UBC Theses and Dissertations

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UBC Theses and Dissertations

Composition of massive from massless bosons Von der Lin, Val Allen 1971

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COMPOSITION OF MASSIVE FROM MASSLESS BOSONS b y VAL VON DER L I N B . S c . ( H o n . ) , The U n i v e r s i t y o f C a l g a r y , 1970 A THESIS SUBMITTED I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n t h e d e p a r t m e n t o f PHYSICS We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A u g u s t , 197-1 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 requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a 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 t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depa rtment The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date i i ABSTRACT M a s s i v e b o s o n s a r e c o n s t r u c t e d f r o m g e n e r a l i z e d b i p a r t i c l e s t a t e s 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 ( " z e r o n s " ) and o f m a s s l e s s v e c t o r p a r t i c l e s ( p h o t o n s ) . The s e l f -c o u p l i n g o f t h e z e r o n f i e l d Q-Q i s d e s c r i b e d by t h e L a g r a n g i a n d e n s i t y g Q c A . The s e l f - c o u p l i n g o f t h e p h o t o n f i e l d i s d e s c r i b e d by t h e d u a l i t y i n v a r i a n t M i s n e r - W h e e l e r L a g r a n g i a n d e n s i t y . D o u b l e t mass s p e c t r a a r e g e n e r a t e d i f t h e c o u p l i n g c o n s t a n t s t a k e t h e a p p r o p -r i a t e s i g n s . E a c h s u c h s p e c t r u m i s f i t t e d t o t h a t o f an e x p e r i m e n t a l l y o b s e r v e d meson d o u b l e t h a v i n g t h e c o r r e c t s p i n and p a r i t y . The c o u p l i n g c o n s t a n t s and t h e c u t - o f f s t h a t a r e i n t r o d u c e d a r e t h e r e b y d e t e r m i n e d u n i q u e l y . The c o u p l i n g c o n s t a n t and t h e c u t - o f f A i n t h e z e r o n model assume t h e v a l u e s g Q - -9•50 and A = 515 n.u. I n t h e p h o t o n m o d e l , b o t h s c a l a r a n d p s e u d o s c a l a r p a r t i c l e s a r e c o n s t r u c t . e d . The r e s p e c t i v e c u t - o f f s and c o u p l i n g c o n s t a n t s a r e A ( + ) = 729 n.u., gw~ 1 . 6 7 x 1 0 " ' ' n.u. and \ = 574 n.u,, g w * 3 . 3 4 X 1 0 " 5 n.u. 1X1 CONTENTS • • • • 11 i i i i v v v i 1 ABSTRACT CONTENTS «••€»••**•••»•*«••&«••••*• L I S T OF FIGURES • • • • • • • • • • • • • • • » * e L»IST OF TABJLES ••••#*«««»»«*»*««e ACKNOWLEDGMENTS INTRODUCTION 1. KINEMATICAL CONSIDERATIONS 2 2. SUMMARY OF FORMALISM 4 3. SELF-COUPLING OF MASSLESS FIELDS 10 « BIFARTIGLE STATES « # « « e « f » t « » * * « » « * « * * « t * « * « * g « * 17 5. CALCULATION OF THE MASS SPECTRUM OF COMPOSITE PARTICLES AS AN EIGENVALUE PROBLEM I : THE _jJ_RON MODEXJ • •e**«»ee*»«»e«ece*e«eeee«»*«®*»#»e« 20 6. CALCULATION OF THE MASS SPECTRUM OF COMPOSITE PARTICLES AS AN EIGENVALUE PROBLEM I I : THE PHOTON IVIODEIJ •••*•••••••••*••»•»•••»••••»•»•»••• 2^ f-7. DISCUSSION OF RESULTS 30 REFERENCES 34 APPENDIX A: ON EQUATION [3.12] 35 APPENDIX B: ON EQUATION [6.1] 39 APPENDIX C: ON EQUATION [6.3] 45 L I S T OF FIGURES G r a p h o f e q u a t i o n [5.141. . . . . G r a p h o f e q u a t i o n [ 6 . 1 9 ] . . . . . V L I S T OF TABLES C o m p a r i s o n o f m o d e l s v i ACKNOWLEDGMENTS I am much i n d e b t e d t o D r . F. A. K a e m p f f e r f o r i n t r o d u c i n g me t o t h i s most i n t e r e s t i n g work and f o r ' h i s g e n e r o u s g u i d a n c e w i t h r e g a r d t o t h i s t h e s i s . I am g r a t e f u l t o my w i f e f o r h e r c o n s t a n t e n c o u r -agement and s u p p o r t . I a l s o w i s h t o t h a n k t h e N a t i o n a l R e s e a r c h C o u n c i l o f Canada f o r a p o s t g r a d u a t e s c h o l a r s h i p . 1. INTRODUCTION The p r o c e d u r e o f u s i n g t h e s e l f - c o u p l i n g o f mass-l e s s f i e l d s t o c o n s t r u c t m a s s i v e b o s o n s has b e e n shown p o s s i b l e i n t h e c a s e o f f e r m i o n s f o r t h e V-A i n t e r -a c t i o n ( K a e m p f f e r 1970) and f o r t h e r e n o r m a l i z a b l e Nambu i n t e r a c t i o n ( E s c h 1 9 7 1 ) . S i m i l a r c a l c u l a t i o n s a r e r e p o r t e d i n t h i s t h e s i s f o r t h e c a s e o f m a s s l e s s b o s o n s . I t i s shown i n s e c t i o n 1 t h a t t h e r e a r e no k i n e -m a t i c a l o b j e c t i o n s t o m o d e l s o f t h i s s o r t . An o u t l i n e o f t h e n e c e s s a r y f o r m a l i s m e m p l o y e d i n t h e q u a n t i z a t i o n o f t h e f r e e z e r o n f i e l d and o f t h e f r e e e l e c t r o m a g n e t i c f i e l d , u s i n g t h e r a d i a t i o n g auge, i s p r e s e n t e d i n s e c -t i o n 2 . I n t e r a c t i o n L a g r a n g i a n s t h a t a r e c h o s e n f o r t h e m a i n c a l c u l a t i o n a r e i n t r o d u c e d and m o t i v a t e d i n s e c t i o n 3« A f t e r i n t r o d u c i n g t h e n o t i o n o f g e n e r a l i z e d b i p a r t i c l e s t a t e s f o r z e r o n s and p h o t o n s i n t h e c e n t e r -of-momentum and d e t e r m i n i n g t h e i r s p i n - p a r i t y a s s i g n -ments i n s e c t i o n 4> t h e c a l c u l a t i o n s o f t h e mass s p e c t r a o f t h e c o m p o s i t e p a r t i c l e s a r e p r e s e n t e d i n s e c t i o n 5 f o r t h e z e r o n model and i n s e c t i o n 6 f o r t h e more i n v o l v e d p h o t o n m o d e l . 1, KINEMATICAL CONSIDERATIONS S i n c e the- t o t a l p r o p e r mass o f a s y s t e m o f f r e e , z e r o mass p a r t i c l e s i s , i n g e n e r a l , n o n z e r o ( T e r l e t s k i i 1968), t h e r e i s no b a s i c k i n e m a t i c a l o b j e c t i o n t o a m o d e l o f m a s s i v e p a r t i c l e s t h a t a r e c o n s i d e r e d t o be c o m p o s i t e s o f m a s s l e s s o n e s . C o n s i d e r N f r e e p a r t i c l e s h a v i n g p r o p e r masses m-^ , m^, . . ., m^, w i t h r e s p e c t i v e e n e r g i e s and momenta [1.13 E i = 7±n±; ELx" 7±mA= E i - i ' 1 = 1 ' 2> * * *' N> where ' X ^ l l - v f ) ~ 2 and v ^ i s t h e v e l o c i t y o f t h e i p a r t i c l e . The sum o f t h e p r o p e r masses o f s u c h i n d i v i -d u a l p a r t i c l e s , . t l . 