UBC Theses and Dissertations

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UBC Theses and Dissertations

The Faddeev-Popov technique in gauge field theories Sharpe, Bruce John 1984

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THE FADDEEV-POPOV TECHNIQUE IN GAUGE F I E L D THEORIES by BRUCE JOHN SHARPE B . S c , The U n i v e r s i t y o f G u e l p h , 1978 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MATHEMATICS Me 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 A p r i l (c) B r u c e J o h n B R I T I S H 1984 S h a r p e , COLUMBIA 1984 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d degree a t t h e 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 a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e head o f my d e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t 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 . D e p a r t m e n t O f Mathematics  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 1956 Main M a l l V a n c o u v e r , Canada V6T 1Y3 Date A p r i l 25, 1984 )E-6 (3/81) i i A B S T R A C T We a d a p t t h e F a d d e e v - P o p o v t e c h n i q u e t o l a t t i c e g a u g e f i e l d t h e o r i e s . O u r • f o r m u l a t i o n s t r o n g l y s u g g e s t s t h a t t h e F a d d e e v - P o p o v f o r m u l a , w h i c h h a s come i n t o d o u b t s i n c e t h e d i s c o v e r y o f G r i b o v a m b i g u i t i e s , i s i n f a c t c o r r e c t . M o r e p r e c i s e l y , we show t h a t G r i b o v a m b i g u i t i e s c a n o c c u r i n t h e l a t t i c e t h e o r y , b u t t h a t " u s u a l l y " t h e y d o n o t a f f e c t t h e F a d d e e v - P o p o v f o r m u l a ; a m e t h o d i s g i v e n f o r d e t e r m i n i n g when t h e l a t t i c e F a d d e e v - P o p o v f o r m u l a i s n o t v a l i d . We a r e a b l e t o a n s w e r i n t h e l a t t i c e t h e o r y many q u e s t i o n s t h a t a r i s e n a t u r a l l y i n t h e c o n t i n u u m t h e o r y b u t w h i c h h a v e r e m a i n e d u n s e t t l e d up t o now. We show t h a t a f o r m a l l i m i t o f t h e l a t t i c e F a d d e e v -P o p o v f o r m u l a y i e l d s t h e u s u a l c o n t i n u u m f o r m u l a . We p r o v e some p a r t i a l r e s u l t s w h i c h b e a r on t h e p r o b l e m o f p r o v i n g a r i g o r o u s c o n t i n u u m l i m i t . i i i TABLE OF CONTENTS A b s t r a c t i i T ab l e of C o n t e n t s i i i L i s t of T a b l e s i v L i s t of F i g u r e s v Acknowledgements v i I n t r o d u c t i o n 1 I. The Faddeev-Popov Techn ique i n Continuum Gauge T h e o r i e s A. Review of the Faddeev-Popov Techn ique 5 B. Proposed Remedies f o r G r i b o v A m b i g u i t i e s 17 II. The Faddeev-Popov Techn ique in L a t t i c e Gauge T h e o r i e s A. D e f i n i t i o n s 23 B. The L a t t i c e Faddeev-Popov Formula 28 C. The Gauge Degree 37 D. V a l i d i t y of the L a t t i c e Formula 46 II I. Continuum Faddeev-Popov Techn ique R e v i s i t e d A. R e l a t i o n Between L a t t i c e and Cont inuum Arguments 58 B. Some Q u e s t i o n s from the Continuum Theory 66 C. A l t e r n a t i v e Geomet r i c I n t e r p r e t a t i o n 73 D. T r u n c a t i o n of the F u n c t i o n a l I n t e g r a l 84 IV, Cont inuum L i m i t A. I n t r o d u c t i o n 90 B. Power C o u n t i n g in QCD 2 102 C. The A c t i o n 107 D. Haar Measure and Lebesgue Measure 115 E. Faddeev-Popov Determinant - Formal L i m i t 123 F. Faddeev-Popov Determinant - T race C l a s s 125 P r o p e r t i e s G. Faddeev -Popov Determinant - D i amagnet i c 142 I n e q u a l i t y V. Open Problems 148 Bi b l i o g r a p h y 151 i v L I S T OF T A B L E S T a b l e 1 S u p e r f i c i a l l y D i v e r g e n t G r a p h s i n QCD2 105 L I S T DF FIGURES F i g u r e 1 Gauge - f i x i n g s u r f a c e and gauge o r b i t s 7 F i g u r e 2 O r i e n t a t i o n numbers f o r t h e map g •+ F ( " a ) 64 F i g u r e 3 S i g n of d e t M ( 9 k a ) 75 F i g u r e 4 F i g u r e f o r Lemma 3.8 77 F i g u r e 5 F i g u r e f o r Theorem 3.9 81 F i g u r e 6 E x a m p l e f o r G r i b o v ' s t r u n c a t i o n scheme 87 ACKNOWLEDGEMENTS I w i s h t o t h a n k my s u p e r v i s o r , L o n R o s e n , who h a s g i v e n me s o much o f h i s t i m e d u r i n g t h e d e v e l o p m e n t o f my t h e s i s a n d who, w i t h k e e n q u e s t i o n i n g a n d c o n s t a n t g o o d h u m o u r , h a s e n c o u r a g e d , i n s t r u c t e d a n d p o i n t e d me i n t h e r i g h t d i r e c t i o n on c o u n t l e s s o c c a s i o n s . I h a v e a l s o b e n e f i t e d g r e a t l y f r o m d i s c u s s i o n s w i t h L e x R e n n e r , J o e l F e l d m a n , N a t h a n W e i s s , S t e v e B r e e n a n d e s p e c i a l l y my c o l l e a g u e s , P e t e r S h a r p e a n d G o r d o n S l a d e , who w e r e a l w a y s r e a d y t o e x p l o r e w i t h me t h e c o n t e n t s o f t h i s t h e s i s a n d o t h e r m y s t e r i e s o f n a t u r e . I N T R O D U C T I O N 1 I N T R O D U C T I O N G a u g e f i e l d t h e o r i e s f o r m t h e c o r e o f o u r t h e o r e t i c a l u n d e r s t a n d i n g o f t h e i n t e r a c t i o n s o f e l e m e n t a r y p a r t i c l e s . T h e o l d e s t a n d m o s t s u c c e s s f u l s u c h t h e o r y i s q u a n t u m e l e c t r o d y n a m i c s w h i c h i s a n e x a m p l e o f a n a b e l i a n g a u g e f i e l d t h e o r y . N o n a b e l i a n g a u g e f i e l d t h e o r i e s , f i r s t ' c o n c e i v e d b y Y a n g a n d M i l l s i n 1 9 5 4 , a r e c u r r e n t l y t h e l e a d i n g c a n d i d a t e s f o r t h e d e s c r i p t i o n o f t h e s t r o n g i n t e r a c t i o n a n d f o r u n i f i e d f i e l d t h e o r i e s . T h e f o r m u l a t i o n o f a g a u g e f i e l d t h e o r y p r e s e n t s a n u m b e r o f s p e c i a l f e a t u r e s a n d c o m p l i c a t i o n s n o t f o u n d i n a n o n g a u g e m o d e l s u c h a s a <Q* t h e o r y . A t t h e v e r y o u t s e t , w h e n o n e a t t e m p t s t o q u a n t i z e a g a u g e f i e l d t h e o r y u s i n g p a t h i n t e g r a l s , o n e e n c o u n t e r s t h e d i f f i c u l t y t h a t g a u g e i n v a r i a n c e l e a d s t o a k i n d o f o v e r c o u n t i n g i n t h e i n t e g r a l s w h i c h r e s u l t s i n i n f i n i t i e s . H o w e v e r , u n l i k e s o m e i n f i n i t i e s , s u c h a s t h o s e r e q u i r i n g r e n o r m a l i z a t i o n s , t h e i n f i n i t i e s d u e t o g a u g e i n v a r i a n c e a r e s o m e w h a t a r t i f i c a l . T h i s i s b e c a u s e p h y s i c a l q u a n t i t i e s c a n i n p r i n c i p a l b e d e r i v e d f r o m a r a t i o o f g a u g e i n v a r i a n t i n t e g r a l s , s o t h e e f f e c t s o f o v e r c o u n t i n g c a n b e e x p e c t e d t o c a n c e l o u t . T h i s c a n c e l l a t i o n c a n b e d e m o n s t r a t e d e x p l i c i t l y i n a n a b e l i a n g a u g e f i e l d t h e o r y b u t s u c h a d e m o n s t r a t i o n i s n o t a s e a s y t o c o m e b y i n t h e n o n a b e l i a n c a s e . A n o t h e r w a y t o v i e w t h e c a u s e o f t h e s e i n f i n i t i e s i n g a u g e t h e o r i e s i s t o o b s e r v e t h a t g a u g e i n v a r i a n c e l e a d s t o I N T R O D U C T I O N 2 t h e a b s e n c e o f an e x p e c t e d d a m p i n g t e r m i n t h e i n t e g r a l s , w h i c h c o n s e q u e n t l y d i v e r g e . In 1 9 6 7 , F a d d e e v a n d P o p o v s h o w e d how t o r e s t o r e t h i s d a m p i n g by w h a t i s l o o s e l y c a l l e d " f i x i n g t h e g a u g e " . T h e F a d d e e v - P o p o v f o r m u l a f o r t h e e x p e c t a t i o n o f a f u n c t i o n o f t h e g a u g e f i e l d s b e c a m e t h e f o u n d a t i o n f o r much o f t h e l a t e r w o r k on n o n a b e l i a n g a u g e f i e l d t h e o r i e s , m o s t n o t a b l y i n t h e p r o o f t h a t t h e y a r e r e n o r m a l i z a b l e . M o r e o v e r , t h e F a d d e e v - P o p o v f o r m u l a s e e m e d t o p r o v i d e a s u i t a b l e s t a r t i n g p o i n t f o r t h e r i g o r o u s c o n s t r u c t i o n o f a n o n a b e l i a n g a u g e f i e l d t h e o r y by s u p p l y i n g a f o r m a l m e a s u r e f o r c o n s t r u c t i n g S c h w i n g e r f u n c t i o n s . U n f o r t u n a t e l y , an i m p o r t a n t a s s u m p t i o n made i n t h e d e r i v a t i o n o f t h e F a d d e e v - P o p o v f o r m u l a was d i s c o v e r e d by G r i b o v i n 1977 t o be i n c o r r e c t i n t h e n o n a b e l i a n c a s e . T h i s d e f e c t , m a n i f e s t e d b y t h e e x i s t e n c e o f " G r i b o v c o p i e s " , h a s b e e n g e n e r a l l y a c k n o w l e d g e d a s a s e r i o u s d i f f i c u l t y e v e n a t a n o n r i g o r o u s l e v e l , a n d many a u t h o r s h a v e s t u d i e d i t s e x t e n t a n d c o n s e q u e n c e s . T h e f u l l i m p l i c a t i o n s o f G r i b o v ' s d i s c o v e r y a r e s t i l l n o t k n o w n , n o r i s i t c l e a r how much o f t h e w o r k b a s e d on t h e F a d d e e v - P o p o v f o r m u l a s h o u l d b e c o n s i d e r e d i n v a l i d b e c a u s e o f i t . A s e c o n d p r o b l e m i n t h e F a d d e e v - P o p o v d e r i v a t i o n was p o i n t e d o u t by H i r s c h f e l d i n 1 9 7 9 . T h i s p r o b l e m , a m a t t e r o f some m i s s i n g a b s o l u t e v a l u e s i g n s , s e e m s a t f i r s t s i g h t t o be l e s s s e r i o u s t h a n G r i b o v ' s a n d h a s n o t r e c e i v e d n e a r l y a s much a t t e n t i o n a s t h e l a t t e r . H o w e v e r , H i r s c h f e l d a r g u e d t h a t i t was t h e k e y t o u n d e r s t a n d i n g t h e e f f e c t o f 6 r i b o v I N T R O D U C T I O N 3 c o p i e s , a n d t h a t i n a s e n s e t h e two e r r o r s c a n c e l l e d e a c h o t h e r o u t . Much o f t h e w o r k d e s c r i b e d i n t h i s t h e s i s g r e w o u t o f an a t t e m p t t o come t o a r i g o r o u s u n d e r s t a n d i n g o f H i r s c h f e l d ' s o b s e r v a t i o n s . I n t h i s t h e s i s we i n v e s t i g a t e t h e F a d d e e v - P o p o v t e c h n i q u e i n l a t t i c e g a u g e f i e l d t h e o r i e s . By w o r k i n g w i t h t h e l a t t i c e a p p r o x i m a t i o n we a v o i d t h e d i f f i c u l t y t h a t i n t h e c o n t i n u u m t h e o r y many o f t h e f u n d a m e n t a l q u a n t i t i e s o f i n t e r e s t a r e n o t w e l l - d e f i n e d . Me show t h a t t h e G r i b o v p h e n o m e n o n c a n o c c u r on t h e l a t t i c e a n d g i v e a r i g o r o u s p r o o f o f a F a d d e e v - P o p o v f o r m u l a w h i c h t a k e s i t i n t o a c c o u n t . O u r m a i n c o n c l u s i o n i s t h a t , b a r r i n g c e r t a i n p a t h o l o g i e s , t h e l a t t i c e F a d d e e v - P o p o v f o r m u l a i s t h e same r e g a r d l e s s o f t h e o c c u r r e n c e o r a b s e n c e o f G r i b o v c o p i e s . T h i s s u g g e s t s t h a t t h e o r i g i n a l F a d d e e v - P o p o v f o r m u l a f o r t h e c o n t i n u u m t h e o r y , w h i c h c a n be o b t a i n e d a s a f o r m a l l i m i t o f t h e l a t t i c e f o r m u l a , i s v a l i d i n s p i t e o f t h e d e f e c t s i n i t s o r i g i n a l d e r i v a t i o n . We b e l i e v e t h a t t h e l a t t i c e F a d d e e v - P o p o v f o r m u l a we h a v e d e v e l o p e d s h o u l d be a u s e f u l t o o l i n t h e r i g o r o u s c o n s t r u c t i o n o f a c o n t i n u u m n o n a b e l i a n g a u g e f i e l d t h e o r y o b t a i n e d by p r o v i n g t h e e x i s t e n c e o f a l i m i t a s t h e l a t t i c e s p a c i n g g o e s t o z e r o . T h e t h e s i s i s o r g a n i z e d a s f o l l o w s . In C h a p t e r I we r e v i e w t h e F a d d e e v - P o p o v t e c h n i q u e i n i t s o r i g i n a l f o r m a n d d i s c u s s t h e i n a d e q u a c i e s o f i t s d e r i v a t i o n . Our f o r m u l a t i o n I N T R O D U C T I O N 4 o f t h e F a d d e e v - P o p o v t e c h n i q u e f o r l a t t i c e g a u g e t h e o r i e s i s g i v e n i n C h a p t e r I I . We u s e t h i s f o r m u l a t i o n i n C h a p t e r I I I t o c l a r i f y s e v e r a l a s p e c t s o f g a u g e - f i x i n g i n t h e c o n t i n u u m t h e o r y . I n C h a p t e r IV we show t h a t t h e c o n t i n u u m F a d d e e v - P o p o v f o r m u l a c a n be o b t a i n e d f o r m a l l y f r o m o u r l a t t i c e v e r s i o n b y t a k i n g t h e l i m i t a s t h e l a t t i c e s p a c i n g g o e s t o z e r o . We a l s o d i s c u s s t h e r e some p a r t i a l r e s u l t s w h i c h b e a r on t h e p r o b l e m o f t a k i n g a r i g o r o u s c o n t i n u u m l i m i t f o r t h e c a s e o f an SU(2) g a u g e t h e o r y i n two s p a c e - t i m e d i m e n s i o n s . I.A REVIEW OF THE FA D D E E V - P O P O V T E C H N I Q U E 5 I . THE FA D D E E V - P O P O V T E C H N I Q U E IN CONTINUUM GAUGE T H E O R I E S I.A R e v i e w o f t h e F a d d e e v - P o p o v T e c h n i q u e L e t G be a c o m p a c t L i e g r o u p o f d i m e n s i o n k a n d E be t h e L i e a l g e b r a o f 6. L e t { t a ^ a = l , 2 , . . . , k b e a b a s i s f o r E w h i c h h a s b e e n n o r m a l i z e d t o s a t i s f y t r ( t a t b ) * - i * a b ( 1 . 1 ) w h e r e t r d e n o t e s t h e character i n some r e p r e s e n t a t i o n o f E. In t h e f o l l o w i n g we u s e t h e c o n v e n t i o n t h a t r e p e a t e d Roman i n d i c e s a , b , . . . a r e summed f r o m 1 t o k. We wor k i n d = s + 1 s p a c e - t i m e d i m e n s i o n s a n d u s e t h e c o n v e n t i o n s t h a t r e p e a t e d Roman i n d i c e s i , j , . . . a r e summed f r o m 1 t o s , w h i l e r e p e a t e d G r e e k i n d i c e s J J , V , . . . a r e summed f r o m 0 t o s . A c l a s s i c a l g a u g e f i e l d A i s an e l e m e n t o f t h e s e t o f f u n c t i o n s f r o m (R d •* E ® I R d . We i d e n t i f y A w i t h a c o l l e c t i o n o f E - v a l u e d f u n c t i o n s ( A Q , A J , . . . , A s ) a n d w r i t e We s h a l l a l s o n e e d t h e f u n c t i o n s p a c e # p t h e s e t o f f u n c t i o n s f r o m IR d -> E . Th e a c t i o n S f o r a E u c l i d e a n p u r e g a u g e f i e l d t h e o r y w i t h g a u g e g r o u p G a n d c o u p l i n g c o n s t a n t % i s S ( A ) = - f $ d d x t r ( F ^ F ^ ) ( 1 . 2 ) w h e r e F ^ = 3 f JA y - 3 y A ^ + A C , A v 3 i s t h e f i e l d s t r e n g t h I . A R E V I E W O F T H E F A D D E E V - P O P O V T E C H N I Q U E 6 t e n s o r . A g a u g e t r a n s f o r m a t i o n i s a f u n c t i o n g t l R d •» 6 w h i c h a c t s o n a g a u g e f i e l d b y % = g A ^ g " 1 • a- igU^g" 1 ) . (1 . 3 ) T h e f u n c t i o n F ^ y t r a n s f o r m s a s Fj , „ < 9 A > • Q F M K ( A ) g " 1 (1.4) s o t h a t t h e a c t i o n i s a g a u g e i n v a r i a n t q u a n t i t y : S ( 9 A ) = S ( A ) . P h y s i c a l q u a n t i t i e s a r e o b t a i n e d i n t h e t h e o r y f r o m t h e e x p e c t a t i o n s o f g a u g e i n v a r i a n t f u n c t i o n s f , t h e e x p e c t a t i o n b e i n g g i v e n f o r m a l l y b y I f ( A ) e " S ( A ) © A < f > = 1 _ _ ( 1 . 5 ) I e " S ( A ) JDA w h e r e t h e © A d e n o t e s t h e ( n o n e x i s t e n t ) p r o d u c t o f L e b e s g u e m e a s u r e s IT IT d A * ( x ) . x « I R D a > f T h e f i r s t d i f f i c u l t y e n c o u n t e r e d i n w o r k i n g w i t h e q . ( 1 . 5 ) i s t h a t t h e g a u g e i n v a r i a n c e o f t h e i n t e g r a n d s l e a d s t o i n f i n i t i e s b e c a u s e o f a k i n d o f o v e r c o u n t i n g i n t h e i n t e g r a l s . I m a g i n e p a r t i t i o n i n g t h e s p a c e A o f g a u g e f i e l d s i n t o o r b i t s < 9 A > a n d c a r r y i n g o u t t h e i n t e g r a l I f ( A ) e " S ( A ) © A o f a g a u g e - i n v a r i a n t f u n c t i o n f b y f i r s t i n t e g r a t i n g o v e r s o m e s u r f a c e Z w h i c h p i c k s o u t o n e r e p r e s e n t a t i v e f r o m e a c h o r b i t a n d t h e n o v e r e a c h s u r f a c e o b t a i n e d b y g a u g e t r a n s f o r m i n g t h e f i e l d s i n Z. We o b t a i n I.A REVIEW OF THE FADDEEV-POPOV TECHNIQUE 7 J ©A f(A) e " S ( A ) = I I f(A) e - S ( A ) dz(A) dg (1.6) where dg and dz(A) are some as yet unspeci f ied measures on the gauge transformations $ and %l r e s p e c t i v e l y . Since the integrand in eq.(1.6) i s constant along each gauge orb i t the in tegra l over dg introduces an i n f i n i t e constant K: J f(A) e " S ( A ) DA = K $ z f(A) e " S ( A ) dz(A) . In a r a t i o of i n t e g r a l s , such as that of eq . (1 . 5 ) , the constant K w i l l cancel out, so we should expect that in t h i s case no i n f i n i t i e s due to gauge invar iance should occur. The r e s t r i c t i o n of the f i e l d s to the surface Z i s known as " f i x i n g the gauge" and Z i t s e l f i s known as the "gauge-fixing surface" . Figure 1 Sauge-fixing surface and gauge orb i t s I.A R E V I E W O F T H E F A D D E E V - P O P O V T E C H N I Q U E 8 T o s e e how t o i m p l e m e n t t h i s i d e a , l e t u s a p p r o a c h t h e p r o b l e m i n a d i f f e r e n t w a y b y r e t u r n i n g t o t h e q u e s t i o n o f m a k i n g s e n s e o f t h e f o r m a l m e a s u r e e - S ( A ) SPA d(i = n • ( 1 . 7 ) J e - S < f i ) JDA W r i t e S = SD + Sj ( 1 . 8 ) w h e r e S0 = -j J d d x trna^A, - a^ ) 2 ] ( 1 . 9 ) c o n t a i n s t h e t e r m s i n S w h i c h a r e q u a d r a t i c i n A, A f t e r f o r m a l l y i n t e g r a t i n g b y p a r t s a n d a p p l y i n g e q . ( 1 . 1 ) we o b t a i n SQ = j I d d x A a ( x ) D j J A J ( x ) ( 1 . 1 0 ) w h e r e D a b, = ("3 2i5 ( i V +  a i i d y ) S a f W e w o u l d I i k e t o i n t e r p r e t t h e f r e e m e a s u r e -S_(A) e 0 JDA . - S n ( A ) ] e 0 2>A ( 1 . 1 1 ) a s a G a u s s i a n m e a s u r e w i t h c o v a r i a n c e D~* a n d t r e a t t h e r e m a i n d e r Sj a s i n t e r a c t i o n t e r m s . H o w e v e r , we e n c o u n t e r t h e d i f f i c u l t y t h a t t h e o p e r a t o r D i s n o t i n v e r t i b l e . I n m o m e n t u m s p a c e we h a v e D j J ( k ) = ( k 2 < ^ y - k k „ ) i a b . ( 1 . 1 2 ; Now 6yV - k ^ k y / k 2 , t h o u g h t o f a s a m a t r i x i n t h e i n d i c e s \i a n d v , i s a p r o j e c t i o n o p e r a t o r , t h e p r o j e c t i o n o n t o t h e " t r a n s v e r s e s u b s p a c e " . T h e k e r n e l o f D a ^ ( k ) i s t h e r a n g e o f t h e p r o j e c t i o n k } i k y / k ' o n t o t h e o r t h o g o n a l I.A REVIEW OF THE F A D D E E V - P O P O V T E C H N I Q U E 9 c o m p l e m e n t , t h e " l o n g i t u d i n a l s u b s p a c e " . T h e f a c t t h a t D i s n o t i n v e r t i b l e i s a d i r e c t c o n s e q u e n c e o f g a u g e i n v a r i a n c e . F o r i f we d e f i n e t h e i n n e r p r o d u c t <• , •> by <A,B> = j j d d x A a ( x ) B j ( x ) , t h e n <A,DA> = S 0 ( A ) . T a k e A t o be i d e n t i c a l l y 0 a n d c o n s i d e r a g a u g e t r a n s f o r m a t i o n g o f t h e f o r m g = e B t w h e r e B e C ™ ( I R d ) a n d t i s a f i x e d e l e m e n t o f t h e L i e a l g e b r a o f G. T h e n f r o m ( 1 . 3 ) ^A^ = - A - 1 S j j B l x ) - t s o t h a t C ^ A ^ A y ] = 0 a n d S j f ^ A ) = 0. T h u s <9A,D9A> = S 0 ( 9 A ) = S ( 9 A ) = 8 ( A ) = 0. S i n c e 9 A * 0, t h e e q u a t i o n < g A , D 9 A > = 0 i m p l i e s t h a t t h e s p e c t r u m o f D c o n t a i n s 0. T h e m e a s u r e ( 1 . 1 1 ) i s s u r e t o l e a d t o i n f i n i t i e s b e c a u s e i t d o e s n o t p r o v i d e e x p o n e n t i a l d a m p i n g i n c e r t a i n " d i r e c t i o n s " , n a m e l y f o r t h e k e r n e l o f D. To r e m e d y t h i s s i t u a t i o n , we w o u l d l i k e t o r e p l a c e S 0 by S f l w h e r e S o = s o + ? a 5 d d k A J ( k ) k(JKK A * ( K ) = S Q - a I d d x t r [ O A^ ( x ) ) 2 ] . ( 1 . 1 3 ) I.A REVIEW OF THE FADDEEV-POPOV TECHNIQUE T h e t e c h n i q u e f o r m a k i n g s u c h a r e p l a c e m e n t was o r i g i n a t e d b y F a d d e e v a n d P o p o v [ F P 3 a n d r e f i n e d b y ' t H o o f t [ t H l ] , We now r e v i e w t h e i r a r g u m e n t i n some d e t a i l s i n c e much o f t h e r e s t o f t h e t h e s i s i s t a k e n up w i t h a n a l y z i n g i t s s h o r t c o m i n g s a n d i m p r o v i n g i t . T h e d i s c u s s i o n w h i c h f o l l o w s i s f a r f r o m r i g o r o u s a n d a numbe r o f c r i t i c i s m s o f i t c o u l d b e made. We s h a l l c o n f i n e o u r c o m m e n t s t o t h o s e w h i c h seem a p p r o p r i a t e t o t h e l e v e l o f t h e a r g u m e n t . We b e g i n b y c h o o s i n g a g a u g e - f i x i n g f u n c t i o n F w h i c h i s a l i n e a r f u n c t i o n m a p p i n g A •* A j . To o b t a i n S a a s i n e q . ( 1 . 13) we w o u l d c h o o s e F (A) = 3 ^ ( L a n d a u g a u g e ) . O t h e r common c h o i c e s i n c l u d e F ( A ) = 3 j A j ( C o u l o m b g a u g e ) F ( A ) = n^A^ ( a x i a l g a u g e ) w h e r e i n a x i a l g a u g e n^ i s a f i x e d v e c t o r , m o s t o f t e n w i t h T\Q = 1 a n d n j = 0. L e t C be a f u n c t i o n i n # j a n d d e f i n e J ( A , C ) b y 1 = d ( A , C ) I 6 ( F ( 9 A ) - C) SDg ( 1 . 1 4 ) w h e r e t h e ^ - f u n c t i o n i s a p r o d u c t o f o r d i n a r y f u n c t i o n s (5(B) = n <S(B a(x>) x , a a n d 2)g i s t h e i n f i n i t e p r o d u c t o f n o r m a l i z e d H a a r m e a s u r e s on 6 JDg = IT dg (x ) . x N o t i c e t h a t b e c a u s e o f t h e r i g h t i n v a r i a n c e o f H a a r m e a s u r e d ( 9 A , C ) " J ( A,C). I.A REVIEW OF THE FADDEEV-POPOV TECHNIQUE To be a b l e t o d e f i n e d (A,C) we of c o u r s e want t he i n t e g r a l i n eq. (1.4> t o be n o n v a n i s h i n g . We s h o u l d at l e a s t r e q u i r e t h a t the argument of t he ^ - f u n c t i o n has a z e r o . T h i s c o n d i t i o n i s known as the " a t t a i n a b i l i t y of a g auge " : g i v e n a gauge f i e l d A and a f u n c t i o n C t h e r e e x i s t s a gauge t r a n s f o r m a t i o n g such t h a t F(9ft) = c. In t he f o l l o w i n g , we s h a l l i g n o r e t he p o s s i b i l i t y t h a t the i n t e g r a l i n e q . ( 1 . 4 ) van i s he s . Suppose f i s a gauge i n v a r i a n t f u n c t i o n . We have $ f ( A ) e " S ( A ) DA = \l 4<A,C) i ( F ( 9 A ) - C) f ( A ) e " S ( A ) DA Dg = l\ <4(9A,C) <S(F(9A) - C) f ( 9 A ) e - S ( 9 A ) ©A JDg (1 .15) where i n t he l a s t l i n e we have used the gauge i n v a r i a n c e of <d(',C), f and S, Make a change of v a r i a b l e s i n the DA i n t e g r a l f rom A t o Qf\, The " Lebesgue measure " DA i s i n v a r i a n t under t h i s t r a n s f o r m a t i o n because i t i s a t r a n s l a t i o n by ( 3 ^g ) g " ^ f o l l o w e d by a u n i t a r y t r a n s f o r m a t i o n g A g - * . We o b t a i n \ f ( A ) e " S ( A ) DA = H d (A ,C ) «5 (F (A) - C) f ( A ) e " S ( A ) DA Dg = I J ( A , C ) <J(F(A) - C) f ( A ) e _ S ( A ) DA (1.16) where we have used t he f a c t t h a t [ Dg = 1. I n t e g r a t e bo th s i d e s of (1.16) a g a i n s t a we i gh t f u n c t i o n E(C) and e v a l u a t e t he tf-function t o o b t a i n I.A REVIEW OF THE FADDEEV-POPOV TECHNIQUE <$ E(C) ©C) <$ f (A) e " S < A ) ©A) = I 4 ( A , F ( A ) ) f ( A ) EoF(A) e ~ S ( A ) ©A. (1.17) By c h o o s i n g F(A) = 3 Aj, and E(C) = e x p ( - a $ d d x t r ( C 2 ) ) we see t h a t we have s ucceeded i n e q . ( 1 . 1 7 ) i n i n s e r t i n g t h e damping f a c t o r needed t o mod i f y S 0 t o S f l . The p r i c e we have had t o pay i n d o i n g so i s t he i n t r o d u c t i o n of the f u n c t i o n J ( A , F ( A ) ) . From eq . (1.14) < 4 ( A , F ( A ) ) - 1 = I <S(F(9A) - F (A) ) ©g. (1.18) To e v a l u a t e d ( A , F ( A ) ) , we assume " u n i q u e n e s s of gauge f i x i n g " : t he o n l y gauge t r a n s f o r m a t i o n g f o r wh ich F(9A) = F (A) i s g = i , the i d e n t i t y . Under t h i s a s s u m p t i o n , o n l y an i n f i n i t e s i m a l ne i ghbou rhood U of fl need be c o n s i d e r e d i n the i n t e g r a l ( 1 . 