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The dressing transformation and its application to a fermion-boson trilinear interaction Hearn, Deborah Jean 1981

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THE DRESSING TRANSFORMATION AND TO A FERMION-BOSON TRILINEAR ITS APPLICATION INTERACTION by DEBORAH JEAN HEARN B . S c , The U n i v e r s i t y of Saskatchewan, 1979 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA J u l y 1981 (c) Deborah Jean Hearn, 1981 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 advanced 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 agr 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 agree t h a t p e r m i s s i o n f o r e x t e n s i v e 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 department 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 . Department o f Physics  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 2075 Wesbrook P l a c e V ancouver, Canada V6T 1W5 Date T a ( v j oV7 , | Q 8 I DE-6 (2/79) i i A b s t r a c t In t h i s t h e s i s , v a r i o u s fermion-boson strong i n t e r a c t i o n p o t e n t i a l s are determined as f u n c t i o n s of the b a s i c fermion-boson t r i l i n e a r vertex f u n c t i o n . Working i n Fock space, we note that the fermion-boson t r i l i n e a r i n t e r a c t i o n does not e x p l i c i t l y i n v o l v e p h y s i c a l p a r t i c l e s . We develop a t r a n s f o r m a t i o n , c a l l e d the d r e s s i n g t r a n s f o r m a t i o n , which a c t s on the fundamental p a r t i c l e c r e a t o r s and a n n i h i l a t o r s . They are transformed i n t o p h y s i c a l p a r t i c l e o p e r a t o r s , and the i n v a r i a n c e p r o p e r t i e s and commutation r e l a t i o n s of the theory are pres e r v e d . A p r e c i s e technique f o r p e r t u r b a t i v e l y determining the d r e s s i n g t r a n s f o r m a t i o n i s formulated, and i s a p p l i e d to some simple models i n f i e l d t heory. The d r e s s i n g t r a n s f o r m a t i o n makes e x p l i c i t the p h y s i c a l p a r t i c l e i n t e r a c t i o n s i m p l i c i t i n the o r i g i n a l t r i l i n e a r i n t e r a c t i o n . When a p p l i e d to the nucleon-pion t r i l i n e a r i n t e r a c t i o n , we f i n d a nucleon mass r e n o r m a l i z a t i o n , a nucleon-pion s c a t t e r i n g term, and a nucleon-nucleon s c a t t e r i n g term present i n the second-order dressed Hamiltonian. Using the NN7C v e r t e x f u n c t i o n d e r i v e d from the Cloudy Bag Model, the nucleon-nucleon c o o r d i n a t e space p o t e n t i a l can be c a l c u l a t e d . We d i s c o v e r t h a t p r o v i d i n g the two nucleons are separated by a d i s t a n c e g r e a t e r than twice the bag r a d i u s , the p o t e n t i a l between them i s given by the one pion exchange p o t e n t i a l m o d i f i e d i n s t r e n g t h by a f u n c t i o n of the bag r a d i u s . i i i TABLE OF CONTENTS A b s t r a c t i i L i s t of Tables v L i s t of F i g u r e s v i Acknowledgements v i i Chapter 1 I n t r o d u c t i o n . 1 Chapter 2 Fermion and Boson Fundamental Dynamical V a r i a b l e s and T h e i r P r o p e r t i e s 4 2.1 The Fundamental Dynamical V a r i a b l e s 4 2.2 Space-Time Transformation P r o p e r t i e s of the Fundamental Dynamical V a r i a b l e s . . . . 6 Chapter 3 The T r i l i n e a r Fermion-Boson I n t e r a c t i o n 10 3.1 R e s t r i c t i o n s Due to C e r t a i n Space-Time Inv a r i a n c e s 10 3.2 The T r i l i n e a r Vertex Function i n the Cloudy Bag Model 19 3.3 P h y s i c a l Bosons and Fermions 21 Chapter 4 The Dre s s i n g Transformation and Some Simple A p p l i c a t i o n s 22 4.1 The General D r e s s i n g Transformation 22 4.2 The S c a l a r F i e l d Model 26 4.3 The Lee Model 33 Chapter 5 Dr e s s i n g the T r i l i n e a r Fermion-Boson I n t e r a c t i o n : The Fermion-Fermion P o t e n t i a l 40 5.1 Dre s s i n g the T r i l i n e a r I n t e r a c t i o n . . 40 5.2 The Second-Order Nucleon-Nucleon P o t e n t i a l . . . 45 5.3 The Nucleon-Nucleon P o t e n t i a l i n the Cloudy Bag Model 50 Chapter 6 Summary and Con c l u s i o n s 54 B i b l i o g r a p h y . 57 Appendix A The Ro t a t i o n M a t r i c e s D^i^ (d p~6 ) 58 Appendix B The S p h e r i c a l Harmonics Y A r r,(g) 60 Appendix C I r r e d u c i b l e Tensor Operators and the Wigner-Eckart Theorem 62 Appendix D Angular Momentum Co u p l i n g . 64 (a) A d d i t i o n of Two Angular Momenta: The Clebsch-Gordan C o e f f i c i e n t and the 3j Symbol. 64 (b) A d d i t i o n of Three Angular Momenta: The 6j Symbol 66 (c) A d d i t i o n of Four Angular Momenta: The 9j Symbol 68 (d) A d d i t i o n of F i v e Angular Momenta: The 12j Symbol . 