UBC Theses and Dissertations

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UBC Theses and Dissertations

On a 1 + 1 - dimensional interacting soliton-fermion system with supersymmetry Keil, Werner H. 1985

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ON A 1 + 1 - DIMENSIONAL INTERACTING SOLI TON-FERMI ON SYSTEM WITH SUPERSYMMETRY by WERNER H. K E I L D i p l o m - P h y s i k e r , T e c h n i s c h e U n i v e r s i t a t C l a u s t h a l , 1 9 8 3 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES Dep a r t m e n t of P h y s i c s We a c c e p t t h i s t h e s i s a s 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 1 , © Werner H. BRITISH COLUMBIA 1 9 8 5 K e i l , 1985 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Physics Department of The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date 26. 7. 85 i i A b s t r a c t A s u p e r s y m m e t r i c i n t e r a c t i n g s o l i t o n - f e r m i o n s y s t e m i n one s p a c e and one t i m e d i m e n s i o n i s i n v e s t i g a t e d . We c o n s t r u c t t h e s o l i t o n s e c t o r o f t h e quantum t h e o r y u s i n g a g e n e r a l i z a t i o n of t h e "method o f c o l l e c t i v e c o o r d i n a t e s " p r e v i o u s l y d e v e l o p e d f o r p u r e l y b o s o n i c t h e o r i e s . A c a n o n i c a l t r a n s f o r m a t i o n l e a d s t o a s e t of " c o l l e c t i v e " f i e l d v a r i a b l e s w i t h c o n s t r a i n t s and t h e t r a n s f o r m e d t h e o r y i s q u a n t i z e d i n t h e c a n o n i c a l way u s i n g D i r a c ' s method f o r c o n s t r a i n e d s y s t e m s . The H a m i l t o n i a n i s e v a l u a t e d i n c o l l e c t i v e c o o r d i n a t e s and t h e e q u a t i o n s o f m o t i o n a r e s o l v e d t o f i r s t o r d e r i n a p e r t u r b a t i v e e x p a n s i o n . We f i n d t h a t the f i e l d e q u a t i o n s a d m i t z e r o - e n e r g y s o l u t i o n s f o r b o t h t h e boson and t h e f e r m i o n f i e l d . The p r e s e n c e o f t h e s o l i t o n has n o n t r i v i a l c o n s e q u e n c e s f o r t h e supersymmetry o f t h e t h e o r y . The supersymmetry a l g e b r a has t o be m o d i f i e d t o i n c l u d e t o p o l o g i c a l c h a r g e s and as a r e s u l t s upersymmetry i s s p o n t a n e o u s l y b r o k e n . I t f o l l o w s t h a t t h e g r o u n d s t a t e i s d o u b l y d e g e n e r a t e . F i n a l l y , t h e z e r o - e n e r g y s o l u t i o n s a r e f o u n d t o be c o n n e c t e d w i t h t h e s y m m e t r i e s o f t h e t h e o r y b r o k e n by t h e s o l i t o n . The boson zero-mode c o r r e s p o n d s t o s p a t i a l t r a n s l a t i o n s , t h e f e r m i o n zero-mode i s a s s o c i a t e d w i t h t h e s u p ersymmetry. i i i Table of Contents A b s t r a c t i i Acknowledgement i v INTRODUCTION 1 1. QUANTIZATION BY COLLECTIVE-COORDINATE METHODS 6 1.1 Q u a n t i z a t i o n a r o u n d a c l a s s i c a l s t a t i c f i e l d 6 1.2 C o l l e c t i v e c o o r d i n a t e s 10 2. QUANTIZATION OF A 1+1-DIMENSIONAL INTERACTING SOLI TON-FERMI ON SYSTEM 13 2.1 C o l l e c t i v e c o o r d i n a t e s f o r a s o l i t o n - f e r m i o n s y s t e m 15 2.2 C o n s t r u c t i o n of t h e H a m i l t o n i a n 23 2.3 The f i e l d e q u a t i o n s i n g ° - o r d e r 31 2.4 The H i l b e r t s p a c e f o r t h e s o l i t o n s e c t o r 38 3. SUPERSYMMETRY FOR A 1+1-DIMENSIONAL SOLITON-FERMION THEORY 40 3.1 The supersymmetry g e n e r a t o r s f o r t h e s o l i t o n - f e r m i o n t h e o r y 40 3.2 S p o n t a n e o u s supersymmetry b r e a k i n g i n t h e s o l i t o n s e c t o r 47 B i b l i o g r a p h y 53 i v Acknowledgement I would l i k e t o t h ank my r e s e a r c h s u p e r v i s o r , D r . Gordon W. Semenoff, f o r h i s h e l p and g u i d a n c e t h r o u g h o u t t h e c o u r s e of t h i s work. H i s p a t i e n t e x p l a n a t i o n s and numerous s u g g e s t i o n s have c o n t r i b u t e d e s s e n t i a l l y t o t h e p r o g r e s s o f t h i s work. I a l s o g r a t e f u l l y a c k n o w l e d g e f i n a n c i a l a s s i s t a n c e i n form of a U n i v e r s i t y of B r i t i s h C o l u m b i a Summer G r a d u a t e F e l l o w s h i p . 1 INTRODUCTION I t has been known f o r a l o n g t i m e t h a t c e r t a i n n o n l i n e a r c l a s s i c a l f i e l d t h e o r i e s p o s s e s s s o l i t o n s o l u t i o n s , by w h i c h we mean n o n t r i v i a l t i m e i n d e p e n d e n t f i n i t e - e n e r g y s o l u t i o n s t o t h e f i e l d e q u a t i o n s ( [ 1 ] ) . D u r i n g t h e l a s t t e n y e a r s c o n s i d e r a b l e i n t e r e s t has a r i s e n i n q u a n t i z e d f i e l d t h e o r i e s w h i c h a d m i t s o l i t o n - l i k e s o l u t i o n s . R e c e n t s t u d i e s have shown t h a t t h e s e s y s t e m s e x h i b i t many u n u s u a l f e a t u r e s w h i c h a r e c l o s e l y r e l a t e d t o n o n t r i v i a l t o p o l o g i c a l p r o p e r t i e s o f t h e f i e l d s . Examples i n c l u d e t h e f r a c t i o n a l quantum numbers f o r v a r i o u s i n t e r a c t i n g s o l i t o n - f e r m i o n s y s t e m s ( [ 2 ] - [ 6 ] and r e f e r e n c e s t h e r e i n ) and t h e Skyrme s o l i t o n model f o r b a r y o n s ( [ 7 ] - [ 9 ] ) . Among t h e s e t h e o r i e s i n t e r a c t i n g s o l i t o n - f e r m i o n t h e o r i e s i n one s p a c e and one t i m e d i m e n s i o n have a t t r a c t e d much a t t e n t i o n . A l t h o u g h t h i s may seem t o be a t b e s t a t o y model, w i t h o u t any p h y s i c a l s i g n i f i c a n c e , t h e r e a r e good r e a s o n s f o r s t u d y i n g i t . L i k e t h e 1 - d i m e n s i o n a l h a r m o n i c o s c i l l a t o r i n quantum m e c h a n i c s , a 1 + 1 - d i m e n s i o n a l f i e l d t h e o r y p r o v i d e s a n o n t r i v i a l s t r u c t u r e w i t h o u t p r e s e n t i n g t o o many t e c h n i c a l d i f f i c u l t i e s . P a r t i c u l a r l y t h e q u e s t i o n o f r e n o r m a l i z a b i l i t y , a c r u c i a l p o i n t f o r e v e r y f i e l d t h e o r y , i s o f no m a j o r i m p o r t a n c e i n t h e models we w i l l . c o n s i d e r . D e s p i t e t h e s i m p l i f i c a t i o n s c e r t a i n b a s i c f e a t u r e s of t h e t h e o r y a r e e x p e c t e d t o r e m a i n v a l i d f o r more c o m p l i c a t e d s y s t e m s , s u c h as gauge t h e o r i e s , i n h i g h e r d i m e n s i o n s . T h i s seems .to be a r e a s o n a b l e a s s u m p t i o n e s p e c i a l l y f o r p r o p e r t i e s 2 t h a t depend o n l y on t h e g l o b a l s t r u c t u r e of t h e t h e o r y . In a d d i t i o n , t h e s e t h e o r i e s a r e known t o d e s c r i b e a c t u a l p h y s i c a l s y s t e m s . S e v e r a l a u t h o r s ([ 4 ] , [ 1 0 ] - [ 1 4 ]) have shown t h a t t h e d y n a m i c s o f e l e c t r o n s i n a l i n e a r l y c o n j u g a t e d d i a t o m i c p o l y m e r i s g i v e n by a 1 + 1 - d i m e n s i o n a l s o l i t o n - f e r m i o n H a m i l t o n i a n . S u m m a r i z i n g t h i s b r i e f d i s c u s s i o n , t h e u s e f u l n e s s o f 1 + 1 - d i m e n s i o n a l t h e o r i e s a s a " p r o v i n g g r o u n d " f o r new i d e a s and methods, as w e l l as a model f o r c e r t a i n c o n d e n s e d m a t t e r s y s t e m s , i s now w e l l e s t a b l i s h e d . F u r t h e r m o r e , w e can c h o o s e t h e a c t i o n f o r t h e s e s y s t e m s t o be s u p e r s y m m e t r i c . S u p e r s y m m e t r i c f i e l d t h e o r i e s a r e known t o p o s s e s s many u n u s u a l p r o p e r t i e s , s u c h a s b o s o n - f e r m i o n symmetry and c a n c e l l a t i o n of d i v e r g e n c e s t o a t l e a s t f i r s t o r d e r . They a r e now b e l i e v e d t o p l a y a m a j o r r o