1 8 ) . For g 6 U we can w r i t e g = e * = 11 + y + 0<y 2> f o r some E - v a l u e d f u n c t i o n y = y a t f l . H a a r m e a s u r e i n U can be w r i t t e n ©g = ©y £ K IT d y a < x ) (1.19) x , a f o r some c o n s t a n t K and F(9A) - F(A) = M (A) y + 0 ( y 2 ) f o r some l i n e a r o p e r a t o r M ( A ) : A j •+ A j ; M(A) i s t he J a c o b i a n at g = 11 of the f u n c t i o n g •* F ( ^ A ) . What we mean by e q . ( 1 . 1 9 ) i s t h a t Haar measure i s a b s o l u t e l y c o n t i n u o u s w i t h r e s p e c t t o the measure i n d u c e d on U by I.A REVIEW OF THE FADDEEV-POPOV TECHNIQUE T7a d y a ( x ) . The c o n s t a n t K r e p r e s e n t s t he Radon-Nikodym d e r i v a t i v e at 11. We t hu s have J ( A , F ( A ) ) - 1 = I <S(M(A)j) ©* = K Idet M (A) I"1 (1.20) where we have used t he i n f i n i t e - d i m e n s i o n a l ana l o gue of t he f o r m u l a J _ <S(Mx) dx = Idet M l " 1 M b e i n g a l i n e a r map f rom IRn •* IRn. On p u t t i n g eq.(1 .20) i n t o e q . ( 1 . 1 7 ) and n o r m a l i z i n g we o b t a i n ] f ( A ) e " S ( A ) ©A < f > = " ^ e ^ ^ r " I Idet M(A)I f ( A ) EoF (A) e " S ( A ) ©A S Idet M(A)I EoF (A) e " S ( A ) ©A The f a c t o r det M(A) i s t he known as t he Faddeev -Popov ^L^MMJl l - For some r e a s o n , pe rhaps because i t was e x p e c t e d t h a t the Faddeev -Popov d e t e r m i n a n t would a lway s be be p o s i t i v e , the a b s o l u t e v a l u e s i g n s i n Idet M(A)| a re ( a l m o s t ) i n v a r i a b l y o m i t t e d w i t h o u t comment i n the l i t e r a t u r e . The r e s u l t i s t he Faddeev -Popov f o r m u l a f o r the e x p e c t a t i o n of a gauge i n v a r i a n t f u n c t i o n : I det M(A) f ( A ) EoF (A) e " S ( A > ©A <{> = ^ . I det M(A) E©F (A) e~S{*> ©A I f we had c a r r i e d out the p r e c e d i n g argument o n l y f o r C = 0 I.A REVIEW OF THE FADDEEV-POPOV TECHNIQUE and had not i n t e g r a t e d a g a i n s t E(C) the r e s u l t would have been the r e s t r i c t e d Faddeev -Popov f o r m u l a S det H(A) f ( A ) <S(F(A)) e - S < A ) ©A <f> = i _ ^ . | det M(A) tf(F(A)) e " b ( f l ' 2>A The Faddeev-Popov f o r m u l a or i t s r e s t r i c t e d v e r s i o n has been the s t a r t i n g p o i n t f o r many i n v e s t i g a t i o n s i n t o n o n a b e l i a n f i e l d t h e o r i e s because i t a l l o w s one to c a s t a n o n a b e l i a n gauge f i e l d t h e o r y i n t o a form much l i k e t h a t of non-gauge f i e l d t h e o r i e s . By c h o o s i n g E and F j u d i c i o u s l y one can o b t a i n an e f f e c t i v e a c t i o n , such as S f l , which l e a d s to a c o v a r i a n c e The Faddeev-Popov d e t e r m i n a n t can be w r i t t e n as an i n t e g r a l over f i c t i t i o u s f e rm ion f i e l d s ( "ghost f i e l d s " ) in a k ind of r e v e r s a l of the Mat thews-Sa l am p r o c e d u r e f o r i n t e g r a t i n g out f e r m i o n s f a m i l i a r from the Yukawa model and quantum e l e c t r o d y n a m i c s . The r e s u l t i n g e f f e c t i v e a c t i o n which i n c l u d e s the ghost f i e l d s i s in a form which i s p a r t i c u l a r l y s u i t a b l e f o r an i n v e s t i g a t i o n of r e n o r m a l i z a t i o n q u e s t i o n s ( t t H l ] , C t H 2 ] ) . To summar ize, the d e r i v a t i o n of the Faddeev-Popov f o r m u l a l a c k s p r o o f s of the e x i s t e n c e and u n i q u e n e s s of the o r b i t - s u r f a c e i n t e r s e c t i o n s and n e g l e c t s to take the a b s o l u t e v a l u e of a J a c o b i a n d e t e r m i n a n t . Of t he se o m i s s i o n s , on l y the un iquenes s q u e s t i o n has r e c e i v e d w idespread a t t e n t i o n in the l i t e r a t u r e . In the a b e l i a n case of quantum e l e c t r o d y n a m i c s , t h i s un iquenes s can be demons t ra ted e x p l i c i t l y . For example, f o r Landau gauge I.A REVIEW OF THE FADDEEV-POPOV TECHNIQUE = a A^ + i A " 1 a 2e where we have w r i t t e n g = e *^ . The c o n d i t i o n <*u^Afj e ^u A f i then reads S-'G = 0. Un iqueness i s o b t a i n e d by r e q u i r i n g t h a t a s o l u t i o n 6 go to 0 at i n f i n i t y . One would i n c o r p o r a t e t h i s r e q u i r e m e n t i n t o the Faddeev -Popov t e c h n i q u e by r e s t r i c t i n g the i n t e g r a t i o n i n e q . ( 1 . 1 4 ) which d e f i n e s d (A,C) to the subgroup of gauge t r a n s f o r m a t i o n s which goes to 11 at i n f i n i t y . Thus when G i s a b e l i a n a gauge f i x i n g c o n d i t i o n such as Landau gauge t o g e t h e r wi th a boundary c o n d i t i o n s e r v e s to p i c k out one r e p r e s e n t a t i v e from each o r b i t . G r i b o v CG] was the f i r s t to i n v e s t i g a t e whether the same ho ld s t r u e when G i s n o n a b e l i a n and d i s c o v e r e d tha t in g e n e r a l i t does n o t . S p e c i f i c a l l y , he c o n s i d e r e d the case of Coulomb gauge wi th G = SU (2 ) , and l ooked f o r s o l u t i o n s g t 11 of the e q u a t i o n 3.9ft. = a . f l ^ (1.21) He d i s c o v e r e d such s o l u t i o n s f o r p a r t i c u l a r c l a s s e s of A. The s o l u t i o n s g c o u l d not be e l i m i n a t e d by r e q u i r i n g t h a t g •» £ at i n f i n i t y . The f i e l d s 9ft which s a t i s f y e q . ( 1 . 2 1 ) have come to be known as G r i b o v c o p i e s and the f a i l u r e of the t r a d i t i o n a l g a u g e - f i x i n g f u n c t i o n s to u n i q u e l y s p e c i f y the f i e l d s i s known as the G r i b o v a m b i g u i t y . S i n g e r LSI has shown tha t the G r i b o v amb i gu i t y i s not I.A REVIEW OF THE F A D D E E V - P O P O V T E C H N I Q U E p e c u l i a r t o t h e C o u l o m b g a u g e c o n d i t i o n o r SU ( 2 ) . He g a v e a r i g o r o u s • f o r m u l a t i o n o f t h e u n i q u e n e s s c o n d i t i o n i n a s e t t i n g w h e r e t h e s p a c e - t i m e m a n i f o l d , w h i c h i s n o r m a l l y IR 4, i s t a k e n t o be S 4 , t h e u n i t s p h e r e i n IR 5. I f 5 d e n o t e s t h e g r o u p o f g a u g e t r a n s f o r m a t i o n s , w h i c h c a n be a n y c o m p a c t L i e g r o u p , t h e n t h e o r b i t s p a c e i s t h e q u o t i e n t A/ $ w h e r e e q u i v a l e n c e i s d e f i n e d t o mean " r e l a t e d b y a g a u g e t r a n s f o r m a t i o n " . S i n g e r p r o v e d t h a t when S i s n o n a b e l i a n i t i s i m p o s s i b l e t o f i x t h e g a u g e i n t h i s s e t t i n g , i n t h e s e n s e t h a t t h e r e i s no c o n t i n u o u s f u n c t i o n s:A/3 •* A w i t h t h e p r o p e r t y t h a t n o s = I , w h e r e tr i s t h e p r o j e c t i o n m A •» A/« a n d I i s t h e i d e n t i t y t r a n s f o r m a t i o n . T a k i n g t h e g a u g e f i e l d s t o be f u n c t i o n s on S 4 r a t h e r t h a t IP.4 c a n be r e g a r d e d a s i m p o s i n g b o u n d a r y c o n d i t i o n s a t » on t h e m . When t h i s r e s t r i c t i o n i s r e m o v e d , i t i s p o s s i b l e t o f i n d a t r u e g a u g e - f i x i n g c o n d i t i o n ( a x i a l g a u g e ) a s we d i s c u s s i n S e c t i o n I . B . I.B REMEDIES FOR GRIBOV AMBIGUITIES I.B Proposed Remedies f o r G r i b o v A m b i g u i t i e s The i n c o m p l e t e n e s s of g a u g e - f i x i n g as d e s c r i b e d in the p r e v i o u s s e c t i o n has been r e g a r d e d as a s e r i o u s prob lem f o r the Faddeev-Popov t e c h n i q u e . I n s tead of hav ing to c o n s i d e r the i n t e g r a l in e q . ( 1 . 1 8 ) o n l y i n a ne i ghbourhood of £ , i t i s n e c e s s a r y to e v a l u a t e i t i n a ne i ghbourhood of each G r i b o v copy ^ A . Each copy c o n t r i b u t e s one d e t e r m i n a n t so t h a t «d(A, F (A)) = . (1.22) Z det M ( 9 k A ) - 1 k E q u a t i o n (1.22) i s somewhat s y m b o l i c , s i n c e in g e n e r a l t h e r e i s a c o n t i n u o u s f a m i l y of G r i b o v c o p i e s (CGI, [ B E P ] ) . In any c a s e , such an e x p r e s s i o n f o r J ( A , F ( A ) ) would be very d i f f i c u l t to work w i t h , s i n c e i t appear s to r e q u i r e some knowledge of a l l the s o l u t i o n s to the c o m p l i c a t e d un iquenes s e q u a t i o n F(9A) = F ( A ) . Moreover , the d e s i r e d r e p r e s e n t a t i o n of d ( A , F ( A ) ) as an i n t e g r a l over ghost f i e l d s i s l o s t . The e x i s t e n c e of G r i b o v c o p i e s shows tha t the Faddeev -Popov t e c h n i q u e does not e n t i r e l y e l i m i n a t e the o v e r c o u n t i n g in gauge i n v a r i a n t i n t e g r a l s . G r i b o v [G] o r i g i n a t e d the i d e a tha t i t might be p o s s i b l e to a v o i d the e f f e c t of the c o p i e s by r e s t r i c t i n g the r e g i o n of i n t e g r a t i o n in some way. That i s , i t may be p o s s i b l e to I.B REMEDIES FOR GRIBOV AMBIGUITIES f i n d a r e g i o n V C A so tha t f o r any gauge i n v a r i a n t f u n c t i o n f (j E(C) DC)(J f (A ) e " S ( A ) DA) = I det M(A) f (A ) EoF(A) e _ S ( A ) DA. J V T h i s i d e a has been t a k i n g up s e v e r a l o t h e r a u t h o r s ( [ B E P ] , [ 6 s3 , [Z2]) but has not en joyed much s u c c e s s because of the d i f f i c u l t y of s p e c i f y i n g the r e g i o n - V in a way t h a t i s both a p p r o p r i a t e and c o n c r e t e . We d i s c u s s t r u n c a t i o n of f u n c t i o n a l i n t e g r a l s i n more d e t a i l in S e c t i o n I I I.D. S i nge r [S3 sugges ted tha t another s o l u t i o n would be to pa t ch t o g e t h e r l o c a l gauge f i x i n g s . He showed tha t Coulomb gauge can be used to d e f i n e a f u n c t i o n s l o c a l l y . The i d e a i s to d e f i n e a p a r t i t i o n of u n i t y s u b o r d i n a t e to a c o v e r i n g on whose members s i s d e f i n e d , and then app l y the Faddeev -Popov t e c h n i q u e on each member. T h i s approach has s u f f e r s from the d i f f i c u l t y of f i n d i n g a c o v e r i n g which can be s p e c i f i e d in a u s e f u l way. The c o p i e s found by G r i b o v had the p r o p e r t y tha t they were " l a r g e " in the sense t h a t a copy would e x i s t i f one of the pa rameter s d e f i n i n g A were l a r g e enough. Hence one of the common r e s p o n s e s to the G r i b o v a m b i g u i t y i s to a s s e r t t h a t i t has no e f f e c t on p e r t u r b a t i o n t h e o r y c a l c u l a t i o n s where the b e h a v i o r of the t h e o r y near A^ = 0 i s i n v e s t i g a t e d . As ment ioned in S e c t i o n I.A, the a x i a l gauges a re examples of t r u e gauges. For example i f n „ A u = 0, the I.B REMEDIES FOR GRIBOV AMBIGUITIES c o n d i t i o n n ^ A ^ = A^ r e q u i r e s tha t n ^ 3 f i ( g " 1 ) = 0 and the c o n d i t i o n t h a t g •* il at i n f i n i t y r e q u i r e s t h a t g = i e ve rywhere . Many a u t h o r s have c o n t i n u e d to i n v e s t i g a t e n o n a b e l i a n gauge f i e l d s by u s i n g the Faddeev-Popov f o r m u l a f o r such a gauge. F i n a l l y , t h e r e a re those approaches to gauge f i e l d t h e o r i e s which do not i n v o l v e gauge f i x i n g and so a v o i d the d e f e c t s of the Faddeev-Popov t e c h n i q u e s . One such approach i s to s tudy a l a t t i c e model ( i n t r o d u c e d by [Ws]; some r e v i e w s a re CD I ] ,CKa ] ,CKo l ] and CKo2]) where the r e g u l a r i z a t i o n of the i n t e g r a l s p r o v i d e d by the l a t t i c e e l i m i n a t e s the i n f i n i t i e s due to gauge i n v a r i a n c e . By s t u d y i n g the cont inuum l i m i t of gauge i n v a r i a n t q u a n t i t i e s o n l y , t h e r e i s no need to f i x a gauge. One d i s a d v a n t a g e of p r o c e e d i n g t h i s way i s tha t the G r e e n ' s f u n c t i o n s of the t h e o r y , on which the s t r u c t u r e of u l t r a v i o l e t r e n o r m a l i z a t i o n s r e s t s , a re not gauge i n v a r i a n t . A second approach i s to q u a n t i z e the c l a s s i c a l t h e o r y i n a d i f f e r e n t way. S t o c h a s t i c q u a n t i z a t i o n [PW],[Z11 i s a t e c h n i q u e which has been advoca ted f o r comput ing p h y s i c a l q u a n t i t i e s w i thout the need f o r gauge f i x i n g te rms . To summar ize, the f a i l u r e to accomodate the G r i b o v a m b i g u i t y has been r e c o g n i z e d as a s e r i o u s sho r t coming of the Faddeev -Popov t e c h n i q u e and many remed ie s have been p r o p o s e d . None of t h e s e i s e n t i r e l y s a t i s f a c t o r y however, s i n c e each r e q u i r e s us to s a c r i f i c e g e n e r a l i t y , c o n c r e t e n e s s I . B R E M E D I E S FOR G R I B O V A M B I G U I T I E S o r • f a m i l i a r i t y . M o r e o v e r , t h e p a t c h - u p s d e s c r i b e d do n o t h e l p u s t o e v a l u a t e t h e s o u n d n e s s o f t h e w o r k b a s e d on t h e F a d d e e v - P o p o v f o r m u l a . T h e m o s t d e s i r a b l e r e s o l u t i o n o f t h i s q u a n d a r y w o u l d b e a c o n v i n c i n g d e m o n s t r a t i o n t h a t t h e F a d d e e v - P o p o v f o r m u l a i s c o r r e c t e v e n t h o u g h i t s d e r i v a t i o n i s n o t . T h e l o n e a d v o c a t e o f t h e p o i n t o f v i e w t h a t G r i b o v c o p i e s do n o t i n v a l i d a t e t h e F a d d e e v - P o p o v f o r m u l a h a s b e e n H i r s c h f e l d C H ] . He was a l s o t h e o n l y a u t h o r t o c a l l a t t e n t i o n t o a n d r e c o g n i z e t h e i m p o r t a n c e o f t h e o m i s s i o n o f a b s o l u t e v a l u e s i g n s on t h e F a d d e e v - P o p o v d e t e r m i n a n t . H i r s c h f e l d r e w o r k e d t h e r e s t r i c t e d F a d d e e v - P o p o v a r g u m e n t i n t h e f o l l o w i n g way. I n s t e a d o f b e g i n n i n g w i t h t h e e q u a t i o n 1 = <d(A> I f ( F ( 9 A ) ) SDg a n d t r y i n g t o p r o v e t h a t 4 ( A ) = d e t M ( A ) , d e f i n e r\(A) by n.(A) = I d e t M(9A) < J ( F ( 3 A ) ) SOg. ( 1 . 2 3 ) L e t < 9 k A > d e n o t e t h e s e t o f G r i b o v c o p i e s , t h a t i s , t h o s e f i e l d s f o r w h i c h F ( 9 k A ) = 0. I f we r e s t r i c t t h e i n t e g r a t i o n i n e q . ( 1 . 2 3 ) t o an i n f i n i t e s i m a l n e i g h b o u r h o o d o f g k , we g e t g k I d e t M(9A) < J ( F ( 9 A ) ) 2>g = K J l U L i — A i _ U k I d e t M ( 9 k A ) | by u s i n g t h e same a r g u m e n t s t h a t l e a d t o e q . < 1 . 2 0 > . T h u s n,(A) = K I s g n d e t M ( 9 k A ) . ( 1 . 2 4 ) k I.B R E M E D I E S FOR GRIBOV A M B I G U I T I E S H i r s c h f e l d made t h e c r u c i a l s t e p o f i d e n t i f y i n g n,<A) w i t h a g e o m e t r i c i n v a r i a n t ( t h e i n t e r s e c t i o n n u m b e r o f t h e g a u g e o r b i t w i t h t h e g a u g e f i x i n g s u r f a c e ) t o a r r i v e a t t h e r a t h e r s u r p r i s i n g c o n c l u s i o n t h a t i s i n d e p e n d e n t o f A. G i v e n t h i s f a c t we c a n m i m i c t h e F a d d e e v - P o p o v a r g u m e n t t o c o n c l u d e t h a t H I f ( A ) e " S ( A ) DA = J d e t M (A) f ( A ) <S(F<A>) e " S ( A ) <DA ( 1 . 2 5 ) f o r a n y g a u g e i n v a r i a n t f u n c t i o n f . T h u s [ d e t M (A) f ( A ) <S(F(A)) e " S ( A ) <f> = J —_- . ( 1 . 2 6 ) I d e t M (A) <S(F(A)) e " 5 ( A ' T h e p r e c e d i n g a r g u m e n t s h o w s how t h e two m a j o r d e f e c t s i n t h e F a d d e e v - P o p o v t e c h n i q u e , t h e n e g l e c t o f G r i b o v c o p i e s a n d a b s o l u t e v a l u e s , h a v e c o m b i n e d a n d c a n c e l l e d e a c h o t h e r . T h e q u e s t i o n o f t h e a t t a i n a b i l i t y o f t h e g a u g e a p p e a r s h e r e a s t h e q u e s t i o n o f w h e t h e r o r n o t n, ~ 0, a n d h e n c e w h e t h e r o r n o t we w e r e j u s t i f i e d i n c a n c e l l i n g i t o u t i n p a s s i n g f r o m e q . ( 1 . 2 5 ) t o e q . ( 1 . 2 6 ) . In t h i s t h e s i s , we t a k e up H i r s c h f e l d ' s s u g g e s t i o n s a n d a p p l y t h e m t o g a u g e f i x i n g i n l a t t i c e g a u g e t h e o r i e s . In S e c t i o n I I I . C we g i v e a r i g o r o u s p r o o f i n t h a t c o n t e x t o f H i r s c h f e l d ' s m a i n c o n c l u s i o n . H o w e v e r , t h e m a i n p a r t o f o u r w o r k i s b a s e d on i d e n t i f y i n g ( t h e l a t t i c e a n a l o g u e o f ) r j ( A ) w i t h a d i f f e r e n t g e o m e t r i c i n v a r i a n t ( t h e o r i e n t e d d e g r e e ) . T h i s i d e n t i f i c a t i o n h a s s e v e r a l a d v a n t a g e s . I t a l l o w s u s t o s i m p l i f y t h e p r o o f t h a t n, i s i n d e p e n d e n t o f A a n d I.B REMEDIES FOR GRIBOV AMBIGUITIES d i s p e n s e wi th c e r t a i n a s sumpt ions tha t must be made to c a r r y out H i r s c h f e l d ' s argument. We are a b l e to o b t a i n the u n r e s t r i c t e d form of the Faddeev-Popov f o r m u l a as e a s i l y as the r e s t r i c t e d fo rm. Moreover , our approach makes i t p o s s i b l e to o b t a i n an e x p l i c i t e x p r e s s i o n f o r r\. T h i s i n t u r n l e t s us e x t r a c t from i t some i n f o r m a t i o n about the o r b i t - s u r f a c e i n t e r s e c t i o n s and i n v e s t i g a t e the q u e s t i o n of when i t v a n i s h e s . I I .A DEFINITIONS CHAPTER II THE FADDEEV-POPOV TECHNIQUE IN LATTICE GAUGE THEORIES 11.A Def i n i t i ons Let us beg in by r e v i e w i n g the framework of s t a n d a r d l a t t i c e gauge t h e o r i e s . Le t A = A ( E , { > ) be a f i n i t e l a t t i c e of p o i n t s x i n d of the form x = (FIQE , . . . , n 5 e ) where the n^ are i n t e g e r s w i th |n^| & N^. The b a s i s v e c t o r of l e n g t h £ in the d i r e c t i o n \i i s denoted by e^. If x and y are a d j a c e n t l a t t i c e p o i n t s ( i . e . , |x-y| = e) we denote the d i r e c t e d bond j o i n i n g x to y by <x,y>. The set of bonds j o i n i n g a d j a c e n t p o i n t s i n A w i l l be denoted by A*. As b e f o r e , the gauge group G i s a compact , connec ted L i e g roup. A l a t t i c e gauge f i e l d i s a map a:A* •* G wi th the p r o p e r t y t h a t a(x,y) = a ( y , x ) _ 1 . (2.1) The fo rma l c o r r e s p o n d e n c e with a cont inuum gauge f i e l d A i s gi ven by a (x ,x±e M ) = e < ± * e y x ± T V , . (2.2) A l a t t i c e gauge t r a n s f o r m a t i o n i s a f u n c t i o n gsA •* G. I t s a c t i o n i s denoted by a •* ^a and i s d e f i n e d by (9a)(x,y) = g(x) a(x,y) g ( y ) - 1 . (2.3) I I . A D E F I N I T I O N S We s h a l l o f t e n f i n d i t c o n v e n i e n t t o c o n s i d e r a g a u g e t r a n s f o r m a t i o n a s a p o i n t i n t h e p r o d u c t m a n i f o l d 5 = IT 6 x« A a n d a g a u g e f i e l d a s a p o i n t i n t h e p r o d u c t m a n i f o l d 5* = IT 6. We d e n o t e t h e L i e a l g e b r a o f $ by £ . <x ,y>e/\* A f u n c t i o n f d e f i n e d on S* i s g a u g e i n v a r i a n t i f f ( 9 a ) = f ( a ) f o r a l l g E 5 a n d a £ » * . T h e a c t i o n f o r t h e l a t t i c e t h e o r y i s a s m o o t h g a u g e - i n v a r i a n t f u n c t i o n S: 5* IR w h i c h i s c h o s e n t o a p p r o x i m a t e t h e c o n t i n u u m a c t i o n S ( A ) = -l- I T r < F ^ ) 2 d d x . (2.4) O f t e n , l a t t i c e q u a n t i t i e s such' a s t h e a c t i o n a r e d e f i n e d i n t e r m s o f a c h a r a c t e r f o r a r e p r e s e n t a t i o n o f 6 a n d s o a r e a r b i t r a r y t o t h e e x t e n t t h a t a c h o i c e o f r e p r e s e n t a t i o n m u s t be made. F o r o u r p u r p o s e s , t h i s c h o i c e i s n o t i m p o r t a n t a n d i t i s c o n v e n i e n t t o s i m p l y i d e n t i f y 6 w i t h o n e o f i t s r e p r e s e n t a t i o n s by a g r o u p o f f i n i t e - d i m e n s i o n a l , u n i t a r y m a t r i c e s . In t h e f o l l o w i n g , we a s s u m e t h a t t h i s i d e n t i f i c a t i o n h a s b e e n made. E x p e c t a t i o n s i n t h e l a t t i c e t h e o r y a r e d e t e r m i n e d by a m e a s u r e d e r i v e d f r o m n o r m a l i z e d H a a r m e a s u r e (J on 6. S i n c e 6 i s c o m p a c t , \i i s t h e u n i q u e m e a s u r e f o r w h i c h ( i ( G ) = 1 a n d I I . A D E F I N I T I O N S 2 5 L *<9> dp«g) = L f ( g h ) d f i ( g ) Q G • L *<hg> d f j ( g ) 6 = J H g " 1 ) d u ( g ) G f o r a n y f e L * < G , d | i ) a n d h E G. T h e m e a s u r e dfi c a n a l s o b e o b t a i n e d f r o m a d i f f e r e n t i a l f o r m E G H V 2 ] . L e t R[g]i6 •* G a n d L C g ] : G -> 6 b e d e f i n e d b y R [ g ] ( h ) * h g L E g ] ( h ) = g h . L e t n b e t h e d i m e n s i o n o f 6 a s a m a n i f o l d . T h e n t h e r e e x i s t s a u n i q u e n - f o r m y o n G w h i c h i s i n v a r i a n t L C g ] * v = R C g ] * v = v f o r a l l g E G a n d s u c h t h a t J " = 1. J G T h e n f o r a l l s m o o t h f u n c t i o n s f o n G we h a v e L e t d g ( x ) a n d d a ( x , y ) b e d f j . T h e n H a a r m e a s u r e o n $ i s t h e p r o d u c t m e a s u r e d g = Tl d g ( x ) a n d o n 5 * i t i s X E A d a = FT d a ( x , y ) . <x , y > E A* We r e q u i r e t h a t t h e a c t i o n S ( a ) b e s u c h t h a t e - S ( a ) £ L 1 ( S * , d a ) . T h e e x p e c t a t i o n o f a f u n c t i o n f E L 2 ( $ * , d a ) i s t h e n d e f i n e d t o b e I I.A DEFINITIONS I , # f ( a ) e " S ( a ) da <f> £ — 5 — : . ( 2 . 5 ) \ # e - S ( a ) da Thanks to the f i n i t e n e s s of Haar measure, the i n t e g r a l s in eq . ( 2 . 5 ) f o r <f> a re w e l l - d e f i n e d i n s p i t e of the gauge i n v a r i a n c e of the a c t i o n . T h i s i s one of the ways tha t the l a t t i c e t h e o r y p r o v i d e s a r e g u l a r i z a t i o n of the cont inuum t h e o r y . We now d e f i n e the l a t t i c e ana logue of the cont inuum damping f a c t o r E « F ( A ) . As we show below i n Theorem 2 . 1 4 , t h e r e a re t e c h n i c a l d i f f i c u l t i e s in t r y i n g to app l y gauge f i x i n g on a l l of A. Thus, l e t Aj C A and d e f i n e * ! s G A 1 . Let £ j denote the L i e a l g e b r a of « j . A l a t t i c e g a u g e - f i x i n g f u n c t i o n i s a smooth map F: $* -> In the examples we d i s c u s s be low, F has the form F(a) (x) = n i ( x , y ) m i K ^ ) . (2 . 6 ) <x,y>e A* where the m(x,y) a re i n t e g e r s . (When G i s n o n a b e l i a n , the o r d e r of the group m u l t i p l i c a t i o n s i n (2 . 6 ) must be s p e c i f i e d . ) We would n o r m a l l y want to choose F so t h a t the l a t t i c e g a u g e - f i x i n g c o n d i t i o n F (a ) = £ y i e l d s a g i v e n cont inuum c o n d i t i o n F(A) =0 i n some sense as e -> 0. In g e n e r a l t h e r e are many f u n c t i o n s tha t w i l l s a t i s f y t h i s c r i t e r i o n . Two examples of gauge f i x i n g f u n c t i o n s which we s h a l l c a r r y a long with us from now on are tho se f o r a x i a l gauge II.A DEFINITIONS and Landau gauge. For the f o r m e r , take Aj = (x £ A: x+e Q E A) (2.7a) and F f t ( a ) ( x ) = a ( x , x + e 0 ) (x £ Aj ) (2.7b) For Landau gauge, t ake Aj = A 0 s {x £ A: |x | < E N ^ } . (2.8a) and F(_(a)(x) = a (x , x + eQ> a (x , x-eQ) . . . a (x , x + e g ) a (x , x - e s ) . (2.8b) The l a t t i c e v e r s i o n of EoF(A) = ef f l$Tr( Vu ) 2 d d , < i s E*F|_(a) with E: •* g i ven by E(c ) = exp C 2 a ^ " 2 E D " 4 E Re Tr ( c ( x ) - f l ) ] . (2.9) X E Aj These c h o i c e s f o r F^, F L and E a re f a i r l y n a t u r a l g i v e n e q . ( 2 . 2 ) . They a re based on o b s e r v a t i o n s such as a (x , x+e^)a (x , x -e^) -11 = e * E A f j ( x + 1 e f i ' e ~ * e A(i ( x " T e f i ' - 11 •v Xe A^ (x + j e ^ ) - Xe. A^ ( x - f e ^ ) * Xt^d^ (x ) . A more d e t a i l e d d i s c u s s i o n of the E •* 0 l i m i t of these f u n c t i o n s i s g i ven i n IV.C. I I.B THE LATTICE FADDEEV-POPOV FORMULA I I.B The L a t t i c e Faddeev-Popov Formula We now d e s c r i b e the l a t t i c e v e r s i o n of the Faddeev-Popov t e c h n i q u e . G iven a g a u g e - f i x i n g f u n c t i o n F and a gauge f i e l d a , d e f i n e {: $j •+ $j by K g ) = r(a)~lf{1a). (2. 