2 ] Hm,- Z_ ( E 2 - p M = ^7l\, i i i x i s a s c a l a r u n d e r L o r e n t z t r a n s f o r m a t i o n s . I t i s , i n g e n e r a l , d i f f e r e n t f r o m t h e s c a l a r [1.3] m s = ( E 2 - p 2 ) * f o r m e d w i t h t h e t o t a l e n e r g y E= 2 ^ _ E j _ a n c * t n e t o t a l momentum p_= T~^R^ °£ a n i s o l a t e d s y s t e m S c o n s i s t i n g o f N s u c h p a r t i c l e s . The s c a l a r m may be r e g a r d e d as t h e t o t a l p r o p e r mass o f t h e s y s t e m b e c a u s e i n t h e c e n t e r o f momentum f r a m e pj=0 i t r e d u c e s t o [1.4] m = H E . = E. s i 1 3. S u b s t i t u t i n g f r o m [1.1] i n t o t l . 4) r e v e a l s t h e n o n - a d d i t i v i t y p r o p e r t y o f t h e p r o p e r m a s s e s , s i n c e now C1.53 m = H ?,ra. > £ m. . s ± i i ± 1 The p r o p e r masses a r e a d d i t i v e o n l y i n t h e s p e c i a l c a s e o f i d e n t i c a l v e l o c i t i e s , v^=v, /} / i= y, e n a b l i n g one t o w r i t e [ 1 . 6 ] m = Z E, ( 1 - V2P = E m . ^ ( 1 - v 2 ) * - Z m. . s i 1 i 1 " i 1 F o r t h e s p e c i a l c a s e o f s y s t e m s o f m a s s l e s s p a r t i c l e s , [1.33 c a n be w r i t t e n i n a c o n v e n i e n t f o r m . F o r s y s t e m s o f o n e , two and t h r e e m a s s l e s s p a r t i c l e s r e s p e c t i v e l y , r-\ ~,2 _ n [1.7b] m 2 = 2 E 1 E 2 ( l - v 1 . v 2 ) and [ l . 7 c ] m 2 = a E ^ g l l - v ^ . v 2 ) - f 2 E 1 E 3 ( l - v 1 . v 3 ) + 2 E 2 E 3 ( l - v 2 - v 3 ) , where e a c h Jv^j = 1. I t i s a p p a r e n t t h a t o n l y i n t h e c a s e when a l l p a r t i c l e s a r e m o v i n g i n t h e same d i r e c t i o n i s t h e t o t a l p r o p e r mass o f t h e s y s t e m a d d i t i v e , t h a t i s , e q u a l t o z e r o . An o r d i n a r y l i g h t beam, f o r e x a m p l e , t h a t e x h i b i t s any d i v e r g e n c e a t a l l h a s a nonzero' p r o p e r mass. O n l y an i n f i n i t e p l a n e wave, u n a t t a i n a b l e i n p r a c t i c e , w o u l d have z e r o p r o p e r mass. 4* 2. SUMMARY OF FORMALISM F r e e raassless p a r t i c l e s o f s p i n 0, c a l l e d " z e r o n s " , a r e d e s c r i b e d by t h e s c a l a r f i e l d o p e r a t o r ( L u r i e 1968) 12.1] o - ( x , t ) = jLT~IIlia(k)e~±kx + b f ( k ) e i k x ] k and i t s h e r m i t e a n a d j o i n t J2.2] ^ ( x , t ) - i r - / L r a + ( k ) e i k x . + b ( k ) e - i k x ] , ° Vv 2k where k x = - k ' X + 1 k\t and |k) = k. The c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s a* , b1" , a and b s a t i s f y t h e c o m m u t a t i o n r e l a t i o n s a l l o t h e r t 3 = 0 . The f i e l d o p e r a t o r s s a t i s f y t h e wave e q u a t i o n 12.4] L j c r ( . x , t ) = D c r + ( x3t ) = 0 . 4-U n l e s s t h e f i e l d i s h e r m i t e a n , cr^ = cr , w h i c h amounts t o h a v i n g a ( k ) = b ( k ) , t h i s d e s c r i p t i o n c a n accommodate t h e p r e s e n c e o f some d i c h o t o m i c a t t r i b u t e o f t h e z e r o n s , s u c h as e l e c t r i c c h a r g e o r h y p e r c h a r g e , i f r e q u i r e d . The f r e e d o m t o p u t a ( k ) = b ( k ) a t any s t a g e w i l l be r e t a i n e d . The f r e e - f i e l d L a g r a n g i a n d e n s i t y LQ t h a t l e a d s t o C.2.4] v i a t h e a c t i o n p r i n c i p l e , [ 2 . 5 1 S J L d^x - 0 , n. i s an a r b i t r a r y 4 - v o l u m e , i s g i v e n by (Roman 1969) [ 2 . 6 ] L Q = ^ c r ^ "o^o^, summation o v e r = 0 , 1 , 2 , 3 . V a r i a t i o n w i t h r e s p e c t t o or g i v e s t h e f i e l d e q u a t i o n s f o r crj" and c o n v e r s e l y . The c a n o n i c a l momentum i s c o n s t r u c t e d f r o m L by o J t h e s t a n d a r d p r e s c r i p t i o n ^Lo and t h e f r e e H a m i l t o n l a n d e n s i t y HQ i s f o r m e d by 12.3] Ho = T r0^0- L0 = i - [ T T 2 + V c r - v c r ] . H i s p o s i t i v e - d e f i n i t e as r e q u i r e d . The e q u a l - t i m e c o m m u t a t i o n r e l a t i o n s a r e 1 2 . 9 ] [ c r ( x ) , c ro(X' ) ] = [ TTQ (x) , TTo (x' )] = 0 , [ 0 3 ( x ) ,Tro(x»)] x = x, = i S ( x - x ' ) , x = ( x , x 0 ) . F r e e m a s s l e s s p a r t i c l e s o f s p i n 1, p h o t o n s , may be d e s c r i b e d by a 3 - v e c t o r o p e r a t o r ( L u r i e 1968) 2 [2.101 A ( x , t ) = jk) ) / — 6 ( k , X ) [ a ( k , X ) e "l k x + af ( k A ) ei k x] , where a+( k , \ ) and a ( k , \ ) a r e c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s r e s p e c t i v e l y o f f r e e p h o t o n s o f momentum k Co) h a v i n g l i n e a r • p o l a r i z a t i o n s l a b e l l e d by TV. A ( x , t ) s a t i s f i e s t h e wave e q u a t i o n C2.111 • A- ( x , t ) - 0 and t h e t r a n s v e r s a l i t y c o n d i t i o n , i n t h e r a d i a t i o n g a u g e , [2.121 yvT\x,t) = 0 . C l a s s i c a l l y , [2.121 and [2.111 a r e e q u i v a l e n t t o M a x w e l l ' s f r e e - f i e l d e q u a t i o n s . The c o m m u t a t i o n r e l a t i o n s o b e y e d b y t h e o p e r a t o r s a(k,Tv.) a n d a+(k,lv) a r e [2.131 [a ( k,\),a - K k J,V)] = £ ^ t . I n t h i s gauge t h e e l e c t r i c f i e l d o p e r a t o r i s - a + ( k , X ) e i k x ] and t h e m a g n e t i c f i e l d o p e r a t o r i s [a(k,l\)e - i k x a + ( k , v ) e x ^ ] , 7. where u s e h a s b e e n made o f t h e r e l a t i o n s [2.161 k y _ ( k , l ) - k e ( k , 2 ) ; k x _.(k,2) = -k e ( k , l ) . O t h e r p r o p e r t i e s o f t h e p o l a r i z a t i o n v e c t o r s t h a t w i l l be u s e d l a t e r a r e [2.173 G ( - k , l ) - - _ ( k , l ) 12.18] _(-k,2) = e(k,2) [2.19] 6 ( - k , X ) . e ( k , V ) = ( - l | " ^ v . S i n c e t h e s t a t e s o f l i n e a r p o l a r i z a t i o n a r e n o t e i g e n s t a t e s o f t h e s p i n , one i n t r o d u c e s s t a t e s o f t r a n s -v e r s e c i r c u l a r p o l a r i z a t i o n b y r o t a t i n g t h e r e f e r e n c e b a s e o f t h e p o l a r i z a t i o n v e c t o r s . The c o r r e s p o n d i n g c r e a t i o n o p e r a t o r s f o r c i r c u l a r l y p o l a r i z e d p h o t o n s t a k e t h e f o r m [2.203 a+(k) = 2"^[a+(k,l)+ia f(k,2)3 f o r r i g h t - h a n d e d p o l a r i z a t i o n and [2.21] b+(k) = 2 " * [ a + ( k , l ) - i s f ( k , 2 ) ] f o r l e f t - h a n d e d p o l a r i z a t i o n . The t o t a l p h o t o n o c c u p a t i o n number o p e r a t o r i s [2.22] N ( k ) = a + ( k ) a ( k ) + b + ( k ) b ( k ) = a + ( k , l ) a ( k , l ) + a - t - ( k , 2 ) a ( k , 2 ) . A s a t i s f a c t o r y L a g r a n g i a n d e n s i t y f o r t h e f r e e , ' M a x w e l l f i e l d - i s ( B j o r k e n , D r e l l 1965) [2.23] L o - ^ E M 2 - E W 2 ) = i - [ r 2-(^A M) 2]. ' o — F r o m t h e c a n o n i c a l 3-momentum, 12.241 Ho - ||^) - A(t" = - E C o \ t h e ( p o s i t i v e - d e f i n i t e ) H a m i l t o n l a n d e n s i t y may be c o