l e i n p a r t i c l e p h y s i c s . A model t h a t c o m b i n e s t h i s u n u s u a l symmetry w i t h a n o n t r i v i a l f i e l d t o p o l o g y ( s o l i t o n s ) i s c e r t a i n l y o f i n t e r e s t . T h i s i s c o n f i r m e d by s e v e r a l a u t h o r s ( [ 1 5 ] - [ 2 0 ] ) who d i s c u s s how t h e p r e s e n c e of t h e s o l i t o n a l t e r s t h e c o n v e n t i o n a l d e s c r i p t i o n f o r f i e l d s w i t h t r i v i a l t o p o l o g y . One of t h e p r o b l e m s o f t h e o r i e s c o n t a i n i n g s o l i t o n s i s t h e f a c t t h a t s t a n d a r d p e r t u r b a t i o n t h e o r y c a n n o t t a k e t h e f i e l d t o p o l o g y - n o n t r i v i a l b o u n d a r y c o n d i t i o n s on t h e s o l u t i o n s of t h e 3 f i e l d e q u a t i o n s - i n t o a c c o u n t . To o b t a i n i n f o r m a t i o n about t h e s e s y s t e m s s e v e r a l d i f f e r e n t t e c h n i q u e s have been d e v e l o p e d , r a n g i n g f r o m new s e m i c l a s s i c a l a p p r o x i m a t i o n methods t o i n d e x t heorems on open s p a c e s ( [ 2 1 ] ~ [ 2 5 ] and r e f e r e n c e s t h e r e i n ) . J a c k i w , G o l d s t o n e , T o m b o u l i s e t a l . ( [ 2 ] , [ 2 6 ] - [ 2 8 ]) d e v e l o p e d a method f o r a s c a l a r f i e l d t h e o r y t h a t a p p r o a c h e s t h e p r o b l e m i n a s t r a i g h t f o r w a r d f a s h i o n w i t h i n t h e framework of H a m i l t o n i a n f o r m a l i s m and c a n o n i c a l q u a n t i z a t i o n . To i n c l u d e t h e c l a s s i c a l s o l i t o n s o l u t i o n i n t h e quantum t h e o r y i n a c o n s i s t e n t way t h e y t r a n s f o r m e d t h e s y s t e m - v i e w e d as a c l a s s i c a l t h e o r y o f c-number f i e l d s - t o a new s e t of f i e l d v a r i a b l e s w i t h c o n s t r a i n t s . T h i s c o n s t r a i n e d s y s t e m c a n t h e n be q u a n t i z e d v i a t h e c a n o n i c a l method, u s i n g t e c h n i q u e s t h a t D i r a c d e v e l o p e d more th a n t h i r t y y e a r s ago ( [ 2 9 ] , [ 3 0 ] ) . An a l t e r n a t i v e a p p r o a c h t o q u a n t i z e c o n s t r a i n e d s y s t e m s i s t h e Feynman p a t h i n t e g r a l method as d e v e l o p e d by F a d d e e v f o r gauge t h e o r i e s ( [ 3 1 ] ) . T h i s v e r y e l e g a n t method i s f a v o r e d by many a u t h o r s ([23 ] , [ 3 2 ] , [ 3 3 ] ) , but i t r e q u i r e s s p e c i a l c a r e i n t h e p r e s e n t c a s e , due t o o p e r a t o r o r d e r i n g p r o b l e m s i n t h e p a t h i n t e g r a l f o r t i m e d e p e n d e n t c a n o n i c a l t r a n s f o r m a t i o n s ( [ 2 6 ] ) . The c a n o n i c a l t r a n s f o r m a t i o n f o r t h e s o l i t o n s y s t e m w i l l l e a d t o a new s e t o f " c o l l e c t i v e v a r i a b l e s " and " i n t e r n a l d e g r e e s of f r e e d o m " . P h y s i c a l l y t h i s can be v i e w e d a s t h e s o l i t o n b e i n g an e x t e n d e d , c l a s s i c a l p a r t i c l e t o w h i c h quantum c o n t r i b u t i o n s a r e a d d e d . F o r p r a c t i c a l c a l c u l a t i o n s t h e s e "quantum f l u c t u a t i o n s " 4 have t o be expanded i n a s u i t a b l e power s e r i e s . I t s h o u l d be e m p h a s i z e d , however, t h a t t h i s method i n v o l v e s no s e m i c l a s s i c a l a p p r o x i m a t i o n but g i v e s a f u l l quantum t h e o r y f o r a c l a s s i c a l c o n s t r a i n e d s y s t e m . In t h e p r e s e n t p a p e r we w i l l d i s c u s s a 1 + 1 - d i m e n s i o n a l i n t e r a c t i n g s o l i t o n - f e r m i o n s y s t e m w i t h s upersymmetry. S p e c i a l e m p h a s i s w i l l be g i v e n t o two p r o b l e m s : (a) how t o e x t e n d t h e q u a n t i z a t i o n method we have j u s t d e s c r i b e d t o i n c l u d e f e r m i o n s , and (b) how does t h e p r e s e n c e of t h e s o l i t o n a f f e c t t h e s upersymmetry of t h e t h e o r y . We w i l l be p r o c e e d a s f o l l o w s : In t h e f i r s t c h a p t e r we g i v e a b r i e f r e v i e w o f t h e " c o l l e c t i v e c o o r d i n a t e q u a n t i z a t i o n method" by J a c k i w e t a l . f o l l o w i n g r e f e r e n c e [26] and [ 2 8 ] . T h i s i s i l l u s t r a t e d w i t h a s c a l a r cft - l'theory, t h e s i m p l e s t t h e o r y t h a t a d m i t s s o l i t o n s o l u t i o n s . I n t h e s e c o n d c h a p t e r we w i l l g e n e r a l i z e t h i s method t o t h e c a s e o f a 1 + 1 - d i m e n s i o n a l i n t e r a c t i n g s o l i t o n - f e r m i o n s y s t e m . We t r a n s f o r m t h e o l d f i e l d o p e r a t o r s t o a s e t of new v a r i a b l e s w i t h c o n s t r a i n t s and show t h a t t h i s t r a n s f o r m a t i o n i s i n d e e d c a n o n i c a l . S u b s e q u e n t l y we e v a l u a t e t h e energy-momentum t e n s o r i n t h e new c o o r d i n a t e s . The r e s u l t w i l l c o n f i r m t h e i n t e r p r e t a t i o n o f t h e new v a r i a b l e s a s " c o l l e c t i v e c o o r d i n a t e s " . 5 U s i n g t h e H a m i l t o n i a n f o r m a l i s m we c o n s t r u c t and s o l v e t h e f i e l d e q u a t i o n s i n a f i r s t o r d e r a p p r o x i m a t i o n . We f i n d t h a t t h e boson and f e r m i o n o p e r a t o r s b o t h c o n t a i n a zero-mode due t o t h e e x i s t e n c e of t h e c l a s s i c a l s o l i t o n s o l u t i o n . A p r e l i m i n a r y d i s c u s s i o n of t h e H i l b e r t s p a c e s t r u c t u r e f o r a t h e o r y w i t h s o l i t o n s c o n c l u d e s t h e s e c o n d c h a p t e r . Our r e s u l t s a r e q u i t e g e n e r a l a l t h o u g h t h e c a l c u l a t i o n s a r e g i v e n f o r t h e s p e c i a l c a s e o f a s u p e r s y m m e t r i c s y s t e m . The t h i r d a n d l a s t c h a p t e r a d d r e s s e s t h e s u p e r s y m m e t r i c p r o p e r t i e s . The p r e s e n c e o f t h e s o l i t o n c o r r e s p o n d s t o t h e e x i s t e n c e of " t o p o l o g i c a l " o r " c e n t r a l c h a r g e s " w h i c h have t o be i n c l u d e d i n t h e supersymmetry a l g e b r a . T h e s e c e n t r a l c h a r g e s b r e a k t h e N=1 supersymmetry down t o an "N=i" s u p e r s y m m e t r y . As a c o n s e q u e n c e t h e g r o u n d s t a t e o f our t h e o r y w i l l c o n s i s t o f a d e g e n e r a t e d o u b l e t . F i n a l l y we w i l l show n o n - p e r t u r b a t i v e l y t h a t i n a s u p e r s y m m e t r i c t h e o r y t h e c l a s s i c a l s o l i t o n s o l u t i o n a l w a y s i m p l i e s t h e e x i s t e n c e of a f e r m i o n zero-mode v i a a supersymmetry t r a n s f o r m a t i o n , whereas t h e b o s o n i c zero-mode i s a s s o c i a t e d w i t h a s p a t i a l t r a n s f o r m a t i o n . 