10) Let £ j denote the L i e a l g e b r a of $ j . The l a t t i c e Faddeev -Popov o p e r a t o r i s the map M < a ) : £ ^ •* £ j d e f i n e d by M(a) = d i f l , (2.11) the d e r i v a t i v e map of i at the i d e n t i t y . We g i v e the m o t i v a t i o n f o r c h o o s i n g t h i s d e f i n i t i o n of M(a) i n the next c h a p t e r and show in Chapter 7 t h a t f o r m a l l y i t y i e l d s the c o r r e c t cont inuum l i m i t . For now, we can see t h a t M(a) i s the r i g h t c h o i c e because of the f o l1owi ng Theorem 2 .1 : ( L a t t i c e Faddeev -Popov f o rmu l a ) There i s a c o n s t a n t , n., depend ing o n l y on F such t h a t f o r any smooth gauge i n v a r i a n t f u n c t i o n f on and any smooth f u n c t i o n E : S + £ , q ( J E ( c ) dc) (J f ( a ) e ~ S ( a ) da) = I det M(a) f ( a ) E©F (a) e " S ( a ) da. (2.12) Hence i f n < $ E ( c , d c ) * 0, I I .B THE LATTICE FADDEEV-POPOV FORMULA I f ( a ) e " S ( a ) da <f> = — — \ e " s < a ) da I det M(a) f ( a ) EoF(a) e " S < a ) da (2.13) I det M(a) EoF(a ) e " S ( a ) da It i s worth remark ing here how Theorem 2.1 " s o l v e s the G r i b o v p r o b l e m " , a l t h o u g h the s t a tement s which a re about to be made are not e x p l a i n e d u n t i l the next c h a p t e r . If t h e r e were no G r i b o v c o p i e s , then n, would be equal to +1 or - 1 . The e x i s t e n c e of t he se c o p i e s has no e f f e c t on e q . ( 2 . 1 2 ) beyond chang ing the v a l u e of n> In a. s e n s e , the v a l u e of q measures the e x t e n t to which G r i b o v c o p i e s l e a d to o v e r c o u n t i n g in f u n c t i o n a l i n t e g r a l s l i k e ( 2 . 1 2 ) . In any c a s e , a change in n, has no e f f e c t on the v a l u e of <f> s i n c e n c a n c e l s in the n o r m a l i z a t i o n , r e f l e c t i n g the f a c t tha t the o v e r c o u n t i n g i s i d e n t i c a l i n the numerator and d e n o m i n a t o r . A d e t a i l e d d i s c u s s i o n of the i m p l i c a t i o n s of Theorem 2.1 f o r the cont inuum Faddeev -Popov f o r m u l a i s g i v e n in Chapter I I I. The t h e o r y of the ( o r i e n t e d ) degree of a smooth map between compact m a n i f o l d s i s c e n t r a l to the proo f of Theorem 2.1 and i t s c o n s e q u e n c e s , so we pause now to rev iew the main i d e a s of t h a t t h e o r y . By n - m a n i f o l d we mean a smooth, r e a l m a n i f o l d of d imens i on n and w i thou t boundary . Let M and N denote n - m a n i f o l d s which are compact , connec ted and o r i e n t e d . (In a l l our a p p l i c a t i o n s , M and N w i l l be the same m a n i f o l d and are a p roduc t of compact , connec ted L i e g roups . Because I I.B THE LATTICE FADDEEV-POPOV FORMULA a l l L i e groups are o r i e n t a b l e [GHV2], the p r e c e d i n g c o n d i t i o n s a re s a t i s f i e d . ) The tangent space to M at x £ M w i l l be denoted by T X ( M ) . Suppose <p:M •* N i s a smooth map and d<px i s i t s d e r i v a t i v e map from ' T X (M) -> T V ( X ) ( N ) . There a re t h r e e e q u i v a l e n t d e f i n i t i o n s which we use f o r the degree of <p, each of which i s based on a d i f f e r e n t theorem. We now b r i e f l y s t a t e t h e s e and l e a v e the d e t a i l s to the r e f e r e n c e s . A p o i n t y E N i s a r e g u l a r v a l u e of <p i f d<px i s s u r j e c t i v e f o r every x E <p"*(y). Suppose y i s a r e g u l a r v a l u e of <p and x E <p _ 1 (y ) . The o r i e n t a t i o n number E(<P,X) of if at x i s +1 i f (p p r e s e r v e s o r i e n t a t i o n at x and -1 i f <p r e v e r s e s o r i e n t a t i o n at x. Let I (<p, (y}) = E E <<p,x). (2.14) XE <p"1 (y) Theorem 1GP]; I(((i,{y>) i s i ndependent of y. • D e f i n i t i o n 1; The o r i e n t e d degree of <p i s deg ((p) = I (<p, (y>) where y i s any r e g u l a r v a l u e of <p. Theorem [ Sp 3: There i s a number, r\, such tha t i f u i s any n - form on N $ ip*u = n I u- (2.15) D M N D e f i n i t i o n 2: D e f i n e deg(<p) = n,. Let H k(M) denote the k t n de Rham cohomology v e c t o r I I .B THE LATTICE FADDEEV-POPOV FORMULA space t6HV2] of M, and <p*:HMN) -» H (M) the map induced by the p u l l b a c k of if. S i n c e H n(M) and H n (N) are o n e - d i m e n s i o n a l v e c t o r spaces and q>* i s l i n e a r , <p* i s j u s t m u l t i p l i c a t i o n by some c o n s t a n t , n,. D e f i n i t i o n 3; D e f i n e deg(<p) = r\. Theorem 2.2 ( [6HV2] ) : The o r i e n t e d degree of a map s a t i s f i e s (1) If <pj and (p2 a re nomotop i c , then deg <pj = (3) If <p i s a d i f f eomor ph i sm , then Ideg <p| = 1. If if i s o r i e n t a t i o n - p r e s e r v i n g , then deg(<p)= l j i f if i s o r i e n t a t i o n - r e v e r s i n g , then deg(<p) = - 1 . (4) If deg(<p> * 0, then if i s s u r j e c t i v e . 0 The proo f of Theorem 2.1 i s based on D e f i n i t i o n 2 of d e g r e e . The next two lemmas a re c o n t a i n e d i n P r o p o s i t i o n XIV, Chapter I of [GHV21 but we i n c l u d e the p r o o f s here because they are fundamenta l f o r what f o l l o w s . Lemma 2.3; Let <p; 3 •* $ and E: $ •* IR be smooth maps and l e t v be the i n v a r i a n t form f o r Haar measure as d e s c r i b e d above. Then deg <P2> (2) deg (<p ^ e> ) = deg(<pj) • deg(<P2>. <f*(£v) (g) = Eoip (g) J (g) v (g) (2.16) where J (g ) = det d(L[<p(g) ] 0 if 0 L [ g ]) £ . I I .B THE LATTICE FADDEEV-POPOV FORMULA P r o o f ; S i n c e <p*(Ev)(g) = tp*E(g) <p%(g) and (p*E(g) = E*(p(g), i t s u f f i c e s to show t h a t <p*v (g) = J (g) v (g ) . (2.17) Let i = L[(p(g)~ 1 ]o<poL[g]. Then <P*v(g) = L [ g - 1 ] * « i * o L [ ( f ( g ) ]*v (g) = L C g " 1 l*o$*y (g) (2.18) where in the l a s t l i n e we have used the l e f t i n v a r i a n c e of v. Now f o r any X j , . . . , X n £ Tg ($) we have L C g " 1 3 *O|*K ( g ) ( X j , . . . , X n ) = i * M l l ) ( d L [ g " 1 ] g X 1 , . . . , d L [ g _ 1 ] g X n which we w r i t e (somewhat l o o s e l y ) as L C g - 1 ] * i * v < g > = I * H l l ) o d L [ g " 1 ] g . (2.19) Because $ ( £ ) = 11, I*v (A) = det d $ f l v ( l l ) so from e q . ( 2 . 1 9 ) L [ g _ 1 ] * i * K ( g ) = det d j £ v ( i ) o d L C g " 1 ] g = det d $ f l L C g " 1 ]* y(g) = det dj f l M g ) . (2.20) Combin ing e q . ( 2 . 2 0 ) w i th e q . ( 2 . 1 8 ) and the f a c t t h a t J (g ) = det d i ^ y i e l d s eq. (2. 17) • Lemma 2.4; Let E s $ -» C and <p: S -> 3 be smooth maps and J ( g ) be d e f i n e d as i n Lemma 2 .3 . Then I Eeip(g) J (g ) dg = deg(<p) \ , E(h) dh. (2.21) I I.B THE LATTICE FADDEEV-POPOV FORMULA P r o o f : By Lemma 2.3 and d e f i n i t i o n 2 of d e g r e e , S Ec<p(g) J (g ) dg = \^ <f*(Ev) => deg (<p) Ev = deg((p) J E(h) dh. 0 If we take ip(g) = F(9a) then J (g ) = det M(9 a ) (see eq . ( 2 . 1 1 ) ) . Thus we have C o r o l l a r y 2.5; Let F: * * •» $ be any smooth map and a E If n,(a) i s the degree of the map if: $ •* $ d e f i n e d by <p(g) = F ( 9 a ) , then f o r a l l smooth f u n c t i o n s Ei $ •* (E det M(9 a ) E ° F ( 9 a ) dg = r\U) \ E(h) dh. (2.22) 0 • S 3 H i r s c h f e l d CH] based h i s work on a q u a n t i t y c l o s e l y r e l a t e d to ( c f . S e c t i o n I I I .C ) . As he p o i n t e d o u t , i t s u t i l i t y depends on Lemma 2.6: The q u a n t i t y r\(a) d e f i n e d in C o r o l l a r y 2.5 i s i ndependent of a. P r o o f ; T h i s i s a consequence of the f a c t tha t degree i s a homotopy i n v a r i a n t . For l e t aj ( i = 0,1) be any two gauge f i e l d s . We show below tha t the maps f j (g) - F ( 9 a j ) ( i = 0 , l ) a re homotop i c . S i n c e n,( a i ) - deg f j , i t then f o l l o w s tha t n ( a o ' = n , ( a i ) « The homotopy i s o b t a i n e d as f o l l o w s . By a s s u m p t i o n , G i s c onnec ted and hence so i s 5*. Thus t h e r e i s a c o n t i n u o u s path a . : C 0 , l ] •* 9* from a^ to a j . We may t ake a, to be smooth (see P r o p o s i t i o n IX, s e c t i o n 1.11 of I I.B THE LATTICE FADDEEV-POPOV FORMULA C6HV13). D e f i n e f . i t O , l ] i S •> » by f t ( g ) = F ( 3 a t ) . Then f^ i n t e r p o l a t e s between f g and f j . It rema ins to show that i t i s smooth. S i n c e F:$* •* S i s smooth i t s u f f i c e s to show tha t ( t , g ) •* 9 a t i s smooth. Now a map i n t o a p roduc t space i s smooth i f and o n l y i f the p r o j e c t i o n onto each component i s smooth. Thus i t i s enough to show tha t f o r each x ,y e A, the map ( t , g ) •* 9 a t ( x , y ) = g(x) a t ( x , y ) g (y ) " * 1 (2.23) i s smooth. By the d e f i n i t i o n of L i e g roup , the m u l t i p l i c a t i o n map y (g ,h ) = gh and the i n v e r s e map v(q) = g - 1 are smooth. A l s o , the p r o j e c t i o n maps g •» g(x) and a •* a ( x , y ) are smooth. Thus the map (2.23) i s a c o m p o s i t i o n of smooth maps and so i s i t s e l f smooth. T h i s shows tha t f^ p r o v i d e s a smooth homotopy between f p and f , . D With the se lemmas beh ind us , we are ready to prove the theorem. The proof i s s i m p l y an a d a p t a t i o n of the o r i g i n a l Faddeev -Popov argument CFP] . P roo f of Theorem 2.1 ; The upshot of the lemmas i s tha t f o r some c o n s t a n t n, depend ing o n l y on F, H J E (h ) dh = I det H(9a) E o F ( 9 a ) dg. (2.24) M u l t i p l y both s i d e s of e q . ( 2 . 2 4 ) by I I .B THE LATTICE FADDEEV-POPOV FORMULA -S (a ) da and app l y F u b i n i ' s theorem to o b t a i n H (fj E(h) dh) ($ ^ f (a) e " S ( a ) da) = (J det M(9 a ) E ° F ( 9 a ) dg) (J # f ( a ) e ' S ( a ) da) * J AJ * det M(9 a ) f ( a ) E ° F ( 9 a ) e " S ( a ) da dg. (2.25) Now we wish to make a "change of v a r i a b l e s " on the r i g h t - h a n d s i d e of e q . ( 2 . 2 5 ) from a to 9 a . More p r e c i s e l y , we are go ing to use the f a c t tha t i f <p i s any smooth f u n c t i o n on f o r any g E To see t h i s , r e c a l l t h a t da i s the p roduc t of the Haar measures da ( x , y ) and tha t Thus the i n t e g r a l over any of the v a r i a b l e s a ( x , y ) on the l e f t - h a n d s i d e of e q . ( 2 . 2 6 ) d i f f e r s from tha t on the r i g h t o n l y by a l e f t and a r i g h t t r a n s l a t i o n of a ( x , y ) . S i n c e d a ( x , y ) i s both l e f t and r i g h t i n v a r i a n t , e q . ( 2 . 2 6 ) h o l d s . Now both f and S a re gauge i n v a r i a n t , so \ . <p(ga) da = J # <p(a) da G G (2.26) 9 a ( x , y ) = g(x) a ( x , y ) g(y) -1 . det M(9 a ) f ( a ) E ° F ( 9 a ) e - S ( a ) da = \ ^ det M(9 a ) f ( 9 a ) E o F ( 9 a ) e - S ( 9 a ) da = \ # det M(a) f (a) E«F (a) e -S (a ) da and e q . ( 2 . 2 5 ) becomes II.B THE LATTICE FADDEEV-POPOV FORMULA 36 q(J E(h) dh)<$ # f (a) e " S ( a ) da) = J J # det M(a) f ( a ) EoF(a ) e ~ S ( a ) da dg = det M(a) f ( a ) E ° F ( a ) e - S < a ) da where we have used the f a c t tha t I I .C THE GAUGE DEBREE II. C The Gauge Degree Lemma 2.6 a l l o w s us to make the f o l l o w i n g d e f i n i t i o n . Def i n i t i on 8 Let F be a gauge f i x i n g f u n c t i o n . The gauge  degree a s s o c i a t e d wi th F i s the number n, which i s the degree of the map g •* F(9a) f o r any a £ As we saw in Theorem 2 .1 , the l a t t i c e Faddeev-Popov f o r m u l a J det M(a) f ( a ) E ° F ( a > e ~ S i a ) da <f> = 5—-I det fl(a) EoF(a) e ~ S ( a ' da i s v a l i d p r o v i d e d tha t the gauge degree i s not z e r o . When the gauge degree i s z e r o , both the numerator and denominator of the r i g h t - h a n d s i d e of the f o r m u l a a re equal to z e r o . That f a c t i s rea son enough f o r us to look f o r a way of d e t e r m i n i n g the gauge d e g r e e , but as we s h a l l see i n Chapter I I I , t h e r e a re o t h e r rea sons as w e l l . For the gauge degree r e f l e c t s to some e x t e n t the n a t u r e and number of o r b i t - s u r f a c e i n t e r s e c t i o n s . Thus i t s v a l u e g i v e s us some i n s i g h t i n t o the G r i b o v c o p i e s f o r t h a t gauge. In Theorem 2.7 below we o b t a i n a f o r m u l a f o r the gauge deg ree which makes i t e f f e c t i v e l y computab le . Let F: 5* •* #i be a gauge f i x i n g f u n c t i o n of the form F ( a ) ( x ) = IT a ( x , y ) m < > < ' > ' ) (2.6) < x , y > E A * where the exponents m(x,y) are i n t e g e r s . For g e * j we I I .C THE GAUGE DEGREE have F ( 9 £ ) ( x ) = n g ( y ) n ( x ' y ) . (2.27) y £ A f o r some i n t e g e r s n ( x , y ) . Let N: ^1 •» 1^ be the l i n e a r map whose matr ix e lement s a re g i ven by N x y • n ( x , y ) . (2.28) The gauge degree i s de te rm ined by the matr ix of exponents N. Theorem 2.7; Let G be a compact , connec ted L i e group w i th rank r. Let F and N be as above. Then the gauge degree a s s o c i a t e d wi th F i s g i ven by q = (det N ) r . (2.29) Remarks; (1) R e c a l l t h a t the rank of a L i e group i s the d imens i on (as a m a n i f o l d ) of a maximal a b e l i a n subgroup. (2) As we noted b e f o r e , t h e r e i s some a m b i g u i t y i n the n o t a t i o n used in e q s . ( 2 . 6 ) and (2.27) when G i s n o n a b e l i a n s i n c e the o r d e r of the f a c t o r s i n the group m u l t i p l i c a t i o n s i s not i n d i c a t e d . As we s h a l l see i n Theorem 2.8 be low, chang ing the o r d e r of these f a c t o r s does not change the gauge deg ree . Thus to a v o i d some clumsy n o t a t i o n ( c f . e q . ( 2 . 2 7 ) w i th e q . ( 2 . 3 0 ) ) we s h a l l use e x p r e s s i o n s l i k e (2.6) and ( 2 . 2 7 ) . Theorem 2.7 i s an immediate consequence of the f o l l o w i n g theorem a p p l i e d to the f u n c t i o n f ( g ) = F ( 9 £ ) . Theorem 2.B: Let G be a compact , connec ted L i e group I I .C THE GAUGE DEGREE w i th rank r. Let D M and ir i 16 D -> G be the c a n o n i c a l p r o j e c t i o n onto the i * n component. Suppose f : G ^ •* G^ has the form m i j , ( T ^ o f M g ) = ' g ^ . . . g. (2.30) f o r i = 1 ,2 , . . . ,D where g denotes (gj «g2 i><» »9n* E ^ and each m^j i s an i n t e g e r . Each j ^ e { 1 , 2 , . . . , D ) and d i f f e r e n t j ^ ' s can have the same v a l u e . D e f i n e n: ; = Z m; j . 1 J < W J > 1 J K Then where N = (nj j ) . deg f = (det N ) r Remarks; 1. In the ambiguous n o t a t i o n of e q . ( 2 . 2 7 ) we would w r i t e (IT, of) (g) = n g " i j f o r e q . ( 2 . 3 0 ) . 2. Theorem 2.8 g e n e r a l i z e s the we l l known r e s u l t tha t the degree of the power map g -> g n on G i s n r . The p roo f of Theorem 2.8 r e q u i r e s some r e s u l t s from the t h e o r y of the cohomology of compact L i e g roups . We g i v e a qu i ck rev iew here l e a d i n g to the main theorem in the s u b j e c t , Theorem 2.9 below. The i n t e r e s t e d reade r i s r e f e r r e d to C6HV2] f o r p r o o f s . The de Rham cohomology a l g e b r a of G i s the graded I I .C THE GAUGE DEGREE a l g e b r a g i ven by the d i r e c t sum n H(G) <= E H k (G) k=0 where n i s the d imens i on of G. We a l s o need the graded a l g e b r a n H +(G> = E H (G) . k = l The Kunneth theorem g i v e s an i somorph i sm H(G D ) = H(G) ® H(G) ® ••• ® H(G) (D c o p i e s ) by the map Xia^ ® a 2 ® ••• ® flpj) = T t j a j > i T 2 a 2 ' " " " n a n " In view of t h i s theorem, we n o r m a l l y i d e n t i f y H(G^) wi th ® D H(G). Let (J:GXG •+ 6 be the m u l t i p l i c a t i o n map fi(g,h) = gh. An element a E H + (G) i s p r im i t i ve i f f i # a = a ® l + l ® a . Let Pg denote the graded subspace of H(G) which c o n s i s t s of the p r i m i t i v e e l e m e n t s . Theorem 2 .9 : Let G be a compact , c o n n e c t e d L i e group wi th rank r. (a) Any p r i m i t i v e e lement of H(G) has odd deg ree . (b) dim Pg = r. (c) The i n c l u s i o n map Pg •» H(G) ex tends to an i somorphi sm * 6 ! A P G + H(G). • ' 0 where APg denotes the e x t e r i o r a l g e b r a over the v e c t o r I I .C THE GAUGE DEGREE space Pg. The p roo f of Theorem 2.8 i s based on the f o l l o w i n g t h r e e lemmas. Lemma 2.10: Let f ^ G ^ 4 6 be the m u l t i p l i c a t i o n map VN(?1i•••I^N' = 9 1 9 2 ' " g N * Suppose a £ H + (6 ) i s p r i m i t i v e . Then « D « liZa = E irfa. i = l P r o o f : We s h a l l need the f o l l o w i n g f a c t . Suppose Mj, Nj ( i = l ,2) a re any m a n i f o l d s and f ^: M j •* a re smooth maps. Let f j ® f 2 : M l x M 2 4 N i x N 2 b e t h e m a p (f i Of 2 ' < x l ' X 2 J = <U(*l} ' f ( x 2 ' } ' I c l a i m tha t ( f j ® f 2 > * ( a 1 ® a 2 > = f *a j ®f2a2-For l e t TTJ and Tfj denote the p r o j e c t i o n maps on N j x N 2 and M j x M 2 r e s p e c t i v e l y . Then, i f aj E H ( N J ) , (f j ® f 2 >* (al<Sa2) - (f j ®f 2 ># <* jf « i 1 " 2 a 2 ' = ( f j ® f 2 ) * a j . ( f j ® f 2 ) # n ^ a 2 = i r j f * a j • i r | f | a 2 = f f a 1 ® f j a 2 - (2.31) We now use i n d u c t i o n on D to prove the lemma. The lemma i s t r u e when D = 2 by the d e f i n i t i o n of p r i m i t i v e e l e m e n t . Note t h a t fi D = v2o(l ® ^ D _ j ) (2.32) I I .C THE GAUGE DEGREE 42 where I:G •* G i s the i d e n t i t y map. Thus f/Ja = ( I ® f / D _ 1 ) * f j j a = ( I ® u D . 1 ) # ( l « a + aftl) i = 1 ®fjo- j « + a ® l by e q . ( 2 . 3 1 ) . By the i n d u c t i o n h y p o t h e s i s , we are f i n i s h e d . • Lemma 2 .11; Let P m : G •* 6 ( m an i n t e g e r ) be the m-power map p.(g) = g m -Then i f a £ H + ( 6 ) i s p r i m i t i v e , pja = ma. P r o o f ; It s u f f i c e s to prove t h i s f o r m ± 0 and m = - 1 , s i n c e P ? m = P V P * . If m i 0 , l e t d m ; G + G m be the d i a g o n a l map <V 9 > = ( g , g , . . . , g ) . Then P m = u m « ^ . By Lemma 2 . 1 0 , and the f a c t t h a t m m . l i = l " ^ / " i ' V * * i = l = mx. If m = - 1 , l e t <p:G •* G*G be g •» ( g , g _ 1 ) . Then U2°<f> i s the c o n s t a n t map 11 so I I .C THE GAUGE DEGREE 43 0 = (fi 2o<p)*a = F>\a + P ? j a = a + P * j a . Hence P ? j a = - a . 0 Lemma 2.12: Let f be as i n Theorem 2.8 and a « H + (G) be p r i m i t i v e . Then * D * ( I N of ) w a = E n ; jTr^ a . j = l J J P r o o f : D e f i n e Q : G D -> GP i by " i i i <n i i m i ' U i " U o m i J „ Q(g) = ( 9 j \ g : 2 9 j P i ) . J l J 2 J p i Then ItjOf = ( J p . oQ. If we l e t 11^:6^ -* G be the c a n o n i c a l p r o j e c t i o n onto the k t n component, then TTI,OO = P. o IT; . Thus k n i i j k J k Pi iu- « Q ) # a = Q # ( E ifja) p i k=l K P i " • = E (jT , ,oQ)*a k = l K I I .C THE GAUGE DEGREE P i = E (P.. . on , >*« k = l " i J k J k Pi ° E ir? P? a k-1 J k m i J k Pi • k-1 1 J k J k D « j -1 1 J J Proof of Theorem 2.8; By d e f i n i t i o n 3 of degree (see S e c t i o n I I . B ) , i t s u f f i c e s to show t h a t f * a = (det N ) r a f o r any nonzero a i n the top cohomology v e c t o r space of G D ( i . e . , a i H D k ( G D ) where k - dim G) . We s h a l l c o n s t r u c t a s u i t a b l e a from the p r i m i t i v e g e n e r a t o r s of H(G). F i r s t , we c l a i m . t h a t t h e r e a re p r i m i t i v e s a j , a 2 i - ' " ' a r E H + (G) so tha t a j 0 ( 2 " ' , a r * s a nonzero e lement of the top cohomology v e c t o r space of 6. F o r , by Theorem 2 .9 , the subspace Pg of p r i m i t i v e e l ement s of H(G) has d imens ion r. Let { a j , . . . , a r > be a b a s i s of Pg The l i n e a r i ndependence of the otj i m p l i e s t h a t (see Lemma VI I I s e c t i o n 4.17 of [GHV21) a 1 « 2 " " ' a r * 0. S i n c e the map ^gsAPg -» H(G) i s an i somorph i sm (Theorem 2 . 9 ) , a^--ar must have top degree s i n c e e v e r y t h i n g in H(G) can be c o n s t r u c t e d from the a^. T h i s I I .C THE GAUGE DEGREE p r o v e s the c l a i m . If 0 £ H d ( G ) , d e f i n e /SD £ H d D ( G D ) by /SD = 0®/3®- • -®/3 (D c o p i e s ) . Set a = a j - a j j . - a j . Note t h a t up to a p o s s i b l e minus s i g n , a e q u a l s iala2-• ' f l r ) D and so i s not z e r o . A l s o , a £ H k D ( G D ) . We now show tha t f # a f = (det N ) a { from which i t f o l l o w s t h a t f # a = (f # a f ) • ( f # a P , ) (f # a ° ) = (det N ) r a . (2.33) Let 0 = a K f o r any k and r e c a l l (Theorem 2 .9 (a ) ) tha t # has odd d e g r e e . Then f # 0 D = f * (ir^p -Tr| TT § ^ > = (f #Tf^|3) • ( f # T f ^ > ( f # T r $ 0 ) . The f a c t t h a t T T * ^ ' T T ^/3 = -iTj/S'Tr*^ and Lemma 2.12 now g i v e f # 0 D = (det N ) f l D . • II.D VALIDITY OF THE LATTICE FADDEEV-POPOV FORMULA II.D V a l i d i t y of the l a t t i c e Faddeev-Popov f o rmu la Now tha t we know how to c a l c u l a t e the gauge degree (Theorem 2.7) we can check in p a r t i c u l a r examples whether or not i t i s z e r o , and hence whether or not the l a t t i c e Faddeev-Popov f o r m u l a i s v a l i d . Some i n s i g h t i n t o the " b a d " case <n = 0) can be o b t a i n e d from Theorem 2 . 1 3 be low. F i r s t , note tha t the Faddeev-Popov de te rm inan t i s most e a s i l y c a l c u l a t e d u s i n g the f a c t tha t i f <p: 3 •* $ maps 11 i n t o 1 , then <p(e*') = 11 + dtp £y + 0 ( y 2 ) . ( 2 . 3 4 ) It i s a l s o c o n v e n i e n t to i n t r o d u c e the s t a n d a r d o r d e r e d b a s i s of IR . The b a s i s e lements ^ x ^ x e A j (which we r e g a r d as f u n c t i o n s from Aj •* IR) are d e f i n e d by b x ( y ) = 6^^ and they are o r d e r e d l e x i c o g r a p h i c a l l y . That i s , b x > by i f f o r some p £ ( 0 , 1 , . . . , s ) X j = yi f o r i < p and X P ;' V Theorem 2 . 1 3 : The gauge degree i s z e r o i f and o n l y i f the Faddeev -Popov d e t e r m i n a n t at 11 i s 0: H = 0 <=> det M ( £ ) = 0. ( 2 . 3 5 ) P r o o f : We s h a l l show tha t det M(!) = (det N ) k ( 2 . 36 ) where k i s the d imens ion of G and N i s the mat r i x I I.D VALIDITY OF THE LATTICE FADDEEV-POPOV FORMULA d e s c r i b e d i n e q . ( 2 . 2 2 ) . The e q u i v a l e n c e (2.35) i s then o b v i o u s s i n c e H • (det N ) r . In the f o l l o w i n g , we s h a l l f r e e l y i d e n t i f y the i s o m o r p h i c v e c t o r spaces R e c a l l tha t where M ( £ ) = dip i <p(g) = F(9f l ) . Let j E €. Then (see e q . ( 2 . 2 7 ) ) y£ A = ff e n ( x , y ) y ( y ) yf /\ = 11+ E n ( x , y ) y ( y ) + 0 ( j r 2 ) . y M A p p l y i n g e q . ( 2 . 3 4 ) , we see tha t d(p a = N ® I (2.37) where I: E •* E i s the i d e n t i t y map. S i n c e the d imens ion of E i s k, det M ( i ) = det . (N ® I) = (det N ) k . • R e c a l l tha t a gauge f i x i n g f u n c t i o n F i s d e f i n e d wi th r e s p e c t to a s u b l a t t i c e Aj C A (see S e c t i o n I I .A ) . The next theorem shows tha t we run i n t o t r o u b l e i f we t r y to f i x the gauge on a l l of A. II.D VALIDITY OF THE LATTICE FADDEEV-POPOV FORMULA Theorem 2.14; If Aj = A then the gauge degree a s s o c i a t e d wi th F i s z e r o : where the n ( x , y ) a re d e f i n e d in e q . ( 2 . 2 7 ) . Thus the k e r n e l of N (see e q . ( 2 . 2 2 ) ) i s n o n t r i v i a l , s i n c e i t c o n t a i n s the f u n c t i o n s on A which are c o n s t a n t . Thus n = (det N ) r = 0. We prove (2.38) assuming tha t 6 i s a b e l i a n . The proo f f o r the n o n a b e l i a n case i s the same except tha t the n o t a t i o n i s c o n s i d e r a b l y more c o m p l i c a t e d . - A => n = 0. P r o o f ! We s h a l l show tha t E n ( x , y ) = 0 ye A (2.38) We have F(a) (x) = n a ( x , y ) ye A m (x, y) and F ( 9 a ) ( x ) = Tl [ g ( x ) a ( x , v ) g ( y ) - 1 l m ( x ' y ) ye A so tha t F(9f i ) (x) = IT g ( x ) m ( ! < ' > ' ) g ( y ) ~ m ( ; < ' y , . y *x From e q . ( 2 . 2 7 ) we see tha t n ( x , x ) e E m(x,y) y*x n (x ,y) = -m(x,y) (y * x ) . Thus II.D VALIDITY OF THE LATTICE FADDEEV-POPOV FORMULA E n (x , y) = n (x , x ) + E n ( x , y ) y£/\ ytA y#x = E m(x,y) - E m (x,y) y£/\ ytA yj*x y#x = 0. The reade r might we l l be wonder ing at t h i s p o i n t whether t h e r e are any ca se s f o r which the gauge degree i s not z e r o . One way to get around Theorem 2.14 i s to take /\j to be a p rope r s u b l a t t i c e of A. We now i l l u s t r a t e how t h i s works f o r a x i a l gauge and Landau gauge. A x i a l gauge i s the e a s i e s t to a n a l y z e by our methods. In t h i s ca se the gauge f i x i n g f u n c t i o n i s ( c f . e q . ( 2 . 