n s t r u c t e d : 12.25] H 0 = ILy A ° - L o = i-(E'*>2+B°'2). A d i f f i c u l t y now a p p e a r s . The u s u a l f o r m o f t h e e q u a l -t i m e c a n o n i c a l c o m m u t a t i o n r e l a t i o n s , n a m e l y I 2 . * 6 j L A ^ i x ) > 1 T 0 j i 3 C i ) j - i o ^ O ^ ' x - X / ) , { i , j M I > 2 , 5 ) i s i n c o n s i s t e n t w i t h t h e t r a n s v e r s a l i t y c o n s t r a i n t s V-A = V-E = 0 i n f o r c e h e r e , a s c a n r e a d i l y be s e e n b y t a k i n g t h e d i v e r g e n c e o f 12.26]. F o r t h e n , a t e q u a l t i m e s , [2.27] [ Y - A f c \ x ) , T T ^ ( x ' ) ] = i O j S ( x - x ' ) $ 0. The c o r r e c t c a n o n i c a l c o m m u t a t i o n r e l a t i o n s a r e g i v e n i n s t e a d b y ( L u r i e 1 9 6 8 ) [2.28] L A ^ x ) , T T . ( x « ) ] t = = t f = i ( g . . - ^ i ] S ( x - x ' ) , 9. where the symbol V represents the o p e r a t i o n [2.291 ^2 l (_ ) = j D ( x - _ M V ( x ' ) d _ f , where : 1 [2.30] D(x-x) = 7 ^ _ ^ j t ^ ( t + | x - _ M )-^(t-|x-xM )] • 1 0 . 3. SELF-COUPLING OF MASSLESS FIELDS The i n t e r a c t i o n L a g r a n g i a n d e n s i t i e s L j t h a t a r e c h o s e n f o r t h i s work t o d e s c r i b e t h e s e l f - c o u p l i n g o f m a s s l e s s f i e l d s e x h i b i t as many o f t h e s y m m e t r i e s o f t h e c o r r e s p o n d i n g f r e e - f i e l d e q u a t i o n s as p o s s i b l e . The wave e q u a t i o n [2 .41 f o r t h e h e r m i t e a n z e r o n f i e l d I s c o n f o r m a l l y i n v a r i a n t and so a c o n f o r m a l l y i n v a r i a n t i n t e r a c t i o n L a g r a n g i a n d e n s i t y i s e m p l o y e d t o d e s c r i b e t h e z e r o n - z e r o n c o u p l i n g . L j = gocA i s t h e o n l y c h o i c e t h a t e x h i b i t s s c a l e i n v a r i a n c e b e c a u s e i t i s t h e o n l y c h o i c e t h a t p r o v i d e s a d i m e n s i o n l e s s c o u p l i n g c o n s t a n t . S c a l e i n v a r i a n c e , i n t h i s c a s e , i m p l i e s f u l l c o n f o r m a l i n v a r i a n c e ( C a r r u t h e r s 1 9 7 1 ) . The f u l l L a g r a n g i a n d e n s i t y L=LQ+ L j i s t h u s t3.1] L = -M_V__co - + ^ o and t h e c o r r e s p o n d i n g H a m i l t o n i a n d e n s i t y I s g i v e n by [3.2] H - ^ ov-L = _ ( v V w 4- 6,2, _ % 0 £ . S i n c e L-r i s a n o n d e r i v a t i v e c o u p l i n g , may be expanded"" i n f r e e c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s a c c o r d i n g t o [2.1] w i t h a ( k ) = b ( k ) . Normal o r d e r i n g o f t h e o p e r a t o r p r o d u c t s i s d e n o t e d by t h e u s u a l d o u b l e d o t n o t a t i o n . 1 1 . The free part becomes, after integration over a l l 3-space, [ 3 . 3 ] W0 = ZZ k a * ( k ) a ( k ) . A truncated version of the interaction Hamiltonlan i s employed by dropping a l l terms that do not contain an equal number of creation and annihilation operators. These p a r t i c l e nonconserving terms w i l l constitute a perturbation. The truncated Hamiltonian density i s [ 3 . 4 ] H j = : ^ J Z L l Z L T Z i T L ( k k t k ! t k t , , ) _ s ^ V k k 1 k" k»! r t ,t ,„ - i l k + k ' - k ' ^ k " 1 ) ^ , + , „t .„ [ a a' a n a , T ! e v + a a t a i t a m c-i(k-k'+k»-k«')-x + a a , t c „ t c _ l l T c-i(~k+k'+k"-k" 1)-x J.' t t „ m t i(k~-k'-k<>+k'n)-x , .+ „ „,t + a a T a " a m e — — ' — + a a M a t T a m T e-i(-k+k'-k»+k»')-x + a a , a n t a m t ei(k+k'-k»-k«»)-x]: where a , T= a ( k n ) e ~ ^ k etc. Integrating over a l l 3 -space, L3.5] Kr = \:HT:dx = : ^ [^(k+k ?-k''-k'") ( a f a^a»'a"' 1 J 1 4V + aa'a'^a"'1") + ^(k-k'+k ! r-k"')(a f a'a'^a'" + aa,ta"a»,t) + S(k-k' -k "+k " T ) ( aa a"ta»' + a + a ' a t t a m t ) ] : . 12. I n a d d i t i o n t o b e i n g L o r e n t z i n v a r i a n t , t h e f r e e M a x w e l l e q u a t i o n s s t i l l h o l d i f t h e f i e l d s E ^ a n d B_<0\ c a l l e d a " d u a l i t y p a i r " , u n d e r g o t h e s o - c a l l e d " d u a l i t y • r o t a t i o n " t h r o u g h a n a r b i t r a r y a n g l e <*- , ( M i s n e r , W h e e l e r 1957) C3.63 / g*<\ I c o s * s i n * \ / E ( ^ j f * ' / \ - s i n c c . cos*/\B , 0y. The i n t e r a c t i o n L a g r a n g i a n d e n s i t y c h o s e n i n t h i s w o rk t o d e s c r i b e a s e l f - c o u p l i n g o f p h o t o n s i s t h e s i m p l e s t , n o n - t r i v i a l , d u a l i t y I n v a r i a n t and L o r e n t z i n v a r i a n t e x p r e s s i o n i n o p e r a t o r s E and B, namely f 3 . 7 l i . _ = i e [ ( B2- E2)2+ 4 ( B - E )2l . - .'. '. The d u a l i t y i n v a r i a n c e o f [3«7] f o l l o w s i m m e d i a t e l y 2 2 f r o m t h e f a c t t h a t B-E and | ( E -B') f o r m a d u a l i t y p a i r . S i n c e t h i s c h o i c e amounts t o t h e i n t r o d u c t i o n o f a d e r i v a t i v e c o u p l i n g , a c o m p l i c a t i o n a r i s e s i n t h e c a n o n i c a l f o r m a l i s m . The c a n o n i c a l 3-momentum f o r t h e t o t a l L a g r a n g i a n d e n s i t y L=Lo+L-j- i s [3.8] TT = + ^ = A+g_A2+2gB ( B • A) - g A B 2 , B=VXA. T h i s e q u a t i o n c a n n o t be s o l v e d f o r A i n c l o s e d f o r m . T h e r e f o r e , A i s exp a n d e d i n powers o f g a c c o r d i n g t o .Co) 13 n,<n) [ 3 . 9 ] A - _ _ _ g " A ^ ' , n O " ' where A = TT - oL/^A i s t h e t i m e d e r i v a t i v e o f t h e ' f r e e f i e l d . To f i r s t o r d e r i n g. [ 3.8] becomes [3.10] _ ° - TTo + rfAD+20i2+2_*',(B%)-B,42_0l. The f u l l L a g r a n g i a n d e n s i t y , a l s o t o f i r s t o r d e r , i s [3.11] L ( 1 ) = „(E ' 2 - B ^ 2 ) + i g [ ( E - 2 - B ^ 2 ) 2 H ( B • £ , ) 2 ] - g E ' £ - gB • ( V X A J and t h e c o r r e s p o n d i n g H a m i l t o n i a n d e n s i t y t a k e s t h e f o r m +4(B • E ) J -g_/A + gB • (VX A ) . I t i s p o s s i b l e t o e x p r e s s t h e l a s t two t e r m s o f [ 3 . 1 2 ] i n t e r m s o f t h e f r e e f i e l d s E f o ) and B ( ° ) . I n c o v a r i a n t n o t a t i o n , t h e E u l e r - L a g r a n g e e q u a t i o n f o r t h e f u l l L a g r a n g i a n d e n s i t y [3.11] i s ( s e e A p p e n d i x A f o r more d e t a i l s ) [3.131 > D LU ) where f C n ^ l r ^ A C n ) P ? I t i s i m p o r t a n t t o n o t e t h a t A°-^0 i n t h e r a d i a t i o n gauge f o r i n t e r a c t i n g f i e l d s . [ 3 . 1 3 ] 1 4 . i s of the form of Maxwell's inhomogeneous equations with.a conserved 4 -current given by By inspection of [ 3 . 1 3 ] , [ 3.15] 1 - -5 L i ^ i i -4x f f ^ + Jr J, where c^h = 0 , h. to be determined. Hence, V X A and A , which follow from fMA(i} are expressible i n terms of free f i e l d s . The l a s t two terms of [ 3 . 