6 1. QUANTIZATION BY COLLECTIVE-COORDINATE METHODS 1.1 Q u a n t i z a t i o n a r o u n d a c l a s s i c a l s t a t i c f i e l d The method d e s c r i b e d i n t h e f o l l o w i n g s e c t i o n c a n be v i e w e d as a f i e l d t h e o r e t i c a n a l o g t o t h e "Born-Oppenheimer" method i n m o l e c u l a r quantum m e c h a n i c s . The e q u a t i o n s o f m o t i o n a r e t r e a t e d as d i f f e r e n t i a l e q u a t i o n s f o r c l a s s i c a l c-number f i e l d s r a t h e r t h a n quantum o p e r a t o r s . T h e s e c l a s s i c a l s o l u t i o n s c a n be v i e w e d as z e r o t h o r d e r a p p r o x i m a t i o n s t o t h e vacuum e x p e c t a t i o n v a l u e of t h e f u l l quantum o p e r a t o r d e s c r i b i n g a c l a s s i c a l e x t e n d e d o b j e c t . The f u l l quantum t h e o r y i s o b t a i n e d by an o p e r a t o r power s e r i e s e x p a n s i o n a r o u n d t h i s c l a s s i c a l f i e l d . We w i l l i l l u s t r a t e t h i s w i t h a s c a l a r f i e l d t h e o r y ( [26 ] , [ 2 8 ] ) . C o n s i d e r t h e L a g r a n g i a n d e n s i t y f o r a r e a l s c a l a r f i e l d c 6 ( x r t ) i n two d i m e n s i o n s where U(<p) i s a p o l y n o m i a l i n <p. S i n c e we a r e i n t e r e s t e d i n a power s e r i e s e x p a n s i o n we assume t h a t U depends on a c o u p l i n g p a r a m e t e r g l i k e U((p) = ^ 2 U(q<p). The s t a n d a r d example we w i l l use f r o m now on i s t h e w e l l known c ^ - s e l f - i n t e r a c t i o n d ) (2) The f i e l d e q u a t i o n f o r t h i s L a g r a n g i a n i s g i v e n by 7 (3) The s o l u t i o n s t o (3) can be c l a s s i f i e d as c o n s t a n t ( x - and t - i n d e p e n d e n t ) , s t a t i c ( t - i n d e p e n d e n t ) and s p a c e - and t i m e -d e p e n d e n t . F o r us o n l y t h e f i r s t two and e s p e c i a l l y t h e s t a t i c s o l u t i o n s , a r e i m p o r t a n t . A c o n s t a n t s o l u t i o n </>0 t o (3) w i t h p o t e n t i a l (2) i s e a s i l y f o u n d : <p0 = ±jjj. We w i l l r e t u r n t o t h i s s o l u t i o n l a t e r on. F o r t h e s t a t i c c a s e t h e f i e l d e q u a t i o n r e d u c e s t o on w h i c h we impose t h e f o l l o w i n g c o n d i t i o n s ( a s y m p t o t i c n o n - t r i v i a l i t y ) 0 0 Then (3) i s i n t e g r a t e d t o or + + c o r> if. ( 4 ) where t h e p r i m e w i l l d e n o t e from now on d i f f e r e n t i a t i o n w i t h 8 r e s p e c t t o t h e e x p l i c i t a rgument. The e n e r g y r e m a i n s f i n i t e i f t h e i n t e g r a t i o n c o n s t a n t i s s e t t o z e r o , so t h e p h y s i c a l c o n s i s t e n c y i s e n s u r e d ( a t l e a s t on t h e c l a s s i c a l l e v e l ) . T h e s e s t a t i c s o l u t i o n s w i t h f i n i t e e n e r g y a r e u s u a l l y c a l l e d s o l i t o n s . An e x p l i c i t example i s p r o v i d e d by our ^ - i n t e r a c t i o n . The s t a t i c s o l u t i o n s a r e t h e w ell-known " k i n k s " 0 L ( x ) = ±™- t a n h m ( x - x 0 ) (5) w i t h x 0 a r e a l p a r a m e t e r d e n o t i n g t h e c e n t e r o f t h e s o l i t o n . We i n o t e t h a t s i n c e V= U 1 i s of 0 ( g _ 1 ) i n g, o u r c l a s s i c a l s o l u t i o n w i l l be o f 0(g~ 1 ) a s w e l l due t o e q u a t i o n ( 4 ) . The same i s t r u e f o r t h e c o n s t a n t s o l u t i o n s . We w i l l now c o n s t r u c t t h e quantum t h e o r y a r o u n d o u r c l a s s i c a l s o l u t i o n s . F i r s t we c o n s i d e r t h e c o n s t a n t s o l u t i o n <j>0. The f i e l d o p e r a t o r i s expanded a s I ( * . t ) - r \ . * I ( x . t ) w i t h <p0 o f 0 ( g _ 1 ) an d t h e quantum c o n t r i b u t i o n of 0 ( g n ) , n > O . The quantum H i l b e r t s p a c e i s t h e u s u a l F o c k s p a c e {|0>,|k>...|k,...k„> . . .} g e n e r a t e d f r o m a vacuum s t a t e |0> whose e l e m e n t s d e s c r i b e b o s o n i c m u l t i p a r t i c l e s t a t e s w i t h momenta k . E x p e c t a t i o n v a l u e s o f $ c a n be e v a l u a t e d by s t a n d a r d p e r t u r b a t i v e t e c h n i q u e s . T h i s p a r t o f t h e t h e o r y w i l l be c a l l e d t h e "vacuum s e c t o r " . 9 To c o n s t r u c t a t h e o r y f o r t h e s t a t i c s o l u t i o n s we have t o i n c l u d e "quantum s o l i t o n s t a t e s " |p> w i t h momentum p, so t h a t a g e n e r a l s t a t e i n our H i l b e r t s p a c e i s now g i v e n by | p , k k „ > . The s o l i t o n i s assumed t o be s t a b l e : < p , 1 , . . . l m | . , # | k , . . . k B > - 0. J a c k i w e t a l . ( [ 2 8 ] ) have shown t h a t t h e s e a s s u m p t i o n s l e a d t o a c o n s i s t e n t t h e o r y , e s p e c i a l l y t h a t a s y s t e m a t i c e x p a n s i o n o f t h e m a t r i x e l e m e n t s i n powers of g i s p o s s i b l e . We w i l l r e t u r n t o t h e H i l b e r t s p a c e s t r u c t u r e l a t e r on. The p a r t o f t h e t h e o r y t h a t t a k e s t h e e x i s t e n c e of t h e s o l i t o n i n t o a c c o u n t i s c a l l e d t h e " s o l i t o n s e c t o r " . As a l r e a d y m e n t i o n e d , we would l i k e t o expand t h e f i e l d o p e r a t o r i n t h e s o l i t o n s e c t o r as $ = <pti + a n a l o g o u s t o t h e vacuum s e c t o r . T h i s i s however, n o t so s t r a i g h t f o r w a r d s i n c e <pti i s n o t t r a n s l a t i o n a l l y i n v a r i a n t . As our example of t h e <t>l ~ t h e o r y has shown t h e r e e x i s t s a whole s e t o f s o l u t i o n s p a r a m e t r i z e d by x 0 . More g e n e r a l l y , w h i l e t h e t h e o r y (1) i s i n v a r i a n t u nder L o r e n t z b o o s t s , t h e s t a t i c s o l u t i o n 0 t l i s n o t , but t r a n s f o r m s i n t o d i f f e r e n t s o l u t i o n s d e p e n d i n g on t h e b o o s t p a r a m e t e r . Thus we a r e l e f t w i t h two p r o b l e m s a) a r o u n d w h i c h <j>^ s h a l l t h e f i e l d be expanded and b) how t o impose t r a n s l a t i o n a l c o v a r i a n c e on The s o l u t i o n t o t h e s e p r o b l e m s i s g i v e n by a t r a n s f o r m a t i o n t o a new s e t o f f i e l d v a r i a b l e s w h i c h e n a b l e us t o c o n t r o l t h e p a r a m e t e r d e p e n d e n c e of 0 . . 10 1 .2 C o l l e c t i v e c o o r d i n a t e s We s t a r t t h e d i s c u s s i o n a t t h e c l a s s i c a l - c-number f i e l d -l e v e l . C o n s i d e r o u r L a g r a n g i a n d e n s i t y 2 • I ^ $ 2 f ~ U ( $ ) w i t h s t a t i c s o l u t i o n s <t>^. The s t a n d a r d c o n o n i c a l v a r i a b l e s a r e the f i e l d <t> and t h e c o n j u g a t e momentum n = 4> t o g e t h e r w i t h t h e u s u a l P o i s s o n b r a c k e t s I $ (*.t) ; £ Lyt)\ - { T T ( - . t ) . F ( j . - t ) J - 0 We now p e r f o r m a c a n o n i c a l t r a n s f o r m a t i o n t o a s e t o f new v a r i a b l e s { * , n,X,P} by d e f i n i n g I ( « . t ) - (« - X(t)) , £ XID , t ) (6a) where M 0 = J d x ^ ' 2 , £(t) = J d x * ' t f ' In o r d e r t o keep t h e number o f i n d e p e n d e n t v a r i a b l e s c o n s t a n t we impose t h e c o n s t r a i n t s r A i J dx £ <j>d = 0 (6c) J i x TT ^ -. 0 (6d) S i n c e we a r e now d e a l i n g w i t h a c o n s t r a i n e d s y s t e m we have t o m o d i f y o u r P o i s s o n b r a c k e t s a c c o r d i n g t o t h e D i r a c p r o c e d u r e ( [ 2 9 ] , [ 3 0 ] ) . G i v e n a s e t of c o n s t r a i n t s on t h e f i e l d v a r i a b l e s i n t h e f o r m 11 hK{<p,ir) = 0 a=1 i • • • t N we impose t h e s e on our s y s t e m by s u b s t i t u t i n g t h e D i r a c b r a c k e t s f o r t h e c o n v e n t i o n a l P o i s s o n b r a c k e t {f,g} f o r any two f u n c t i o n s f , g . A d o p t i n g t h i s p r o c e d u r e f o r our c a s e we o b t a i n and a l l t h e r e m a i n i n g b r a c k e t s v a n i s h . The d e s c r i b e d p r o c e d u r e d e f i n e s a c a n o n i c a l t r a n s f o r m a t i o n f r o m our s t a n d a r d v a r i a b l e t o a s e t of new " c o l l e c t i v e c o o r d i n a t e s " w i t h c o n s t r a i n t s . We c a n now c o n s t r u c t t h e energy-momentum t e n s o r , t h e L o r e n t z g e n e r a t o r s and t h e e q u a t i o n s o f m o t i o n f o r our new v a r i a b l e s s i m p l y by s u b s t i t u t i n g ( 6 a ) - ( 7 b ) i n t o t h e o l d t h e o r y ( [ 2 6 ] ) . S i n c e t h e c a l c u l a t i o n and r e s u l t s a r e r a t h e r l e n g t h y we w i l l n o t r e p r o d u c e them h e r e . We n o t e however, t h a t the t o t a l momentum of t h e s y s t e m t u r n s out t o be j u s t P. Thus X, i t s c o n j u g a t e v a r i a b l e , can be p h y s i c a l l y i n t e r p r e t e d a s t h e c e n t e r - o f - m a s s c o o r d i n a t e o f t h e s o l i t o n . F u r t h e r m o r e we t a k e {X,P} = 1 (7b) The q u a n t i z a t i o n o f o u r