7 ) ) F Q < a)(x ) = a (x , x + e Q ) . We use e q . ( 2 . 3 4 ) to c a l c u l a t e M(a). Let * = { * < x , } K t A j E f l ' a 6 * * a n d * < 9 ) = F A ( a ) " l F A ( 9 a ) • Then $ (e* ) (x ) = F A < e * a ) (x) F A ( a ) ( x ) " 1 = e* u , a ( x , x + e 0 ) e " * ( x + e 0 > a (x , x +e 0 ) - 1 = (I+y (x) ) a (x, x + e 0> ( I -y (x+e 0 ) ) a (x , x + e 0 ) - 1 + 0<y 2 ) = 1 + y(x) - a ( x , x + e 0 ) j ( x + e 0 ) a ( x , x + e 0 ) _ 1 + 0 ( y 2 ) = I + y(x) - a d C a f x ^ + e ^ J I y f x + eQ) + 0 ( y 2 ) where a d ( a ) : E -» E f o r a £ 6 i s the a d j o i n t map ad(a) = ay a - 1 . A p p l y i n g e q . ( 2 . 3 4 ) , we deduce tha t II.D VALIDITY OF THE LATTICE FADDEEV-POPOV FORMULA M ( a ) x x = I M u , x , x + e 0 = - adCa (x ,x + e 0 ) 3 M(a ) X y = 0 i f y*x or x + e^. In the s t a n d a r d o r d e r e d b a s i s of IR 1 b x + e 0 > b x so the matr ix M(a) i s upper t r i a n g u l a r wi th d i a g o n a l e lement s equa l to one. Thus det M(a) = 1 f o r a l l a £ 5*. Theorem 2.13 t e l l s us t h a t i n p a r t i c u l a r , r\ t 0 f o r a x i a l gauge. A r i g o r o u s v e r s i o n of the l a t t i c e Faddeev-Popov f o r m u l a f o r a x i a l gauge has a l s o been g i ven by COS]. They showed tha t f o r any gauge i n v a r i a n t f u n c t i o n f S f A ( a ) e - s A ( a ) da <f> • i I e " S ( a ) da where f A and S A a re o b t a i n e d from f and S by s e t t i n g a l l arguments a(x,x+e,j) to 11. We can o b t a i n t h i s f o r m u l a from e q . ( 2 . 1 3 ) by c h o o s i n g E (c J = <$(c), where by 6(c) we mean Tfx (c x ) the l a t t e r i - f u n c t i o n be ing the one a p p r o p r i a t e to Haar measure COS]. We now tu rn to the more c o m p l i c a t e d Landau gauge. G iven x e A, d e f i n e a d ( i ^ ) = ad C a (x , x+eQ> a (x , x -e^) a (x , x+e j) • • >a (x , x ie^j) ] s o , f o r example, a d d ) = ad C a (x , x+ e,^ ) a (x , x -e^) a (x , x+ e j) ] a d ( - l ) = a d C a ( x , x + e 0 ) a ( x , X - B Q ) a ( x , x + e j ) a ( x , x - e j ) ]. II.D VALIDITY OF THE LATTICE FADDEEV-POPOV FORMULA For F|_ as g i v e n by e q . ( 2 . 8 b ) and g = e* t 3 we have Fj_(a) ( x ) " 1 F L ( 9 a ) <x) = (I + xMx)) a (x ,x + e 0 ) (I - y(x + e 0 >) x(I + y ( x ) ) a ( x , x - e 0 ) (I - y ( x - e 0 ) ) x . . . x(I + y ( x ) ) a ( x , x - e s ) (I - y ( x - e s ) ) xCa (x ,x+e 0 ) . . . a ( x , x - e s ) ] - 1 + Q ( y 2 ) = I + (I - a d ( - s ) + Cad ( + (<)+ ad( - f j ) ] } y(x) - ad( + (j) yfx+e^) - ad( - f i ) y l x - e ^ ) + 0 < y 2 ) . (2.39) The matr ix e lement M ( a ) x y i s the c o e f f i c i e n t of y (y) i n the p r e c e d i n g e x p r e s s s i o n . Thus M ( a ) x > ( = 1 - a d ( - s ) + E [ad( + fi) + a d ( - u ) l s U = 0 M ( a , x , x ± e f J a - a d , ± ( j ) • ( 2 - 4 0 ) M ( a ) x y = 0 i f I x - y l > e. For the gauge degree we need o n l y c o n s i d e r M( l l ) . From eq . (2.40) we have M ( f l ) x x = 2 d I M < i ) „ „+_ = - I (2.41) « , n _ c M< l l ) K y = 0 i f I x - y l > e. Thus M ( i ) = E 2 I (2.42) where d\ i s the f i n i t e - d i f f e r e n c e L a p l a c i a n with D i r i c h l e t I I.D VALIDITY OF THE LATTICE FADDEEV-POPOV FORMULA boundary c o n d i t i o n s on A t6RS2 ] ,C6J3 . The e i g e n v a l u e s of A\ are CGJ] K » E 4e i s i n M k . . T r / 4 N , , ) (2.43) K u=0 U U with E { 1 , 2 , . . . , 2 N ^ - 1 > . No e i g e n v a l u e i s z e r o , so det Mil l) * 0 and the gauge degree n, # 0 f o r t h i s c h o i c e of A j . We have j u s t seen tha t by c h o o s i n g boundary c o n d i t i o n s c a r e f u l l y i t i s not too hard to a r r ange t h a t n # 0. However, i t would be b e t t e r to have more freedom to make t h i s c h o i c e . For example, one n a t u r a l way to d e f i n e a gauge f i x i n g f u n c t i o n on a l l of A i s to impose p e r i o d i c boundary c o n d i t i o n s . If we had done t h i s f o r Landau gauge i n e q . ( 2 . 8 b ) , the d i f f e r e n c e would be tha t we would have dp, the L a p l a c i a n with p e r i o d i c boundary c o n d i t i o n s , i n s t e a d of A\ i n e q . ( 2 . 4 2 ) . In t h i s c a s e , det M(fi) = 0 because the k e r n e l of dp c o n t a i n s the c o n s t a n t f u n c t i o n s and so i s non t r i v i a 1 . Th i s example p r o v i d e s an i l l u s t r a t i o n of Theorem 2.14 and at the same t ime sugge s t s a way to c i r c u m v e n t i t . If the c o n s t a n t f u n c t i o n s a re the sou rce of the t r o u b l e , why not f a c t o r them out at the b e g i n n i n g ? The f o l l o w i n g lemma and theorem are a p a r t i a l i m p l e m e n t a t i o n of t h i s i d e a . Suppose tt i s a normal subgroup of $ and l e t Cg] denote the image of g E S under the c a n o n i c a l p r o j e c t i o n Lemma 2.15; Suppose F: $* •* $ i s such t h a t f o r any a E $* II.D VALIDITY OF THE LATTICE FADDEEV-POPOV FORMULA the map <pa; •* $/H 5 a(Cg]) = t F ( ' a ) ] (2.44) i s w e l l - d e f i n e d . Then n, = deg(!J>a) i s the same f o r a l l a. F u r t h e r m o r e , f o r any smooth f u n c t i o n E on $/h' and gauge i n v a r i a n t f u n c t i o n f on * * , n,(J E (Cc ] ) dCc]) (I # f (a) e - s < a ) da) = det M(a) f ( a ) E ICF (a ) } ) e " S ( a ) da where M(a) = d&£ f o r e<[gl) = J a ( t g ] ) v a ( C i ] ) _ 1 • P r o o f ; F i r s t note tha t the map [g] •* M( y a) i s -v g; w e l l - d e f i n e d . For suppose t h a t [ g j ] = Cg23 . M( a) i s the d e r i v a t i v e at 11 of the map e i (Chl ) = [F( *a) ] CF( 1 a )3 l. S i n c e Choj] = [ f ^ J and <pa i s w e l l - d e f i n e d , C F ( 9 l a ) ] = [ F ( 9 2 a ) 3 and h g 1 h g <-> [F( *a) J = CF( * a ) ] . Thus 6j = e 2 and M ( 9 l a ) = M ( ? 2 a ) . App ly Lemma 2.4 wi th * r e p l a c e d by and <p by (pa to o b t a i n J det M( g a) E ( [ F ( g a ) ] dCg] = d e g f J j [ E ( C h 3 ) d t h l . (2.45) The p roo f tha t deg('p,) i s i ndependent of a i s the same II.D VALIDITY OF THE LATTICE FADDEEV-POPOV FORMULA as tha t of Lemma 2.6. The rema inder of the proof of the p r e s e n t theorem runs p a r a l l e l to the p roo f of Theorem 2 .1 : n<$ E ( C c l ) d C c l ) ( | # f ( a ) e " S ( a ) da) = $ \ # det fi(9a) f ( a ) E<[F ( Q a>] ) e " S ( a ) da d t g ] = [ [ # det fi(9a) f ( 9 a ) E ( [ F ( 9 a ) ] ) e " S ( 9 a ) da d i g ] • S , i d e t M(a) f ( a ) E ( C F ( a ) ] ) e " S ( a ) da d t g ] = J * det M(a) f ( a ) E ( I F ( a ) ] ) e " S ( a ) da. D Theorem 2.16: Let 6 be a b e l i a n . Us ing the n o t a t i o n of Lemma 2 .13, l e t M(a) be the Faddeev-Popov o p e r a t o r a s s o c i a t e d wi th F. Suppose the k e r n e l of M(fl) c o i n c i d e s wi th the L i e a l g e b r a of U. Then r\ t Q and 5 det M(a) f ( a ) E ( t F ( a ) ] ) e " S ( a ) da <f> = — 5 — q T T i • ( 2 , 4 6 ) \ det M(a) E ( £ F ( a ) ] ) e'b{a) da P r o o f : Let J = Because 6 i s a b e l i a n we have 6 ( [h ] ) = <p(Chg]) y ( [ g ] ) _ 1 = [F ( h 9 a ) F ( 9 a ) _ 1 3 = I F ( h i ) ] (2.47) so tha t M(a) i s i ndependent of a; we s h a l l denote i t s i m p l y by M. By t a k i n g E = 1 in eq.<2.22) we see tha t det M = deg if. We a l s o see from e q . ( 2 . 4 7 ) tha t 9.(th]) = <p(Lh]) so M = d$£ = d(f£ and c o n s e q u e n t l y II.D VALIDITY OF THE LATTICE FADDEEV-POPOV FORMULA deg ip = det dip^. Let 7 denote the L i e a l g e b r a of ft. The L i e a l g e b r a of i s i s o m o r p h i c to €/J [BHV23. Let fig) => F ( g f l ) . Then <p i s the map induced by ip. By h y p o t h e s i s ? i s the k e r n e l of the l i n e a r map dip^, so tha t t h e r e i s an i nduced map (d<pfl> i€/7 •* €/7 which i s n o n s i n g u l a r . But under the i somorph i sm of L i e a l g e b r a s j u s t men t i oned , <d<Q^) i s mapped i n t o d(p£, so tha t the l a t t e r i s n o n s i n g u l a r . Thus deg (<p) = det (d(p^) # 0. D I c o n j e c t u r e tha t Theorem 2.16 i s a l s o t r u e when G i s n o n a b e l i a n . However, as we s h a l l see be low, i t i s d i f f i c u l t to s a t i s f y the hypo these s of Lemma 2.15 when 6 i s n o n a b e l i a n . Let us now app l y Theorem 2.16 to the example tha t i n t r o d u c e d i t . Suppose we wish to use Landau gauge when G = U ( l ) and A i s the p e r i o d i c l a t t i c e A * { (n,-,£ , . . . , n sE ) : -N^ £ n^ £ N^) wi th boundary p o i n t s i d e n t i f i e d . As we d i s c u s s e d b e f o r e Lemma 2.15, M ( i ) = -e 2<dp i n t h i s c a s e . The k e r n e l 7 of dp c o n s i s t s of the c o n s t a n t f u n c t i o n s , i . e . , $ = {y;A->E I y(x) = c f o r some c c E , a l l x e A ) . T h i s can e a s i l y be seen by t a k i n g the F o u r i e r t r a n s f o r m , Let A = { ( n 0 b 0 , . . . , n s b s ) : = — , - N ^ n ^ N ^ - l ) . The f u n c t i o n s {e k > k e J ( g i ven by e " k < x ' = e i k * x form a II.D VALIDITY OF THE LATTICE FADDEEV-POPOV FORMULA b a s i s f o r CA and we have the r e l a t i o n s f<k> = E f(x) ek(x) x £ A * <x> = S T L - T 2 e„(x) f (k) TT(2N > v k p H k£/\ A s h o r t c a l c u l a t i o n shows tha t " 4 e k = V k where * k = 4 E " 2 E s in 2 ( fk E ) . u = 0 If f e Ker<4' then <d|f (x) = 0 => s-^ - — r E l i . e j (k) = 0 p TT < 2N ) k k k => f ( k ) = 0 f o r a l l k * 0 s > f < x) = •—-J——• f (0) TT(2N ) i . e . , f i s a c o n s t a n t f u n c t i o n . The subgroup tt c 3 which has 7 f o r i t s L i e a l g e b r a i s the group of c o n s t a n t f u n c t i o n s A •* G. S i n c e G i s a b e l i a n , tt i s a normal subgroup. The map <pa of Lemma 2.15 i s w e l l - d e f i n e d f o r any F s $ * •* S. To see t h i s , note f i r s t tha t f o r any h £ tt and a £ S* h a = a. If I g j ] = t. g 2 ^  t n e n t h e r e i s an h e tt such t h a t g j = g 2 n a n d i p a ( [ g j ] ) = C F ( 9 l a ) ] I I.D VALIDITY OF THE LATTICE FADDEEV-POPOV FORMULA = C F ( 9 2 h a ) ] = C F ( 9 2 a ) ] Thus Theorem 2.16 a p p l i e s and we have an Faddeev-Popov f o r m u l a f o r the a b e l i a n p e r i o d i c l a t t i c e model . The p r i c e we have had to pay to overcome the n = O p rob lem i s tha t now the damping f a c t o r E i s a f u n c t i o n of CF (a ) ] i n s t e a d of F ( a ) . The f u n c t i o n E sugges ted i n e q . ( 2 . 9 ) i s not of t h i s fo rm. It i s more d i f f i c u l t to a p p l y Lemma 2.15 to a n o n a b e l i a n model . The subgroup ti c o n s i s t i n g of the c o n s t a n t f u n c t i o n s of S i s the n a t u r a l c h o i c e to make, but i t i s not a normal subgroup so i s a homogeneous space r a t h e r than a L i e g roup. More s e r i o u s l y , the map <pa i s not w e l l - d e f i n e d in g e n e r a l when G i s n o n a b e l i a n . I I I.fi RELATION BETWEEN LATTICE AND CONTINUUM CHAPTER III CONTINUUM FADDEEV-POPOV TECHNIQUE REVISITED I I I .A R e l a t i o n Between the L a t t i c e and Continuum Arguments Much of the d i s c u s s i o n i n the p h y s i c s l i t e r a t u r e of the G r i b o v phenomenon in cont inuum t h e o r i e s i s based on i m p l i c i t a n a l o g i e s between f i n i t e - and i n f i n i t e - d i m e n s i o n a l f i e l d t h e o r i e s . One of the main purposes of the work d e s c r i b e d i n t h i s t h e s i s i s to c l a r i f y these d i s c u s s i o n s , f i r s t of a l l by p u t t i n g the f i n i t e - d i m e n s i o n a l ( l a t t i c e ) t h e o r y on a f i r m f o u n d a t i o n . We have done so i n Chapter II. In t h i s s e c t i o n we take the next s t e p , which i s to make more e x p l i c i t the p a r a l l e l s between the cont inuum Faddeev-Popov t e c h n i q u e d e s c r i b e d i n Chapter I and the l a t t i c e Faddeev -Popov t e c h n i q u e i n Chapter II. S p e c i f i c a l l y , we r e d e r i v e C o r o l l a r y 2.5 and Lemma 2.6 in a h e u r i s t i c f a s h i o n and in a way which mimics the cont inuum argument. Of c e n t r a l i n t e r e s t i s the Faddeev-Popov de te rm inan t i t s e l f . In the cont inuum t h e o r y , i t a r i s e s as a J a c o b i a n y f o r a change of v a r i a b l e s y •* F ( e a) on the L i e a l g e b r a of the gauge group. We show below ( C o r o l l a r y 3.2) tha t the l a t t i c e Faddeev-Popov d e t e r m i n a n t i s the J a c o b i a n f o r the g change of v a r i a b l e s g •+ F( a) on the gauge g roup. To b e g i n , l e t us d e r i v e a change of v a r i a b l e s f o r m u l a I I I .A RELATION BETWEEN LATTICE AND CONTINUUM •for i n t e g r a l s taken wi th r e s p e c t to Haar measure. Theorem 3.1 below g e n e r a l i z e s the f a m i l i a r f o r m u l a f o r Lebesgue measure on IRn $ Eo<p<x) | J ( X ) I dx = \ E l y ) dy (3.1) U J<p(U) where J i s the J a c o b i a n d e t e r m i n a n t J (x ) = det dip x . (3.2) Iheorem 3 .1 ; Let S be a L i e group (not n e c e s s a r i l y compact) and U be an open subset of G. Suppose tha t E:G •» IR i s a smooth f u n c t i o n wi th compact suppor t and <p:U •* 6 i s a di f f eomorphi sm onto i t s image. Then J E(h) dh = [ E o f ( g ) |J(g) | dg (3.3) J<p(U) *U i n which dg and dh denote l e f t Haar measure on G and J i s the J a c o b i a n d e t e r m i n a n t J (g ) = det d j £ (3.4) where i = L C ^ I g ) " 1 ] « f o L E g ] , (3.5) Moreover , J (g ) j* 0 f o r any g £ U and J E c ip ( g ) dg = I _ l ^ i _ d h . (3.6) U V<U) |J (if 1 (h)) I P r o o f ; We f i r s t c l a i m tha t J (g ) >.0 i f and o n l y i f <f p r e s e r v e s o r i e n t a t i o n at g. We o r i e n t G by c h o o s i n g a p o s i t i v e o r i e n t a t i o n c l a s s 0 + of T f l (G ) and d e c l a r i n g t h a t a b a s i s B of Tg(G) i s p o s i t i v e l y o r i e n t e d i f and o n l y i f d L t g - 1 ] g B e 0 + . In o t h e r words, l e f t t r a n s l a t i o n i s an o r i e n t a t i o n p r e s e r v i n g map. I I I .A RELATION BETWEEN LATTICE AND CONTINUUM C o n s e q u e n t l y , <p p r e s e r v e s o r i e n t a t i o n at g i f and o n l y i f I p r e s e r v e s o r i e n t a t i o n at £ . S i n c e $(fi) = I, $ p r e s e r v e s o r i e n t a t i o n at 11 i f and o n l y i f det dig > 0, t h a t i s , i f and o n l y i f J ( g ) > 0. T h i s p roves the c l a i m . W r i t e U as the d i s j o i n t un ion of U + and U_ where U + = {g E u I J (g ) > 0 ) , U_ = (g E u I J (g ) < 0 ) . If u i s any compac t l y s u p p o r t e d n - form (n = dim 6) then 3 \ i + > ( U + ) \ if u = - \ w U_ <f(U_) s i n c e <p p r e s e r v e s o r i e n t a t i o n on U + and r e v e r s e s o r i e n t a t i o n on U_ CGP]. Take u = Ev and app l y Lemma 2.3 to o b t a i n , \ ^ E * <p(g) |J (g )| dg = - \ ^ E « <p(g) J ( g ) dg = - { V * ( E y ) U_ = [ E(h) dh U_ and s i m i l a r l y f o r the i n t e g r a l over U + . F i n a l l y , s i n c e (p i s a di f f eomor ph i sm tp(U) i s the d i s j o i n t un ion of <p(U + ) and <p(U_). E q u a t i o n (3.3) i s o b t a i n e d by add ing the i n t e g r a l s over U + and U_. The "moreover " i s a consequence of the f a c t tha t J (g ) = 0 o n l y i f d i f i i s s i n g u l a r , but i t never i s because if and LCh l f o r any h are di f f eomorphi sms. E q u a t i o n (3.6) i s o b t a i n e d by r e p l a c i n g the f u n c t i o n E I I I .A RELATION BETWEEN LATTICE AND CONTINUUM in eq . (3. 3) by E/ IJ o<p~11. 0 C o r o l l a r y 3.2; The FP d e t e r m i n a n t , det M(a) , i s the J a c o b i a n de te rm inan t J ( l l ) f o r the change of v a r i a b l e s <p(g) = F ( ?a ) . P r o o f : By e q s . ( 3 . 4 ) and ( 3 . 5 ) , J ( f l ) = det di& where K g ) = L C i p d n " 1 ] o ip o L [ 1 ] (g ) = F ( a ) " 1 F ( 9 a ) . T h i s f u n c t i o n f i s the same as the f u n c t i o n $ d e f i n e d i n eq . ( 2 . 1 0 ) . Thus by e q . ( 2 . 1 1 ) , det M(a) = det d i f l = J ( i ) . 0 We now app l y t he se r e s u l t s to the FP t e c h n i q u e . The o r i g i n a l FP argument, e x p r e s s e d i n the n o t a t i o n of the l a t t i c e t h e o r y , i s based on the f a l s e e q u a l i t y ( c f . e q . ( 1 . 14) ) det M(a) J. <5 (F <9a) c " 1 ) dg = 1. (3.7) Let us beg in i n s t e a d by d e f i n i n g the q u a n t i t y n,(a,c) s J det M(9 a ) 6 ( F ( 9 a ) c " 1 ) dg. (3.8) The i - f u n c t i o n in e q s . ( 3 . 7 ) and (3.8) i s the one a p p r o p r i a t e to Haar measure, as d e s c r i b e d by COS]. However, we are a r g u i n g somewhat i n f o r m a l l y here and do not r e q u i r e a n y t h i n g from 6 o the r than tha t i t behave as a i - f u n c t i o n s h o u l d : I E(h) 6(he-1) dh = E ( c ) . The c o n n e c t i o n between q ( a , c ) and the gauge degree n, I I I .A RELATION BETWEEN LATTICE AND CONTINUUM g i ven by e q . ( 2 . 1 3 ) can be made by t a k i n g E(h) = <S(hc~*) i n the l a t t e r e q u a t i o n . Let us suppose tha t c i s a r e g u l a r v a l u e of (p(g) = F ( 9 a ) . Then by the i n v e r s e f u n c t i o n theorem CGPJ, ip i s a l o c a l d i f f e o m o r p h i s m at each g E (p~*(c). S i n c e S i s compact , we c o n c l u d e t h a t <p~* i s a f i n i t e s e t . Let (p - 1 (c) = <9! 192 ' • ' • >9n } and l e t U^ be a ne i ghbourhood of g k on which tp i s a d i f f e o m o r p h i s m . We can then use Theorem 3.1 and r e w r i t e e q . ( 3 . 8 ) as n , n,(a,c) = E t det M( 9 a) tf(F(9a)c_1) dg k = l u k . s i A ? i j ^ : ^ i ^ S { h c - h dh k = 1 v ( U k ) |det M ( V ~ 1 < h , a ) | . J det M ( 9 k a ) k = l Idet ( 9 k a ) I = £ sgn det M ( 9 k a ) (3.9) k = l where sgn(x) = x / t K I f o r x t 0. In o rde r tha t e q . ( 3 . 8 ) be a f r u i t f u l r ep l acement f o r e q . ( 3 . 7 ) , we need to show t h a t n,(a,c) i s i ndependent of a and c. Of c o u r s e , we have a l r e a d y seen t h i s s i n c e n.(a,c) i s the degree of <p(g) = F ( 9 a ) . However, t h e r e i s some i n t u i t i v e i n s i g h t to be ga ined by p u r s u i n g the matter aga in in the p r e s e n t c o n t e x t . As we saw in the proo f of Theorem 3 .1 , I I I .A RELATION BETWEEN LATTICE AND CONTINUUM sgn det M( K a > = e (<p, g k ) (3.10) where e(<p,g k) i s the o r i e n t a t i o n number of if at g k . Thus from e q . ( 3 . 9 ) n. (a, c) = E g ^ f " 1 <c) e (V ,g k) = I (<p, <c>) = deg if (3.11) where we are u s i n g d e f i n i t i o n 1 of degree (see S e c t i o n I I .B ) . C o n s i d e r how we would o b t a i n n,(a,c) i n a s i m p l e example where <p:U(l) -> U ( l ) . S i n c e U ( l ) £ S 1 ( the u n i t c i r c l e ) we can p l o t the graph of if i n a u n i t square wi th o p p o s i t e edges i d e n t i f i e d . (See F i g . 2) The o r i e n t a t i o n number £ ( t p , g k ) i s +1 or -1 depend ing on whether the s l o p e of if i s p o s i t i v e or n e g a t i v e when i t i n t e r s e c t s the l i n e de te rm ined by c. The homotopy i n v a r i a n c e of I(<p,{c>) and i t s i ndependence of c are i n t u i t i v e l y o b v i o u s from t h i s p i c t u r e . If if (and hence i t s graph) or c changes in a c o n t i n u o u s manner, the p o i n t s of i n t e r s e c t i o n are c r e a t e d or d i s a p p e a r i n p a i r s w i th o p p o s i t e o r i e n t a t i o n number. Hence the sum over g k i n (3.11) remains the same. We are now in a p o s i t i o n to d i s c u s s in more d e t a i l the remarks made a f t e r the s ta tement of Theorem 2.1. If t h e r e were no G r i b o v c o p i e s , then cp - 1 (c) would c o n t a i n o n l y one p o i n t g. There would be o n l y one term i n the sum (3.9) and c o n s e q u e n t l y ln . (a ,c )| = 1. E q u a t i o n (3.7) i s c o r r e c t I I I .A RELATION BETWEEN LATTICE AND CONTINUUM 64 F i g u r e 2 O r i e n t a t i o n numbers f o r the map g •+ F ( 9 a ) I I I .A RELATION BETWEEN LATTICE AND CONTINUUM in t h i s ca se (up to a s i gn ) and. i s e q u i v a l e n t to e q . ( 3 . 8 ) . When t h e r e a re G r i b o v c o p i e s , t h e r e a re s e v e r a l terms i n the sum ( 3 . 9 ) , one f o r each copy. The q u a n t i t y n then measures not the t o t a l number of c o p i e s , but the net number or a l g e b r a i c sum of the number of c o p i e s , with the s i g n of a c o p y ' s c o n t r i b u t i o n de te rm ined by the s i g n of the Faddeev -Popov d e t e r m i n a n t at t h a t p o i n t . I I I .8 SOME QUESTIONS FROM THE CONTINUUM THEORY I I I .B Some Q u e s t i o n s from the Continuum Theory The l a t t i c e Faddeev -Popov t e c h n i q u e we have d e s c r i b e d in Chapter II a l l o w s us to a n a l y z e the b e h a v i o r of gauge f i e l d o r b i t s and g a u g e - f i x i n g s u r f a c e s i n more d e t a i l than i n the cont inuum model . In t h i s s e c t i o n we a d d r e s s the l a t t i c e ana logues of some of the i s s u e s r a i s e d i n Chapter I f o r the cont inuum t h e o r y . The t o p i c s d i s c u s s e d a r e : ( i ) the a t t a i n a b i l i t y of gauge c o n d i t i o n s ( i i ) the e x i s t e n c e of G r i b o v c o p i e s ( i i i ) the gauge i n v a r i a n c e of the Faddeev-Popov determi nan t . Not s u r p r i s i n g l y , the d i s c u s s i o n r e l i e s h e a v i l y on our knowledge of the gauge deg ree . Throughout t h i s s e c t i o n we use the usua l n o t a t i o n of F:#* -> 3 f o r the g a u g e - f i x i n g f u n c t i o n and q f o r the gauge degree a s s o c i a t e d with F. ( i ) A t t a i n a b i l i t y of gauge c o n d i t i o n s : F i x a gauge f i e l d a e $* and l e t c e $. The q u e s t i o n to be i n v e s t i g a t e d i s , does t h e r e e x i s t a gauge t r a n s f o r m a t i o n g £ 3 such tha t F(9a) = c? (3.12) As we d i s c u s s e d in Chapter I, i t i s impor t an t to have an a f f i r m a t i v e answer i n o rder to c a r r y out the Faddeev-Popov argument. I I I .B SOME QUESTIONS FROM THE CONTINUUM THEORY We do not have to look f a r f o r an example where e q . ( 3 . 1 2 ) does not h o l d . For example, l e t A be a p e r i o d i c l a t t i c e A - (x = (n^e , . . . , n s e ) I -NJJ i & N }^ w i th o p p o s i t e boundary p o i n t s i d e n t i f i e d , F ( a ) ( x ) = a(x,x+eQ) and take c = 11. F ix a E S* and l e t us seek g E $ such t h a t F(9a) (x) = 11 tha t i s , such tha t g (x ) a (x,x + e 0 ) g ( x + e 0 ) - 1 = £ (3.13) f o r a l l x £ A. Suppose we had such a g. M u l t i p l y the e q u a t i o n s (3.13) t o g e t h e r f o r a l l x in A wi th a f i x e d v a l u e of X Q . We o b t a i n g (x) a (x , x + e,j) g (x + e^) - *g (x + e 0 ) a (x + e 0 , x + 2e,j) * ••• » g ( x - e 0 ) a ( x - e 0 , x ) g ( x ) _ 1 = £ . (3.14) On s i m p l i f y i n g and r e a r r a n g i n g (3.14) we f i n d t h a t a (x ,x + eQ)a(x+eQ,x + 2eQ> ••• a (x - e Q , x ) = £ . (3.15) Of c o u r s e , e q . ( 3 . 1 5 ) i s not t r u e in g e n e r a l so tha t even a x i a l gauge can be u n a t t a i n a b l e . However, we a l r e a d y know tha t a x i a l gauge on a p e r i o d i c l a t t i c e i s p a t h o l o g i c a l because i t s gauge degree i s ze ro (see Theorem 2 .14 ) . The next theorem shows tha t these f a c t s are c l o s e l y c o n n e c t e d . Theorem 3.3 ; If the gauge degree a s s o c i a t e d wi th F i s not I I I .B SOME QUESTIONS FROM THE CONTINUUM THEORY z e r o , then F(9a) = c has a s o l u t i o n in g e 3 f o r any a E c £ Proof •  The gauge degree i s the degree of the map <p(g) r F ( 9 a ) . Any map with nonzero degree i s s u r j e c t i v e (see Theorem 2 . 2 ( 4 ) ) . • In p a r t i c u l a r , f o r a x i a l and Landau gauge as g i ven by e q s . ( 2 . 7 ) and (2.8) the gauge can always be a t t a i n e d . ( i i ) E x i s t e n c e of G r i b o v c o p i e s : G a u g e - f i x i n g in the l a t t i c e model c o u l d not be c o n s i d e r e d a good gu ide to the cont inuum t h e o r y i f G r i b o v c o p i e s never o c c u r r e d on the l a t t i c e . L o v e l a c e CL3 has shown t h a t t h e r e can be gauges w i thout c o p i e s i n the l a t t i c e t h e o r y . However, we can see tha t t h e r e a re gauges where c o p i e s occur from Theorem 3.4: Let r| be the gauge degree a s s o c i a t e d with th gauge f i x i n g f u n c t i o n F. F i x a gauge f i e l d a E $ . Suppose c = F(a ) i s a r e g u l a r v a l u e of <p(g) = F(aa). Then the number of s o l u t i o n s i n g £ S of F ( « a ) = c i s at l e a s t InI• In p a r t i c u l a r , i f InJ > 1, then f o r a lmost a l l c £ the e q u a t i o n F<9 a ) = c has m u l t i p l e s o l u t i o n s . P r o o f : Suppose f i r s t tha t c i s a r e g u l a r v a l u e of ip. Use d e f i n i t i o n 1 of degree (see S e c t i o n I I.B) to w r i t e I I I .B SOME QUESTIONS FROM THE CONTINUUM THEORY H = I (<p,{c>) = E E <<p,g) g £ ( p _ 1 (c) f o r <p(g) = F ( 9 a ) . Then ln.1 * E |E <<p,g) I £ E 1 ge V~ 1 (c) g E<P _ 1 <c) which p roves t h a t t h e r e a re at l e a s t I n, I G r i b o v c o p i e s . Suppose InJ > 1. By S a r d ' s theorem CGP], a lmost a l l c £ $ a re r e g u l a r v a l u e s of (p and as above, any such c has m u l t i p l e s o l u t i o n s . • C o r o l l a r y 3 .5; G r i b o v c o p i e s occur i n Landau gauge (eq. ( 2 . 