1 2 ] become [ 3 . 1 6 ] - g E V V gjf-(s:xAco) - - g l E ^ - B ^ +YA°-Y) [3.17] ^ - J:H ( 1 ):dx - : j L i l ^ + B " 2 ) + i g { ( B ' ^ - E 6 ' 2 ) ^ ^ ^ ^ where VA° '"^  E < o )has vanished upon Integration. At this stage i t i s legitimate to expand the Hamiltonlan i n terms of the f r e e - f i e l d operators given i n t2.141 and [2.15]. Retaining only p a r t i c l e conserving terms, the f i e l d products that occur i n the interaction part are given by <o4 1 S V ^ T /kk'k"k'» k,K kJV k«V' k ' " , T v m e(k,iv). e ( k ' ,V )e(k»,V*)- e(k"',Tv."') 15. [ a a < a " V " % i ( ^ ' - ^ ' ' ' K * + a V a - a ^ ei(-k+k'+k"-k»')-x + a a ,t a n t a m e i(k-k'-k»+k"')-x + a t a' t a"a"'e i ( -^ , + k" + k"')-x + a a , t a „ a m t ei(k-k'+k"-k'-').x + a t a , a „ t a m ei(-k+k»-k»+k»')-x-j . s k,\ k',\' knX k'^Tf y • L D ( - l ) - X - X , - ^ m e ( k , 3 - x ) . c _ ( k . , 3 ~ T v M € (k»,3-\")- e(k»" ;3-V")[. • , where [«••] t ^ l e s a m e s e t °£ o p e r a t o r s as in £3.18], kk'k«k'» [ 3 . 2 0 ] : B M 2 E ^ 2 : - : I Z . TH. / V ~ ' k3Tv k«,V k",Tv!' k"',^' " x o ( - l ) " ^ " v e ( k ^ ) - e ( k » , T V ) e (k»,3-KM-£lk"< J 3--T\ m) [ .. .J :, [3.21] - : ^  H ^ Z Z ^ / 1 ! ^ v kf\ k' ,T\! k",^' k»!,t' ( - D ^ - ^ G l k ^ - l v J . ^ k ' ^ - V ) £(k» , V 1 ) - e(k t",Tv' n)[ • ..] : and [_3.22] :(E-B + B-E ) : = : - 3 / / / / 7 6 V k-,K k'X k»,tC knViT " 16. [ e ( k , A ) « i ( k ' , 3 - A t ) ( - l ) 3 ~ > 1 ' + c(k,3-X).£(k', v)(-i) 3-A] [£(k", x")-£(k'" J 3 - X m ) ( „ i ) 3 - A m + £ ( k » , 3 - \ " ) - e ( k m , A " ' ) i - l ) 3 ' A " j [ . . . ] :, where a " = a ( k " , X«)e""1 '-"I t , e t c . The f r e e H a m i l t o n i a n i s [ 3 . 2 3 ] 140 = -i {(E*}2+Bl°)2)dx: = £1 k a + f k ^ M k , ^ ) . 1 7 . 4 . BIPARTICLE STATES Under t h e o p e r a t i o n o f p a r i t y P, t h e h e r m i t e a n z e r o n f i e l d t r a n s f o r m s a c c o r d i n g t o [ 4 . 1 ] Per ( x ) P " 1 - s c ^ ( - x , t ) , where s = l f o r s c a l a r tr and s = - l f o r p s e u d o s c a l a r C . o c o U s i n g [ 2 . 1 1 w i t h a ( k ) s = b ( k ) , i t f o l l o w s t h a t [ 4 . 2 ] P a + ( k ) P - 1 = s a t ( - k ) . C o n s i d e r now t h e b i p a r t i c l e s t a t e | B ( k ) ) = | l ^ , l _ _ ^ c o n s i s t i n g o f two z e r o n s h a v i n g e q u a l and o p p o s i t e momenta k and - k . Thus, u n d e r P, [ 4 . 3 ] P a + ( k ) a t ( - k ) ( 0 > s p | l ,1 j,> = [Pa+ (k )P" 1] [Pa+ ( - k ) P " 1 ] [P10 > ] = s 2 | l k , l _ k > , where P | 0 ) = | 0 ) by a s s u m p t i o n . The t w o - z e r o n s t a t e j B ( k ) ) i s t h u s an e i g e n s t a t e o f p a r i t y w i t h e i g e n -v a l u e + 1 . C o n s i d e r s t a t e s o f two photons h a v i n g momenta k and - k . E i g e n s t a t e s o f t h e p a r i t y o p e r a t o r a r e s o u g h t . W i t h c i r c u l a r l y p o l a r i z e d p h o t o n s i t i s p o s s i b l e t o c o n s t r u c t t h e f o l l o w i n g f o u r h e l i c i t y s t a t e s : [ 4 . 4 ] | l k > R » l _ k > R ) = | R R > = at(k)a t(-k)10>, £4'^ l ^ R ' 1 - ! ^ ) = 1 R L > = a t(k)b t(-k)10>, IS. [ 4 . 6 ] j l k ^ ^ - k , ? . ) ~^LR^ = ^ ( k J a ^ - k ) |0>, f 4 * 7 ^ ^ k , L j l - k , L > s i L L > = b t ( k ) b t ( _ k ) \ 0 ) . Under P, the mixed states go into themselves, |RL)->IRL) , I LR)->|LR>and \RBHLL> and lLL)-»|KR> . The t r i p l e t of states |RL>, |LR) and \RR) + )LL) are therefore eigenstates of the parity operator with even parity and the singlet state fRR) - |LL> i s an eigenstate with odd pa r i t y . A two-photon state IRR> + ILL) has therefore the appropriate quantum numbers of a massive, neutral scalar p a r t i c l e i n the center-of-momentum; the state (RR> - |LL) has the appropriate quantum numbers of a massive, neutral pseudoscalar p a r t i c l e . Litner of tne states | RL") or 1LR) has the quantum numbers of a massive, neutral, spin 2, even parity p a r t i c l e . A more general b i p a r t i c l e state may be constructed by superposition: [ 4 . S ] JB> = ZZ f ( k ) | B ( k ) > , k where f(k) i s an arbitrary amplitude function. One finds here a direct analog to" the two-electron state, Cooper pair, formalism used i n the BCS theory of super-conductivity (Cooper 1956). a n a u l a r aaraentiaa c>er>e/-bene: on tbe for-; o.;;' b ( h K f o r example« i l l ; , ) .^ay i t s e l f be ereei; ahel e:e fee u s u a l see o f s p h e r i c a l hare^onice. veij.ee ^ e e e ee erpenfooetaone o f the orb' r ; ~- - -• - ~ • ( - j n - j ^ , , , s p h e r i c a l . : v arooe^.a.c eeee no eaeea f an a- f u n c t i o n o n l y of j k j ,. the o r b i t a l a n g u l a r ea.,lentaae of !?/ e: e e r o c The s t a t e 1B> has the eeee p a r i t y aeac;lgn;aent as t h a t of |B{k}> i f f f ( k ) i s even arid h a s the o p p o s i t e p a r i t y assignment i f f f ( k ) i s odeu 2 0 . 5. CALCULATION OF THE MASS SPECTRUM'OF COMPOSITE PARTICLES AS AN EIGENVALUE PROBLEM I : THE ZERON MODEL O p e r a t i n g t h e f u l l H a m i l t o n l a n ]i g i v e n b y [3.3] a n d £3.5] o n t h e t w o - z e r o n s t a t e | B ( o J ) = [ l ,1 ^ , i t i s f o u n d t h a t [5 . 1 3 %)B( a)> = 2 q | B ( a ) > - M o y ~ i | B ( k ) > , V k kq w h e r e u s e h a s b e e n made o f i d e n t i t i e s o f t h e f o r m [ 5 . 2 ] S ( k+k ' -k»-k« ' ) a f a t +a na«'lB(a)> = [S(k'" - q j o(k"+q.) + £(k«" + a ) S ( k J ' - a J ] |B(kJ) . T h u s |B(g_)/ i s n o t a n e i g e n s t a t e o f t h e H a m i l t o n l a n . H o w e v e r , b y i n t r o d u c i n g t h e s u p e r p o s i t i o n g i v e n i n [_4.8]j n a m e l y [5.3] IB) - JZ f ( a ) lB ( a ) > , w h e r e t h e unknown a m p l i t u d e f u n c t i o n f(gj i s t o be d e t e r m i n e d , t h e n [_5.ll b e c o m e s [5.4] *|B> = H L 2 q f ( a ) | B ( a ) > - — f ( a ) | B ( . k ) > ] a V k k q o r , i n t e r c h a n g i n g k a n d q_ i n t h e s e c o n d s u m m a t i o n , [5.5] X)B) = U L 2 q f ( q . ) - M s H ± f (k)] JB(c,)> . a V k k q 2 1 . I t f o l l o w s t h a t I B ) i s an e i g e n s t a t e . o f t h e H a m i l t o n i a n w i t h e n e r g y e i g e n v a l u e E p r o v i d e d t h a t f ( p j s a t i s f i e s 15.6] f l c j - A ) ILil (2q-E)q V k o r , i n i n t e g r a l f o r m , [ 5 . 7 ] f ( o J = — I S , ( ^ d k . (2q-E)(2TT)-jq ] k The s o l u t i o n o f t h i s i n t e g r a l e q u a t i o n i s , by i n s p e c t i o n , [ 5 . 8 ] f ( k ) = f ( - k ) = A / ( E - 2 k ) k , A i s a c o n s t a n t . The s p h e r i c a l symmetry i m p l i e s t h a t I B> , l i k e l B ( q J ) , i s a s t a t e o f e v e n p a r i t y . S u b s t i t u t i n g [ 5 . 