t h e o r y i n t h e new c o l l e c t i v e v a r i a b l e s i s s t r a i g h t f o r w a r d . We s i m p l y f o l l o w t h e c a n o n i c a l 12 q u a n t i z a t i o n scheme: t h e f i e l d v a r i a b l e s a r e now o p e r a t o r s i n a H i l b e r t s p a c e , t h e b r a c k e t s c o r r e s p o n d t o c o m m u t a t o r s . As we can see from ( 6 a-f) t h i s g i v e s p r e c i s e l y our d e s i r e d r e s u l t . The f i e l d o p e r a t o r s a r e g i v e n by t h e s t a t i c c-number p a r t #cl p l u s an a d d i t i o n a l quantum p a r t The t r a n s l a t i o n a l n o n i n v a r i a n c e has been t a k e n c a r e of by t h e i n t o d u c t i o n o f a " p o s i t i o n o p e r a t o r " X. To m a i n t a i n c o n s i s t e n c y we had t o i n t r o d u c e c o n s t r a i n t s and change t h e c a n o n i c a l momenta and t h e commutation r e l a t i o n s i n a n o n t r i v i a l way. We w i l l o mit a d e t a i l e d d i s c u s s i o n of t h e p u r e l y b o s o n i c t h e o r y s i n c e we a r e g o i n g t o e x t e n d t h e s e methods t o t h e more g e n e r a l c a s e of an i n t e r a c t i n g f e r m i o n - b o s o n t h e o r y . 13 2. QUANTIZATION OF A 1+1-DIMENSIONAL INTERACTING SOLITON-FERMION SYSTEM. We w i l l c o n s i d e r t h e 1 + 1 - d i m e n s i o n a l f i e l d t h e o r y d e s c r i b e d by t h e s u p e r s y m m e t r i c L a g r a n g i a n d e n s i t y w i t h <(> a r e a l s c a l a r f i e l d , \p a M a j o r a n a 2 - s p i n o r (a complex 2 - s p i n o r s u b j e c t t o t h e M a j o r a n a c o n d i t i o n <// = C i / / T , £ t h e c h a r g e c o n j u g a t i o n m a t r i x and T d e n o t i n g m a t r i x t r a n s p o s i t i o n ) and V(</>) t h e boson i n t e r a c t i o n t e r m d i s c u s s e d i n t h e p r e v i o u s c h a p t e r . . We n o t e t h a t t h e c h o i c e of a M a j o r a n a f e r m i o n p r o v i d e s t h e supersymmetry o f t h e L a g r a n g i a n , we a r e i n t e r e s t e d i n . I t d o e s not however, a f f e c t t h e g e n e r a l i t y o f our q u a n t i z a t i o n method, but imposes o n l y t e c h n i c a l r e s t r i c t i o n s i n t h e s e n s e t h a t o u r a n t i c o m m u t a t o r b r a c k e t s have t o be c o m p a t i b l e w i t h t h e M a j o r a n a c o n d i t i o n . I t i s s t r a i g h t f o r w a r d t o r e p e a t t h e f o l l o w i n g . c o n s t r u c t i o n f o r D i r a c f e r m i o n s w i t h s t a n d a r d a n t i c o m m u t a t o r s . The r e s u l t s w i l l r e m a i n unchanged and some o f them a r e f a m i l i a r i n t h e c o n t e x t o f c h a r g e f r a c t i o n i z a t i o n i n p o l y m e r s . The M a j o r a n a c o n d i t i o n m e r e l y removes t h e c h a r g e d e g r e e of f r e e d o m i n D i r a c f e r m i o n s and l e a v e s us w i t h . t h e s i m p l e s t p o s s i b l e s p i n o r . We w i l l now examine how t o g e n e r a l i z e o u r r e s u l t s f r o m t h e p r e v i o u s c h a p t e r t o t h e L a g r a n g i a n ( 8 ) . F i r s t we n o t e t h a t t h e c l a s s i c a l p a r t 0 ( g ~ 1 ) o f t h e t h e o r y r e m a i n s unchanged i f we t a k e 1 4 t h e i n t e r a c t i o n t e r m V" (</>) = V(<p) p r o p o r t i o n a l t o q<p so t h a t V i s a t l e a s t o f 0 ( g ° ) . Our s t a n d a r d </>'-model, f o r example f u l f i l l s t h i s c o n d i t i o n . The L a g r a n g i a n (8) l e a d s t o t h e o p e r a t o r f i e l d e q u a t i o n s £ X V - V ( * ) Y • 0 . In t h e l o w e s t o r d e r , 0 ( g " 1 ) , we can s e t t h e c l a s s i c a l f e r m i o n f i e l d - v i e w e d as a Grassmann s p i n o r f i e l d - t o be z e r o . The D i r a c e q u a t i o n c o u p l e s t h e f e r m i o n s t o t h e <9(g°)-term V ' ( 0 t ) so t h a t \p has t o be a t l e a s t o f 0 ( g ° ) . F u r t h e r m o r e t h e i n t e r a c t i o n between bosons and f e r m i o n s i s seen t o be a t l e a s t 0 ( g ) . Thus t h e c l a s s i c a l e q u a t i o n s of m o t i o n i n 0(g~ 1) r e m a i n unchanged. On t h e f i r s t quantum l e v e l 0(q°) t h e f e r m i o n s a r e c o u p l e d t o t h e s t a t i c c-number p o t e n t i a l V ' ( 0 t L ) . I n t e r a c t i o n s between quantum bosons and f e r m i o n s w i l l o c c u r i n h i g h e r o r d e r s . In t h e vacuum s e c t o r t h e 0(g°)-approximation i s t r i v i a l s i n c e i t c o r r e s p o n d s t o a f r e e f i e l d t h e o r y . In t h e s o l i t o n s e c t o r , however, n o n t r i v i a l e f f e c t s w i l l o c c u r due t o o u r s o l i t o n b a c k g r o u n d p o t e n t i a l V'(<£ t V). I t s u f f i c e s i n g e n e r a l t o c o n s i d e r t h e quantum o p e r a t o r s o n l y up t o 0 ( g ° ) . We can now i n t r o d u c e t h e c o l l e c t i v e p o s i t i o n c o o r d i n a t e X and s h i f t t h e boson o p e r a t o r by t h e c l a s s i c a l s o l u t i o n # . 15 I t s h o u l d be m e n t i o n e d a t t h i s p o i n t t h a t t h e L a g r a n g i a n (8) i s known t o p o s s e s s f e r m i o n zero-modes ( [ 2 ] , [ 2 0 ] ) w h i c h can be i n t e r p r e t e d as t h e "supersymmetry p a r t n e r " o f t h e s o l i t o n <pti. I t i s p o s s i b l e t o d e f i n e c o l l e c t i v e c o o r d i n a t e s a n d c a n o n i c a l t r a n s f o r m a t i o n s b a s e d on t h e s e f e r m i o n modes a n a l o g o u s l y t o t h e boson p a r t of t h e t h e o r y . We w i l l r e t u r n t o t h i s p o s s i b i l i t y when we d i s c u s s t h e supersymmetry p r o p e r t i e s of o u r t h e o r y . 2.1 C o l l e c t i v e c o o r d i n a t e s f o r a s o l i t o n - f e r m i o n s y s t e m . As we d i s c u s s e d b e f o r e , t h e L a g r a n g i a n (8) a d m i t s a s e t o f s t a t i c , f i n i t e e n e r g y s o l u t i o n s {4>(L ( x - x 0 ) | x 0 e R} t o t h e c l a s s i c a l e q u a t i o n o f m o t i o n w i t h <L (* • * • ) • , ^ (* • t 00 ) • 0 so t h a t t h e e q u a t i o n of m o t i o n i s i n t e g r a t e d t o (we have a r b i t r a r i l y f i x e d our s i g n f o r V ) . The quantum t h e o r y d e s c r i b e d by t h e L a g r a n g i a n (8) h a s t h e s t a n d a r d c a n o n i c a l v a r i a b l e s w i t h t h e f o l l o w i n g e q u a l - t i m e commutation r e l a t i o n s i [ T («), £ (j)] • S ( » - j ) ( 9 a ) 1 6 • i § c o , n ^ l • [ IT co , * ( 3 ) ] • [ iW , V C j U • [ i r w . * f t y 1 • 0 ( 9 b ) T h e a n t i c o m m u t a t o r s f o r f e r m i o n s a r i s e f r o m t h e f a c t t h a t V>'s a r e M a j o r a n a f e r m i o n s , t h a t i s , s u b j e c t t o t h e c o n s t r a i n t $ = t \pJ. T h u s we a r e d e a l i n g w i t h a c o n s t r a i n e d s y s t e m a n d we w i l l i m p o s e t h e c o n s t r a i n t s o n o u r t h e o r y b y t h e D i r a c p r o c e d u r e , a l r e a d y m e n t i o n e d i n C h a p t e r 1 . T h e c h a r g e c o n j u g a t i o n m a t r i x C, d e t e r m i n e d b y t h e c o n d i t i o n l f z'{ » - r i s g i v e n b y T h e M a j o r a n a c o n d i t i o n r e a d s w h i c h g i v e s f o r t h e c o m p o n e n t s t , . <0 - 0 • f . 17 On t h e c l a s s i c a l l e v e l we have t o t a k e D i r a c b r a c k e t s i n s t e a d of t h e u s u a l P o i s s o n b r a c k e t s {,} ( [ 2 9 ] , [ 3 0 ] ) : - { * ( « ) , < ( • > } h ( » ) . f ( i " ) j " ( H O . ^ j ) } where so t h a t our new b r a c k e t s a r e g i v e n by - U x - j ) and a n a l o g o u s l y { ^ . n ^ l , . - sc-j) The t h e o r y i s q u a n t i z e d by p o s t u l a t i n g t h e c o r r e s p o n d e n c e between D i r a c b r a c k e t s a nd a n t i c o m m u t a t o r s w h i c h e n s u r e s t h e c o m p a t i b i l i t y o f a n t i c o m m u t a t o r s and c o n s t r a i n t s . From t h i s s e t of c a n o n i c a l v a r i a b l e s we t r a n s f o r m t o a new s e t o f " c o l l e c t i v e " and " i n t e r n a l " v a r i a b l e s {$,n,#,#^,X,P} by 18 d e f i n i n g £ ( x . t ) - ^ t ( x . x(t)) • i ( x - / ( t ) , t ; ( 1 0a ) T T ( y . t ) = T T ( x - X ( t ) , t j ( t ) , f ^ (X-X(t)) ( 10b ) w i t h { H O . * f t y j • S('-j) ( 1 0 c ) a n d t h e e q u a l - t i m e c o m m u t a t i o n r e l a t i o n s a r e [ T ( o , t ( 3 ) j . .