen as f o l l o w s . I f one t a k e s \-nr"L.-) the terms i n the m a t r i x elements which c o n t a i n the ^ f i e l d are of the form where th e (r^ are g i v e n by (G70). The m a t r i x elements i n v o l v i n g two component n e u t r i n o s which appear i n the p r e v i o u s C hapters c o n t a i n terms of the form 121 where the are g i v e n by (g68). I t i s n e c e s s a r y t o show t h a t (G-73) and (g74-) are e q u a l . One can show t h a t cr s a t i s f i e s t he same e q u a t i o n t h a t '/i does so t h a t one may t a k e ( C i l S ) 7l = c r 2 ^ / which i m p l i e s t h a t Upon s u b s t i t u t i o n o f (g76) i n t o (g74-), one o b t a i n s But, (die) [il' <rx*> „ s tf') - U T - ^  V " < r ' V + J T F u r t h e r , Then (G77) becomes T h e r e f o r e , up t o an unimportant s i g n , t h e m a t r i x elements (G73) and (g74-) are e q u a l . 

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