8 ] i n t o r c y i /\1 OI ffomral n o n n n r l i f . " i n n L s • • J J — ~ - ^ - C J • •-• •-• •• - - — J k^(E - 2 k ) 3 g Q Thus, i n p o l a r c o o r d i n a t e s , [ 5. 1 0] \ I [ s i n 8 d 6 d ^ d k = 4 T r f [ p _ 1 i T I S ( E - 2 k ) ] dk -H i E -2k J (E - 2 k ) - 3 g G where a c u t - o f f X has b e e n i n t r o d u c e d i n t h e k - i n t e g r a t i o n and P d e n o t e s t h e p r i n c i p a l v a l u e . I n t e g r a t i n g , [5.11] i l n ( l - 2 V E ) + iTT - 27T 2/3g 0. I f t h e a s s u m p t i o n i s made t h a t t h e s t a t e s a r e q u a s i -22, s t a t i o n a r y , t h e n d i s c r e t e , c o m p l e x e n e r g y e i g e n v a l u e s E = ^ + ijS w i l l be o b t a i n e d , where fb c o r r e s p o n d s t o t h e w i d t h o f t h e e n e r g y l e v e l °<, . S u b s t i t u t i n g E = + i p i n [5.11] and s e p a r a t i n g i n t o r e a l a n d i m a g i n a r y p a r t s g i v e s 15.12] and 15.13] i l n 1-i t an -1 2\< 2>P> \ oc 2+p> 2-2W. 4 \ 2 f i 2 U 2 + / £ ) 2 -2JI 3g + nrr «= 0 From [5.13], £ = 0 f o r a l l n, so t h a t [5.12] r e d u c e s t o 2 i l n I 1-x 1 =15- , x = 2\/*L. [5.141 3e E q u a t i o n [5.14] i s d i s p l a y e d g r a p h i c a l l y i n f i g u r e 1. +1- 1 f ( x ) ! ! 1 1 1 < 1 1 1 1 1 0 V -s-1 X 1 X 1 \ 1 / \ I / \ 1 / f ( x ) - l n | l \ 1 / \ / S o l u t i o n s - x / V i A / -x-F i g u r e 1 \ 1 / \ 1 / \ 1 / \ 1 / : G raph o f e q u a t i o n [5.14] * X g o < 0 23. The g r a p h r e v e a l s t h a t o n l y one v a l u e o f x w i l l s a t i s f y [5.14] i f g Q > 0 b u t t h a t two s o l u t i o n s e x i s t i f g Q < 0. I n t h i s c a s e , '[5.15] I n i l - x - j ) = l n ( x 2 - l ) , x± = 2"K/«^i, i = 1,2. •The o n l y p a r t i c l e s so f a r e x p e r i m e n t a l l y o b s e r v e d w i t h t h e s p i n - p a r i t y a s s i g n m e n t o f 0 , a s w o u l d be r e q u i r e d in t h i s c a s e , i s t h e T|o+(1373 n.u.)-if\ +( 2080) d o u b l e t ( P a r t i c l e D a t a Group 1970). I f ^  a n d ^ a r e f i t t e d t o t h e s e p a r t i c l e s r e s p e c t i v e l y , t h e n g Q and Tv c a n be d e t e r m i n e d u n i q u e l y . I n t r o d u c i n g R == °l2^oCl* L5»15] becomes r 5.161 x. - R ( 2-R). •X W i t h R« r3/2 and h e n c e x^ -=3/4 f o r t h e c h o s e n d o u b l e t , t h e c u t - o f f iv t a k e s t h e v a l u e L5.173 X= x - ^ / 2 ~ 515 n.u. and t h e c o u p l i n g c o n s t a n t becomes 2 2 * 6. CALCULATION OF THE MASS SPECTRUM-OF COMPOSITE PARTICLES AS AN EIGENVALUE PROBLEM I I ; THE PHOTON MODEL I f t h a i n t e r a c t i o n H a m i l t o n i a i i [ 3 . 1 7 ] , e x p r e s s e d i n t e r m s o f t h e e x p a n s i o n s [ 3 . 1 8 ] - [ 3 . 2 2 ] , i s o p e r a t e d ,on e i t h e r t h e e v e n p a r i t y s t a t e | B ( q J ) + =|RR) + | LL> o r t h e odd p a r i t y s t a t e |B(q_)>_ = |RR> - I L L ) , t h e r e s u l t f o r b o t h , a f t e r i n t e g r a t i o n o v e r a l l s p a c e , i s [.see A p p e n d i x B] [ 6 . 1 ] (:H t1B(CL)> + :dx = - J l ] kq<4 + [£(k,l)• £( a,l)] 2 -f-[£(k,l)-e(o_,2)] 2 + [£(k , 2 ) - £ ( a , l ) ] 2 + [ e ( k , 2 ) - € ( a , 2 ) ] 2 j!B(k)> ±. U s i n g t h e r e p r e s e n t a t i o n , 16.2] • e ( k , l ) = ( l - i c | ) " * ( S 2 , - ^ l f O ) , k = k/k; e ( k , 2 ) - ( l - k 2 ) " * ! ^ ^ , ^ ^ , ^ 2 - ! ) , t h e n [ 6 . 1 ] , t o g e t h e r w i t h t h e f r e e p a r t , c a n be e x p r e s s e d w h o l l y i n t e r m s o f k_ and q_ as [ s e e A p p e n d i x C] [6.31 K l B ( a)> ±-2q| B ( a ) > ± + JLj2_kq[5+ (k •£) 2 ] ( B ( k ) > ± . 2V ^ I n t r o d u c i n g , as i n [ 5 . 7 ] , t h e s u p e r p o s i t i o n [ 6 . 4 ] lB>+«n f ( a ) | B ( a ) \ , a 2 5 . w i t h t h e a m p l i t u d e f ( o j t o be d e t e r m i n e d , t h e n 16.51 KlB>+= LL\2qf + -£-T~_ f (k)kq[5 + (|-£) 2]]iB ( pj> 4 - a 2 V jT -• T h i s e q u a t i o n i s o f t h e f o r m "K l B ) + = E | B ) + p r o v i d e d •[6.6] f ( q j 1 } k q f ( k ) C 5 + ( k - a ) 2 ] 2V(E-2q) P u t t i n g (£,'CL) = c o s T , where T 1 i s t h e a n g l e b e t w e e n k and g_, t h e n [6.6] becomes, i n i n t e g r a l f o r m , [6.7] f(a) = ^ [k3f ( k ) s i n 0 ( 5 + c o s 2 V )d9de$dk. 2(21T)3(E-2q) ) I f ( c o s K > c o s f i , c o s y ) and ( c o s < !, cos p>f, c o s V ) a r e t h e d i r e c t i o n c o s i n e s o f k and o_ r e s p e c t i v e l y , t h e n [6.8] c o s V = cos<tf.cos<<T+cos(5cosp.'+cosVcos Y' . These d i r e c t i o n c o s i n e s a r e d e f i n e d i n t e r m s o f 0 and $ by [6 .9] cos<<- = sinOcosc/; cos(3 = s i n O s i n c / ; c o s t = cos© and s i m i l a r l y f o r t h e p r i m e d a n g l e s . P e r f o r m i n g t h e i n t e g r a t i o n i n [6.7] o v e r t h e a n g u l a r c o o r d i n a t e s , IT ZK [6.10] I J s i n 9 ( 5 + c o s 2 ^ ) d 6 d ^ = 5 ( / f 7 l ) + ^ ( c o s 2 e ' c o s 2 0 ' + s i n 2 0 ' s i n 2 ^ ' + c o s 2 e ' ) = 5(471 ) + ^ | ( c o s 2 o C H - c o s 2 p ' + c o s 2 ) f ' ) - 1 6 U T T / 3 ) , t h e i n t e g r a l e q u a t i o n r e d u c e s t o 26. [ 6 . i i i f ( a ) = 2gq 3 T I 2 ( E ~ 2 q ) i k^f(k)dk, w h e r e , as i n s e c t i o n 5 , T\_ i s a c u t - o f f p a r a m e t e r t o be d e t e r m i n e d . The s o l u t i o n o f C 6 . l l ] i s o f t h e f o r m 16 .12] f ( k ) «= f ( - k ) = A q / ( E - 2 k ) , A i s a c o n s t a n t , w h i c h when s u b s t i t u t e d i n t o 16 . 1 1 ] y i e l d s t h e e i g e n -v a l u e c o n d i t i o n 16.13] k4 d k _ 3^\2 E-2k 4 0. P e r f o r m i n g t h e i n t e g r a t i o n as b e f o r e , [6.141 Tv4+iE37v + i E 2 T v 2 + | E K 3 + i E 4 l n ( l - 2 ^ ) +£iHE 4 12T\' g S e p a r a t i n g [ 6 . 1 4 ] i n t o r e a l and i m a g i n a r y p a r t s , r 2 [ 6 . 1 5 ] ^ [ ( 3 ( ^ - / J 2 ) + 2 < t 2 / i ] + ^ ( i T \ 2 + l p . T v 3 + i ^ ( ^ 2 - ( i 2 ) l n ^ - T ^ y j K W ) 2 K2 - p 2 ) 2 - 4 ° L 2 p 2 + 2Tin 1 K2 - ( 5 2 ) 2 - 4 1 2 p 2 tan' 0 •1 2X(i OL2+ fi 2 - 2 T v o l and [ 6 . 1 6 ] k _ 4 + i o L 2 - fi 2) -2 £21 + i K 2 ( d 2 - (32) + f * K 3 2 i + 8 (^ 2-(3 2 ) - 4 o c 2 ( b 2 2 X o L \ " + 4 X 2 f i 2 n ol2+{32 ( o L2+ ^ )2 -dft(dL 2 - (52) t a n " - 2A(Md 2 - / i 2 ) n TT ^2 -2"notfl( o L 2 - f 2 . 2 ) = - 4JL_C ; n , i s an i n t e g e r . ' J 4g 2 7 . Assuming that the state i s stable with respect to the interaction considered here so that [ 3=0, then [6.15] and [6 .16] become [6.171 X k + U ^ H ^ 2 + i A H * 4 l n -j 2JV " 4g 2 "and [6.18] =0, i s a n integer. If d 4 >0, then n s n-^ +n2+2 = 0 and corresponds to a groundstate; f o r higher n, (3 i1 0 and the state decays Introducing x = 2"N/CL i n 16.17], [6.19] ix f+ix3+ix2+x+ln|l-x| = . g°C T h i s e q u a t i o n i s d i s p l a y e d gi'apuxeaxxy x n xxg g<0 -» x f(x)= ix4+|x3+ix2 +x+ln|l-x| -- g > 0 Solutions Figure 2'. Graph of equation [6.193 2 8 , The g r a p h r e v e a l s t h a t i n t h i s c a s e i t i s p o s s i b l e t o accommodate a d o u b l e t s p e c t r u m i f g > 0 . P u t t i n g di^ and <^2 as t h e two p o s s i b l e mass v a l u e s , t h e n f r o m [6.171 '[6.20] i x 3 ( R ~ l ) + i b c 2 ( R2~ l ) + x-^ ( R3- l } ' + R4l n ( l - x 1 / R ) - I n l x ^ - l ) - 0, where R = ^ ^ l a n c * x l ~ 2"N/<*.