[ i w . | ( , ) ] • [ I ( « k n 3 ) ] • [ I w . ^ f ( j ) j (1 1a) ( 1 1 c ) (1 1d) 19 i [ ? ( 0 , X ( t ) j . 1 f o r *,n,it$* . The c o n s t r a i n t s a r e t a k e n t o be J §(„,!) « ^(,)TT(*.t) • 0 (12) as b e f o r e . A E x p r e s s i o n s o f t h e f o r m <t> ( x - X ( t ) , t ) a r e a l w a y s u n d e r s t o o d as a f o r m a l power s e r i e s e x p a n s i o n i n X . The c o n s t r a i n t (12) g u a r a n t e e s t h a t t h e number o f i n d e p e n d e n t v a r i a b l e s i s c o n s e r v e d and w i l l be u s e d l a t e r t o e l i m i n a t e u n p h y s i c a l z e r o - modes. I t r e m a i n s t o show t h a t t h e t r a n s f o r m a t i o n d e f i n e d by (10) and (12) i s c o m p a t i b l e w i t h our s t a n d a r d c a n o n i c a l f o r m a l i s m , t h a t i s , l e a v e s t h e o r i g i n a l commutators u n c h a n g e d . The v e r i f i c a t i o n i s s t r a i g h t f o r w a r d b u t somewhat l e n g t h y . To s i m p l i f y our n o t a t i o n we w i l l d r o p t h e arguments i n t h e o p e r a t o r s whenever t h e meaning i s o b v i o u s ; an i n t e g r a l s i g n d e n o t e s s p a t i a l i n t e g r a t i o n /dx. We w i l l g i v e t h e e x p l i c i t c a l c u l a t i o n o n l y f o r t h e two l e s s o b v i o u s c a s e s [ H 0 . 1 U } ) ] a n d [ TT(0, £ (^)J 20 [ £M),{ |4f, PJ j I'i} L ' 1 • T f A t « 1 V T7 U s i n g our r e l a t i o n s (11 a - f ) we o b t a i n X = 0 Aini) ( [ ' £ ( , . * ) , ? J U s i n g t h e f o r m a l power s e r i e s f o r \}/ t h e commutator c a n be e v a l u a t e d n > o n ' n > o CD * 5 (-0 f 21 H E z 1 A • J 0 IT R e w r i t i n g t h e f e r m i o n commutator w i t h t h e h e l p of t h e M a j o r a n a c o n d i t i o n g i v e s f t y | f (3), f (*-*)] 07 1 / 6t LJf- . ( . P , A A , jT [ , T T (V. t r 0 22 * [ TT ( x - X ) , J ( j - ' x ) ] (x-X) A g a i n u s i n g t h e power s e r i e s f o r o u r commutator we o b t a i n A 15 P. K C-x) J • ; ^ („.x) [ ?. I (.-x) ] • i i'(,.x) F u r t h e r m o r e t h e commutator , $(«j-X')J r e d u c e s t o I'd) [ f ( 0 , I c , - x ) ] - i ' ( 0 • c-o ( U y x - 0 - i o i ' o o f ' t ( r j f ) w h i c h s i m p l i f i e s t h e t e r m s t o 1 ' 4 ( y - X ) 1 I z 4* (* - x) 23 The c a l c u l a t i o n s f o r t h e o t h e r commutators a r e s t r a i g h t f o r w a r d and need not be r e p r o d u c e d h e r e . Thus we have a c o n s i s t e n t c a n o n i c a l t r a n s f o r m a t i o n t o . a s e t of quantum c o l l e c t i v e v a r i a b l e s . We c a n now p r o c e e d t o c o n s t r u c t t h e H a m i l t o n i a n and t h e e q u a t i o n s of m o t i o n . 2.2 C o n s t r u c t i o n of t h e H a m i l t o n i a n The energy-momentum t e n s o r d e n s i t y , T 0, , v = 0,1 f o r our L a g r a n g i a n (8) i s g i v e n by where 3 0 = f t ' 3 1 = f x * We o b t a i n (a) f o r t h e e n e r g y d e n s i t y 24 (b) f o r t h e momentum d e n s i t y where T 0 I has been s y m m e t r i z e d i n $ _ cf We w r i t e (a) and (b) as T0I • : ( TT • $ ' T T + ; f f' ) (14) To c o n s t r u c t t h e H a m i l t o n i a n H = Jdx T oo i n our new v a r i a b l e s we s i m p l y s u b s t i t u t e (10) i n t o ( 1 3 ) . The most d i f f i c u l t p a r t i s t h e e v a l u a t i o n of ^ J j x T T A s i n c e t h e t r a n s f o r m a t i o n ( 1 0 b ) . f o r n i s h i g h l y n o n l i n e a r . To r e d u c e t h e b u l k o f o u r c a l c u l a t i o n s we w i l l a g a i n d r o p a r g u m e n t s and t h e i n t e g r a t i o n measure and d e f i n e i r ( « . t ) • i r ( » - x ( t ) , t ) - i ( ^ ( - / w ) ( ? « . ( r ; . ;*•*•) A , W i t h t h i s n o t a t i o n n 2 g i v e n by IT - -if' - i ( i ^ j + ^ a; T • T a ^ + c, ^ f ) 25 IT X - IE U s i n g t h e b a s i c c ommutation r e l a t i o n > . ^ h r [M* '<•«•*>] * i r i (15) we r e w r i t e II2 i n t e r m s of commutators and a n t i c o m m u t a t o r s : ft, • CA. i - - i ( T <j>t' ( £ + 0 • ( - - x { IT L, £ i - : [ The e x p r e s s i o n f o r I i s more c o m p l i c a t e d : 1 , 1 CI TT 1 A. CX a a <£a + cx, <f>ci 4 i a = ^ L <x a * r\, a a, • ^cl x J • ^ 6i r + 4 CD n ^ so t h a t ( <f>tl* Zl • ce' ^ * + a a + aa" + J c ~ r~ • * _ A0<1 a - <- a hi hi 26 + 2 * X T A Z ^ + A T ^ and we have t h e f o l l o w i n g p r e l i m i n a r y e x p r e s s i o n f o r n 2 : The f i r s t f o u r t e r m s a r e i n t h e i r f i n a l f o r m . The n e x t two terms c a n be e v a l u a t e d u s i n g ( 1 5 ) : - f rk ' 4 ~ 1 I • r ^ " ' ^ ' 4 " -<p„ a- = - o co • - A ou A L A ' J AT A • I *. a. A T a and f u r t h e r m o r e , A * T L 1 " A T + X (AT)* 1 ' 1 * ' AT (AT)* W 4. ' CL 4. AT O a " a. A* I (AT) 4 ^* 27 T o g e t h e r w i t h t h e n e x t two t e r m s i n our H 2 - e x p r e s s i o n we have a, To c o m p l e t e o u r c o m p u t a t i o n o f n 2 i t r e m a i n s t o e v a l u a t e h b r . * J a n d l i ^ ' * ] ' w h e r e 0/ ( ?• M * f - z f ' f ))TT so t h a t t h e commutators a r e r e d u c e d t o - X 7 U s i n g t h e power s e r i e s e x p a n s i o n I! i r ^0 • I ±^2 t h e commutator becomes 0 rt . t> and s i n c e 28 we have A * S r i IT L A T ' Ii 00 A n M. hr: n -1 ( A T J J 1 , « C L Summing up a l l t h e s e terms g i v e s t h e f i n a l r e s u l t f o r I I 2 : t -( A T r . (*-X(i)j * (A* O 1 , 4 The p u r p o s e o f t h i s r a t h e r l e n g t h y c a l c u l a t o n was t o t r a n s f o r m n 2 i n t o a f o r m t h a t p e r m i t s i n t e g r a t i o n o v e r x. E a c h f a c t o r depends o n l y on x-X and commutes w i t h X. T h u s , we can t r a n s l a t e our i n t e g r a t i o n v a r i a b l e x t o x + X and use our c o n s t r a i n t s ( 1 2 ) : dx T dx TT dx 7T <jf> a + a. 29 I • A I M * ,l I A £ / , i / / 4 C W ) * where we have u s e d t h e f a c t t h a t t h e c l a s s i c a l s o l i t o n mass M 0 i s The r e m a i n i n g terms a r e e a s i l y e v a l u a t e d Thus t h e t o t a l H a m i l t o n i a n i s g i v e n by H Z * ( v c ^ t i ) # y £ - *• fy f )) d e ) 4 4 ( M a ) S i n c e we want p e r t u r b a t i v e e x p a n s i o n s i n g f o r our q u a n t i t i e s l e t us c o n s i d e r t h e o r d e r s o f g i n v o l v e d . We want o n l y o r d e r s up t o 0 ( g ° ) f o r our o p e r a t o r s , t h a t i s , we have $,\p,n,\p , P , X ^ g 0 , and ^ - g " 1 as b e f o r e . Thus we have M 0~g~ 2, S/M 0~g hence ( M 0 + £ ) - 1 ~ g", n>2. 30 F u r t h e r m o r e we c a n expand V 2 and V a r o u n d . Then o u r H a m i l t o n i a n H decomposes a s f o l l o w s H = M 0 + H 0 + H x where (17) w i t h H 0 d e s c r i b e s e x a c t l y t h e s i t u a t i o n we d i s c u s s e d a t t h e b e g i n n i n g o f t h i s c h a p t e r : a boson s y s t e m w i t h " p o s i t i o n d e p e n d e n t mass" V 2 " ( ^ ) and a f e r m i o n s y s t e m i n a s t a t i c b a c k g r o u n d p o t e n t i a l . T h i s c o n f i r m s our e a r l i e r s t a t e m e n t t h a t a t t h e 0 ( g ° ) l e v e l t h e s o l i t o n s e c t o r o f o u r t h e o r y i s a l r e a d y n o n t r i v i a l . F o r t h e vacuum s e c t o r <p0 = c o n s t . , on t h e o t h e r hand, (17) d e s c r i b e s j u s t a f r e e s o l i t o n and boson f i e l d . H x d e s c r i b e s i n t e r a c t i o n s i n h i g h e r o r d e r i n g. I t s e f f e c t s c a n be i n c l u d e d u s i n g s t a n d a r d p e r t u r b a t i o n t h e o r y w i t h t h e u n p e r t u r b e d H a m i l t o n i a n H 0 . The momentum o p e r a t o r P =Jdx T 0 1 i s e a s i l y e v a l u a t e d . S u b s t i t u t i n g (10) i n t o (14) and p r o c e e d i n g a s b e f o r e we o b t a i n P = P (18) so t h a t t h e t o t a l momentum o f t h e s y s t e m i s g i v e n by t h e 31 c o l l e c t i v e momentum P, i n agreement w i t h t h e p u r e boson c a s e . We n o t e t h a t o u r H a m i l t o n i a n (16) i s i n d e p e n d e n t o f X, hence commutes w i t h P. We can d i a g o n a l i z e H and P s i m u l t a n e o u s l y and t a k e P t o be a c-number. 