8 ) ) . P r o o f : We have c a l c u l a t e d the gauge degree in S e c t i o n II.D to be n. = det ( E 2 ^ ) . From e q . ( 2 . 6 ) , the e i g e n v a l u e s of *^d/\) are 5 y a k = E H where k = 2s i n ( k u rr / 4N^ ) f o r k^ e {1 , 2 , . . . , 2N ( J -1}. By the a r i t h m e t i c - g e o m e t r i c mean i n e q u a l i t y , ak i (s + 1) ( TT & ) 1 / ( s + 1 ) K 0*u*s M so I n I = TT cu k K i TT (s + 1) TT (TT (3? ) 1 / ( s + 1 ) . k k u f By g a t h e r i n g t o g e t h e r common f a c t o r s we have I I I .B SOME QUESTIONS FROM THE CONTINUUM THEORY n (IT /3k2 )^<s + D x n . . . IT ( ^ . . . $2 + k (/ P k 0 k s 0 S k 0 % k 5 % T H n ^ ). Consequen t ! y f V k u " i n (s+i) n ( n a? ). k p k M Thus in o rder to p rove tha t I n, I > 1, i t c e r t a i n l y s u f f i c e s to show t h a t II fit > 1. In f a c t we now show tha t k u f n $ \ t i = Ny. (3 . 16) Cons i der n A • 2 , kit. „ . , kit. . , (2N -k) it. II 4 s i n * ! — - ) = II 4 sI n (—) s l n ( — — ) U H 2 N - 1 4 N H M 2 N - 1 4 N 4 N IT 4 s i n ( ~ ) c o s ( ~ - ) 4 N 4 N n 2 s i n ( ~ ) . ( 3 . 17 ) U k i 2 N - l 2 N W r i t e the p o l y n o m i a l P (z ) = ( z 2 N - l ) / ( z - 1) as p roduc t of i t s l i n e a r f a c t o r s to o b t a i n P(z) = If (z - e 2 n i k / 2 N ) . U ' k * 2 N - l Thus l i m . . . P(z) = 2N = TT (1 - e 2 v i k / 2 H ) U k * 2 N - l n e i t i k / 2 N ( e - i t i k / 2 N _ & n i k / 2N } H k * 2 N - l I I I .B SOME QUESTIONS FROM THE CONTINUUM THEORY n e " k / 2 N ( - Z i s i n < ! £ > > . (3.18) Tak ing the modulus of both s i d e s of e q . ( 3 . 1 8 ) y i e l d s 2N = IT Uki2N-l 2 s i n ( 2N •). (3.19) E q u a t i o n (3.18) y i e l d s the d e s i r e d e q . ( 3 . 1 6 ) . • ( i i i ) Gauge i n v a r i a n c e of the Faddeev-Popov d e t e r m i n a n t : In the o r i g i n a l d e r i v a t i o n of the Faddeev -Popov d e t e r m i n a n t i t appeared t h a t i t was gauge i n v a r i a n t . The e x i s t e n c e of G r i b o v c o p i e s makes t h i s d o u b t f u l . However, one s t i l l sees arguments in the l i t e r a t u r e which make use of t h i s supposed i n v a r i a n c e . (See f o r example the d i s c u s s i o n of the S I a v n o v - T a y l o r i d e n t i t i e s in C IZD. ) We now show tha t i n the l a t t i c e t h e o r y , gauge i n v a r i a n c e h o l d s o n l y i n ve ry s p e c i a l c a s e s . Of c o u r s e , one such case i s tha t where 6 i s a b e l i a n , f o r then M(a) i s i ndependent of a ( c f . e q s . ( 2 . 1 0 ) and ( 2 . 1 1 ) ) . Theorem 3.6: Suppose the gauge group G i s n o n a b e l i a n . Then the Faddeev-Popov d e t e r m i n a n t det M(a) i s gauge i n v a r i a n t i f and on l y i f i t i s a c o n s t a n t and tha t c o n s t a n t i s 0, 1 or - 1 . P r o o f : Take E to be the c o n s t a n t f u n c t i o n 1 in e q . ( 2 . 2 2 ) to o b t a i n If det M(a) were gauge i n v a r i a n t , then we would have q = $ det M(9 a ) dg. I I I .B SOME QUESTIONS FROM THE CONTINUUM THEORY n. = det M(a) •for every - f i e l d a , and the Faddeev -Popov de te rm inan t would be a c o n s t a n t . Moreover , suppose tha t k i s the d imens ion of G. From e q s . ( 2 . 2 9 ) and (2.36) we have (det N ) r >= (det N ) k But the rank r of G i s the same as the d imens ion of G i f and o n l y i f G i s a b e l i a n . Hence i t must be tha t det N = 0 or Idet N| = 1. Q In p a r t i c u l a r the Faddeev-Popov d e t e r m i n a n t f o r Landau gauge i s not gauge i n v a r i a n t . I I I .C ALTERNATIVE INTERPRETATION OF GAUGE DEGREE I I I .C A l t e r n a t i v e I n t e r p r e t a t i o n of the Gauge Degree H i r s c h f e l d ' 5 s tudy CH] of G r i b o v c o p i e s i s s i m i l a r to ours in many r e s p e c t s , but he o f f e r s a d i f f e r e n t i n t u i t i o n based on the geometry of o r b i t - s u r f a c e i n t e r s e c t i o n s . In t h i s s e c t i o n we r e v i e w tha t geomet r i c i n t u i t i o n and g i v e r i g o r o u s p r o o f s of H i r s c h f e l d ' s main c o n c l u s i o n s w i t h i n the framework we have d e v e l o p e d . Let us c o n s i d e r aga in e q . ( 3 . 9 ) n.(a,c) = $ ^  det M(9a) o M F ^ a c " 1 ) ) dg = E sgn det M ( 9 k a ) (3.20) k=l where the sum i s over those g k £ < such tha t F ( 9 k a ) = c. One can p i c t u r e the s i t u a t i o n when t h e r e a re G r i b o v c o p i e s as i n F i g . 3. The gauge o r b i t 0 = ( 9 a I g £ $) has s e v e r a l p o i n t s of i n t e r s e c t i o n with the g a u g e - f i x i n g s u r f a c e 2 = (b £ $* I F(b) = c>. H i r s c h f e l d argues tha t g , sgn det M( a) measures the sense of t r a v e l of the gauge gi. o r b i t as i t pas ses through the s u r f a c e at a. Thus in g < g ? F i g . 3 sgn det M( 'a ) = +1, sgn det M( ' a ) = - 1 , and so on. As i n S e c t i o n I I I .A, the p i c t u r e makes i t c l e a r why n,(a,c) i s independent of a and c. If a or c changes i t s v a l u e in a c o n t i n u o u s manner, the gauge o r b i t 0 and the gauge f i x i n g s u r f a c e w i l l be deformed in a c o n t i n u o u s I I I .C ALTERNATIVE INTERPRETATION OF GAUGE DEGREE manner. Any p o i n t s of i n t e r s e c t i o n which are c r e a t e d or d i s a p p e a r d u r i n g t h i s t r a n s f o r m a t i o n do so i n p a i r s w i th o p p o s i t e v a l u e s of sgn det M( K a ) so t h a t the sum in e q . ( 3 . 2 0 ) does not change. To make t h i s argument more p r e c i s e , we aga in make use of some n o t i o n s of o r i e n t e d i n t e r e c t i o n t h e o r y CGP1. Our goal i s to show tha t in the absence of p a t h o l o g i e s , n J a t C ) i s the o r i e n t e d i n t e r s e c t i o n number of the map f ( g ) = 9a wi th Z. The c o n c l u s i o n t h a t n ( a i c > i s i ndependent of a i s then a consequence of the f a c t tha t the o r i e n t e d i n t e r s e c t i o n number i s a homotopy i n v a r i a n t . However, i t would take more work u s i n g t h i s approach to show t h a t q ( a , c ) i s i ndependent of c . Suppose X, V and Z a re o r i e n t e d m a n i f o l d s w i thout boundary , X i s compact , Z i s a c l o s e d s u b m a n i f o l d of V and dim X + dim Z = dim Y. A smooth map f :X •+ Y i s t£ii]sjvjBj^sa_l_ to Z i f f o r every x e f - 1 ( Z ) d f x T x ( X ) + T f ( x ) (Z) = Tf ( x ) (Y). (3.21) A compactness argument shows t h a t f " * ( Z ) i s a f i n i t e set in t h i s c a s e . Moreover , because dim X + dim Z = dim V, the sum in e q . ( 3 . 2 1 ) i s a d i r e c t sum d f x T x ( X ) 0 T f ( K ) ( Z ) - T f ( x ) ( Y > (3.22) and d f x i s an i somorph i sm onto i t s image. Thus d f x and the g i ven o r i e n t a t i o n of T X ( X ) i nduce an o r i e n t a t i o n on d f x T x ( X ) which a l ong wi th the g i ven o r i e n t a t i o n of T f ( X ) ( Z ) d e t e r m i n e an o r i e n t a t i o n of T ^ ( y j ( Y ) . If t h i s I I I .C ALTERNATIVE INTERPRETATION OF BAUSE DEGREE F i g u r e 3 S ign of det fi ( K a ) o r i e n t a t i o n of T f ( x ) ( V ) ag rees with the g i ven o r i e n t a t i of lf{x) then the o r i e n t a t i o n number £ ( f , x ) of f at i s +1; o t h e r w i s e f ( f , x ) = - 1 . Def i ni t i on; Let X, Y, Z and f be as above. The i n t e r s e c t i o n number of f wi th Z i s the i n t e g e r I ( f , Z) = 2 £ (f , x ) . (3. x £ f - 1 (Z) Theorem 3 . 7 ; ([GP]) Homotopic maps have the same i n t e r s e c t i o n number. • We now resume our p roo f of the c l a i m tha t q ( a , c ) i I I I .C ALTERNATIVE INTERPRETATION OF GAUGE DEGREE an o r i e n t e d i n t e r s e c t i o n number. We use the s t a n d a r d c r i t e r i o n IGP] t h a t Z = F"*(w) i s a s u b m a n i f o l d i f w i s a r e g u l a r v a l u e of F. The next theorem shows the r e l a t i o n between the + and s i g n s of F i g u r e s 2 and 3. Lemma 3.8; Let X, Y and W be o r i e n t e d m a n i f o l d s w i thout boundary , X compact and dim X = dim W. Let f :X •» Y and F:Y -> W be smooth maps ( F i g . 4 ) . Suppose t h a t w e W i s a r e g u l a r v a l u e of F and l e t Z = F~*(w). Suppose tha t f i s t r a n s v e r s a l to Z. Then Z can be o r i e n t e d in such a way tha t f o r a l l x « f _ 1 ( Z ) , the o r i e n t a t i o n number e ( f , x ) of f at x i s +1 i f and o n l y i f F o f p r e s e r v e s o r i e n t a t i o n at x. P r o o f ; We o r i e n t Z by the pre image o r i e n t a t i o n . A d i s c u s s i o n of pre image and d i r e c t sum o r i e n t a t i o n s can be found in CGP], but we rev iew the e s s e n t i a l s t e p s f o r the case at hand. Let us use the n o t a t i o n 0 + ( V ) to denote the c l a s s of p o s i t i v e l y o r i e n t e d bases of an o r i e n t e d v e c t o r space V. For z £ Z, l e t H be a subspace of T 2 ( Y ) such tha t H 0 T z ( Z ) n T z (Y) . (3.24) We f i r s t o r i e n t H and then use the d i r e c t sum o r i e n t a t i o n to o b t a i n an o r i e n t a t i o n of T Z ( Z ) . S i n c e w i s a r e g u l a r v a l u e of F, we have d F z T 2 ( Y ) = T W (W) . (3.25) But dF , T, (Z) = {()} so in f a c t I I I .C ALTERNATIVE INTERPRETATION OF GAUGE DEGREE « X I I I .C ALTERNATIVE INTERPRETATION OF GAUGE DEGREE d F z H n T W (W) . (3.26) The o r i e n t a t i o n of T W(W) p r o v i d e s an o r i e n t a t i o n of d F z H , which i n t u r n p r o v i d e s an o r i e n t a t i o n f o r H de te rm ined by d e c l a r i n g B E 0 + (H ) i f and o n l y i f d F z B £ Q + (T W <W) ) . By t r a n s v e r s a l i t y and c o m p l e m e n t a r i t y of d i m e n s i o n s , d f J x ( X ) ® T 2 ( Z ) = T Z ( Y ) so tha t i t i s l e g i t i m a t e to take H = d - f x T x <X). To summarize, we have o r i e n t e d Z i n such a way tha t B 2 £ 0 + ( T 2 ( Z ) ) i f and on l y i f t h e r e e x i s t s Bj £ D + < d f X T K ( X ) ) such t h a t Bj ® B 2 £ 0 + ( T z ( Y ) ) . The proo f i s now j u s t a matter of u n r a v e l l i n g t h i s t a n g l e of d e f i n i t i o n s . Let x £ f _ 1 ( Z ) , B{ £ 0 +<T x<X>) and £ 0 + ( T „ ( Z ) ) . Then •• y Fcf p r e s e r v e s o r i e n t a t i o n at x < = > d ( F o f ) x Bj £ 0 + ( T w ( W ) ) < = > d f x Bj £ 0 + ( d f X T X (X)) ( d e f i n i t i o n of 0 + ( d f x T x ( X ) ) ) < = > dfjjBj « B 2 £ 0 + ( T z ( Y ) ) ( d e f i n i t i o n of 0 + ( T z ( Y ) ) ) <=> £ ( f , x ) = +1 ( d e f i n i t i o n of o r i e n t a t i o n number). D Theorem 3.9: F ix a £ * * and l e t f : S •* $ be the map which gauge t r a n s f o r m s a: f ( g ) = 9*. Suppose the g a u g e - f i x i n g f u n c t i o n F : $ •+ $ has c as a r e g u l a r v a l u e so tha t Z = F _ 1 ( c ) i s a s u b m a n i f o l d of F u r t h e r suppose tha t f i s t r a n s v e r s a l to Z. I I I .C ALTERNATIVE INTERPRETATION OF GAUGE DEGREE Then n,(a,c), as g i ven by e q . ( 3 . 2 0 ) , i s the o r i e n t e d i n t e r s e c t i o n number of f w i th Z. P r o o f ; We s h a l l show tha t Fof p r e s e r v e s o r i e n t a t i o n at a p o i n t g £ 5 i f and o n l y i f sgn det M( g a) = +1. Then by Lemma 3.8 n,(a,c) = E sgn det M ( 9 k a ) k = E e ( f , g ) k = I ( f , Z ) . We f i r s t o r i e n t 3 i n the f o l l o w i n g s t a n d a r d way. Choose an o r i e n t a t i o n c l a s s 0 + < £ ) f o r the L i e a l g e b r a € of 3. For any p o i n t g E 3 d e f i n e 0 + ( T g < $ ) ) = d L C g ] £ < 0 + ( € ) ) where LCg] (h ) = gh. F ix g £ 3 and suppose B £ Q*(€). Let <p s F o f . Then Fof p r e s e r v e s o r i e n t a t i o n at g < = > <d<pg d L C g J f l H B ) £ 0 + ( T ( p ( g ) (S) ) < = > (dLCcp (g) D" 1 d(pg d L C g l f l M B ) £ D + ( £ ) < = > d j £ (B) £ 0 + <€) where i = L[<p(g) _ *]o(poL[g]. S i n c e d $ f l : £ •> € , d I a <B) £ 0 + < £ ) i f and o n l y i f sgn det d $ f l = +1. But (see e q . ( 2 . 1 D ) d | f i = H ( 9 a ) I I I .C ALTERNATIVE INTERPRETATION OF GAUGE DEGREE so the c l a i m i s p r o v e n . • Theorems 3.9 and 3.7 a l l o w us to c o n c l u d e t h a t q ( a , c ) i s i ndependent of a and hence a l l o w another approach to p r o v i n g the l a t t i c e FP f o r m u l a . However, the a s sumpt ions of Theorem 3 .9 , tha t c i s r e g u l a r v a l u e of F and t h a t f i s t r a n s v e r s a l to Z, can be d i f f i c u l t to check in r e a l i s t i c examples . C a l c u l a t i o n s i n some e s p e c i a l l y s i m p l e ca ses suggest t h a t F~* (c ) i s " u s u a l l y " a s u b m a n i f o l d and tha t the more r i s k y a s sumpt ion i s t h a t f i s t r a n s v e r s a l to Z. A h i n t tha t t h e r e i s some c o n n e c t i o n between n o n t r a n s v e r s a l i n t e r s e c t i o n s and gauge deg rees tha t a re z e r o may be seen in the f o l l o w i n g theorem. Theorem 3.10; Suppose a £ $*, F: 5* •+ 3 and suppose c i s a r e g u l a r v a l u e of F ( F i g 5 ) . Then the gauge o r b i t f ( g ) = 9 a n a s a n o n t r a n s v e r s a l i n t e r s e c t i o n wi th Z = F _ 1 ( c ) at g i f and o n l y i f the FP d e t e r m i n a n t det M(9 a ) = 0. P r o o f ; Let k = dim S, m = dim 3* and g £ F _ 1 ( Z ) . Note tha t det M( 9 a) = 0 i f and o n l y i f d ( F o f ) g i s s i n g u l a r , because M(9 a ) = d L C F o f ( g ) _ 1 ] F o f (g) * d ( F « f ) g ° d L L g 3 fl and dLCh l i s a b i j e c t i o n f o r any h £ S. Suppose f i r s t tha t det M(9a) * 0. We need to show t h a t d f g T g ( $ ) ® T 2 ( Z ) = T Z ( S * ) (3,27) I I I . C A L T E R N A T I V E I N T E R P R E T A T I O N OF GAUGE D E G R E E • U *5 I I I .C ALTERNATIVE INTERPRETATION OF GAUGE DEGREE ( the d i r e c t sum r e f l e c t s the f a c t tha t dim Z = m - k ) . We s h a l l show ( i ) dim d f g T g ( $ > = k ( i i ) d f g T g ( S ) n T 2 ( Z ) = ( 0 ) . If dim d f g T g < < k then dim d ( F ° f ) g T g < $ ) < k. But t h i s i s i m p o s s i b l e because det M(9a) * 0 so d ( f « f ) g i s an i s omorph i sm. Thus ( i ) i s p r o v e n . As f o r ( i i ) note tha t d F 2 T 2 < Z ) = {0} because FIZ i s c o n s t a n t . In p a r t i c u l a r , d F 2 ( d f g T g ( 5 ) n T 2 ( Z ) ) = ( 0 ) . If dfgTg (S ) n T Z ( Z ) t (01, then a n o n t r i v i a l subspace of d f gTg ( S ) would be mapped to (0> by d F 2 . But t h i s i s i m p o s s i b l e because d ( G * f ) g i s an i somorph i sm. Thus we have shown tha t e q . ( 3 . 2 7 ) h o l d s and f i s t r a n s v e r s a l at On the o the r hand, suppose t h a t e q . ( 3 . 2 7 ) h o l d s . Because c i s a r e g u l a r v a l u e of F, d F 2 s u r j e c t i v e . App ly d F 2 to both s i d e s of e q . ( 3 . 2 7 ) . Because dF z (Z) = (0> we o b t a i n d F 2 d f g T g < « > = T £ < » ) Hence d ( F * f ) g i s onto and n o n s i n g u l a r , so det Mt^a) * 0. D To summar ize, l e t us compare the d i f f e r e n c e s between H i r s c h f e l d ' s i d e n t i f i c a t i o n of q as an o r i e n t e d i n t e r s e c t i o n number and our i d e n t i f i c a t i o n of n, as an o r i e n t e d d e g r e e . Theorem 3.9 shows t h a t , g i ven c e r t a i n I I I .C ALTERNATIVE INTERPRETATION OF GAUGE DEGREE a s s u m p t i o n s , these p o i n t s of view a re e q u i v a l e n t . However, w h i l e q as an o r i e n t e d i n t e r s e c t i o n number has a more d i r e c t geomet r i c i n t e r p r e t a t i o n ( F i g . 3 ) , n as a degree i s t e c h n i c a l l y s i m p l e r and more g e n e r a l : n, i s the degree even w i thout the a s sumpt ions of r e g u l a r i t y and t r a n s v e r s a l i t y made i n Theorem 3 . 9 j the f a c t t h a t n ' ^ i O i s i ndependent of c i s e a s i l y demons t ra ted o n l y wi th t h i s i n d e n t i f i c a t i o n . I I I.D TRUNCATION OF THE FUNCTIONAL INTEGRAL I I I .D T r u n c a t i o n of the F u n c t i o n a l I n t e g r a l S e v e r a l a u t h o r s , b e g i n n i n g wi th G r i b o v CGI, have c o n s i d e r e d the p o s s i b i l i t y t h a t the e q u a t i o n ($ ^ E t c ) Dc) (J f ( a ) e " S ( a ) da) = I det M(a) f ( a ) E©F (a) e " S ( a ) da , (3.28) which i s f a l s e when V = $* u n l e s s n, = 1, might n e v e r t h e l e s s be t r u e i f V i s taken to be an a p p r o p r i a t e subset of G r i b o v has argued tha t such a t r u n c a t i o n c o u l d p r o v i d e a mechanism f o r quark c o n f i n e m e n t . T h i s i d e a has r e c e i v e d some suppor t from Bender , Eguch i and Page l s CBEPJ but has been argued a g a i n s t by G r e e n s i t e LBsl. In any c a s e , the s u g g e s t i o n tha t t r u n c a t i o n of the f u n c t i o n a l i n t e g r a l c o u l d be a way to improve the Faddeev-Popov argument has been pursued by some a u t h o r s [ Z 2 3 , [ G s ] . From our p o i n t of view such an improvement i s not needed, but t r u n c a t i o n c o u l d c o n c e i v a b l y p r o v i d e another way to dea l with the prob lem of a gauge degree t h a t i s z e r o . Another goal of t r u n c a t i o n i s to p r o v i d e a d e n s i t y i n the Faddeev -Popov f o r m u l a which i s m a n i f e s t l y p o s i t i v e so as to f a c i l i t a t e n u m e r i c a l s t u d i e s of the l a t t i c e e q u a t i o n s . In t h i s s e c t i o n we examine in the c o n t e x t of l a t t i c e gauge t h e o r i e s some of the s u g g e s t i o n s made f o r a c c o m p l i s h i n g t h i s t r u n c a t i o n . The i d e a i s to choose V i n I I I.D TRUNCATION OF THE FUNCTIONAL INTEGRAL such a way tha t i f a £ V then any G r i b o v copy of a l i e s o u t s i d e V. R e c a l l t h a t the gauge degree n measures the net number of t imes tha t the f u n c t i o n <p(g) = F(^a) wraps S around i t s e l f . The hope i s tha t r e s t r i c t i n g the gauge f i e l d s to l i e in V w i l l r e s t r i c t if i n such a way tha t i t s image c o v e r s S e x a c t l y once . The e f f e c t would be to r e p l a c e n, by 1 so t h a t e q . ( 3 . 2 8 ) would h o l d . To b e g i n , l e t V be a measurab le subset of $* and f o r any a £ d e f i n e U(a) = (g £ $ I Q a E V>. (3.29) It i s easy to see tha t ( i ) i £ U(a) <=> a £ V ( i i ) U(a) = U ( 9 a ) g . Def i ne 6(a) = I det M( 9 a) E ° F ( 9 a > dg ^ ( a ) = S ^ 1 U ( a ) ( 9 ) d e t M ( 9 a ) EeF(9a) d o (3.30) where 1Q denotes the c h a r a c t e r i s t i c f u n c t i o n of the set Q . Now g £ U(a) <=> g £ U ( 9 a ) g (by ( i i ) ) < = > 11 £ U ( 9 a ) <=> 9 a £ v (by ( i ) ) . Hence l ( j ( a ) < 9 ) = ^ V ( 9 a ' a n d 5 0 6(a) = l v ( g a ) det M(9 a ) E « F ( 9 a ) dg. (3.31) From (3.31) and the r i g h t i n v a r i a n c e of Haar measure we see I I I .D TRUNCATION OF THE FUNCTIONAL INTEGRAL t h a t 6 i s gauge i n v a r i a n t . Let us app l y the Faddeev -Popov t e c h n i q u e . We suppose tha t 9(a) * 0 f o r any a £ S*. Then I f ( a ) e " S ( a ) da = H Z 7 - T f ( a > l y < 9 a ) det H(9a) E ° F ( 9 a ) e - S ( a ) da dg w (a) I Z 7 - T f ( a ) V 3 ' d e t M ( a ) E ° F ( a ) e" w (a) S(a) da = J ^ r - r^<a> det M(a) E«F (a) e " S ( a ) da. (3.32) j\j 6(a) The d i f f i c u l t y , of c o u r s e , i s f i n d i n g the a p p r o p r i a t e subset V so tha t 6(a) i s equa l to \ E(c> dc or at l e a s t i s i ndependent of a. G r i b o v [GJ makes the f o l l o w i n g c h o i c e : V = (a £ S* I M(a) has no n e g a t i v e e i g e n v a l u e s } . In the case of Landau gauge, where M ( £ ) i s the D i r i c h l e t L a p l a c i a n , V c o n t a i n s a ne i ghbourhood of fi. It i s d o u b t f u l t h a t G r i b o v ' s e s s e n t i a l l y a n a l y t i c c o n d i t i o n c a p t u r e s the geomet r i c i d e a tha t i s needed to d e f i n e V. To see what can go wrong, c o n s i d e r the graphs of the two f u n c t i o n s F ( 9 a )^ and F ( 9 a 2 ) shown in F i g . 6. By a p p l y i n g the change of v a r i a b l e s f o r m u l a d i s c u s s e d i n S e c t i o n 11.A, we o b t a i n 6(a<) = S det M(9a) E ° F ( 9 a ) dg 1 ° U ( a j ) = [ E ( c ) dc w h i l e I I I.D TRUNCATION OF THE FUNCTIONAL INTEGRAL d ( a 9 ) = S det M(9 a ) E o F ( 9 a ) dg 2 J U ( a 2 ) = I E t c ) dc + I E (c ) dc S* E t c ) dc + I Etc ) dc 50 tha t in g e n e r a l 6 ( a j ) * 8 ( a 2 ) . As G r i b o v r e c o g n i z e d , i t i s q u i t e p o s s i b l e tha t an e x t r a c o n d i t i o n on V i s needed to ensure tha t F ( 9 a ) r e s t r i c t e d to Uta) i s o n e - t o - o n e . F i g u r e 6 Example f o r G r i b o v ' s t r u n c a t i o n scheme G r e e n s i t e CGsl works with the r e s t r i c t e d Faddeev-Popov f o r m u l a where E i s a rf-function. He a r r i v e s at the e x p r e s s i o n I f<a> •S(a; da = 5 A — det Mta) <S(F(a)) f t a ) e " S ( a ) da (3.33) J N (a) I I I .D TRUNCATION OF THE FUNCTIONAL INTEGRAL where N(a) i s the number of G r i b o v c o p i e s of the f i e l d a. E q u a t i o n (3.33) i s q u i t e c l o s e to our e q . ( 2 . 1 2 ) . The rea son t h a t N(a) appear s i n s t e a d of n, i s t h a t in d e r i v i n g e q . < 3 . 3 3 ) , G r e e n s i t e n e g l e c t e d to take the a b s o l u t e v a l u e of the Faddeev-Popov d e t e r m i n a n t when making h i s change of var i ab1es. G r e e n s i t e argues f o r t r u n c a t i o n from e q . ( 3 . 3 3 ) , but says no more than t h a t V i s a r e g i o n which c o n t a i n s one and o n l y one i n t e r s e c t i o n p o i n t of each o r b i t w i th the gauge f i x i n g s u r f a c e F(a) = fl. If we add the c o n d i t i o n tha t det M(a) > 0 at each such i n t e r s e c t i o n p o i n t then 6(a) = 1 as d e s i r e d . However, t h i s d e s c r i p t i o n of V i s too i m p r e c i s e . A more c o n c r e t e d e s c r i p t i o n of a c h o i c e of V i s g i ven by Zwanziger [Z I . H i s c h o i c e of V i s based on a p a r t i c u l a r gauge, the "background gauge" . A f i e l d a i s i n V i f i t i s on the g a u g e - f i x i n g s u r f a c e and Mia) i s p o s i t i v e d e f i n i t e . The o b j e c t i o n s tha t were r a i s e d above f o r G r i b o v ' s V app l y here as we l l and Zwanziger i s not a b l e to show tha t h i s c h o i c e of V r e s u l t s in 6(a) be ing i ndependent of a. To summar ize, the a t tempt s to a v o i d the G r i b o v prob lem by t r u n c a t i n g the f u n c t i o n a l i n t e g r a l have had on l y l i m i t e d s u c c e s s . E i t h e r the d e s c r i p t i o n of the r e g i o n V i s too i m p r e c i s e to be u s e f u l or the c h o i c e of V does not seem l i k e l y to work. I I I.D TRUNCATION OF THE FUNCTIONAL INTEGRAL D e s p i t e the s u p e r f i c i a l appea l of t r u n c a t i o n , a c o n s i d e r a t i o n of d iagrams such as F i g . 