-|_. F o r t h e e v e n p a r i t y s t a t e | B ( = jRR) + {LL/) , t h e f i t i s made t o t h e 7 \ Q + d o u b l e t as i n t h e z e r o n m o d e l . E q u a t i o n L 6 . 2 0 ] becomes [6.21] f x3 + 1 0 x 2 + 3 8 x x + 81 In(.!-!>:_,_) - 16 l n ( x] L- l ) = The s o l u t i o n t o t h r e e s i g n i f i c a n t f i g u r e s i s x-) - 1 .06. Hence t h e c u t - o f f i s 116.22] \ w = - 729 n . u . = 372 Mev and t h e c o r r e s p o n d i n g v a l u e f o r t h e c o u p l i n g c o n s t a n t li [6.23"] g ^  2.80 * 1 0 " 1 0 n . u . ( u n i t s o f l e n g t h4) , p P u t t i n g g = g c + ) f o r s t a n d a r d u n i t s , 16.24] g( + )« 1-67 X 10~ 5 n . u . « 7 . 8 x l 0 ~ 4 3 e r g - c m3. . F o r t h e odd p a r i t y s t a t e |B(oJ>_ = \RR) - I L L ) , i t i s p o s s i b l e t o make a f i t t o a number o f p a r t i c l e s . 29. The most n a t u r a l choice i s the ^ O _(1078 n.u. ) - ^ _ ( 1 8 7 0 n.u.) doublet. The c u t - o f f and the coupling constant take the values ' [6.25] 574 n.u. ^ 293 Mev ' and [6.26] g c _ ^ 3.34 x 1 0 " 5 n.u. 30. 7. DISCUSSION OF RESULTS The i n t e r a c t i o n s u s e d h e r e a r e o f t h e " d i r e c t " . p a r t i c l e i n t e r a c t i o n t y p e . T h a t i s , no i n t e r m e d i a t e b o s o n mechanism i s s u p p o s e d . I t i s a p p r o p r i a t e t h e r e -f o r e t o compare t h e z e r o n i n t e r a c t i o n w i t h t h e d i r e c t , w eak, 4 - f e r m i o n i n t e r a c t i o n , e v e n t h o u g h t h e c o u p l i n g c o n s t a n t i n t h e z e r o n model was f o u n d t o be o f t h e o r d e r o f 10, a number u s u a l l y a s s o c i a t e d w i t h t h e s t r o n g i n t e r a c t i o n . I n o r d e r t o compare i n t e r a c t i o n s c h a r a c t e r i z e d by c o u p l i n g c o n s t a n t s o f d i f f e r e n t d i m e n s i o n s , a d i m e n s i o n l e s s measure M i s s o u g h t f o r e a c h i n- 1 -c r a ^ " t i o ^ T+-. -j « n n c<?i v>l ft t o construe*"- ^ u c ^ w h e r e s i n c e t h e r e e x i s t n a t u r a l i n t r i n s i c l e n g t h s i n t h e t h e o r y , n a m e l y t h e r e c i p r o c a l s o f t h e i n v a r i a n t c u t - o f f s . The c o r r e c t c o m b i n a t i o n o f t h e c o u p l i n g c o n s t a n t s and t h e i r a s s o c i a t e d c u t - o f f s t h a t g i v e d i m e n s i o n l e s s numbers a r e l i s t e d i n t a b l e I f o r e a c h m o d e l , i n c l u d i n g "che n e u t r i n o m o d e l s o f K a e m p f f e r (1970) and E s c h (1971). The a p p r o p r i a t e c o m b i n a t i o n i s o f t h e f o r m M = gX f o r e a c h model shown e x c e p t f o r t h e z e r o n m o d e l , w h i c h h a s a c o u p l i n g a l r e a d y d i m e n s i o n l e s s . 31. T a b l e I : C o m p a r i s o n o f M o d e l s M o d e l I n t e r a c t i o n L a g r a n g i a n J P o f t h e v . Composite"" Momentum C u t - o f f (n.u.) C o u p l i n g C o n s t a n t (n.u.) M Z e r o n (cr ~ c r ) v o o 4 . So^o 0 + 5 1 5 9 . 5 0 X 1 0 5 9 . 5 0 P h o t o n M i s n e r -W h e e l e r 0 + 7 2 9 1 . 6 7 8 . 8 7 0 " 5 7 4 3 . 3 4 1 0 . 9 N e u t r i n o v e e V - A 0 " 5 9 3 . 5 . 2 0 1 8 . 6 N e u t r i n o ( v - ^ i ) • ' M e ' Nambu 0 + 7 3 9 4 - 7 7 26.0 E a c h 0 c o r r e s p o n d s t o t h e ^ Q + d o u b l e t and e a c h 0 c o r r e s -ponds t o t h e n - d o u b l e t . The v a l u e s o f M r e v e a l t h a t t h e i n t e r a c t i o n s s t r e n g t h s o f a l l t h e m o d e l s a r e o f t h e same o r d e r o f m a g n i t u d e . ( I t s h o u l d be n o t e d t h a t t h e v a l u e s o f M r e f l e c t t h e c h o i c e o f t h e n u m e r i c a l f a c t o r s s u c h as. £ and J~2 e t c . i n t h e i n t e r a c t i o n L a g r a n g i a n s . ) These i n t e r a c t i o n s t r e n g t h s may be compared w i t h t h a t o f t h e weak i n t e r a c t i o n p r o v i d e d one knows t h e i n t r i n s i c l e n g t h "N a s s o c i a t e d w i t h t h e weak i n t e r a c t i o n . Then M , = w weak 3 A 1 0 " 1 2 K 2 . 3 2 . The ~y-7 i n t e r a c t i o n r e m a i n s one o f t h e f u n d a -m e n t a l u n m e a s u r e d q u a n t i t i e s i n p a r t i c l e p h y s i c s . However, t h e measurement o f t h e t o t a l If-7 c r o s s -, s e c t i o n may be w i t h i n t h e s c o p e o f c u r r e n t t e c h n i q u e s ( S t o d o l s k y 1971). I n t h e l o w e n e r g y r e g i o n ( « 1 M e v ) , t h e p r o b a b l e u p p e r l i m i t t o t h e c r o s s - s e c t i o n , b a s e d o n p o n q.e.d. c a l c u l a t i o n s , i s o f t h e o r d e r o- ~2.5 x lO"*' cm ( K u n s z t e t . a l . 1970). I t i s p o s s i b l e , i n t h i s l o w e n e r g y l i m i t , t o s i m u l a t e p h o t o n - p h o t o n s c a t t e r i n g phenomena b y a n e f f e c t i v e n o n - l i n e a r i n t e r a c t i o n L a g r a n g i a n d e n s i t y o f t h e f o r m [7.1] H = f ( E 2 - B 2 ) 2 + 7 ( B - E ) 2 ] , 1 45 J where <X i s t h e f i n e s t r u c t u r e c o n s t a n t e^/l+W 4 and m I s e l e c t r o n r e s t mass. The f o r m a l s i m i l a r i t y o f Lj- t o t h e M i s n e r - W h e e l e r L a g r a n g i a n d e n s i t y t 7 . 2 ] Lx = t g [ ( E 2 - B 2 ) 2 + 4 ( B - E ) 2 ] e n a b l e s one t o make a c r u d e o r d e r o f m a g n i t u d e e s t i m a t e o f t h e e n e r g y domain f o r w h i c h t h e g o f t h e p h o t o n m odel i s c o m p a t i b l e w i t h t h e a f o r e m e n t i o n e d c r o s s - s e c t i o n . U s i n g ( a d a p t e d f r o m J a u c h and R o h r l i c h 1955), [7.3] d ^ l O G 2 ^ 6 , 2 4 where G may c o r r e s p o n d t o e i t h e r - 2 0 L /1+5 m of [7.1] o r t o the :,v; of [70,0] . TOoo O 4 10'" 2 , ; S V , i f 1 0 " A .'ie-Ao>,<i hee. Ones t h i s direct pho'oon i n t e r -a c t t i o .v: TO?."'.