2.3 The f i e l d e q u a t i o n s i n q ° - o r d e r G i v e n t h e H a m i l t o n i a n H 0 t h e e v o l u t i o n e q u a t i o n f o r any f u n c t i o n f of t h e f i e l d v a r i a b l e s i s g i v e n by t h e c a n o n i c a l H a m i l t o n i a n f o r m a l i s m : In p a r t i c u l a r we o b t a i n (1 ) A A. (19) r A IT (2) A and + A 32 or (20) (3) A L H.. A. I A | j A . A Using the representation 7 ° = a 2 , 7*«/aJ f 7 5=-a,, for the Dirac matrices we can reduce the (pseudo) scalars to A . ^ A A t t 2 - A t S - ' Using the anticommutation. relations (8) we obtain A , A A A i A A V y* v . ^ ] r " -f - A | A , which y i e l d s , for the fermionic equation of motion i - - v ( f c ) i , i 1 l 33 w h i c h i s w r i t t e n i n m a t r i x f o r m as X • V'(*u) u « 0 (21a) o r ( ^ t i - i f (o x • v'(<U)) V • o (21b) S i n c e our H a m i l t o n i a n H 0 d o e s n o t i n v o l v e P o r X, t h e i r e q u a t i o n s of m o t i o n a r e t r i v i a l (4) X = P = 0 (22) The e q u a t i o n s (20) and (21) a r e f a m i l i a r i n th e c o n t e x t o f c h a r g e f r a c t i o n a l i z a t i o n i n 1 + 1-dimensional s y s t e m s ( [ 3 ] , [ 3 4 ] ) . T h i s shows t h a t o u r method i s i n d e e d q u i t e g e n e r a l and not r e s t r i c t e d t o M a j o r a n a f e r m i o n s . We w i l l c o m p l e t e t h e d i s c u s s i o n o f o u r quantum s y s t e m by c o n s t r u c t i n g an e x p l i c i t s o l u t i o n t o (20) and (21) . U s i n g t h e " a n s a t z " A A 'tut $ (x.-t) * </> CO t v ( * . t ) . cKOt t o s e p a r a t e t h e t i m e dependence we o b t a i n (" • 1 V*"(0 ) fo>) ' » h > ) <23) • « r"W (24) We n o t e t h a t t h e f e r m i o n i c s o l u t i o n s c a n be e x p r e s s e d i n terms of 34 t h e b o s o n i c one: ( 2 1 ) 0 -3* v'UJJ ^  • « £ I f »e d e f i n e ^ , i ^ . y ' ( < ) J ) t h e f i r s t e q u a t i o n t u r n s i n t o t h e b o s o n e q u a t i o n f o r $ 2 i (- V * v"ifc) • v'( i • « £ s o t h a t f 00 - $ (25) p r o v i d e s a s o l u t i o n f o r ( 2 1 ) . Thus i t s u f f i c e s t o s o l v e t h e bo s o n e q u a t i o n ( 2 0 ) . The f i e l d e q u a t i o n s a r e now l i n e a r , h e n c e we c a n e x p a n d t h e f i e l d o p e r a t o r s i n t e r m s o f c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s . We o b t a i n f o r t h e bo s o n o p e r a t o r * ( - * ) - L L ( r , , w r % . r w - 4 ; ; (26) w i t h £ n a s o l u t i o n t o e q u a t i o n (20) a n d w k « f k 4 • 1 • T n e s u m £ r u n s o v e r b o t h t h e d i s c r e t e and t h e . c o n t i n u o u s p a r t o f t h e s p e c t r u m . The ( n o r m a l i z e d ) z e r o - e n e r g y s o l u t i o n ft. * (27) 35 i s e x c l u d e d f r o m $ b e c a u s e of t h e o r t h o g o n a l i t y r e l a t i o n ( 1 2 ) , hence we have t h e c o m p l e t e n e s s r e l a t i o n f o r p r o p e r l y n o r m a l i z e d £ k . The e x p a n s i o n f o r t h e f e r m i o n o p e r a t o r r e q u i r e s somewhat more c a r e . E q u a t i o n (24) a d m i t s t h e z e r o - e n e r g y s o l u t i o n t W " H "I* ( " f V(*J ) (o) (28a) w i t h N a n o r m a l i z a t i o n c o n s t a n t . U s i n g e q u a t i o n (4) -0 t L ' = V ( 0 c l ) - and t h e change o f v a r i a b l e s d0 c L = 0 t L'dx t h i s c a n i n t e g r a t e d t o (28b) To e l i m i n a t e t h e zero-mode we p r o c e e d a n a l o g o u s l y t o t h e boson c a s e . F i r s t we r e p l a c e t r a n s f o r m a t i o n e q u a t i o n (10c) by f(*.t) . X (x-X(M) a + ? ( « - * W . O ( 2 9 ) w i t h c o n s t r a i n t \ i* X f f • o and new a n t i c o m m u t a t i o n r e l a t i o n s 3 6 [ f.w. v^ i • K - i.^co<t,)) <30b> ( 3 0 c ) i M • { • { ^ . M • • o ( 3 0 d ) I t i s s t r a i g h t f o r w a r d t o v e r i f y t h a t t h i s t r a n s f o r m a t i o n i s c a n o n i c a l , t h a t i s , i t p r e s e r v e s t h e M a j o r a n a a n t i c o m m u t a t o r s f o r \p and i//t a i s a H i l b e r t s p a c e o p e r a t o r a s s o c i a t e d w i t h t h e zero-mode, i// i s now o r t h o g o n a l t o \j/0 and c a n be expanded i n terms of s o l u t i o n s o f (21) w i t h f e r m i o n c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s \ , ^ (31) k w i t h a s o l u t i o n t o (24) g i v e n by (25) and <»>„ •-/ k * * ' . The commutation r e l a t i o n s f o r f . j a r e f i x e d by ( 3 0 ) . F o r o u r s t a n d a r d ^"-model t h e f i e l d e q u a t i o n s c an be s o l v e d a n a l y t i c a l l y . U s i n g t h e c l a s s i c a l s o l u t i o n ( 5 ) , <f> = m-tanh mx ( f o r s i m p l i c i t y we have t a k e n x o=0,g=1) t h e s o l u t i o n o f (23) has been e v a l u a t e d ( [ 1 3 ] ) ! ( * ) • - : — * x - ) i k ' , I c * T R 37 and e q u a t i o n (24) y i e l d s t h e c o r r e s p o n d i n g s p i n o r s o l u t i o n s E q u a t i o n (26) and (31) g i v e t h e f u l l boson and f e r m i o n o p e r a t o r s i n 0 ( g ° ) f o r our L a g r a n g i a n (8) w i t h p o t e n t i a l H a v i n g t h u s c o m p l e t e d t h e c o n s t r u c t i o n of t h e f i e l d o p e r a t o r a comment on t h e z e r o-modes i n o u r t h e o r y seems t o be i n o r d e r . We have seen t h a t , i n f i r s t o r d e r p e r t u r b a t i o n t h e o r y , a b o s o n i c and a f e r m i o n i c zero-mode a p p e a r s as a d i r e c t c o n s e q u e n c e o f t h e e x i s t e n c e of t h e s o l i t o n . The b o s o n i c " t r a n s l a t i o n mode" can be v i e w e d a s t h e i n f i n i t e s i m a l l y t r a n s l a t e d s o l i t o n . L a t e r we w i l l show t h a t i n a s u p e r s y m m e t r i c t h e o r y t h e f e r m i o n i c zero-mode can be i n t e r p r e t e d as g e n e r a t e d by an i n f i n i t e s i m a l supersymmetry t r a n s f o r m a t i o n . To c o m p l e t e t h e d i s c u s s i o n of t h e s o l i t o n s e c t o r o f our t h e o r y we have t o f i n d t h e H i l b e r t s p a ce f o r o u r o p e r a t o r s . 38 2.4 The H i l b e r t s p a c e f o r t h e s o l i t o n s e c t o r The H i l b e r t s p a c e f o r t h e vacuum s e c t o r i s t h e s t a n d a r d one f o r a f r e e f i e l d t h e o r y . I t c o n s i s t s of t h e Fo c k s p a c e {|k,...k n ;q,...q m>} f o r m u l t i p a r t i c l e s t a t e s w i t h momenta k; and q. g e n e r a t e d from a vacuum s t a t e | 0 t > » | 0 ^ > by t h e b o s o n - and f e r m i o n o p e r a t o r s b ( k ) , f ( q ) . In t h e s o l i t o n s e c t o r we have t o e n l a r g e t h i s p r o d u c t s p a c e by a s e t of e x t r a s t a t e s |P> t o t a k e t h e s o l i t o n i n t o a c c o u n t ( [ 2 8 ] ) . These s t a t e s form t h e quantum m e c h a n i c a l r e p r e s e n t a t i o n s p a c e f o r t h e c e n t e r - o f - m a s s o p e r a t o r s P,X. They a r e r e q u i r e d t o be s u m u l t a n e o u s l y e n e r g y and momentum e i g e n s t a t e s (32) T h i s i s a l w a y s p o s s i b l e s i n c e o u r H a m i l t o n i a n (16) i s i n d e p e n d e n t of X. We n o t e f u r t h e r m o r e t h a t i n f i r s t o r d e r 0 ( g ° ) t h e s o l i t o n r e m a i n s s t a t i c (X=0), so we s e t P 4=0, E(0)=M o f o r t h e g r o u n d s t a t e . Thus t h e H i l b e r t s p a c e f o r t h e s o l i t o n s e c t o r i s g i v e n by th e p r o d u c t s p a c e { | P 4 ; k , ... k r;q, . . .q w>] w i t h vacuum s t a t e |0 4 > » | 0 k > » | 0 4 >. The p r o b l e m o f p o s s i b l e d e g e n a r a c i