6 l e ad s one to the c o n c l u s i o n tha t i t i s r a t h e r u n n a t u r a l and c o n s e q u e n t l y d i f f i c u l t to f o r m u l a t e . We do not b e l i e v e t h a t i t h o l d s much p romi se f o r m o d i f y i n g e i t h e r the l a t t i c e or cont inuum v e r s i o n of the Faddeev-Popov argument. IV.A INTRODUCTION CHAPTER IV CONTINUUM LIMIT IV.A I n t r o d u c t i o n One of the major m o t i v a t i o n s f o r d e v e l o p i n g the l a t t i c e t h e o r y as d e s c r i b e d i n the p r e c e d i n g c h a p t e r s i s to p r o v i d e a t o o l f o r the r i g o r o u s c o n s t r u c t i o n of a cont inuum n o n a b e l i a n gauge f i e l d t h e o r y . One c o u l d hope to c a r r y out t h i s c o n s t r u c t i o n by o b t a i n i n g the Schwinger f u n c t i o n s of the cont inuum t h e o r y as a l i m i t as t •* 0 of a p p r o p r i a t e l a t t i c e f u n c t i o n s . The p h i l o s o p h y beh ind the approach we have taken i s to t r y to b r i n g a gauge f i e l d t h e o r y i n t o a form which re semb le s a nongauge f i e l d t h e o r y such as the P(<p) or Yukawa mode l s , so as not to have to depar t too d r a s t i c a l l y from the t e c h n i q u e s used i n t h e i r c o n s t r u c t i o n . In p a r t i c u l a r , we wish to a v o i d the n e c e s s i t y of working at a l l t imes o n l y with gauge i n v a r i a n t f u n c t i o n s . By b r e a k i n g gauge i n v a r i a n c e , we can use n o n i n v a r i a n t o b j e c t s l i k e Schwinger f u n c t i o n s as i n t e r m e d i a t e c o n s t r u c t s and in the end r e s t r i c t our a t t e n t i o n to the p h y s i c a l l y more r e l e v a n t gauge i n v a r i a n t q u a n t i t i e s . In t h i s s e c t i o n , we su rvey some of the i s s u e s i n v o l v e d i n t a k i n g the cont inuum l i m i t of the l a t t i c e model deve l oped in the p r e c e d i n g c h a p t e r s , c o n c e n t r a t i n g on the example of quantum chromodynamics in two s p a c e - t i m e d imens ion s (QCD2) with gauge group S U ( 2 ) . We then i n t r o d u c e the p a r t i a l IV.A INTRODUCTION r e s u l t s we have been a b l e to o b t a i n and which a re d e s c r i b e d i n the r e m a i n i n g s e c t i o n s of t h i s c h a p t e r . We a l s o d e s c r i b e the prob lems we have not been a b l e to s o l v e and i n d i c a t e what work rema ins to be done. (a) Continuum t h e o r y Let us beg in by w r i t i n g down the formalism of cont inuum SCDj. The a c t i o n f o r t h i s t h e o r y i s S(A,Y,Y> = S g ( A ) + S m ( Y , y ) = J £ g d 2 x • J £ m d 2 x (4.1) where ^g = " F * r ^r yyryv' F f iv = " ^ c A u + ^ tA^ tAy ] £ m = Y(a + m + A) Y in which Z s t ands f o r F, C^y^, the y^ be ing a r e p r e s e n t a t i o n of the C l i f f o r d a l g e b r a The a c t i o n (4.1) i s i n v a r i a n t under the combined l o c a l gauge t r a n s f o r m a t i o n s Y(x) •* g(x) Y(x) V (x ) •* g (x ) *Y (x ) <\ (x) •» g (x )CA^(x) + X ' 1 S^lqix)'1. B e f o r e gauge f i x i n g , the e x p e c t a t i o n of a f u n c t i o n f ( A , y , Y ) of the gauge f i e l d A and the matter f i e l d s Y, Y i s g i v e n IV.A INTRODUCTION by the •formal e x p r e s s i o n \ f (A ,Y ,Y> e " S ( A ' y ' y > ©A ©Y ©Y <f > = I _ . J g - S (A , y ,Y ) <£|A J)y J)y We d i d not e x p l i c i t l y i n c l u d e matter • f ie ld s such as Y, Y in our d i s c u s s i o n of gauge f i x i n g , but we c o u l d have w i thout chang ing any of the r e s u l t s . For i f f ( A , Y , Y ) i s a gauge i n v a r i a n t f u n c t i o n , we can imag ine c a r r y i n g out the f e rm ion i n t e g r a t i o n ©Y ©Y to o b t a i n S f (A ) e " S g ( A ) ©A <f> = rr-nrr-  f 4 - 2 ) \ e - s g ( A ) ©A where f (A ) i s a gauge i n v a r i a n t f u n c t i o n of A. The Faddeev-Popov t e c h n i q u e can then be a p p l i e d as b e f o r e . If we employ gauge f i x i n g u s i n g Landau gauge to make the r ep l a cement S n ( A ) •> S n _(A) = S „ ( A ) - fl [ d 2 x t rC O..A.. (x) ) 2 1 g g»a g •> H H the q u a d r a t i c p a r t of the gauge f i e l d a c t i o n becomes \ \ Aj <x ) C-^i^y + d - a ) dydyU^f]* (x) d 2 x . ( 4 . 3 ) The s i m p l e s t v e r s i o n of ( 4 . 3 ) i s o b t a i n e d by making the c h o i c e o=l, which y i e l d s what i s known as Feynman gauge i n which the c o v a r i a n c e i s (-d) ~^(iySab, The Faddeev-Popov o p e r a t o r M (A) o p e r a t e s on L 2 (IR2) ®fR 2 ®E and i s g i ven by M(A)B = -<JB - aa^IAp.B]. Thus the fo rma l cont inuum e x p r e s s i o n f o r the e x p e c t a t i o n of a gauge i n v a r i a n t f u n c t i o n in Feynman gauge IV.A INTRODUCTION i s \ det M(A) f (A , Y ,Y ) e 9 » * m ©A ©Y ©Y <f > = _ . - S . , (A) - Sm(Y,Y> J det M(A) e 9,1 m ©A ©Y ©Y (4.4) We want to modi fy (4.4) a l i t t l e b e f o r e d i s c u s s i n g how to o b t a i n i t as a cont inuum l i m i t . The matter f i e l d s Y,Y appear i n QCD very much as they do in QED (CWC], f.W]) and we s h a l l drop them so as to be a b l e to c o n c e n t r a t e on those t h i n g s tha t a re s p e c i a l to the n o n a b e l i a n c a s e . The model tha t rema ins i s sometimes r e f e r r e d to as a "pure gauge t h e o r y " . There i s l i t t l e hope of making sense of the Faddeev-Popov d e t e r m i n a n t u n l e s s we n o r m a l i z e i t by w r i t i n g (1(A) = <-d) (I + L (A)) and c a n c e l l i n g the common f a c t o r det (-<d) top and bottom in ( 4 . 4 ) . Let us w r i t e s g , l = s o + S I where S 0 (A ) = { I A j (x ) (-6) i tfab A J (X ) d 2 x Sj (A) = X \ t r 3 AyCAj,, A „ J d 2 x + X 2 \ t r [ A^, A y J 2 d 2 x . A l though we have not r e f l e c t e d i t in our n o t a t i o n , we shou ld keep i n mind tha t we are work ing in Feynman gauge. Thus the fo rma l e x p r e s s i o n we wish to make r i g o r o u s sense of i n pure QCD2 wi th Feynman gauge i s IV.ft INTRODUCTION „ -at\HI \ f(A> d e t ( l + L ( A ) ) e 1 du < f > - . (4.5) I d e t ( l + L ( A ) ) e 1 du i n which we are w r i t i n g dji(A) f o r the •formal measure e - S o ( A ) DA \ e - S o ( A ) DA which can be i n t e r p r e t e d as Gaus s i an measure with c o v a r i a n c e ( - d ) - 1 i 5 ^ v ( J a b . At t h i s p o i n t we do not r e q u i r e tha t f be gauge i n v a r i a n t , but r a t h e r we have i n mind t h a t f (A ) i s , f o r example, a p o l y n o m i a l in A. To f o l l o w the t r a d i t i o n s of c o n s t r u c t i v e f i e l d t h e o r y , we s h o u l d f i r s t a t tempt to c o n s t r u c t the f i n i t e volume v e r s i o n of ( 4 . 5 ) . We put the t h e o r y in the f i n i t e volume A C IR* by t a k i n g the measure du to be Gaus s i an measure wi th c o v a r i a n c e (-^/\) _ 1 * p K ^ a b w n e r e _ ^ / \ * 5 * n e L a p l a c i a n o p e r a t o r with D i r i c h l e t boundary c o n d i t i o n s on A [-GRS23. A l s o , the i n t e g r a l s d e f i n i n g Sj w i l l now extend on l y over A i n s t e a d of IR2. It i s s t i l l too much to expect to be a b l e to work with e q . ( 4 . 5 ) as i t s t ands because of u l t r a v i o l e t d i v e r g e n c e s tha t must be removed by r e n o r m a l i z a t i o n . That i s , we must expec t to have to make the r e p l a c e m e n t s det (I + L) •> d e t r e n (I + L) SI * s I , r e n < 4 " 6 ) by s u b t r a c t i n g c o u n t e r t e r m s . IV.A INTRODUCTION (b) L a t t i c e t h e o r y We now t u r n to the l a t t i c e v e r s i o n of e q . ( 4 . 5 ) . For the gauge i n v a r i a n t a c t i o n we choose the a c t i o n due to Wi l son LWs] S j ( a ) = - j %' 2 t~ 2 E Re t r [U p (a> - £ ] where the sum ranges over a l l p l a q u e t t e s ( i . e . , e l e m e n t a r y l a t t i c e square s ) P and Up(a) i s the p l a q u e t t e v a r i a b l e Up(a) = a (x , x +e^) a (x +e^, x -t-e^ + e y ) a (x+e^+e^, x + e y ) a (x + e y , x ). The Faddeev-Popov d e t e r m i n a n t det M E (a ) tha t a r i s e s from the gauge f i x i n g o p e r a t i o n i s g i ven by e q . ( 2 . 4 0 ) . We n o r m a l i z e M £ and d e f i n e L £ by <-^)( I + L £ (a)) = ME (a) . Thus the l a t t i c e c o u n t e r p a r t to e q . ( 4 . 5 ) i s c - S . ( a ) [ f . ( a ) det (1 + L. (a) ) e £ da < f * \ ' — t - S E ( a ) • ( 4 ' 7 J d e t ( l + L £ ( a ) ) e E da (c) Cont inuum l i m i t : a sugges ted approach We are now in a p o s i t i o n to pose the two fundamenta l prob lems of the cont inuum l i m i t . Prob lem 1: What a re the a p p r o p r i a t e r e n o r m a l i z a t i o n s to make i n eq . (4 .6 ) ? Prob lem 2: How can we o b t a i n the r e n o r m a l i z e d e q . ( 4 . 5 ) as the l i m i t of the r e n o r m a l i z e d e q . ( 4 . 7 ) ? With r ega rd to Prob lem 1, i t i s n a t u r a l to beg in wi th those r e n o r m a l i z a t i o n s sugges ted by p e r t u r b a t i o n t h e o r y and power c o u n t i n g . We d i s c u s s t he se m a t t e r s in S e c t i o n IV.B. IV.A INTRODUCTION We now g i v e an o u t l i n e of a p o s s i b l e approach to s o l v i n g Prob lem 2. It i s d o u b t f u l tha t the scheme we are about to d e s c r i b e can be c a r r i e d out in as s i m p l e a form as we a re s u g g e s t i n g . N e v e r t h e l e s s , i t g i v e s us a p l a c e to s t a r t and any p r o g r e s s made here w i l l c e r t a i n l y be h e l p f u l in e v e n t u a l l y o b t a i n i n g the s o l u t i o n to Problem 2. To b e g i n , we shou ld f i r s t break up the l a t t i c e a c t i o n somehow i n t o terms c o r r e s p o n d i n g to f r e e and i n t e r a c t i o n p a r t s S E = S 0 , E + S I , E ' We d i s c u s s f u r t h e r below how one might want to make t h i s s p 1 i t - u p. As in p r e v i o u s l a t t i c e models i n quantum f i e l d ' t h e o r y ( [GRS1], [BFS1 - 31, CWCD) we c o n s i d e r the l a t t i c e t h e o r y to be embedded i n the cont inuum t h e o r y : we w r i t e + XE A,, , (x±}e,.> a (x ,x + e ( i ) = e 2 V (4.8) as in e q . ( 2 . 2 ) and r e g a r d A f / ( £ ( y ) a s a cont inuum f i e l d A^ smeared with some smooth a p p r o x i m a t i o n x £ ) y to the d - f u n c t i o n wi th c o n c e n t r a t e d at y. For example, in [GRS11 f £ y i s chosen in such a way t h a t J A j ( f € | X ) A ; < f £ i y > du = 6^ Sab ( - 4 ) ( x , y ) . The c h o i c e of l a t t i c e embedding i s to a l a r g e degree mere ly a matter of t e c h n i c a l c o n v e n i e n c e and f o r the purposes of t h i s d i s c u s s i o n , we s h a l l assume tha t the CGRSl l embedding i s the one we are u s i n g . IV.A INTRODUCTION Another a spec t of the l a t t i c e embedding to c o n s i d e r i s the r e l a t i o n between the f r e e l a t t i c e measure - S . ( a ) e 0 da J e 0 da to the cont inuum Gaus s i an measure du. Haar measure da on * * i s e q u i v a l e n t to the measure o b t a i n e d from Lebesgue measure on a subset of the L i e a l g e b r a € * . That i s , f o r some f u n c t i o n R£ ( the Radon-Nikodym d e r i v a t i v e ) we have -S_(a) - S - ,<e £ ) e 0 da e ° ' £ R (A ) dA , -S . (a ) £ * A e \ e ° da . - S n e ( e e ) J \ e ° ' £ R£ (A £ ) dA e i n which dA £ i s Lebesgue measure on € . If the f a c t o r R£ were not p r e s e n t , the n a t u r a l c h o i c e f o r S n . would be S O . E ' ^ ^ 6 ' = 1 J £ * FTJ,£(X) ' - ^ V a b A 5,£ ( x ' which i s the f i n i t e - d i f f e r e n c e a p p r o x i m a t i o n to S 0 ( f l ) = \ \ A A j ( x ) ( - 4 A ) ^ K 4 A B A b ( x ) d 2 x . With the [GRS1] embedding and t h i s c h o i c e of S 0 ) £ we have f o r any i n t e g r a b l e f u n c t i o n F ( A £ ) . -S- . ( A F ) [ F (A , ) e ° ' £ £ dA. J F ( A £ ) du - £  I e ° ' £ e dA £ Thus EAA. E AA T EAA, St ( ( e £ ) I f £ (e £ ) d e t ( I + L £ (e £ ) ) e 1 ' £ du < f E > £ = rxw • EAA E S i . ( e £ ) I det (I + L e (e £ ) ) e 1 ' £ dy IV.A INTRODUCTION The next task i s to r e n o r m a l i z e the l a t t i c e q u a n t i t i e s . Assuming t h i s has been done, we would have a comp le te s o l u t i o n to Problem 2 i f we c o u l d show t h a t the f o l l o w i n g converge i n a l l L.P(dfi) (p * 1 ) : EAA C ( i > f £ <e e ) •» f (A) ( i i ) d e t r e n <I+L£ <e £ )> -» d e t r e n (I + L (A)) EAA, ST , r p n ( e £ ) -ST r e n ( A ) ( i i i ) e 1 ' £ » r e n •* e 1 ' r e n ( i v) R£ (A £ ) •* c o n s t a n t . (d) Cont inuum l i m i t : d i s c u s s i o n and p a r t i a l r e s u l t s So as not to m i s l e a d the r e a d e r , l e t us p o i n t out r i g h t away tha t t h e r e a re s e r i o u s d i f f i c u l t i e s i n the p r e c e d i n g scheme. F i r s t of a l l , the l i m i t ( i v ) i s p r o b a b l y not t r u e in g e n e r a l . For example, f o r the case of SU(2) i n two d i m e n s i o n s we show in S e c t i o n IV.D the fo rma l l i m i t R£ (A ) •» exp <-j A 2 E $ I Af, (x ) I 2 d 2 x ) . (4 . 9 ) T h i s a s u r p r i s i n g and unwelcome r e s u l t f o r s e v e r a l r e a s o n s . However we argue in IV.D tha t the fo rma l l i m i t (4 . 9 ) i s m i s l e a d i n g and tha t i n f a c t R£ makes no c o n t r i b u t i o n i n the cont inuum l i m i t . A second d i f f i c u l t y with the scheme o u t l i n e d in p a r t (c) above i s r e l a t i n g the a c t i o n S £ ( a ) which i s based on gauge f i e l d s in the gauge group to S 0 ( A ) where the gauge f i e l d i s in the L i e a l g e b r a . The i n t u i t i o n i s t ha t IV.A INTRODUCTION the l a t t i c e a c t i o n ought to p r o v i d e Gaus s i an damping f o r the l a t t i c e gauge f i e l d s which a re f a r from 11 so tha t i t i s the f i e l d s near 11 which d e t e r m i n e the cont inuum l i m i t . However, the p e r i o d i c n a t u r e of the group i n t e g r a l s can s p o i l t h i s i n t u i t i o n . The prob lem i s comparab le to t r y i n g to o b t a i n 2 \ f ( x ) e ' x 1 2 dx (4.10) JIR as a l i m i t as e 0 of J 1 " f (x) e - ( l - co s(k £ x))/ E 2k 2 d X i ( 4 i U ) -7i/e For any k, we have the p o i n t w i s e conve rgence of e -( l -cos(kEx))/e 2k 2 t £ ) e - x 2 / 2 > N o r e o v e r ) w h e n k = j the d e s i r e d conve rgence (4.10) •* (4.11) can be shown u s i n g the dominated conve rgence theorem t o g e t h e r wi th the e s t i m a t e 2 ? 1 - cos EX i — r f o r x £ [ - i r / e , TT/E ] . ir*-However when k = 2 t h i s argument f a i l s . The d i f f e r e n c e i s that when k = 1, the exponent has a un ique maximum at x = 0, whereas when k = 2 t h e r e a re a d d i t i o n a l maxima at x = i.E/2 and the e x p o n e n t i a l p r o v i d e s no damping near the b o u n d a r i e s of the r e g i o n of i n t e g r a t i o n . In g e n e r a l , (4.11) w i l l not converge (depending on what f i s ) . What a l l t h i s has to do with i n t e g r a l s in l a t t i c e gauge t h e o r i e s i s s p e l l e d out i n S e c t i o n IV.C. In b r u t a l l y s i m p l i f i e d terms the ana logy i s t h i s . E q u a t i o n s (4.10) and IV.A INTRODUCTION 100 (4.11) a re the i n t e g r a l s f o r cont inuum and l a t t i c e e x p e c t a t i o n s r e s p e c t i v e l y . The term cos(kEx) appear s in the l a t t e r i n s t e a d of x 2 because the l a t t i c e gauge f i e l d i s in the gauge group r a t h e r than the L i e a l g e b r a . The r e g i o n of i n t e g r a t i o n [ - t r / e , i r / E ] a r i s e s from e x p r e s s i n g Haar measure on the group as Lebesgue measure on a subset of the L i e a l g e b r a . Va lues of k g r e a t e r than 1 occur becau se , f o r example, the p l a q u e t t e v a r i a b l e s c o n t a i n a p roduc t of f o u r gauge f i e l d s , c o r r e s p o n d i n g to k = 4, and the gauge i n v a r i a n t a c t i o n f a i l s to have a un ique minimum. One c o u l d hope tha t f i x i n g the gauge would r e s u l t in an a c t i o n which does have a un ique minimum, a l t h o u g h t h e r e i s no reason to expec t t h i s a p r i o r i . What does happen i n g e n e r a l i s tha t gauge f i x i n g r educe s k, but not a l l the way to 1. T h i s i s an impor tan t p o i n t because i t i s d i f f i c u l t to see how to get conve r gence of the f r e e measure w i thout t h i s un ique minimum p r o p e r t y . Let us now d i s c u s s the r e s u l t s we have o b t a i n e d in c a r r y i n g out the program d e s c r i b e d in (c) above. With r e g a r d to ( i i ) , we show i n S e c t i o n IV.E the fo rma l E * A , convergence of L £ (e e ) -> L ( A ) . In S e c t i o n IV.F we show t h a t the cont inuum Faddeev-Popov o p e r a t o r L(A) i s in the C a r l eman c l a s s e s 2^+<S f o r a l l S y 0 and f o r a lmost a l l A. T h i s means in p a r t i c u l a r tha t the c u t o f f d e t e r m i n a n t d e t 3 < I + L ( A ) ) i s w e l l - d e f i n e d a . e . and c o n s t i t u t e s the f i r s t s tep in r e n o r m a l i z i n g the Faddeev-Popov d e t e r m i n a n t . IV.A INTRODUCTION In two o the r gauge f i e l d models which have been c o n s t r u c t e d to d a t e , QED 2 CWC.W] and H i g g s 2 CBFS1-33, a l a t t i c e l i m i t was employed which a l l o w e d the p r o v i n g of d i a m a g n e t i c i n e q u a l i t i e s . In our c a s e , a d i a m a g n e t i c i n e q u a l i t y would be Idet< I+L £ (a ) )I i 1 (4.12) f o r a l l a and e. I n e q u a l i t i e s of t h i s s o r t were c r u c i a l in e s t a b l i s h i n g conve rgence and L.P p r o p e r t i e s of d e t e r m i n a n t s in QED 2 and H i g g s 2 and w i l l no doubt be i m p o r t a n t in QCD 2 as w e l l . We prove e q . ( 4 . 1 2 ) in S e c t i o n IV.G, but f o r a o n e - d i m e n s i o n a l model o n l y . IV.B POWER COUNTING IN QCD 102 IV.B Power Coun t ing i n QCD 2 We now g i v e a s k e t c h of what r e n o r m a l i z a t i o n s we shou ld expec t to have to make in pure QCD2 to e l i m i n a t e u l t r a v i o l e t d i v e r g e n c e s . Our r e a s o n i n g i s based on a g e n e r a l power c o u n t i n g f o r m u l a which we now deve lop f o l l o w i n g llll. Let G be a Feynman graph w i thout any i n t e r n a l l o o p s . Suppose G has : L i ndependent momentum i n t e g r a t i o n s Ig i n t e r n a l boson l i n e s Eg e x t e r n a l boson l i n e s Ip i n t e r n a l f e r m i o n l i n e s Ep e x t e r n a l f e r m i o n l i n e s I n d e r i v a t i v e s on i n t e r n a l l i n e s E n d e r i v a t i v e s on e x t e r n a l l i n e s V v e r t i c e s Then the s u p e r f i c i a l degree of d i v e r g e n c e i n d d imens ions i s u(G) = dL - 2 I B - Ip + I D and the i n t e g r a l c o r r e s p o n d i n g to the graph G w i l l d i v e r g e i f u (G) i 0. Momentum c o n s e r v a t i o n at the v e r t i c e s i m p l i e s L - I B + I F - V + 1 so u(G) = d - dV + ( d - 2 ) I B + ( d - l ) I F + D. Each v e r t e x v a r i s e s from an i n t e r a c t i o n term T v i n the L a g r a n g i a n . Let B y = number of bosons in T.y F y = number of f e r m i o n s in T, IV.B POWER COUNTINE IN DCD 103 D y = number of d e r i v a t i v e s in T y . Because we have assumed t h e r e a re no l o o p s i n 6, each i n t e r n a l l i n e i s i n c i d e n t on two v e r t i c e s and each e x t e r n a l l i n e i s i n c i d e n t on one v e r t e x . Thus *B = i £ B v " 1 E B v Ip = | E F V - jEp v ID = ED y - E D . v P u t t i n g t he se r e l a t i o n s i n t o the p r e v i o u s e x p r e s s i o n f o r d) (6) y i e l d s u(G) = d - dV + j ( d - 2 ) ( E B v - Eg) + f l d - l ) ( E F v - Ep) v v + 2 D y - E D v where u v = | ( d - 2 ) B v + i ( d - 1 ) F v + D v - d. In the d = 2 case we are go ing to c o n s i d e r , these e x p r e s s i o n s reduce to u(6) = 2 + Ew v - fEp - E D (4.13a) v <->v B K + D v " 2 " (4.13b) To app l y the power c o u n t i n g f o r m u l a s (4.13) to GCD2 we take the c o n v e n t i o n a l s tep CR,IZ3 of w r i t i n g the Faddeev -Popov d e t e r m i n a n t as an i n t e g r a l over a p a i r of an t i commut ing " g h o s t " f i e l d s n,» h a s f o l l o w s , \ rj(x) M (A) n,<x> d 2 x -det M(A) = J e d n d h IV.B POWER COUNTING IN QCD . I n(x) (-j)n(x) - An(x)3 , . [A..(x) ,n(x)] d 2 x -= J e f H dqdn,. For purposes of power c o u n t i n g the ghost f i e l d s behave as bosons because of the r\<-d)r\ te rm. The i n t e r a c t i o n term ndptAp,nl i s a v e r t e x wi th t h r e e bosons and one d e r i v a t i v e . When the gauge-ghos t i n t e r a c t i o n i s added to those i n (4.1) we have a t o t a l of f o u r i n t e r a c t i o n s . Let us use the f o l l o w i n g n o t a t i o n to denote the d i f f e r e n t f i e l d s = gauge f i e l d = ghost f i e l d + = d e r i v a t i v e of ghost f i e l d = d e r i v a t i v e of gauge f i e l d The v e r t i c e s a re (3A )A 2 A 4 n3(An) If we l e t N y denote the number of v e r t i c e s of type v, then from (4.13) the power c o u n t i n g f o r m u l a f o r pure QCD2 i 5 u(G) = 2 - N 3 - 2 N 4 - N g - E D . T a b l e 1 shows the p o s s i b l e connec ted graphs with n o n n e g a t i v e degree of d i v e r g e n c e . Those graphs with no e x t e r n a l l i n e s (vacuum graphs) can IV.B POWER COUNTING IN QCD 105 Tab le 1 S u p e r f i c i a l l y d i v e r g e n t graphs i n QCD2 IV.B POWER COUNTING IN QCD 106 be r e n o r m a l i z e d " t r i v i a l l y " by c o n s t a n t c o u n t e r t e r m s . Graphs wi th l o o p s , which we have been i g n o r i n g from the b e g i n n i n g , a re e l i m i n a t e d by Wick o r d e r i n g . The r e m a i n i n g n o n t r i v i a l c o u n t e r terms a re tho se r e q u i r e d to r e n o r m a l i z e the graphs each of which has u(G) = 0. These graphs appear to r e q u i r e a gauge f i e l d mass c o u n t e r t e r m (as i n the Yukawa m o d e l ) , but such a term would v i o l a t e gauge i n v a r i a n c e . However, gauge i n v a r i a n c e in the form of the S Iavnov-Tay1 or i d e n t i t i e s i m p l i e s t h a t these g r a p h s , when added t o g e t h e r , are l e s s d i v e r g e n t than power c o u n t i n g sugge s t s (see C IZ1) . T h e i r e f f e c t i v e degree of d i v e r g e n c e i s n e g a t i v e . Note t h a t t h i s improved degree of d i v e r g e n c e i s a r e s u l t of c a n c e l l a t i o n s between terms t h a t a r i s e from det ri(A) and e . It i s f o r t h i s reason t h a t we s a i d in S e c t i o n IV.A tha t i t might be too o p t i m i s t i c to expec t those two f a c t o r s to be in L.P(du) s e p a r a t e l y . Thus we have seen t h a t t h e r e are no n o n t r i v i a l r e n o r m a l i z a t i o n s r e q u i r e d in QCD2- However, the c a n c e l l a t i o n s due to gauge i n v a r i a n c e can be expec ted to be s u b t l e and q u i t e n o n t r i v i a l to e x h i b i t i n a r i g o r o u s c o n s t r u c t i o n . It i s c l e a r l y impor t an t to have a gauge i n v a r i a n t r e g u l a r i z a t i o n of the t h e o r y , a f a c t which argues f o r u s i n g a l a t t i c e model . IV.C THE ACTION 107 IV.C The A c t i o n In t h i s s e c t i o n we show t h a t the •formal l i m i t of the l a t t i c e a c t i o n in Feynman gauge (as d e s c r i b e d i n IV.A) y i e l d s i t s expec ted cont inuum c o u n t e r p a r t . To c o n v e r t t h a t f o rma l l i m i t i n t o a r i g o r o u s l i m i t , i t i s f i r s t n e c e s s a r y to ensure t h a t i t i s the gauge f i e l d s near the group i d e n t i t y which make the most i m p o r t a n t c o n t r i b u t i o n to the cont inuum l i m i t . In the l a t t e r p a r t of t h i s s e c t i o n we d i s c u s s some t e c h n i c a l prob lems we have encoun te red in t r y i n g to a c c o m p l i s h tha t g o a l . (a) Formal l i m i t We have seen tha t the a c t i o n f o r l a t t i c e Feynman gauge i s S(a) = -\\~2 £ d _ 4 E p Re t r C U p ( a ) - 113 - J T 2 e d _ 4 E x Re t r C V x ( a ) - 113 where U p ( a ) i s the p l a q u e t t e v a r i a b l e Up(a) = a (x , x + e^)a(x+e^, x + e^ + e y )a (x+e^+ey,x+e^)a (x + e y , x ) and V x ( a ) i s the " c r o s s " v a r i a b l e V x<a) = a ( x , x + e 0 ) a ( x , x - e ^ ) • • • a ( x , x + e ^ ) a ( x , x - e ^ ) . We take the fo rma l l i m i t by u s i n g the r e l a t i o n ±eAA. . ( x+ ie , . ) a ( x , x ± e ( i ) = e V 1 V . (4.14) The p r o o f s of the conve r gence of the Wi l son a c t i o n and the IV.C THE ACTION damping term to t h e i r cont inuum c o u n t e r p a r t s a re very s i m i l a r . S i n c e the former i s we l l -known (see f o r example CKo23), we g i v e the p roo f f o r the l a t t e r o n l y . Let A a : IR d •* IR be smooth wi th compact suppor t i n A and l e t a be the l a t t i c e gauge f i e l d o b t a i n e d from A by eq . (4. 14) . C o n s i d e r the damping term exp ( - A " * £ -2.d-4 E x Re t r [ V x (a) - £ ] ) . We c l a i m tha t t r C V x ( a ) - fU = | X 2 e d " 4 t r C 3 £ A { i ( x ) 3 2 + 0 ( E 5 ) (4.15) where From the B a k e r - C a m p b e l 1 - H a u s d o r f f f o r m u l a C Sp J e E A e e B = e e(A+B) + | e2 [ A , B ] + 0 ( E 3 ) we have a ( x , x + e u ) a ( x , x - e „ ) S i n c e by a s sumpt ion 3 U A i s c o n t i n u o u s CA f i (x+|e ) J ) ,A^(x = 0(E ) so t h a t a ( x , x + e u ) a ( x , x - e „ ) - e (4.16) Consequent 1y IV.C THE ACTION 109 A e 2 E.. 3f.A..