~i l e a d to cross-secthcac s h i g h e r then, rhosa preciso i'.ctcl s o l a J 3 v on the b a s i s ox tho v i r t u a l pnoi troninm exchange The photon model oh s t i n g u i s b e s i t s e l f from the o t h e r models by the f a c t t h a t the c a n o n i c a l f o r r a l i s n waa c a r r i e d out u s i n g p o t e n t i a l s, tho d e r i v a t i v e s of yah1' ch are the f i e l d s * The- 5 n t o r a c t i o n L a g r a n g i a n may bo expres-sed i n a " i n f i e l d " form s i m i l a r to the o t h e r s by i n t r o -d u c i n g the combinations 17,5] • F s-, E + i B and F' s E -:LB. For t h e n , [7.23 can be v / r i t t e n i n the now m a n i f e s t l y d u a l i t y i n v a r i a n t form [7.63 L I - £g(£ f*F +) (£•£)> where v a r i a t i o n i s nov/ ta k e n vsith r e s p e c t to F end p} T h i s model i s not e q u i v a l e n t to the photon moch.l p r e ~ sented in. t h i s work* D i f f i c u l t lew encountered w i t h I t have not been overcome t 3 4 . REFERENCES B j o r k e n , J.D. and D r e l l , S.D., ( 1 9 6 5)., R e l a t i v e s t i c Quantum f i e l d s . M c G r a x ^ - H i l l Book Co., N.Y., 70. * C a r r u t h e r s , P., ( 1 9 7 1 ) , P h y s . R e p o r t s 1C, 1. C o o p e r , L.N., ( 1 9 5 6 ) , P h y s . Rev. 104_, 1 1 8 9 . E s c h , R. J . , Can. J . P h y s . , ( t o be p u b l i s h e d ) . J a u c h , J.M. and R o h r l i c h , F., ( 1 9 5 5 ) , The T h e o r y o f P h o t o n s a n d E l e c t r o n s . A d d i son-We s i e y P u b l . Co., I n c . , M a s s . , 294. K a e m p f f e r , F.A., ( 1 9 7 1 )> U n p u b l i s h e d l e c t u r e n o t e s , 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 . K u n s z t , Z., M u r a d y a n , R.M., T e r - A t o n y a n , V.M., (1970) Comm. J I N R , Dubna E 2 , ( 5 4 2 4 ) . L u r i e , D., ( 1 9 6 8 ) , 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 . N.Y., 97, 144. 176. " M i s n e r , C. and W h e e l e r , J . , ( 1 9 5 7 ) , Ann. o f P h y s . 2, 525. P a r t i c l e D a t a G r o u p , ( 1 9 7 0 ) , Rev. Mod. P h y s . 4_2, 6 7 . Roman, P., ( 1 9 6 9 ) , I n t r o d u c t i o n t o Quantum F i e l d T h e o r y . J . W i l e y and S o n s , I n c . , N . I . S t o d o l s k y , L., ( 1 9 7 1 ) , P h y s . Rev. L e t t . 2 6 , 404. T e r l e t s k i i , Y.P., ( 1 9 6 S ) , P a r a d o x e s i n t h e T h e o r y o f R e l a t i v i t y . Plenum PresTJ N.Y., h~2~. 3 5 . APPENDIX A: ON EQUATION [3.12] I n t e r m s o f t h e u s u a l a n t i s y m m e t r i c , s e c o n d r a n k t e n s o r , t A . l ] f ' ~ = ^ A " - ^ A ^ = ^ A * 3 , t h e n ( J a u c h and R o h r l i c h , 1955) [A.2] ( B 2-E 2) 3 I± = i £ ^ f ^ , [A.3] B-E = I 2 = i e ^ r " ^ , I A . 41 I 3 s + I 2 = i f ^ r ' f ^ f ^ , where ^ i s t h e t o t a l l y a n t i s y m m e t r i c t e n s o r f o r w h i c h £ 1234 = > s o t^la't' t h e i n t e r a c t i o n L a g r a n g i a n d e n s i t y r r> T\ ^^-^ ..rv<i> + o t i o <= [ A . 5 ] L x - - i g [ i ( f ^ f n 2 - f ^ ^ . S i n c e [A.6] _ _ i f r r v f w n - gr * f ^ i % , M ^ A p ) and [ A . 7 ] £ ( f ^ f ' " ) = 4 f ^ 0 , f ^ x 2 _ rAr r * t h e n t h e c a n o n i c a l momentum t e n s o r f o r t h e t o t a l L a g r a n g i a n d e n s i t y L = L Q + L j i s 3 6 . [ A .8] T T Ps — = - i g ^ f ^ f ^ f ^ - 8 f ^ f ^ f _ ] . The E u l e r - L a g r a n g e e q u a t i o n •U . 9 ] o .thus becomes CA. io ] - v\ a Y + B J W ) where [A.11] = i ^ ^ M ^ - ^ ^ £ ^ \ . To z e r o t h o r d e r i n g, [A*10] becomes [A.12] \f"**?- 0 , w h i c h a r e f r e e M a x w e l l e q u a t i o n s as r e q u i r e d . To f i r s t o r d e r i n g, LA.13] = - P / [ f ^ ^ E q u a t i o n [A.13] i s o f t h e f o r m o f M a x w e l l ' s inhomogeneous e q u a t i o n s w i t h a c o n s e r v e d 4 - c u r r e n t g i v e n by [A.14] a ' = -Z^LZ^I,? f - 4 f f iju«J. The d i v e r g e n c e c o n d i t i o n = 0 i s s a t i s f i e d due t o t h e a n t i s y m m e t r y o f fC e , , r t / J. F u r t h e r m o r e , f c 0 d < 3 s a t i s f i e s t h e r e m a i n i n g two (homogeneous) M a x w e l l e q u a t i o n s e x p r e s s e d as [A.15] V f - K + KKX = D' 37. By i n s p e c t i o n o f [A.131? [ A . 1 6 ] f ^ = - h C ^ ^ +2f"~ f ' ^ i % +).^, where h ^ i s an a n t i s y m m e t r i c t e n s o r t o be d e t e r m i n e d s u b j e c t t o t h e c o n d i t i o n S^h = 0. A l t h o u g h t h e " d i m e n s i o n s o f h ^ a r e t h o s e o f E 3 o r B 3 , s u b s t i t u t i n g [ A . l 6 ] i n t o [A.15] r e v e a l s t h a t h * p s a t i s f i e s a l s o t h e homogeneous M a x w e l l e q u a t i o n s . The o n l y n o n - t r i v i a l s o l u t i o n i s h ^ = c f < o ) < * ^ , where c i s a c o n s t a n t h a v i n g t h e d i m e n s i o n s o f l / g . I n o r d e r t h a t t h e i n t e r a c t i o n p a r t c o n s i s t o n l y o f t e r m s q u a d r i l i n e a r i n t h e f r e e f i e l d s , t h e s i m p l i f y i n g a n d c o n s i s t e n t c h o i c e o f t h e t r i v i a l s o l u t i o n h — 0 i s a d o p t e d . S i n c e [A.17] V k-Vj t h e n [A. 183 {V*£\ = + Z E ' ^ E p B 1 * ' + E^B'p and s i m i l a r l y f o r (V Y A M 2 and ( v K A W ) 3 . A l s o , IA.19] • vV'V? -" ^  X j + 2 W j / so t h a t = -(B f o , 2-E ( o ) 2)f ( ; k + 2 f S / ^ J d,k - 1,2,3, - ( B 6 1 2 - E ^ 1 2 ) B*> + 2^Brt 2 - 2B^ ( t f + z f ) 33. LA.20J ^ - ( B w 2 - E f c > 2 ) E ^ + 2 E > ° 2 . - 2 E ^ ( B ( f - f B f ) Col (ol Ul Co}. • + 2 B 1 ( E 2 B 2 + E 3 B 3 ) and s i m i l a r l y f o r A 0-^A /By and A^-dA / 3 z . * #c o j> o The t e r m s -gE^- A 1 + g i f ? ( v X Ac°) c a n be e x p r e s s e d i n t e r m s o f E M a n d Bte\as [A.21] -gE^A + g ? ( i x f ) - - g ( E M 4 - B ( " 4 f - V A ^ f ) . CO to1, S i n c e ^ A Q - E v a n i s h e s upon i n t e g r a t i o n , t h e l o n g i t u d i n a l p a r t o f E c° , w h i c h c o u l d n o t be s u p p r e s s e d s i n c e j < 0 ' ^ ^ 0 , does n o t c o n t r i b u t e t o t h e H a m i l t o n l a n . The c o r r e c t H a m i l t o n i a n t o f i r s t o r d e r i n g i s [ A . 2 2 ] X= :|(I(E" 2 +B f c' 2) + * g [ ( B * 2 - E * ^ ' + 4 ( B - r , ) 2 ] } d x : H i g h e r o r d e r c a l c u l a t i o n s a r e p o s s i b l e i n p r i n c i p l e . I f f% + g f ^ + g 2 f i s s u b s t i t u t e d f o r f ^ i n [A. 1 0 ] , t h e n c> f i s e x p r e s s i b l e i n t e r m s o f a c u r r e n t ' c o n t a i n i n g t e r m s i n v o l v i n g fM^ and f " ' ^ . The d i v e r -g e n c e o f s u c h j c m ^ a l w a y s v a n i s h e s as r e q u i r e d . 3 9 APPENDIX B: ON EQUATION £6.1] E x p r e s s i n g t h e o p e r a t o r s a( k ,\) i n t e r m s o f a ( k ) a n d b ( k ) , t h e e x p a n s i o n E 6 . 1 S ] , i n t e g r a t e d o v e r a l l 3 - s p a c e , i s [B.1] ( : ^ 4 : d x « i-^r YZTZ IZ I k k ? k » k i T t 1 6 k 5 ? L k\ii k",y? k"',iv" ^ i ( - l ) r G ( k , X )• e ( k ' , V ) e ( k » , y ) . 