e s o f t h e g r o u n d s t a t e i s u s u a l l y a s s o c i a t e d w i t h t h e e x i s t e n c e of t h e f e r m i o n i c zero-mode. As d i s c u s s e d i n t h e p r e v i o u s c h a p t e r t h e f i r s t - o r d e r f i e l d 39 equation (24) admits a zero-energy solution so that the fermion A operator \p has to be expanded as V(*.t) - t OO £ » ? ( » . t ) . The zero-mode operator a does not have an interpretation as a creation operator but is only required to provide a representation of the algebra In a theory with Dirac fermions these relations are substituted by the standard anticommutator and the zero-mode operator w i l l commute with the Hamiltonian. The representation of a and a* is then given by a doubly degenerate ground state |0+> ([3]) with oJ I 0 +> - 0 a, 1 0. > * O | 0. > •- I 0 +> a l 0 + > -- 1 0 . > This is not v a l i d for the present case. The Hamiltonian (16) A ^ A contains b i l i n e a r s ^ V and the resulting crossterms w i l l not vanish under commutation with a. The conventional view that the existence of a zero-energy mode signals degeneracy does not hold and no d e f i n i t e statement can be made at t h i s point. The solution to this problem i s given by the supersymmetry of the theory. The supersymmetry operators w i l l turn out to possess just the right properties to provide the expected degeneracies for the sol i t o n states. We w i l l return to these questions at the end of the following chapter. 40 3. SUPERSYMMETRY FOR A 1+1-DIMENSIONAL SOLITON-FERMION THEORY One o f t h e most r e m a r k a b l e f e a t u r e s of our t h e o r y i s the f a c t t h a t i t p o s s e s s e s supersymmetry, i . e . t h e r e e x i s t c o n s e r v e d c u r r e n t s whose c o r r e s p o n d i n g c h a r g e s f o r m the g e n e r a t o r s of a s u persymmetry a l g e b r a . S u p e r s y m m e t r i c quantum f i e l d t h e o r i e s a r e known t o p o s s e s s u n u s u a l p r o p e r t i e s and t h i s t h e o r y i s no e x c e p t i o n . D u r i n g t h e l a s t few y e a r s s e v e r a l p a p e r s have been p u b l i s h e d ( [ 1 5 ] - [ 2 0 ] ) c o n c e r n i n g t h e quantum c o r r e c t i o n s t o t h e s o l i t o n mass and t h e s a t u r a t i o n of t h e Bogomolny bound f o r t h e e n e r g y . I t has been shown, a t l e a s t i n f i r s t o r d e r , t h a t the s o l i t o n mass r e c e i v e s no quantum c o r r e c t i o n s and t h a t t h e Bogomolny bound r e m a i n s s a t u r a t e d ( [ 2 0 ] ) . Here we w i l l c o n c e n t r a t e on t h e s p o n t a n e o u s b r e a k i n g of supersymmetry i n t h e s o l i t o n s e c t o r . 3.1 The supersymmetry g e n e r a t o r s f o r t h e s o l i t o n - f e r m i o n t h e o r y . We r e c a l l b r i e f l y t h e d e f i n i t i o n o f an (N=l) supersymmetry a l g e b r a ( [ 3 5 ] ) : F o r a g i v e n a c t i o n S t h e supersymmetry a l g e b r a i s g e n e r a t e d by a s e t o f o p e r a t o r s 0^ , Q r * i.^f)^ , ^ , o , i t o g e t h e r w i t h the a n t i c o m m u t a t i o n r e l a t i o n s Q r o (33a) (33b) I ) - o (33c) 41 w i t h P e t h e energy-momentum o p e r a t o r . F o r t o p o l o g i c a l l y n o n t r i v i a l f i e l d c o n f i g u r a t i o n s , e .g. s o l i t o n s , (33b) has t o be m o d i f i e d t o i n c l u d e " t o p o l o g i c a l c h a r g e s " ( [ 1 6 ] ) . The a c t i o n S = JdS 2(*,t ) c o r r e s p o n d i n g t o our L a g r a n g i a n d e n s i t y (8) p o s s e s s e s c o n s e r v e d c u r r e n t s j r , ( jf<fr t ; V ( o » ) T r ^ (34) and t h e c o r r e s p o n d i n g " s u p e r c h a r g e " Q * |j« j 6(*,t) » ( Q ° ) g e n e r a t e s a supersymmetry a l g e b r a w i t h t o p o l o g i c a l c h a r g e s T, i . e . i t s components g e n e r a t e a supersymmetry a l g e b r a w i t h { Q. . Q , J • -if,t \ * i ; T y r [ Q«. T j • [ Q. . ?, ] -o <35b) where T can be i n t e g r a t e d t o z e r o i f t h e f i e l d <pA i s a s y m p t o t i c a l l y t r i v i a l , t h a t i s , <pcl(x=t<x> ) = <f>0. A s o l i t o n f i e l d however, w i t h 0 ( x = t » ) = 0 t has t o i n c l u d e t h e s u r f a c e t e r m T. To v e r i f y (35) we w i l l e v a l u a t e t h e a n t i c o m m u t a t o r d i r e c t l y u s i n g t h e f o l l o w i n g i d e n t i t y f o r two boson and f e r m i o n o p e r a t o r s B,b and f , F : 42 { F b , b f } = F f [ B , b ] + { F , f } b B The g e n e r a t o r s (37) a c a n be v i e w e d a s c o m p o n e n t s o f t h e f o l l o w i n g m a t r i x p r o d u c t Q. i. L Kij'f.* * f r A ' - i s v(r)) d* ZL 5>pj J y ' « |i« E L r |. Now a p p l y ( 3 7 ) t o t h e a n t i c o m m u t a t o r The c o m m u t a t o r s a r e e v a l u a t e d t o be ( a ) o S * I,. V [ f U ) > ( j ) ] - i l r $,„ [ v(4>), ity] 43 (b) S O that the boson product H L y can be reduced to • Ux-O [ ( *-yV *<yX*)) (y"f - / V V O ) ) * - i ; ^ T(«) - v'it) Ho) J IK J ,t 1 and for the fermion part frj, ur..ifj • - * fp 7 ; . ^ - * * r r ; . * v(+) . v'(*) f'r f r . f f , \ Thus we have as a preliminary result 44 * V ( * Y < r , . / ' ( • ) O V ) , ( / » , . - V'( <t>) S(o) S.^  As t h e l a s t t erm i n d i c a t e s t h e r e a r e p r o b l e m s w i t h t h e o p e r a t o r o r d e r i n g a r i s i n g f r o m terms l i k e \p7 (x)i//(y)-» ^ 6 ( 0 ) f o r x - * y . F o r m a l l y , however, a l l t h e s e terms w i l l c a n c e l . T h a t we have i n d e e d found o u r e q u a l i t y c a n be e a s i l y s een we c h o o s e our f a m i l i a r r e p r e s e n t a t i o n 7 0 = a 2 , 7 1 = / 03, 7 5 = " and e v a l u a t e t h e e l e m e n t s of our m a t r i x (ML,. = {Q. ,Q.}. We o b t a i n f o r t h e p a r t i c u l a r t e r m s (b) \ . m+ H;' + , 1 , • \ 45 ( c ) * f *(o) 3 L ( d ) ^7) • (ye^) , o - \ + , •x'x 0 / + 1 w h i c h g i v e s t h e f i n a l e x p r e s s i o n f o r t h e a n t i c o m m u t a t o r s Ho) i v'u) (^ 7 - ) - U 7 ' * ^ 7 ) 46 whereas t h e r i g h t hand s i d e of e q u a t i o n (37) g i v e s « . ft - X 1 «\ eo 1 - I 1 72 « = o , fl - f 0 | 6 r r" 7 s -r and t h e supersymmetry r e l a t i o n (35b) i s p r o v e d . The v e r i f i c a t i o n of t h e o t h e r two r e l a t i o n s i s s t r a i g h t f o r w a r d . I n t h e n e x t s e c t i o n we w i l l i n v e s t i g a t e t h e a c t i o n of t h e Q's on t h e s o l i t o n g r o u n d s t a t e . 47 3.2 S p o n t a n e o u s supersymmetry b r e a k i n g i n t h e s o l i t o n s e c t o r H a v i n g e s t a b l i s h e d t h e e x i s t e n c e on an N=1 supersymmetry f o r our a c t i o n we w i l l now show t h a t t h e symmetry i s s p o n t a n e o u s l y b r o k e n due t o t h e p r e s e n c e of c e n t r a l c h a r g e s . A g e n e r a l d i s c u s s i o n of supersymmetry b r e a k i n g c a n be f o u n d i n r e f e r e n c e s [35] and [ 3 6 ] . We w i l l m o d i f y t h e i r a r guments f o r our p r e s e n t c a s e . F i r s t we d e r i v e t h r e e e q u a t i o n s from our c e n t r a l s upersymmetry r e l a t i o n ( 3 5 a ) . M u l t i p l i n g by 7 0 on b o t h s i d e s and t a k i n g t h e t r a c e g i v e s t h e w e l l known f o r m u l a H" • i ( {Q..