(x) + 0 (e 3 > V x ( a ) = e M M f (4.17) i n which the f u n c t i o n r e p r e s e n t e d by O(e^) i s i n the L i e a l g e b r a . Now 6 i s a u n i t a r y group ( that i s , we have assumed we are work ing i n a u n i t a r y r e p r e s e n t a t i o n of G ) so t h a t any e lement A of the L i e a l g e b r a E of G i s a n t i h e r m i t i a n . It f o l l o w s t h a t Re t r ( A ) = 0. Hence f o r any A,B E E 2 3 Re t r C e e A + £ B ] = Re t r t i + E 2 A + e 3 B + | E 4 A 2 + 0 ( E 5 ) 3 = Re t r C i + | E 4 t r ( A 2 ) + 0 ( E 5 ) 3 . Consequent 1y, - A - 2 E D - 4 E x Re t r C V x ( a ) - 113 = - j E y .. E D t r [ 3 E A u ( x ) 3 2 + 0 ( E ) ' " i C MM •» - f j t r C 3 „ A „ (x) 3 2 d d x as E •» 0. (b) Unique minimum of l a t t i c e a c t i o n We d i s c u s s e d in S e c t i o n IV.A the need to have an e f f e c t i v e a c t i o n S(a) which has a un ique minimum at a = 11 in o rder t h a t i t be the f i e l d s near 11 which d e t e r m i n e the cont inuum l i m i t , the f i e l d s away from 11 be ing e x p o n e n t i a l l y damped. T h i s r e q u i r e m e n t can be met by making the a p p r o p r i a t e c h o i c e s of the g a u g e - f i x i n g f u n c t i o n F, the damping term E and the s u b l a t t i c e A j . However, t h e r e •are o t h e r c o n s i d e r a t i o n s as w e l l . S p e c i f i c a l l y we want to IV.C THE ACTION 110 choose F, E and A± so t h a t the •fo l lowing c o n d i t i o n s h o l d : ii) the e f f e c t i v e a c t i o n EoF(a ) e ' S ( a ) has a un ique minimum at a = 1. ( i i ) the gauge degree q * s n o n z e r o , ( i i i ) the q u a d r a t i c p a r t of the e f f e c t i v e a c t i o n l e a d s to a t r a c t a b l e Gaus s i an measure in the cont inuum l i m i t . The r a t h e r vague c o n d i t i o n ( i i i ) i s meant to exp re s s our d e s i r e to o b t a i n from the l a t t i c e a c t i o n a w e l l - d e f i n e d Gaus s i an measure f o r the n o n i n t e r a c t i n g pa r t of the gauge f i e l d measure. To i l l u s t r a t e what we have i n mind, l e t us c o n s i d e r an example which i s i d e a l from the p o i n t of view of c o n d i t i o n ( i i i ) . If we expand the Wi l son a c t i o n to l e a d i n g o r d e r i n e and second o rde r in A, we get S(a) = -{ E £ d t r C ( 3 £ A y - 3 £ A ) ( x + { e u + \ e y ) I 2 + 0 ( A 3 ) . x£ A * (4.18) The damping term f o r Feynman gauge EoF (a) = e x p ( - X ' 2 e d ~ 4 E Re t r CV x ( a ) - HI) (4.19) x which to the same a p p r o x i m a t i o n i s EoF(a ) = exp (- E e d t r [ 3 £ A ( x ) ] 2 + 0 ( A 3 ) ) . (4.20) X,(J We now impose D i r i c h l e t boundary c o n d i t i o n s by t a k i n g the IV.C THE ACTION range of the sums i n e q s . ( 4 . 1 6 ) and (4.19) to be a l l p l a q u e t t e s and p o i n t s i n the i n f i n i t e l a t t i c e e Z d , but w i th the gauge f i e l d s t a k i n g the v a l u e £ on a l l bonds not i n A, so t h a t t h e r e i s o n l y a f i n i t e number of nonzero terms in each sum. By combin ing e q s . ( 4 . 1 8 ) and (4.20) and p e r f o r m i n g a summation by p a r t s we o b t a i n the q u a d r a t i c p a r t of the e f f e c t i v e a c t i o n : -j E E d t r C OjAy -aJf l j , ) (K + j e^-t-^e y ) ] 2 - E e d t r [ 3 ^ (x ) ] 2 = - E E d t r [ A v ( x + f e y ) • (-Sjsjj) A y < x + £ e y ) ) = \ E E d A ^ x + fe^) ( - 4 ) t f a b ^ y A y ( x + i e y ) where i s the l a t t i c e L a p l a c i a n with DBC. The q u a d r a t i c form i s tha t g i ven by the o p e r a t o r - J ^ < S a b < S v i which sugge s t s t h a t the n o n i n t e r a c t i n g l a t t i c e measure conve rge s to Gaus s i an measure wi th c o v a r i a n c e (-V _ 1*abV-We now r e t u r n to the q u e s t i o n of c h o o s i n g the components of the l a t t i c e model . It has tu rned out to be u n e x p e c t e d l y d i f f i c u l t to do t h i s and we have not yet been a b l e to f i n d c h o i c e s which s a t i s f y more than two of the t h r e e c o n d i t i o n s ( i ) - ( i i i ) . To i l l u s t r a t e the p rob lem, l e t us c o n s i d e r aga in the e f f e c t i v e a c t i o n d e s c r i b e d above which s a t i s f i e s c o n d i t i o n ( i i i ) . We c l a i m t h a t i t a l s o s a t i s f i e s c o n d i t i o n ( i ) . For G c o n s i s t s of u n i t a r y m a t r i c e s so |tr bl i t r fl f o r any b £ G. Thus S e f f has a minimum when Up(a) = fl and IV.C THE ACTION 112 V x ( a ) = 11 f o r a l l P and x. It i s not hard to c o n v i n c e o n e s e l f , by drawing d iagrams f o r sma l l l a t t i c e s , t h a t t h i s can happen o n l y when a = i . However, to o b t a i n the damping term ( 4 . 1 9 ) , we would need to f i x the gauge on a l l of A ( tha t i s , take Aj = A) wh i ch , by Theorem 2.14, would r e s u l t in the gauge degree be ing z e r o , v i o l a t i n g ( i i ) . Suppose we modi fy the p r e c e e d i n g example by c h o o s i n g Aj as i n ( 2 . 8 a ) . We then s a t i s f y ( i i ) but v i o l a t e ( i ) . F o r , a ga in by c o n s i d e r i n g d iagrams of sma l l l a t t i c e s , i t i s not d i f f i c u l t to f i n d a gauge f i e l d a such tha t a ( x , y ) = +11 and such t h a t each Up(a) and V x ( a ) depends on an even number of bonds where a ( x , y ) has the v a l u e - £ . An example i s shown in the d iagram below. The heavy do t s i n d i c a t e the l a t t i c e s i t e s i n A j i the gauge f i e l d s have the v a l u e 11 on the bonds l a b e l l e d by " + " and - £ on the bonds l a b e l l e d For such a f i e l d we have U p ( a ) = U p ( £ ) and V x ( a ) = V x <f l ) , 5 0 * n e m i n i m u , n * s n o * u n i q u e . F i n a l l y , j u s t to i l l u s t r a t e tha t c o n d i t i o n s ( i ) and ( i i ) a re not m u t u a l l y e x c l u s i v e , we p r e s e n t an example of a l a t t i c e model which s a t i s f i e s both ( i ) and ( i i ) . C o n s i d e r IV.C THE ACTION 113 the l a t t i c e A shown below. We take S(a) to be the Wi l son a c t i o n f o r the p l a q u e t t e s shown, wi th the u n d e r s t a n d i n g tha t the gauge f i e l d has the v a l u e £ on tho se bonds wi th a s t r o k e drawn through them (on the boundary' of A). T h i s a c t i o n i s i n v a r i a n t under gauge t r a n s f o r m a t i o n s g f o r which g(x) - £ f o r x e 3A. Take Aj = A 0 as i n (2.8) but l e t the g a u g e - f i x i n g f u n c t i o n F be a map i n t o G where Aj i s the s u b l a t t i c e c o n s i s t i n g of the p o i n t s which a re c i r c l e d . D e f i n e F ( a ) ( x ) = V x ( a ) f o r x e A j . • 1 • I • I • I • O — O — O — o — o — o To s e e - t h a t ( i i ) i s s a t i s f i e d , c o n s i d e r the c o n d i t i o n f o r the e x i s t e n c e of gauge c o p i e s V x ( ? £ ) = £ f o r a l l x e A j . (4.21) By c o n s i d e r i n g (4.21) f i r s t f o r a l l the p o i n t s x at the lower boundary of Aj and then f o r the p o i n t s on the next -row up and so on, we see tha t (4.21) can h o l d o n l y when g = fl. Thus t h e r e a re no G r i b o v c o p i e s in t h i s gauge and I ri I = 1. S i m i l a r c o n s i d e r a t i o n s show that i f V x ( a ) = fl and Up(a) = £ f o r a l l x and P then a = £ so ( i ) h o l d s . The d e f e c t of t h i s model i s t ha t the e f f e c t i v e a c t i o n has an IV.C THE ACTION 114 a s y m m e t r i c a l and awkward form and i t i s not obv i ou s what the c o r r e s p o n d i n g Gaus s i an measure shou ld be. We do not b e l i e v e t h a t the c o n d i t i o n s ( i ) - ( i i i ) a re i n h e r e n t l y i n c o m p a t i b l e , but n e i t h e r does i t seem l i k e l y tha t they w i l l be s a t i s f i e d ve ry e a s i l y . P r o b a b l y the r e s o l u t i o n of t h i s d i f f i c u l t y l i e s i n o b t a i n i n g ( i i ) and ( i i i ) at the expense of ( i ) and then somehow e l i m i n a t i n g the i n v a r i a n c e which the e x i s t e n c e of m u l t i p l e minima i n d i c a t e s s t i l l rema ins a f t e r b r e a k i n g gauge i n v a r i a n c e wi th the damping te rm. To c a r r y out such an e l i m i n a t i o n , i t would be n e c e s s a r y t h a t gauge i n v a r i a n t f u n c t i o n s a l s o sha re the r e s i d u a l i n v a r i a n c e . That t h i s might be so i s sugge s ted by the f o l l o w i n g h e u r i s t i c c o n s i d e r a t i o n s . When the gauge group i s a b e l i a n , an argument [WC] based on the S t o n e - W e i e r s t r a s s theorem shows tha t a c o n t i n u o u s gauge i n v a r i a n t f u n c t i o n f ( a ) i s de te rm ined by the Wi l son loop' v a r i a b l e s , which a re p r o d u c t s of the gauge f i e l d s on l a t t i c e bonds f o rm ing a c l o s e d l o o p . By a l a t t i c e v e r s i o n of S tokes theorem, these loop v a r i a b l e s can be w r i t t e n in terms of the p l a q u e t t e v a r i a b l e s Up (a ) . Thus f o r any gauge f i e l d a f o r which U p ( a ) = Up(H) f o r a l l p l a q u e t t e s P we have f ( a ) = f ( l l ) , which i s the d e s i r e d i n v a r i a n c e . IV.D HAAR MEASURE AND LEBESGUE MEASURE 115 IV.D Haar Measure and Lebesgue Measure In t h i s s e c t i o n we i n v e s t i g a t e the r e l a t i o n between l a t t i c e Haar measure on the gauge group and cont inuum " Lebe sgue " measure on i t s L i e a l g e b r a by examin ing the cont inuum l i m i t of the Radon-Nikodym d e r i v a t i v e R £ ( A £ ) (see S e c t i o n IV .A ) . In Theorem 4.1 below we f i n d t h a t , c o n t r a r y to what one might e x p e c t , R £ does not become n e g l i g i b l e in the fo rma l cont inuum l i m i t . Let a be the 2*2 P a u l i s p i n m a t r i c e s „2 = i _ To i g - L l 0 and put t = - i f f / 2 . G iven A e IR° w r i t e A = A ' t . Then 0 - i 1 0 1 0 0 -1 f o r any a e SU(2) t h e r e i s an A £ IR such tha t a = e H = c o 5<|A|/2)J l + 2A • t s i n (| AI / 2) . If da denotes Haar measure on SU (2 ) , then f o r any f e L 1 ( S U ( 2 ) , d a ) we have the f o l l o w i n g e x p r e s s i o n f o r the Haar i n t e g r a l of f ( [ W i l , p.152) \ f ( a ) da = ir~ 2 J •» f ( e A ) s ( ( A | / 2 ) 2 dA J S U ( 2 ) J | A M 2 ¥ where s (x ) = 5 1 n x . Thus i f we use the r e l a t i o n x £ AA„ c ( x + r e „ ) a lx .x+e^) = e P i 6 * P (4.22) the l a t t i c e measure d a = TT da (x, x +e„ ) x ,u IV.D HAAR MEASURE AND LEBESGUE MEASURE 116 can be w r i t t e n (up to a n o r m a l i z a t i o n c o n s t a n t ) da = R£ (A £ ) JT dA„ £ (x + j e ) x,u K ' where R £ (A £ > = n n s(eA|A (x) I/2) 2 l 2 T r / E ^ ( I A £ (x)|) x £ A £ u H ' H ' (4.23) and *K = *<x : |x| £ K ) . Theorem 4 . 1 ; Let A C TR*\ F i x A^ such t h a t each A* £ C ™ ( A ) . D e f i n e A„ . ( x ) = A.,<Cx]> where Cx] denotes the p o i n t i n the l a t t i c e A £ n e a r e s t to x. Then R (A) -> e x p ( - ^ - E $ IA ( x ) | 2 dx) 12 ^ J A ^ as e •» 0. D Proof •  Choose £ so sma l l t ha t (I I „, < 2ir/e A. Then l og R £ ( A £ ) =2 E log C s (£ A JA (x ) 112) ]. x ,u Now s (x) = 1 - \- x 2 + 0 ( x 4 ) 6 and l og (1 + x) = x + 0 (x 2 ) so £ 2 A 2 | A U , (x) I2 . l og R £ ( A £ ) = 2 E Clog (1 £ x f ) + o ( e * n x,u l u > Thus IV.D HAAR MEASURE AND LEBESGUE MEASURE E 2 A 2 | A U (x) I 2 . • 2 • E C £ + 0<E4) ] x , u 2 4 = - ~ E e 2 IA £ ( x ) I2 + 0 (e 2 ) 1 1 X , u H ' l o g R £ ( A £ ) 4 " M A I V X ) | 2 d x as E •» 0. The l i m i t o b t a i n e d i n Theorem 4.1 i s p u z z l i n g , and e s p e c i a l l y so i s the appearance of what seems to be a mass term f o r the gauge f i e l d in two d i m e n s i o n s . Such a term i s gauge v a r i a n t and does not appear in the usua l f o r m u l a t i o n of the cont inuum t h e o r y . It a r i s e s not from any p h y s i c a l i n p u t , but s i m p l y from the f a c t tha t we have used Haar measure as an a p p r o x i m a t i o n to Lebesgue measure. If t h i s term c o u l d be shown to p e r s i s t i n a more r i g o r o u s form of the cont inuum l i m i t , i t would i n d i c a t e a s e r i o u s d e f i c i e n c y i n tha t a p p r o x i m a t i o n . However we now argue tha t Theorem 4.1 i s m i s l e a d i n g b e c a u s e , in gauge i n v a r i a n t i n t e g r a l s at l e a s t , the Radon-Nikodym d e r i v a t i v e does not a c t u a l l y occur i n q u i t e the same form as ( 4 . 2 3 ) . That c o n c l u s i o n i s based on the f o l l o w i n g theorem. Theorem 4 .2 : Let f be a gauge i n v a r i a n t f u n c t i o n in a two d i m e n s i o n a l l a t t i c e t h e o r y . Then t h e r e i s a f u n c t i o n f j such tha t IV.D HAAR MEASURE AND LEBESGUE MEASURE 118 J f ( a ) da = I f j ( U ) dU where U = (Up(a ) }p C / ^ and dU i s the p roduc t of Haar measures fl d U p . PE A B e f o r e p r o v i n g Theorem 4.2 l e t us e x p l a i n how i t r e s c u e s us from Theorem 4 . 1 . With our s t a n d a r d a p p r o x i m a t i o n ( e q . ( 4 . 1 4 ) ) a = e £ * A we have E 2 X F U V _ ( P ) U p ( a ) = e P " ' E where ^ y , £ i s a f i n i t e d i m e n s i o n a l a p p r o x i m a t i o n to the cont inuum f i e l d s t r e n g t h t e n s o r F ^ . When we use the v a r i a b l e s F u y , e a s * n e c o o r d i n a t e s of the g roup , the Radon-Nikodym d e r i v a t i v e f o r Haar measure i s R £ ( 1 ) ( F £ ) = ^ s ( c 2 X | F ( I V I E ( P ) i / 2 ) 2 l 2 , / « X < i ^ r 1 i < p , i > -In t h i s case the f o rma l l i m i t i s l o g R £ ( 1 ) ( F E ) = 2 E log (1 - — - I F ^ ^ t P ) ! 2 + 0 ( E 6 ) ) A 2 12 E 2 E E 2 IF.., , (P) 8 2 + 0 ( E 4 ) •» 0 as E -> o s i n c e Ee 2 I F „ „ . I 2 •» t « F . , y » 2 d 2 x . Hence R E 1 ' -> 1. We d i s c u s s below the s i t u a t i o n f o r gauge v a r i a n t i n t e g r a l s . We now t u r n to the p roo f of Theorem 4 .2 . G iven a f i n i t e l a t t i c e A d e f i n e a t r e e to be a set of bonds of A which c o n t a i n s no c y c l e s ( c l o s e d l o o p s ) . The f o l l o w i n g lemma i s v a l i d in any number of d i m e n s i o n s . IV.D HAAR MEASURE AND LEBESGUE MEASURE 119 Lemma 4 . 3 ; Let T be a t r e e . Then f o r any gauge f i e l d a t h e r e i s a gauge t r a n s f o r m a t i o n g such t h a t 9 a ( X ) y ) B j f o r a l l <x,y> « T. Moreover , g(x) » i when the ve r tex x l i e s o u t s i d e T. P r o o f ; The p roo f i s by i n d u c t i o n on |T| = the number of bonds i n T. The case |T| = 1 i s t r i v i a l . Suppose the lemma h o l d s whenever |T| < N and l e t T be a t r e e with I T I = N. Let <x,y> E T be a bond wi th the p r o p e r t y tha t one of i t s e n d p o i n t s , y s ay , i s the e n d p o i n t of no o the r bond i n T. Such a bond <x,y> must e x i s t f o r o t h e r w i s e T would c o n t a i n a c y c l e . App ly the i n d u c t i o n h y p o t h e s i s to f i n d the gauge t r a n s f o r m a t i o n g f o r the t r e e T - {<x,y>>. Set g(y) = a ( x , y ) g ( x ) ~ * . Then 9 a ( x , y ) = g ( x ) a ( x , y ) g ( y ) _ 1 = 1 . D Now suppose as u sua l tha t A i s a r e c t a n g u l a r l a t t i c e and l e t T be the t r e e c o n s i s t i n g of the bonds shown i n the d iagram be low, i . e . , T i s the set of a l l h o r i z o n t a l bonds and a l l bonds on the r i g h t - h a n d s i d e of A. Let a|T denote <a(x,y)}< x y > e j , the set of bond v a r i a b l e s f o r bonds i n T. We wish to make a change of v a r i a b l e s from the bond v a r i a b l e s (a(x,y)><• H ^ y > c A to a|T U { U p } p c A , a subse t of the bond v a r i a b l e s t o g e t h e r wi th the p l a q u e t t e v a r i a b l e s . ( S i nce we are d i s c u s s i n g the p l a q u e t t e v a r i a b l e s Up t h e m s e l v e s and not j u s t Tr(Up) we need to make some IV.D HAAR MEASURE AND LEBESGUE MEASURE 120 c o n v e n t i o n about the o r d e r i n which the bonds i n P a re m u l t i p l i e d t o g e t h e r to p roduce Up.) A The change of v a r i a b l e s j u s t d e s c r i b e d i s an i n v e r t i b l e t r a n s - f o r m a t i o n . F o r , g i ven the v a l u e s of a ( x , y ) on T and the p l a q u e t t e v a r i a b l e s Up we can u n i q u e l y d e t e r m i n e a ( x , y ) on a l l of A as f o l l o w s . For each p l a q u e t t e i n the column on the r i g h t - h a n d s i d e of A the v a l u e s of a on t h r e e out of f o u r of the p l a q u e t t e ' s bonds a re known, s i n c e the se bonds a re in T. We a l s o know Up and hence can d e t e r m i n e the v a l u e of the f o u r t h bond. But tha t means tha t f o r the next column of p l a q u e t t e s to the l e f t , we have de te rm ined t h r e e out of f o u r of i t s bonds, and so on. P roo f of Theorem 4 .2 : Let T be as above. From the p r e c e d i n g d i s c u s s i o n we know t h a t f o r some f u n c t i o n f 2 we have f ( a ) = f 2 ( a | T , U ) . IV.D HAAR MEASURE AND LEBESGUE MEASURE G iven a, l e t g be the gauge t r a n s f o r m a t i o n g i v e n by Lemma 4.3 so tha t f ( a ) = f ( 9 a ) = f 2 ( £ , 5 U ) . ( 4 .24 ) ' For each bond <x,y> which i s not i n T use the t r a n s l a t i o n i n v a r i a n c e of Haar measure to w r i t e da (x , y) = dUp where P i s the p l a q u e t t e which has <x,y> as i t s l e f t boundary . D e f i n e d a T = 17 d a ( x , y ) . Then u s i ng (4.24) <x,y>eT we have I f ( a ) da = J f ( a ) dU d a T «= \ f 2 ( H ,9U) dU d a T i n which g depends on a T > Now 9 U p = g ( x ) U p g ( x ) _ 1 , so by the i n v a r i a n c e of Haar measure, J f 2 ( H , 9 U ) dU = I f 2 ( £ , U ) dU. (4.25) D e f i n e f j t - ) = f 2 ( i , 0 . Then by (4.25) we have I f ( a ) da = I f j ( U ) dU d a T = J f M U ) dU. D Let us now c o n s i d e r how the p r e c e d i n g argument would be d i f f e r e n t i n the gauge v a r i a n t l a t t i c e t h e o r y . In the g e n e r a l ca se of an i n t e g r a l where the i n t e g r a n d depends on (a (x , y) } < x ^ y>>E j we c o u l d expec t Theorem 4.1 to have an e f f e c t in the cont inuum l i m i t . However, i n our l a t t i c e model the a c t i o n i s a f u n c t i o n of Up and V x , the p l a q u e t t e v a r i a b l e s and the c r o s s v a r i a b l e s . If we c o u l d show t h a t the change of v a r i a b l e s ( a (x ,y )> •> IV.D HAAR MEASURE AND LEBESGUE MEASURE 1 2 2 ^ U P J P c A u * v x J x « A i s w e l l - d e f i n e d we c o u l d w r i t e I f ( a ) da = J f j ( U , V ) dU dV. Once aga in Theorem 4.1 would be an i n a p p r o p r i a t e gu i de to the cont inuum l i m i t because e 2 * F u v , (P ) Up = e f y ' e e 2 !U E .A.. . (x) V x = e P * J » E Both s e t s of v a r i a b l e s F.,u 3,.A„ . have an E* dependence i n s t e a d of the e dependence of the v a r i a b l e s A ^ ) £ . By the same argument used i n the gauge i n v a r i a n t c a s e , the Radon-Nikodym d e r i v a t i v e makes no c o n t r i b u t i o n i n the fo rma l cont inuum l i m i t . It i s p l a u s i b l e t h a t the change of v a r i a b l e s (a) -» <U,V> i s w e l l - d e f i n e d s i n c e , h e u r i s t i c a l 1 y, U * F^ y i s the t r a n s v e r s e p a r t of the gauge f i e l d and V * S^A^ i s the l o n g i t u d i n a l p a r t . However, in t r y i n g to implement t h i s i d e a , we run i n t o the same s t u m b l i n g b l o c k as d e s c r i b e d i n S e c t i o n IV.C in c o n n e c t i o n with the un ique minimum prob lem. We saw t h e r e tha t t h e r e a re gauge f i e l d s a not everywhere 11 f o r which Up(a) and V x ( a ) a re everywhere 11. Hence the change of v a r i a b l e s Ca> •+ (U,V} i s not i n v e r t i b l e . T h i s appear s to be ma in l y a t e c h n i c a l prob lem and does not c o n t r a d i c t the i n t u i t i o n j u s t d e s c r i b e d . We c o n c l u d e t h a t , at l e a s t at a fo rma l l e v e l of argument, the Radon-Nikodym d e r i v a t i v e does not make a c o n t r i b u t i o n to the cont inuum l i m i t . IV.E FADDEEV-POPOV DETERMINANT - FORMAL LIMIT 1 2 3 IV.E Faddeev-Popov De terminant - Formal L i m i t The l a t t i c e Faddeev -Popov d e t e r m i n a n t f o r Landau gauge M£ (a) was shown in S e c t i o n II.D to be s M ( a ) y v = 11 - a d ( - s ) + E Cad( + u) + a d ( - u ) ] M ( a > x , K ± e M = " a d ( ± f ) ( 2 . 4 0 ) M ( a ) x y = 0 i f I x - y l > E where a d ( ± u ) = a d ( a ( x , x + e ^ ) a ( x , x - e ^ ) a ( x , x + e j ) . . . a ( x , x ± e ^ ) ) . The cont inuum Faddeev -Popov o p e r a t o r M ( A ) o p e r a t e s on L 2 ( I R 2 ) ® I R 2 ®E and i s g i ven by M (A ) B = -6% - X ^ C A ^ . B I . We now show t h a t i f B a e c"(/\> then E ~ 2 M £ < e e * A ) B •» M ( A ) B. We have from e q . ( 2 . 4 0 ) t ha t E " 2 ( M £ ( a ) B ) ( x ) = E " 2 E Y M ( a ) x y B(y) = E " 2 t i l - a d ( - s ) ] B(x) + E " 2 EJJ ad( + r ) [B(x) - Btx + e ^ H + E " 2 E ^ ad(-f/) [B(x) - B l x - e ^ ) ] = e " 2 [fl - a d ( - s ) ] B(x) - E " 1 E l ad(+y) a jB fx+fe^) + e " 1 E ^ ad(- f i ) 3 j B ( x - f e ^ ) . ( 4 . 2 6 ) By e q . ( 4 . 1 4 ) IV.E FADDEEV-POPOV DETERMINANT - FORMAL LIMIT a f x ^ + e ^ a ^ x - e ^ ) = IL + 3kE 2 S^A^x ) + Q ( e 3 ) w i th the r e s u l t f o r terms i n (4.26) be ing ( i ) E - 2 [ £ - a d ( - s ) ] B(x) = -A E C 3 £ i A ( i ( x ) , B ( x ) ] + 0(e) ( i i ) - E _ 1 ad( + u) 3 E B (x + j e f i ) = - s - ^ j B t x + l e ^ ) - A C A ^ x + fe^) .ajBtx+Je,, + 0<e ) ( i i i ) e " 1 ad ( -u ) 3 jB (x -^e^ ) = e " 1 3Jj,B(x-fe^) * 0 ( e ) . P u t t i n g t he se e q u a t i o n s i n t o e q . ( 4 . 2 6 ) we o b t a i n e _ 2 M £ ( a ) B ( x ) = ( - A [ 3 £ A ^ (x) ,B(x) ] - 3 £ 3 E , B ( x ) - A [ f l t l ( x + | e ( J ) , 9 j B ( x + y e J J ) ] ) + 0(e) -* - J B ( x ) - A C^Sp C A p (x ) ,B(x) ] as E -» 0. IV.F FADDEEV-POPOV OPERATOR - TRACE CLASS PROPERTIES 125 IV.F Faddeev-Popov Opera to r - T r a c e C l a s s P r o p e r t i e s Let A C IR2 be the r e c t a n g l e A = (x e IR2 I 1^1 i (i • 1,2) and <dA be the L a p l a c i a n o p e r a t o r wi th D i r i c h l e t boundary c o n d i t i o n s on A. The u n n o r m a l i z e d Faddeev-Popov o p e r a t o r M(A) i s g i ven by M(A)B = -d^B - M [ A p . B ] . De-fine the n o r m a l i z e d o p e r a t o r L(A) by fl(A) = (-<J A)[I + L (A) ]. In t h i s s e c t i o n we show tha t t h e r e i s a H i l b e r t space on which L(A) i s compact and in the Car leman c l a s s C 2 + £ f o r a l l E > 0 f o r a lmost a l l A. T h i s means i n p a r t i c u l a r t h a t d e t 3 ( I + L ( A ) ) i s w e l l - d e f i n e d and c o n s t i t u t e s the f i r s t s tep in o b t a i n i n g a w e l l - d e f i n e d r e n o r m a l i z e d Faddeev -Popov d e t e r m i n a n t . For a d e f i n i t i o n of the Car leman norms I -1p and d e t p see C S i l ] . Our b a s i c approach i s taken from the t e c h n i q u e s used in the Yukawa model (CSeD.CBR]) . The gauge group G can be any compact L i e g roup. If we take a b a s i s ( t a } f o r i t s L i e a l g e b r a E we have " i ' V • f a b c *c which d e f i n e s the s t r u c t u r e c o n s t a n t s f a j , c > If we w r i t e A = A a t "u"-a IV.F FADDEEV-POPOV OPERATOR - TRACE CLASS PROPERTIES 126 and L (A)B = L ( A ) a b B b t a we have L a b = - A f c b a i - A A r l a ^ A j . (4.27) The n o t a t i o n 3 ^ * i s used to emphas ize tha t the o p e r a t o r c o m p o s i t i o n i s i n t e n d e d . We now d e f i n e the H i l b e r t space h' on which the Faddeev -Popov o p e r a t o r a c t s . Let (•,•) denote the i n n e r p r o d u c t on L 2 ( A , d x ) and l e t tfi be the c o m p l e t i o n of 2 1 D ( ( - « d ^ ) 5 ) w i th the i n n e r p r o d u c t I <f , g ) » = ( f , ( - A A ) i q ) . i ' We d e f i n e tt = Hi. ® E. 2 The f i e l d A a i s the Gaus s i an random p r o c e s s with c o v a r i ance < A a ( f f l ) Aj (g y ) > = Sab i-4A)-lq¥\. We l e t dp denote the c o r r e s p o n d i n g measure, which i s the f r e e measure on the gauge f i e l d s in Feynman gauge. The main p o i n t of t h i s s e c t i o n i s to prove Theorem 4.4 ; The o p e r a t o r L d e f i n e d i n e q . ( 4 . 2 7 ) i s i n C 2 + £ a . e . ( d p ) f o r a l l £ > 0. The approach we take i s to show t h a t 2 + E J |L(A) I dp < a. IV.F FADDEEV-POPOV OPERATOR - TRACE CLASS PROPERTIES To b e g i n , we d i s c u s s the o p e r a t o r (-<dA)~* 3^ a p p e a r i n g i n i , eq .<4 .27 ) . Let D £ i - d ^ t , m* = i n f s p e c ( - d A ) > 0, D Q = ( - J + m£ )J where 6 i s the L a p l a c i a n wi th f r e e boundary c o n d i t i o n s . Lemma 4 . 