6(k'",Tv m) [ ( - D X + V [ b«Tb'«'rbb' + ( - 1 ) X + V b ^ b ^ a a ' + ( - 1 ) K , ' + W n a - i + a ^ b b ' + ( . ^ ^ M - V W & t , t a , „ t a a q S (k+k ' -k»-k'i') + i - i ) ^ " + " K " , ^ b t b ' t b " b " ' + ( - l ) V ' + K T % ^ ^ a " a - + { - 1 ) ^+^U ^ a ^ a ^ + ( - 1 ) X + V a t a ' t b ' ' b " ' ] ^ ( - k - k ' + k W ' ) + ( ~ 1 ) X ' + K " [ b ^ b ' i ^ b ' b " + ( - l ) K l + K V b " ' T a ' a » + a « ' V V b » + + a t a m r a i a » ] £ ( - k + k J + k ^ k , M ) ( - l ) K + K m L b ' r b " t b b m + ( - l ) K + V n b i t b I I t a a n , + ( „ i ) ^ f + K " a , t a r , t b b , r t + ( „ 1 ) X + K , + V » + X ' » art a „ - r a a t „ - J ^ ( k - k ' - k " + k ) + ( - i ) ^ + K n [ b ' T b m T b b „ + ( - i ) K + K " b t r b l I I r a a l l + ( - D K ' + K m a « t a n f r b b t f + {.i)^^u+^\n^,taa,q S(-k+k'-k»+k"') + ( - l ) x , + v \ b tb» tb'b ' i ' + ( - l ) K t b + b « t a . a n , + ( - i ) U X V a « t b t b t „ + a " r a ' t a a ' " ] 5(k-k'+k"-k'»)-] :, 4 0 . where t e r m s t h a t l e a d t o 1RL) and jLR ) s t a t e s , o r t o z e r o , when a p p l i e d t o ILL}- o r |RR)- s t a t e s a r e d r o p p e d . A p p l y i n g t h e above e x p r e s s i o n on (LL) and 1RR) s t a t e s l e a d s t o c o n d i t i o n s on t h e k ' s , p e r m i t t i n g a s e p a r a t i o n o f t h e f - f u n c t i o n s . F o r e x a m p l e , i n t h e f i r s t t e r m , k = ± g_, I k' = + q_ so t h a t k" = - k ' " . The t e r m ( k k f k " k < " ) 2 becomes 2 2 k q i n e a c h c a s e . P e r f o r m i n g t h i s o p e r a t i o n , [ B . 2 ] (:E" 4:|RR)dx = t :E^ 4: |LL> dx - ± zL IZZ U {l2G ( a,\).fc ( - a,Tv.)e ( k , v O-e^ ( I L L ) + 1 RR) ) ] + [ 2 G ( k A ) - & ( - k , T v ) e ( a , V ! ) - f e ( - a ^ ' ' ) |LL> + 1 RR) )] +L fc(k,\). £(-a,V ) e(q_,V')- &(-k,x"') + G ( k , X ) . e ( q _ , V ) e ( - q , l v " ) - £ (-k,K'" (-1 ) ^ + K n ' i " ( X + K ' + 1 V " + X m ) [ ( - 1 ) I L L ) f ^ ^ x + ^ + X n - f X - n l R R ) ] + L e ( - a , X ) . £(k,\')e(-k,V')-e(a,Tv" f) + e(a ,x).e(k,A ,)G(-k,-^ ,')^£(-q. 5V , ,)] ( _ i ) M V + v . + V " l R R ) ] + l e(-a,\)> e ( k , V )e ( a,V')-e(-k,x»») + e ( s > ) ' e(k , x M e(- a,V ! ) - f e(-k,X» f)] ( _ 1 ) x + x « + V ' + K » ' l R R > ] + L e ( k } X ) . e ( a , v ) e ( - k , X « ) . V M ) + e ( k , K ) - V ) e ( - k , \ " ) - §.(a,V")] ( - i ) v + x V { K + v + v ' + v ^ 41. = y 2 C . k q ( ILL) + )RR>) + ^  ZZZ fef&(kA)-€ . ( - a > i)f ( a , i)-e(-k , x " ' ) [ | L L > + ( - i ) * + x " i R R > ; j + l'^'™" G(k,X)' G ( - q , 2 ) e ( q . 5 2 ) ' G(-k,V") [|LL> + k q j ( - l ) * ' ^ J o v k a l l X L L e ( - a , 2 ) - 6 ( k , v )e(a.}2)- e ( - k 5X»') - e i - ^ D - e f j c A ' ) §. (*>!)• £(-k,vn)][!LL> + k a l l X kq [ ( - l ) 7 v + V ' i " > v e ( k , X ) . e ( o , , 2 ) e ( - k , X " ) - § . ( -a,2) - £(k>X). e (a , l)e ( -k ,x , ,)-e ( -q.,l)][|LL> + ( - D ^ ' l R R ) ] ] = ZZ kq( ILL) + IRR)) + ZZZZ- kq[ |LL> + IRR>) + 1(1LL> - lRR))e f 2 )+i( ILL) - \ RR) )e + ( |LL> + \ RR) )£ c 4 )j , where CB.3] e ( 1 3 s e ( k , l ) . e ( - a , 2 ) e ( a , 2 ) - e ( - k , l ) - <L(k,D- 6(-a,l)e(q_,l)-e(-k,l) CB.4] £ f 2 )s &(k,2)- e(-a>2)£(a,2)- e ( - k,l) - e(k,2).e ( - a,l)e ( a,i)*e(-k,i) LB.5] e^3) s e ( k , D - €(-a,2)e.(a»2)-§.(-k,2) - € ( k , l)-e ( - a , l ) § ( a , l)-e ( - k,2) 42. [ B . 6 ] £ ( 4 ) a 6(k,2). £(-a,2)6(a,2)-£(-k,2) - &(k,2)- e ( - a , ' i ) e ( e ( - k , 2 ) . T h e r e f o r e , [ B . 7] j : E ^ 4 : |RR)'dx = j : E ( o ) 4: ]LL> dx ^ ZZ kq( ILL> + \ RR) ) + ~y ZZ. kq [ (- S ( J ) + i€<2> + i e O + % ) l I i > + ( " € Q - I G C 2 ) - ^ 3 ) + t 4 ) n ] ^ ] -A s i m i l a r c a l c u l a t i o n f o r B (^ 4 g i v e s [B. 8] {: B ( 6 ) 4:1 RR) dx - f: B f° 4: | LL> dx = -y IZ kq(lLL> + IRR>) + ^  ZZ k q [ ( -" 1 6 C 2 ) ~ i € (3) + GC4) ) I LL) + (- ^ + i + i € ^ + 6 ^ ) | RR)]. I n t h e c a l c u l a t i o n f o r E'°^B<a)2, d i f f e r e n t r e s u l t s a r e o b t a i n e d when t h e o p e r a t i o n i s on |RR)> f r o m t h o s e o b t a i n e d when t h e o p e r a t i o n i s on ILL/. F o r t h i s c a s e , t h e r e s u l t s are: LB.9] \:^2BUSZ: |RR> = - ~i ZZ kq( \LL> + IRR>) k • + \h ZZ kq"[ ( ^ + i e C 2 ) + i e-C3) - & C A ) ) !LL> JK and 4 3 . IB.10] (:E W 2B^ 2: 1 LL) dx = -~IZkq( I LL> -h 1 RR> ) V k ' - ^ H kq [ ( ^  + i ^ 2 ) + i & C 3 ) " 6^4) } | L L > k F o r t h e l a s t p r o d u c t , (BCt? ECa>)2, i t i s f o u n d t h a t [ B . l l ] j:(BC'" E ^ ) 2 : |RR)dx = - j :(B^ E < O )) 2\LL>dx = ^  ZZ kq( |RR> - IL L ) ) + ^ Z Z ^ ^ Q J + 1 ^ + "eW | L L > + ( ~ €U> + 1 G ^ + ^ - % ) 1R>1 • I t f o l l o w s t h a t i f }(_ o p e r a t e s o n t h e e v e n p a r i t y s t a t e ILL) + I R R ) , t h e n no c o n t r i b u t i o n i s o b t a i n e d f r o m t h e p s e u d o s c a l a r p a r t (B"• E ( ^ ) 2 o f t h e i n t e r a c t i o n . On t h e o t h e r h a n d , i f K o p e r a t e s on t h e odd p a r i t y s t a t e |LL) - IRR) , no c o n t r i b u t i o n i s o b t a i n e d f r o m t h e E"'4 and t e r m s . The c r o s s - t e r m E (" I 2B^ 2 g i v e s c o n t r i b u t i o n i n b o t h c a s e s . G a t h e r i n g t h e r e s u l t s t o g e t h e r , one o b t a i n s : [ B .12] J:Hj: ( |RR> — \LL) ) d x = ftjlB ( a j > ± • - ^  ZZ kq|B(A)>± t ^ Z ; kq(€(4)-etl) ) | B ( a )> ± , k k where € ^ - & ^ = [ e (k,l)-£(q.,l)] 2 + [£(k,2)-6(c,,l)] 2 + [&(k,D- €(a>2')] 2 + [e(k,2)-£(a.,2)] 2 • s P(k,a). 44 • I t i s t h e d u a l i t y i n v a r i a n c e o f t h e i n t e r a c t i o n L a g r a n g i a n t h a t a c c o u n t s f o r t h e f a c t t h a t |B( <!))> + = ^ - j - |B(Q_)) _ . E q u a t i o n £6.1] f o l l o w s i m m e d i a t e l y f r o m [ B . 1 2 ] . 45. A P P E N D I X C: ON E Q U A T I O N 16.31 In equation [6.1], consider the terms [ c i ] P(k,gJ = t & ( k , l ) - e ( a , l ) ] 2 f [ 6(k,l)- e( a,2)] 2 + [e(k,2)- e ( a , l ) ] 2 + L e ( k , 2 ) . e(c,, 2)] 2 Using the representation 16.2], [C.2] P(k,qJ = " ^ { U 2 q | k 2 + k 2 q 2 q 2 - ^q^q^^ )k 2 + ( k f k ^ 2 + k 2 k f ^ l ~ 2 k 1 k 2 k 2 q 1 q 2 ) q 2 + ( k 2 q 2 + 2 k 1 k 2 q 1 q 2 + k 2 q 2 ) k 2 q 2 + k 2 k 2 q 2 q 2 + 2 k 1 k 2 k 2 q 1 q 2 q 2 + 2 (k 2-k 2) ( q 2~q 2 J k j k ^ q - j + k 2 k 3 q 2 q 2 + 2 (k 2-k 2) ( q 2-q 2 Jk^k^q^q,, + ( k 2 - k 2 ) 2 ( q 2 - q 2 ) 2 ] , where A = k 2 q 2 ( k 2 - k 2 ) ( q 2 - q 2 ) . Since [ C 3 ] ( k 1 q 1 + k 2 q 2 ) 2 = (k 2-k 2) (q 2-q 2) - I k - ^ - k ^ ) 2 then [C.4] P(k,p,) = h Y ( k 2 q ^ k 1 q 2 - k 2 q 1 ) 2 + q 2k| ( k ^ - k . o ^ ) 2 + k 2 q 2 ( k i q i + q 2 k 2 ) 2 + k 2 q 2 ( k i q i + k 2 q 2 ) 2 + 2 k 3 q 3 ( k 2 - k 2 ) ( q | - q 2 ) ( k i q i + k 2 q 2 ) + ( k 2 - k 2 ) 2 ( q 2 - q 2 ) 2 ] - 7 5 - [ ( k r k 2 ) ( q r q 2 ) [ ( k - - a ) 2 + k 2 q 2 ] ] . A / \ = 1 + k*2_* 

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