<M + tQ- -M ) : i Q-4 ) (38) F u r t h e r m o r e we deduce d i r e c t l y f r o m (35a) C L * • H - T (39) Q,4 - H * T (40) I t i s o b v i o u s f r o m e q u a t i o n (38) t h a t H i s p o s i t i v e . E q u a t i o n (39) i m p l i e s t h e n t h e quantum Bogomolny bound on H H > T (41) w h i c h i s i n t e n d e d t o h o l d i n t h e weak s e n s e . We w i l l now examine t h e s e o p e r a t o r e q u a t i o n s f o r our s o l i t o n g r o u n d s t a t e |0>. O b v i o u s l y we have t h e a l t e r n a t i v e Q-|0>*0, 48 w h i c h means t h a t supersymmetry i s s p o n t a n e o u s l y b r o k e n , o r Q-|0>=0, i n w h i c h c a s e supersymmetry i s p r e s e r v e d . I t i s o b v i o u s f r o m e q u a t i o n s (38) and (40) t h a t i s a l w a y s b r o k e n . I t i s n o t so e a s y t o d e c i d e , however, whether Q 0 r e m a i n s u n b r o k e n o r , e q u i v a l e n t l y , whether t h e Bogomolny bound (41) on H i s s a t u r a t e d . On t h e c l a s s i c a l l e v e l 0(g-') t h e s o l u t i o n i s e a s y : t h e c e n t r a l c h a r g e T = j A * ^ t l f ' r e d u c e s t o M 0 = j <*•* <t>a 1 , t h e H a m i l t o n i a n a c t s on a p u r e s o l i t o n s t a t e |P=0> w i t h H|P=0> = M o|P=0> ( c f . ( 3 2 ) ) and t h e bound i s s a t u r a t e d . The q u e s t i o n w hether t h i s r e s u l t w i l l h o l d t o h i g h e r o r d e r s i n p e r t u r b a t i o n t h e o r y h as been d i s c u s s e d by s e v e r a l a u t h o r s ( [ l 5 ] - [ 2 0 ] ) . I t has been f o u n d t h a t t o f i r s t o r d e r i n p e r t u r b a t i o n t h e o r y b o t h H and T r e c e i v e i d e n t i c a l quantum c o r r e c t i o n s , so t h a t t h e c o m b i n a t i o n H-T r e m a i n s z e r o . T h i s r e s u l t h o l d s a l r e a d y on t h e d e n s i t y l e v e l f o r H and T ( [ 1 9 ] , [ 2 0 ] ) . Imbimbo and Mukhi have a r g u e d q u a l i t a t i v e l y t h a t t h i s w i l l r e m a i n v a l i d t o a l l o r d e r s i n p e r t u r b a t i o n t h e o r y ( [ 1 9 ] ) . T h e i r argument i s b a s e d on t h e a b s e n c e o f G o l d s t o n e f e r m i o n s i n l o w e s t o r d e r p e r t u r b a t i o n z e r o . The f e r m i o n zero-mode t h a t a p p e a r s ( c f . s e c t i o n 2.3) c o r r e s p o n d s t o t h e b r e a k i n g o f t h e Ch-symmetry. To g e n e r a t e more G o l d s t o n e modes, t h e h i g h e r o r d e r c o r r e c t i o n s would have t o r e n o r m a l i z e t h e masses of t h e o t h e r modes e x a c t l y t o z e r o , a p r o c e s s t h a t seems v e r y u n l i k e l y . 49 We c a n now r e f o r m u l a t e t h e p r o b l e m o f s p o n t a n e o u s symmetry b r e a k i n g i n t h e s o l i t o n . C o n s i d e r t h e new H a m i l t o n i a n H' d e f i n e d by H' = H - T. T h i s c o r r e s p o n d s t o a s h i f t of t h e e n e r g y s p e c t r u m by a c o n s t a n t and does n o t a f f e c t t h e d y n a m i c s o r t h e H i l b e r t s p a c e s t r u c t u r e of our s y s t e m . T h i s new H a m i l t o n i a n has v a n i s h i n g vacuum e n e r g y i n agreement w i t h more g e n e r a l r e s u l t s i n s u p e r s y m m e t r i c f i e l d t h e o r i e s , and e q u a t i o n s (39) and (40) can be r e w r i t t e n as Qo' * H (42) Q* • H' W T ( 4 3 ) The Q 0-symmetry i s o b v i o u s l y c o n s e r v e d f o r t h e g r o u n d s t a t e whereas t h e 0.,-symmetry r e m a i n s s p o n t a n e o u s l y b r o k e n . In o t h e r words, t h e N=1 supersymmetry i s b r o k e n down t o an "N=i" symmetry due t o t h e p r e s e n c e of a c e n t r a l c h a r g e T. We c a n now c o m p l e t e t h e d i s c u s s i o n o f t h e p o s s i b l e d e g e n e r a c i e s of t h e s o l i t o n - s e c t o r g r o u n d s t a t e t h a t we s t a r t e d i n s e c t i o n 2.4. The supersymmetry g e n e r a t o r s Q 0 and Q, b o t h commute w i t h t h e H a m i l t o n i a n a c c o r d i n g t o ( 3 5 c ) . Q, d o e s n o t a n n i h i l a t e t h e g r o u n d s t a t e w h i c h c a n be t a k e n as a d e g e n a r a t e d o u b l e t |0 t> w i t h a, fo +> ~ I0_> , Q, |0.> - | 0<> (44) a,4 l o4 > - T i os > Thus t h e s p o n t a n e o u s b r e a k i n g o f supersymmetry p r o v i d e s t h e 50 d e g e n e r a c y of t h e g r o u n d s t a t e f a m i l i a r from a n a l o g o u s t h e o r i e s w i t h D i r a c f e r m i o n s ( [ 3 ] , [ 3 4 ] ) . We w i l l c o n c l u d e t h i s d i s c u s s i o n w i t h a n o t e on t h e c o n n e c t i o n m e n t i o n e d e a r l i e r between s o l i t o n s and boson and f e r m i o n zero-modes i n a s u p e r s y m m e t r i c t h e o r y . So f a r , t h e c o l l e c t i v e c o o r d i n a t e f o r m a l i s m has been b a s e d on t h e t r a n s l a t i o n a l symmetry o f our t h e o r y . The s o l i t o n p a r t 0 t l(x-X) of t h e b o s o n o p e r a t o r was i n t e r p r e t e d as t h e quantum r e p r e s e n t a t i v e of a s e t o f c l a s s i c a l s o l u t i o n s 4> t l(x-x 0). S i n c e t h e c l a s s i c a l s o l u t i o n s a r e n o t t r a n s l a t i o n a l l y i n v a r i a n t t h e y have t o be l a b e l l e d by t h e t r a n s l a t i o n p a r a m e t e r x 0 w h i c h i s t h e n c o n s i d e r e d a s a c o l l e c t i v e f i e l d v a r i a b l e and X i s t h e c o r r e s p o n d i n g quantum o p e r a t o r . F u r t h e r m o r e t h e s o l i t o n 0 c t i n d u c e s t h e z e r o - e n e r g y t r a n s l a t i o n mode <f>lL v i a an i n f i n i t e s i m a l t r a n s l a t i o n . Hence t h e boson zero-mode can be v i e w e d as a c o n s e q u e n c e o f t h e b r o k e n t r a n s l a t i o n a l symmetry. Now c o n s i d e r t h e i n f i n i t e s i m a l supersymmetry t r a n s f o r m a t i o n on t h e f i e l d s c o r r e s p o n d i n g t o t h e c o n s e r v e d c h a r g e s Q 0 and ( [ 3 5 ] , [ 3 7 ] ) 5 (45a) (45b) w i t h e an a n t i c o m m u t i n g M a j o r a n a 2 - s p i n o r . I f we a p p l y t h e s e t r a n s f o r m a t i o n s - w i t h t i m e d e p e n d e n t t - p a r a m e t e r - t o t h e 51 c l a s s i c a l s t a t i c s o l u t i o n s we o b t a i n t h e new s o l u t i o n s (46a) - ta ( o ) 4 6 ( * ) ( 4 6 B ) 6i//o(/ i s - up t o a n o r m a l i z a t i o n c o n s t a n t - t h e z e r o - e n e r g y mode ( 2 8 ) , now g e n e r a t e d by an i n f i n i t e s i m a l supersymmetry t r a n s f o r m a t i o n ( [ 3 8 ] ) . We c a n now q u a n t i z e t h e s e " s u p e r t r a n s l a t e d " c l a s s i c a l s o l u t i o n s as b e f o r e by t r e a t i n g t h e t r a n s f o r m a t i o n p a r a m e t e r E.(t) as a c o l l e c t i v e c o o r d i n a t e and t r a n s f o r m i n g t h e f i e l d s t o i UO • ta (*) + £ Ut) (47) T- (*.t) • SYa («) tCO • ( 4 8 ) w i t h c o n s t r a i n t ) <U V7 • 0 . T h i s s y s t e m c a n be q u a n t i z e d e i t h e r c a n o n i c a l l y as f o r t h e c o l l e c t i v e p o s i t i o n c o o r d i n a t e o r by t h e Feynman p a t h i n t e g r a l f o r c o n s t r a i n e d s y s t e m s ( [ 3 1 ] ) . A d e t a i l e d d i s c u s s i o n can be f o u n d i n t h e work o f B a a k l i n i ( [ 3 9 ] , [ 4 0 ] ) who, i n a d d i t i o n , g e n e r a l i z e d t h e c o l l e c t i v e c o o r d i n a t e method t o t h e c a s e where b o t h b o s o n i c and f e r m i o n i c p a r a m e t e r s a r e p r e s e n t . 52 To summarize t h e r e s u l t s , we have shown t h a t t h e e x i s t e n c e of z e r o - e n e r g y s o l u t i o n s i s due t o s y m m e t r i e s of t h e L a g r a n g i a n w h i c h a r e b r o k e n by t h e p r e s e n c e of t h e s o l i t o n . The zero-modes a r e g e n e r a t e d by a p p l y i n g an i n f i n i t e s i m a l symmetry t r a n s f o r m a t i o n t o t h e s o l i t o n 0cL, i n p a r t i c u l a r , a t r a n s l a t i o n f o r t h e boson-mode and a s u p e r s y m m e t r y t r a n s f o r m a t i o n f o r t h e f e r m i o n mode. In a d d i t i o n , t h e p a r a m e t e r s a s s o c i a t e d w i t h t h e s e t r a n s f o r m a t i o n s can be u s e d a s c o l l e c t i v e c o o r d i n a t e s f o r t h e q u a n t i z a t i o n of t h e t h e o r y . C o n v e r s e l y , i t has been a r g u e d t h a t t h e e x i s t e n c e o f zero-modes g i v e s i n f o r m a t i o n a b o u t t h e symmetry p r o p e r t i e s of a t h e o r y , i n p a r t i c u l a r , t h a t i n a t h e o r y w i t h D i r a c f e r m i o n s t h e p r e s e n c e o f f e r m i o n zero-modes i m p l i e s t h e e x i s t e n c e o f a " h i d d e n " supersymmetry. ( [ 3 4 ] ) . 53 BIBLIOGRAPHY 1] A.C. 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