5 ; (a) If the o p e r a t o r A > 0 and 0 < fl < 1| then -0 « s u ^ . » + _ ! ( 4 < 2 0 ) rr J 0 (b) For a l l a £ CO,2] t h e r e i s a c o n s t a n t c = c (o ) so t h a t D " a = c D ~ a . (4.29) P r o o f : (a) T h i s f o l l o w s from the s p e c t r a l theorem and the f a c t t h a t f o r X > 0 » - I \ 4 ft o <A + t) s i n IT0 Thus f o r any f e L 2 <f,A~0f) = \ X~& d ( f , E ^ f ) (i(A) w C ( A ) ^ - 1 1 j " t " * <f f (A + t ) " 1 * ) dt " o by F u b i n i ' s theorem. The r e s u l t f o r ( f ,A~ "g ) then f o l l o w s from the p o l a r i z a t i o n i d e n t i t y . (b) We have the form i n e q u a l i t i e s CGRS23 IV.F FADDEEV-POPOV OPERATOR - TRACE CLASS PROPERTIES "4l > m o which imply t h a t f o r a l l t * 0, <-d A + t) i + m 2 + 2 t ) . So by a theorem i n CK1 (p. 330) <-d A + t ) " 1 * 2 ( - d + m 2 + 2 t ) _ 1 . (4. In p a r t i c u l a r , D" 2 * 2 D " 2 . From ( a ) , f o r a < 2 (f , D " « f ) = j " t " a / 2 ( f , ( - ^ A + t ) - J f ) dt * K j " t " a / 2 ( f , ( - 6 + m 2 + t ) _ 1 f ) dt i K' ( f , D ; a f ) . Lemma 4 .6 ; If a * 1, D " a 3 ( i ex tends from C*(A) to a bounded o p e r a t o r on tfi. 2 P r o o f ; D~" i s bounded on tf± f o r any $ > 0 so i t 2 s u f f i c e s to p rove the theorem f o r the ca se a-l. By Lemma 4.5(b) D " 1 i K D g 1 . Thus i f f e C ^ A ) we have i • i K O ^ f . D " 1 * ^ ) = K ( f , f ) IV.F FADDEEV-POPOV OPERATOR - TRACE CLASS PROPERTIES 129 = K | D " 2 f ! 2 ? £ K I D ~ 2 | 2 | f | 2 0 2 We now d e f i n e the o p e r a t o r <-d A)~* to be the un ique bounded e x t e n s i o n of D ~ 2 3 ^ | C ™ ( A ) . We now r e t u r n to the Cp p r o p e r t i e s of L. These a re based on the x - s p a c e b e h a v i o r of the i n t e g r a l k e r n e l s of ( - ^ A ) ~ a ( x , y ) and - — ( - d A ) ~ a ( x , y ) which a re g i ven by the 3x ^  next two lemmas. The l e t t e r s K, Kj and so on used below denote d i f f e r e n t c o n s t a n t s on d i f f e r e n t o c c a s i o n s . However, at a l l t i m e s , these c o n s t a n t s depend at most on A. In p a r t i c u l a r , they a re i ndependent of x ,y £ A and of m. Lemma 4 .7 ; The i n t e g r a l k e r n e l of C a obeys K | x - y r 2 ( 1 " a ) (0 < a < 1) 0 £ C a ( x , y ) £ KI l n l x - y l | (a = 1) f or a l 1 x , y £ A. Lemma 4 .8 ; The f i r s t d e r i v a t i v e of C a obeys | 3 ( J C a ( x , y M £ K | x - y | ' 2 < 1 " a , _ 1 , 0 < a < 1 f or a l l x , y £ A. Lemmas 4.7 and 4 .B, which are proven below, a l l o w us to p rove the main r e s u l t Theorem 4 . 9 ; For each e > 0, L £ C 2 + e a . e . IV.F FADDEEV-POPOV OPERATOR - TRACE CLASS PROPERTIES 130 P r o o f ; Let du be the Gaus s i an measure a s s o c i a t e d wi th A3, and d e f i n e the l " l p ( q norms i n the usua l way: If B i s a map from the gauge f i e l d s A a i n t o the space of bounded o p e r a t o r s on ft', | B | p | q = (\ J B l J d u ) 1 ^ . By complex i n t e r p o l a t i o n C S i l ] and H o l d e r ' s i n e q u a l i t y , f o r a l l 6 > 0 B L « 2 + £ , 2 + £ * « D " 2 E i L » 2 , 2 U > ( 2 ~ £ > i l - l 4 , 4 so i t s u f f i c e s to show <i) B D " a L g 2 , 2 < 0 0 (0 < a < j ) ( i i ) I D * L I 4 | 4 < » (a < i ) to c o n c l u d e t h a t IL12*+e * s f i n i t e a . e . # + Let denote the a d j o i n t on h' and the a d j o i n t on L 2 ( A ) ® E . Then i n g e n e r a l B* » D _ I B + D so (D - ( J L ) * ( D " d L ) = D " 1 L + D 1 _ 2 6 L . A d i r e c t computa t i on then shows I D _ a L i | = K k l J t D _ 1 ( x , y ) A j ( y ) D - 3 ' 2 - 0 (y, z) /\3 f sy^ x D " 3 / 2 _ a ( z ,x) A a ( x ) dx dy dz so tha t I D~ aL || 2 - K J D _ 1 ( x , y ) D " 2 ( x , y ) I j— D " 3 / 2 - a (y, z ) | • A 3 <>yu * l r ^ - D " 3 / 2 _ a ( z , x ) I dx dy dz (4.31) Now app l y Lemmas 4.7 and 4.8 to e q . ( 4 . 3 1 > . We o b t a i n IV.F FADDEEV-POPOV OPERATOR - TRACE CLASS PROPERTIES |D~ f lL|?, 2 i K J I x - y l " 1 | l n — ^ | x | y - z | -3/2+a. | z _ x ( - 3 / 2 + a d K d - y d Z i (4.32) R e c a l l Lemma C6 of [BR] , which says t h a t •\ d 2 z I z - x J - S l z - y l ' * i K | x - y | 2 " ^ " ^ (4.33) i f 0 £ 0,y il and /) + % > 2. T h i s e s t i m a t e , combined wi th I I n — — - | * K, I x - y l " * |x-y| 6 f o r a l l S > 0 and some y i e l d s , from e q . ( 4 . 3 2 ) | D " a L l | 2 - K ! I x - y | ~2+2a-<S d x d y < w ' A 2 i f we chose 6 < 2a. T h i s p rove s ( i ) above. The proo f of ( i i ) i s s i m i l a r , be ing based on the e x p r e s s i o n \~1iv » _\ n" 1 4 * K $ D (x j,x2> D " 1 ( x 4 , x 5 ) ' A 6 , J _ D - 3 / 2 + 0 ( x , _ L . D - 3 / 2 + f l ( , 3 x ^ 2 3 3 x J 3 4 x J _ D - 3 / 2 + fl| , _3 D - 3 / 2 + 0 ( x , 3 x § 5 6 3 x J 6 1 x U ^ d ^ D - 2 ( x 2 , x 4 ) D " 2 ( x 5 , x 1 ) + * M y l S r > D ~ 2 ( > < 2 ' ! < 5 ) D " 2 ( x 4 , X j ) + <5 f i^<S^D~ 2<x 2,x 1) D " 2 ( x 4 , x 5 ) 3 ^ d x j , We now t u r n to the p r o o f s of the e s t i m a t e s used above. IV.F FADDEEV-POPOV OPERATOR - TRACE CLASS PROPERTIES 132 Proo f of Lemma 4 .7 : The e s t i m a t e f o r a = 1 f o l l o w s the we l l - known b e h a v i o u r of the D i r i c h l e t G r e e n ' s f u n c t i o n i n two d imens i on s (see e . g . , [ CH ] ) . For 0 < a < 1, we make use of a Wiener i n t e g r a l r e p r e s e n t a t i o n of C a ( x , y ) . By the s p e c t r a l theorem and the f a c t t h a t f o r A > 0 a = J r 0 0 ta-l „ - . U J T ° = =~-r- \ t " ' e dt (4.34) T(a) ''ci we have t h a t the d i s t r i b u t i o n a l k e r n e l of (-<d^)" a i s g i v e n by ( - J A ) " a ( x , y ) = fT^ fS™*0"1 e J A t ( K , y ) d t . (4.35) Let u. „ u . t denote Wiener measure c o n d i t i o n e d to s t a r t at x at t ime 0 and be at y at t ime t , w i th no rma l i zat i on S y I t • ( 2 . t ) - « e - ' " - V l 2 ' 2 « . Then (see CSi 23) e^Sx.y) = JQ d p 0 ) K ) y ; 2 t where Q = (u I u ( s ) e A f o r a l l s e [ 0 , t ] > . T h i s e x p r e s s i o n shows c l e a r l y t h a t 0 i e ^ f x . y ) £ ( 4 » t > _ 1 e ~ | x ~ y | 2 / 4 t (4.36) P u t t i n g e q . ( 4 . 3 6 ) i n t o e q . ( 4 . 3 5 ) , we o b t a i n 0 i C a ( x , y ) 4 _ J f " t a - l - d / 2 e l x - y | 2 / 4 t d t 4irT(a) J o i K | x - y | 2 a " 2 by s c a l i n g t •* | x - y | 2 t . • IV.F FADDEEV-POPOV OPERATOR - TRACE CLASS PROPERTIES 133 The p roo f of Lemma 4.B i s c o n s i d e r a b l y more c o m p l i c a t e d . The s t r a t e g y i s to use Lemma 4.5 (a ) to e x p r e s s C^ i n terms of the mass ive G r e e n ' s f u n c t i o n s CjjJ(x,y), and then d i f f e r e n t i a t e both s i d e s of e q . ( 4 . 2 B ) . Thus the proo f i s reduced to e s t a b l i s h i n g e s t i m a t e s f o r d C jJ (x ,y ) . These a re o b t a i n e d from the method of images e x p r e s s i o n f o r Cjjj. We s h a l l p rove Lemma 4.8 in a s e r i e s of f u r t h e r lemmas. Lemma 4 .10 : Let C m be the G r e e n ' s f u n c t i o n f o r -d + itr w i th f r e e boundary c o n d i t i o n s . Then f o r some c o n s t a n t s K j , K 2 and a l l x f IR2 |3 f J C ( n ( x )| i m e " m | x | [ K 1 ( m | x | ) " i + K 2 (m 1 x I) ~ 1 3 . (4.37) P r o o f : By p e r f o r m i n g one p i n t e g r a t i o n i n the x - d i r e c t i o n , we o b t a i n C ^ x ) = — l — - | f d 2 p (211)* p z + m 2 oo p -| i (k) m I x | = K [ — dk (4. 38) J o f<k) where u(k) = ( k 2 + 1)1. (4.3?) Note tha t f o r some c o n s t a n t s a, b > 0 1 + a k 2 , k = 1 fj(k) i (4.40) 1 + bk, k i 1. Now from (4.38) 3 „ C m ( x > = Km -2- \ e " f ( k , m | x | dk. (4.41) IV.F FADDEEV-POPOV OPERATOR - TRACE CLASS PROPERTIES 134 A s c a l i n g argument combined wi th e q . ( 4 . 4 0 ) shows j 1 e - ( i(k )m|x l d k i e -m|x| j 1 g-Ek 2 m|x| d k o o £ K (m|x|)"i e " 1 " 1 " 1 . (4.42) S i m i l a r l y , \ * e - j i (k) ml x f d k < K ( m | x | , - l g - n l K l , (4,43) E q u a t i o n s (4.41) through (4.43) g i v e the d e s i r e d r e s u l t . • By the method of images C A ( x , y ) = E < - l > | n | C m ( x - p n y ) n £ Z 2 where |n| = In^l + I n 21 and ( P n y ) ^ = ( - ! ) " " Lemma 4. 111 (a) l x - p n y | * |x-y| f o r a l l n £ Z 2 . (b) If | n y | i 2 f o r some v, then | x - p n y | * KIn| f or a l 1 x , y £ A. P r o o f : (a) I c l a i m t h a t in f a c t I ( x - p n y ) M l » I ( x - y ^ l f o r each p. If In^l = 0 t h i s i s o b v i o u s ; i f In^l i 2 then t < x - P n V V fe |np|Jp " 1 V " 1 V i ( In^l - 1) lu s i n c e Ix^I, l y ^ l £ 1^/2 f o r x, y £ A. IV.F FADDEEV-POPOV OPERATOR - TRACE CLASS PROPERTIES 135 Suppose In^l = 1. Then we must show £ (x -y) p * l j , + t (x+y)^ (4.44) f o r a l l c h o i c e s of e , v e {-1,1}, i . e . , ( e - H X j , - < £ + v ) y ^ * 1 . (4.45) But o n l y one term on the l e f t - h a n d s i d e of e q . ( 4 . 4 5 ) i s n o n z e r o , and i t s a b s o l u t e v a l u e i s bounded by 2(1^/2) = 1^. (b) Suppose I n y I * 2. Then I ( x - p n y ) y I i In y 11 y - Ix y | - f y y I * ( I n y I - 1) 1 y * K( | n y | + 1) . (4.46) If the o t h e r component of n, n y . s ay , s a t i s f i e s | n y ' | i 2, then the theorem i s p roved by summing e q . ( 4 . 4 6 ) . If |n y • | i 1 then l x - p n y I * I ( x - p n y ) I i K( | n y | + 1) i Kin I. • Lemma 4.12: The s e r i e s E ( - 1 ) , n 1 — C m ( x - p n y ) (4.47) n e Z 2 ax" m " converge s a b s o l u t e l y and u n i f o r m l y f o r x, y such tha t J x — yI i s bounded below u n i f o r m l y . Thus ^ - C j ( x , y ) = E n ( - l ) | n | — ; • C f f l ( x - p n y ) (4.48) nil.7 an" m " " " „ ^ 2 ' " ' ax" f or a l 1 x , y e A. P r o o f ; Combin ing Lemmas 4.10 and 4.11 we have I-*- C m ( x - p n y ) | i Kj e - K 2 ' n | . Sx^ IV.F FADDEEV-POPOV OPERATOR - TRACE CLASS PROPERTIES 136 A l though the s e r i e s (4.47) i s a b s o l u t e l y c o n v e r g e n t , to o b t a i n the d e s i r e d e s t i m a t e s from e q . ( 4 . 4 6 ) , we need to t ake i n t o account the c a n c e l l a t i o n s a r i s i n g from the a l t e r n a t i n g s i g n s of the te rms . For t h i s p u r p o s e , l e t us i n t r o d u c e some n o t a t i o n . If f : Z d •* IE, l e t dp* (n) = f ( n + e^) - f ( n ) where e^ i s the v e c t o r in Z d wi th 1 in the component and 0 in the o t h e r s . Let d{6) = • • • 4 d . Lemma 4 .13 : If f : Z d •» E i s such tha t E | f ( n )| < » , n e Z d then E ( - 1 ) | n | f ( n ) = E J ( d ) f ( n ) . (4.49) n e Z d n £ 2 Z d + l P r o o f : A s t r a i g h t f o r w a r d i n d u c t i o n on d. • Lemma 4.14: If n e 2 2 d + 1, M < d ' ^ C < " ( X " P n y ) l ' K S u p l s y M l b „ l l D r D d " - D l c m ( a n + s ) l (4.50) where a n. v - y y + < n „ + { > l b = x + kl D y - - 3 ~ . P r o o f : I c l a i m tha t IV.F FADDEEV-POPOV OPERATOR - TRACE CLASS PROPERTIES b j d < d ) C m < x - p n y ) - If J d S j O d . . . D 1 C , ( i n + i ) . (4.51) 1 i J = • 0 j For example, ^ l C m ( x - p n y ) n+ei = C f f l ( x - ( - l ) M y - t n + e j J l ) - C m ( x - ( - 1 ) n ( y - n l ) ) • C m ( ( - l ) n + e i x - y + ( n + e 1 ) l ) - C m ( ( - 1 ) n x - y + n l ) ' C m ( a n n + b i « i > " C,(a{»» - b ^ ) where a n j i " " V l + < n l + I> l i a n ! v = " x v " V, + " v 1 * " " * 1 ' Thus b ^ l C m ( x - P n y ) " \ . b d s l ¥i««n • » 1 > ' D l The g e n e r a l f o r m u l a e q . ( 4 . 5 2 ) i s proved by i n d u c t i o n s i m i l a r l y . A s h o r t c a l c u l a t i o n u s i ng the r u l e s f o r d i f f e r e n t i a t i n g i n t e g r a l s shows tha t j ( d ) J _ C m ( x - p n y ) = - i - d ( d ) C m ( x - p n y ) U j = d " b j The theorem f o l l o w s from t h i s e q u a t i o n , by t a k i n g K = TT I 2b : I. • l » j = d J Lemma 4. 15: For a l l x E (R2 D l C m < a n + s > IV.F FADDEEV-POPOV OPERATOR - TRACE CLASS PROPERTIES | D f J D 2 D 1 C f f l ( x ) | i < K 11 K | ~ 31 + K 2 l x l " 2 + K 3 m i | x r 5 / 2 + K 4 n 3 | x r 3 / 2 + K 5 m 3 / 2 | x | " 3 / 2 + K 6 m 5 / 2 | x | " i ) e" f f l | x The Kj a re independent of |x| and m. P r o o f : S t a r t from e q . ( 4 . 3 B ) „ oo p - u (k)m I x I C . ( x ) = K I — dk (4.38) m J o p(k) and c a l c u l a t e D j C ^ x ) - -Km ^ - $ " e - M k ) m | x | d k DoD,C.(x) = Km ^ f - [* (fc) m ( x I { _ L + t l { k ) } d k , I x Without l o s s of g e n e r a l i t y we may assume the JJ = 2. Now 2 .00 X. 2 X . X n X < X 9 D D.C m !x ) = Km [ {t — l — = j^/( k ) »J } 2 1 m J o ) x , 2 | x ( 4 ( x 7 7 ^ 2 X , X' » [ J _ . + „ , « , „ ] - l i ! l e - ( i < k ) m | x | d f c i I x | K j x i 4 Usi ng I x „ I £ I x I we o b t a i n 00 K l K 2 ,, , ^ ' D 2 D l C m ( x , l - m S„ I 7 ~ ? 2 + T7T ' J ( k , m + |X o x| + K 4 m 2 p ( k ) 2 ] e ~ f J < k , , n | > i | dk. (4.52) Us ing e q . ( 4 . 4 0 ) we f i n d \ l u ( k ) n e -F<k )m|x l d k t e - m U I j 1 e - a k 2 m | x | d k o o i K ( m | x | ) ' l e " m ( x 1 and IV.F FADDEEV-POPOV OPERATOR - TRACE CLASS PROPERTIES 139 [ * u ( k ) n e " * J ( k ) m | K | dk i K e-<<>lxl t " k n e " b k m | x | dk 1 i 1 i K (m I x I ) ~ n _ 1 e " m | x l . That i s , j " u ( k ) n e - M k ) m l x l d k  J o H * [ K j ( m | x | ) " i + K 2 ( n | x I ) " " " 1 ] e ~ m , x | . <4.53) E q u a t i o n (4.53) a p p l i e d to e q . ( 4 . 5 2 ) g i v e s the d e s i r e d r e s u l t . • Lemma 4 .16 : The bound I — C | J(x,y)| i (Kj ml | x -y | + K j l x - y l " 1 ) e " f f l | ! < " > ' 1 + f(m) 3xf h o l d s u n i f o r m l y f o r x, y e A where f i s a f u n c t i o n such tha t S00 m 1 " 2 * f (m) dm < co. J o P r o o f : The i d e a i s to s p l i t up the method of images sum (4.48) i n t o two p i e c e s . One p i e c e i n v o l v e s terms wi th sma l l n, and i s e s t i m a t e d by the known behav i ou r of 3|jC f f i (x,y). The sum of the r e m a i n i n g terms i s shown to be bounded by the f u n c t i on f ( m ) . From Lemmas 4.12 and 4 .13 , we have 3 - C ^ x . y ) = E d (2) S ^ l x - p ^ ) . (4.54) n € 2 Z 2 + l Let Q = <n e 2Z 2+1 s In yI < 2 f o r a l l v>. Then by combin ing e q s . ( 4 . 1 4 ) and Lemma 4 . 1 1 ( a ) , we have IV.F FADDEEV-POPOV OPERATOR - TRACE CLASS PROPERTIES H O J 4 ( 2 ) W x - p n y ) u i m e ~ m | x " V | [ K j ( m | x - y | ) ~ t + K 2 < m | x - y I ) 1 ] . (4.55) To e s t i m a t e E r e c a l l Lemmas 4.13 and 4.14. A l s o note t h a t i f n E Q c , then | n y | * 3 f o r some c and so | a n + s| i K|n| (4.56) which can be seen from an argument s i m i l a r to t h a t of Lemma 4 . 1 1 ( b ) . Combin ing these f a c t s y i e l d s H < 2 V f l . ( * - P n y > ' * ( E K.m j | n | J ) e " K m | n | (neQ c ) (4.57) j - l J where O j , fij * 0 and 0j + a j - 2. We now " s t e a l " a l i t t l e b i t from the e x p o n e n t i a l term i n (4.57) v i a e -Km|n| _ e -Km|n|/2 e -Km|n|/2 £ K. ( m i n i ) " £ j e " K m | n | / 2 . (4.58) j We choose 6j so t h a t « j + 2 ( l - a ) - E j > 0 8j + E j > 2. T h i s i s p o s s i b l e because 1 - a > 0 and « j + #j > 2. From e q . ( 4 . 5 7 ) , e q . ( 4 . 5 8 ) and the f a c t tha t | n | » < oo n E Q c i f y > 2 we o b t a i n E M ( 2 , 3 u C m ( x - p n y ) I £ ( E K j i / - " * j ) e " K m . (4.59) n E Q c -> = 1 Take f(m) to be the r i g h t - h a n d s i d e of e q . ( 4 . 5 9 ) . IV.F FADDEEV-POPOV OPERATOR - TRACE CLASS PROPERTIES E q u a t i o n s (4.55) and (4.59) g i v e the d e s i r e d bounds. • P roo f of Lemma 4 .8 : By Lemma 4.5 (a ) C a ( x ,y> = S i n ^ t " t " a (-4* + t ) _ 1 ( x , y ) dt TT o = K m 1 " 2 * c j (x,y> dm n where we have made a change of v a r i a b l e s t •» m to o b t a i n the l a s t l i n e . A dominated convergence argument u s i n g the bounds of Lemma 4.16 a l l o w s us to d i f f e r e n t i a t e under the i n t e g r a l to o b t a i n 3,,Ca<x,y> = K j " m 1 _ 2 a a^C^x.y ) dm. Us ing Lemma 4.16 a g a i n , we o b t a i n i a ^ C a ( x , y ) | i K 1 | x - y | ' i j " m 3 / 2 * 2 a e. - m I x - y I d m + K 2 I x - y l " 1 j " m 1 " 2 0 e - m l x - y l d i n + ^ °° m 1 _ 2 a f (m) dm. J o By s c a l i n g m •+ m/|x-y|, the f i r s t two terms are bounded by K | x - y | " 3 + 2 a . The t h i r d term i s bounded by a c o n s t a n t , which can be bounded by K ' | x - y | ~ 3 + 2 a . • IV.G FADDEEV-POPOV DETERMINANT - DIAMAGNETIC INEQUALITY IV.G Faddeev-Popov Determinant - D i amagnet i c I n e q u a l i t y In t h i s s e c t i o n we prove Theorem 4 .17 ; Let M(a) be the Faddeev-Popov o p e r a t o r f o r Landau gauge ( e q . ( 2 . 4 0 ) ) in d = 1 d i m e n s i o n . Then Idet M(a) | £ det H(fl). (4.60) The i n e q u a l i t y (4.60) i s of a c l a s s known as d i a m a g n e t i c i n e q u a l i t i e s in which a d e t e r m i n a n t or p a r t i t i o n f u n c t i o n in an e x t e r n a l f i e l d i s bounded by i t s v a l u e i n z e r o e x t e r n a l f i e l d . The term " d i a m a g n e t i c " i s used because one of the f i r s t i n e q u a l i t i e s of t h i s t ype e x p r e s s e d the d i a m a g n e t i c re sponse of a system of s p i n l e s s p a r t i c l e s to an e x t e r n a l magnet ic f i e l d CS i21. D iamagnet i c i n e q u a l i t i e s have been proven f o r E u c l i d e a n l a t t i c e f e r m i o n s wi th a n t i p e r i o d i c boundary c o n d i t i o n s [BFS1] , [W] and f o r l a t t i c e bosons f o r a wide v a r i e t y of boundary c o n d i t i o n s CBFS1], The proo f we g i v e f o r QCDj i s r a t h e r c lumsy and does not r e a d i l y extend to the two (or h i g h e r ) d i m e n s i o n a l c a s e . The impor tance of Theorem 4.17 i s to sugges t tha t a d i a m a g n e t i c i n e q u a l i t y f o r the Faddeev -Popov d e t e r m i n a n t does in f a c t e x i s t . P roo f of Theorem 4 .17; Labe l the l a t t i c e p o i n t s as shown o « — - . — . — . . , — . — , — , . — . o 0 1 3L Y\ r u i and w r i t e U (a )^ j f o r a d ( a j j ) . We s h a l l u s u a l l y omit the IV.G FADDEEV-POPOV DETERMINANT - DIAMAGNETIC INEQUALITY 143 argument a and j u s t w r i t e U j j . For i n t e r i o r p o i n t s i , the Faddeev -Popov o p e r a t o r M(a) i s g i ven by M ( a ) u - 1 • U U M ( a , i , i + 1 = _ u i , i + l M ( a ) i , i - 1 " - u i , i + l u i , i - l ' In matr ix •form, M(a) i s 1 + U 1 2 " U 1 2 • u 2 3 u 2 1 i + u 2 3 - u 2 3 o " u k , k + l u k , k - l 1 + u k , k + l " u k , k + l ^ - U n , n + l u n , n - l 1 + u n , n + l _ Because G i s compact and a •* ad(a) i s a • f i n i t e - d i m e n s i o n a l r e p r e s e n t a t i o n of G, each U j j i s a u n i t a r y m a t r i x . Note a l s o tha t U j j = ( U j j ) ~ * and U ^ (11) = I. We s h a l l pe r fo rm a s e r i e s of m a n i p u l a t i o n s (a r e d u c t i o n to t r i a n g u l a r form) on the matr ix M(a) to show t h a t det M(a) = d e t ( E k V k (a)) where each V k ( a ) i s a p roduc t of U j j ( a ) ' s . In p a r t i c u l a r , V k ( H ) = I. Thus by L i e b ' s i n e q u a l i t y C S i l ] we have | d e t ( E k V k ( a ) ) | 2 < d e t ( E k I V k (a > I) d e t ( E K |V k ( a ) *| ) = d e t ( E k I ) 2 IV.6 FADDEEV-POPOV DETERMINANT - DIAMAGNETIC INEQUALITY 144 = d e t ( E k V k ( £ ) and e q u a t i o n (4.60) f o l l o w s . To b e g i n , m u l t i p l y M(a) on the l e f t by the b l ock d i a g o n a l mat r i x J21 '32 0 0 u n + l ,n The r e s u l t i s M u 2 I + i J21 u 3 2 + i 0 - l " u k , k - i u k + i , k + 1 • u n , n - l u n + l , n + 1 and det M = det <TT Uj i + det M r ( 4 . 6 i : Nex t , r e p l a c e row(k) in Mj by row(k) + r o w ( k - l ) , s t a r t i n g wi th row(2 ) . The r e s u l t i s M 2 IV.6 FADDEEV-POPOV DETERMINANT - DIAMAGNETIC INEQUALITY 145 u 2 1 +i -u 21 J32 -1 o o u « , k - i - 1 0 u n + l , n + 1 and x det M 2 = det Mj. (4.62) We now s t a r t the r e d u c t i o n p r o c e s s p r o p e r . M u l t i p l y M 2 on the r i g h t by 1 + U O 12 o which has the e f f e c t of m u l t i p l y i n g the second column on the r i g h t by 1 + U j 2 » The r e s u l t i s M3 IV.G FADDEEV-POPOV DETERMINANT - DIAMAGNETIC INEQUALITY 146 u 2 1 +i -u+u2 1) u 3 2<i +u 1 2) -1 o 0 u k , k - l - 1 U n + l , n + 1 and < det M 3 = d e t ( l + U 1 2 ) det M 2 > (4.63) To comp le te the f i r s t s tage of the r e d u c t i o n , r e p l a c e column(2) by column(2) + c o l u m n ( 1 ) . The r e s u l t i s M 4 u1 2 +i ! 0 - 4 — - -O 1 ; u32(i+u12)+i -1 O 1 • l 0 ••" u k + l , k -1 1 ; i i • • • 0 V i ,n + l and det M 4 = det M 3 . (4 .64) Let M 2 denote the matr ix c o n s i s t i n g of the 1 ower r i g h t b l o c k of M 4 , as shown. Then det M 4 = d e t ( l + U 1 2 ) det M 2 (4 . 65) which combined with (4.63) and (4.64) y i e l d s IV.G FADDEEV-POPOV DETERMINANT - DIAMAGNETIC INEQUALITY 147 d e t ( l + U 1 2 ) det M 2 = de t (1+U 1 2 > det M 2 . (4.66) We c l a i m t h a t i t i s p e r m i s s i b l e to c a n c e l the f a c t o r det d + U 1 2 ) i n e q . ( 4 . 6 6 ) . For i f det (1+U 1 2 ) = 0, r e p l a c e U 1 2 by a sequence u j 2 ' such tha t u{ 2* •* U j 2 and det U ^ ' # 0. We can c a n c e l the f a c t o r det u{2> f o r each n and take the l i m i t a f t e r to o b t a i n det M 2 = det Mj. (4.67) Note tha t M 2 has the same form as M 2 but has one l e s s row and co lumn. We r e p e a t the r e d u c t i o n (n-1) t imes to o b t a i n a matr ix M 2 n * with o n l y one e n t r y which i s a sum of p r o d u c t s of the and det M j n ) = det M 2- By e q . ( 4 . 6 1 ) we thus have det M(a) = det(Tt Ut i + 1 > det M | n ) i ' * = det (TT U i | i + 1 ) d e t ( E TT U j j ) = det (E Vi, ( a ) ) . 0 V. OPEN PROBLEMS 148 V. Open Prob lems We now g i v e a summary of some of the more impor t an t q u e s t i o n s l e f t u n r e s o l v e d i n t h i s t h e s i s and of the p o s s i b l e ways i n which the r e s u l t s we have g i ven c o u l d be e x t e n d e d . S e v e r a l of t h e s e prob lems and e x t e n s i o n s a re r e l a t e d and a l l a re d i r e c t e d toward the ach ievement of the main g o a l , which i s to g i v e a r i g o r o u s c o n s t r u c t i o n of QCD in two (or more) d imens i on s by p r o v i n g the O s t e r w a l d e r - S c h r a d e r axioms ho ld f o r the cont inuum l i m i t of a gauge v a r i a n t l a t t i c e a p p r o x i m a t i o n . Problem 1 - Unique minimum f o r the l a t t i c e a c t i o n s F i n d a c h o i c e of s u b l a t t i c e and g a u g e - f i x i n g f u n c t i o n F f o r which the gauge v a r i a n t l a t t i c e a c t i o n i s r e a s o n a b l e and has a un ique minimum at a = fl. (Cf . S e c t i o n IV.C) F a i l i n g t h i s , f i n d a way to take the cont inuum l i m i t which does not r e l y on the i n t u i t i o n tha t f i e l d s f a r from 11 are be ing damped out by the a c t i o n . It i s p o s s i b l e t h a t the f a i l u r e to s o l v e t h i s problem i n d i c a t e s t h e r e i s some p h y s i c s go ing on t h a t has not been taken i n t o a c c o u n t . F i n d out what i t i s . Problem 2 - D i amagnet i c i n e q u a l i t y . " Prove the d i a m a g n e t i c i n e q u a l i t y ( S e c t i o n IV.G) f o r the Faddeev -Popov d e t e r m i n a n t in h i g h e r d imens i on s and f o r a V. OPEN PROBLEMS 149 wider c l a s s of boundary c o n d i t i o n s . Problem 3 - Zero gauge d e g r e e : F i n d a way to a v o i d the prob lem of z e r o gauge degree when A i = A. For example, extend Theorem 2.16 ( S e c t i o n II.D) to the n o n a b e l i a n c a s e . Problem 4 - T r u n c a t i o n of the f u n c t i o n a l i n t e g r a l : We were not o p t i m i s t i c about the p r o s p e c t s f o r t r u n c a t i n g the f u n c t i o n a l i n t e g r a l ( S e c t i o n I I I .D ) , but in any case the framework we have deve loped shou ld be adequate to prove or d i s p r o v e the f o l l o w i n g theorem ( c f . e q . ( 3 . 2 8 ) ) : There e x i s t s a subset V of the l a t t i c e gauge f i e l d s such tha t the Faddeev-Popov de te rm inan t i s p o s i t i v e on V and f o r any gauge i n v a r i a n t f u n c t i o n f (J # E t c ) dc) (\ f ( a ) e " S ( a ) da) = [ det M(a) f ( a ) E°F (a) e ~ S ( a ' da. V Prob lem 5 - R e n o r m a l i z a t i o n and gauge i n v a r i a n c e : By u s i n g a l a t t i c e v e r s i o n of the S I a v n o v - T a y l o r i d e n t i t i e s or some o t h e r means, demons t r a te the c a n c e l l a t i o n s between the l a t t i c e v e r s i o n s of the s u p e r f i c i a l l y d i v e r g e n t graphs a l l u d e d to in S e c t i o n IV.B. Problem 6 - Gauge i n v a r i a n c e and p l a q u e t t e v a r i a b l e s : Prove or d i s p r o v e tha t the e q u a t i o n \ f ( a ) da = I f j (U) dU V. OPEN PROBLEMS 150 e x p r e s s i n g the i n t e g r a l of a gauge i n v a r i a n t f u n c t i o n i n terms of an i n t e g r a l over p l a q u e t t e v a r i a b l e s h o l d s i n more than two d i m e n s i o n s . (Cf . Theorem 4 . 5 , S e c t i o n IV.D.) BIBLIOGRAPHY 151 BIBLIOGRAPHY CBEP3 C M . Bender , T. Eguch i , and H. 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