70 Appendix E T w o - P a r t i c l e Operators i n Fock Space 72 Appendix F C a l c u l a t i o n of the T r i l i n e a r Vertex F u n c t i o n f o r the Cloudy Bag Model 76 Appendix G An Exact D r e s s i n g Operator f o r the Lee Model 79 Appendix H A Dr e s s i n g Transformation f o r a G e n e r a l i z e d Fermion-Boson T r i l i n e a r I n t e r a c t i o n 82 (a) The G e n e r a l i z e d Fermion-Boson T r i l i n e a r I n t e r a c t i o n 82 (b) D r e s s i n g the G e n e r a l i z e d I n t e r a c t i o n . . . . 87 Appendix I Some P r o p e r t i e s of B e s s e l Functions 94 Appendix J One Pion Exchange P o t e n t i a l s (OPEP) 98 Appendix K Dr e s s i n g a Poincare I n v a r i a n t System.. 100 V L i s t of Tables Table I Values of the Constant C L M ^ ' 49 Table I I Matrix Elements of the S c a l a r OPEP ['U'MM^^^XPEP- • 99 v i L i s t of F i g u r e s F i g u r e 1 The T r i l i n e a r F-B I n t e r a c t i o n . . . . 11 F i g u r e 2 The S c a l a r F i e l d Model I n t e r a c t i o n 27 F i g u r e 3 The Fermion-Fermion P o t e n t i a l i f c F ( k , l k / ,K) 30 F i g u r e 4 The Fermion-Boson P o t e n t i a l ^ P 8 ( k , k ' ,K) 30 F i g u r e 5 The Lee Model I n t e r a c t i o n 34 F i g u r e 6 The N-V P o t e n t i a l ^ 38 F i g u r e 7 The V- 6- P o t e n t i a l 38 F i g u r e 8 The N-6> P o t e n t i a l tf8 38 F i g u r e 9 The IL -N P o t e n t i a l IT™ M.^MM'^"{k, k' ,K) 44 3 y l w m, fviim — — — F i g u r e 10 The N-N P o t e n t i a l lJ~NtJ '>** ( k - k / ) 44 F i g u r e 11 The Two-Fermion P o t e n t i a l ^ s j f / ^ / k , ^ ,K) 75 F i g u r e 12 The G e n e r a l i z e d T r i l i n e a r F-B I n t e r a c t i o n 83 F i g u r e 13 The Fermion-Boson P o t e n t i a l l/ha ( kr k ' ,K) 90 F i g u r e 14 The Fermion-Fermion P o t e n t i a l t ^ p C k - k ' ) 90 F i g u r e 15 The NA.7C T r i l i n e a r Vertex 93 v i i Acknowledgements I would f i r s t l i k e to thank my s u p e r v i s o r , Dr. J . Malcolm McMillan, f o r h i s ex t e n s i v e input of ideas i n t o t h i s p r o j e c t and h i s constant encouragement and i n t e r e s t throughout i t s p r o g r e s s . Secondly, I would l i k e to thank Serge Theberge, Dr. A.W. Thomas, and Dr. G. M i l l e r f o r p r o v i d i n g a p r e p r i n t of t h e i r paper on the Cloudy Bag Model. L a s t l y , I thank the N a t u r a l Sciences and E n g i n e e r i n g Research C o u n c i l f o r t h e i r f i n a n c i a l support. 1 Chapter j_ I n t r o d u c t i o n The problem of f i n d i n g a u s e f u l and ac c u r a t e theory of the strong i n t e r a c t i o n s of nucleons and pions i s an ongoing one. C e r t a i n l y one important advance i n t h i s area was the concept of one boson exchange p o t e n t i a l s , implying that the u n d e r l y i n g fermion-boson strong i n t e r a c t i o n i s a t r i l i n e a r one. These one boson exchange p o t e n t i a l s p r o v i d e the b a s i s f o r the f u n c t i o n a l form of the phenomenological nucleon-nucleon p o t e n t i a l s . In t h i s t h e s i s , we develop a technique f o r f i n d i n g p o t e n t i a l s f o r d i r e c t fermion-fermion, boson-boson, and fermion-boson i n t e r a c t i o n s , as w e l l as boson p r o d u c t i o n on two fermions. These i n t e r a c t i o n p o t e n t i a l s are determined i n terms of the fermion-boson t r i l i n e a r vertex f u n c t i o n . They can serve as the b a s i s f o r a phenomenological strong i n t e r a c t i o n Hamiltonian f o r systems of pions and nucleons at intermediate e n e r g i e s . Using an approach which takes the fundamental dynamical v a r i a b l e s of the theory to be the elementary fermion and boson c r e a t o r s F T and B T, d i s c u s s e d i n Chapter 2, the t r i l i n e a r f u n c t i o n h depends on the momenta i n v o l v e d . We show i n Chapter 3 that r e q u i r i n g t h i s i n t e r a c t i o n to be i n v a r i a n t under c e r t a i n space-time t r a n s f o r m a t i o n s g r e a t l y r e s t r i c t s the vertex f u n c t i o n h. The Cloudy Bag Model i s then used to determine a s p e c i f i c form f o r the strong i n t e r a c t i o n v e r t e x f u n c t i o n . One d i f f i c u l t y with t h i s fermion-boson t r i l i n e a r i n t e r a c t i o n i s that i t does not e x p l i c i t l y i n v o l v e p h y s i c a l p a r t i c l e s s i n c e F^|0> i s not an eigenket of the Hamiltonian. In i n t e r a c t i o n i n v o l v e s the i n t e g r a l where the vertex 2 Chapter 4 we formulate a technique, c a l l e d the d r e s s i n g t r a n s f o r m a t i o n , f o r t r a n s f o r m i n g the elementary p a r t i c l e c r e a t o r s i n t o p h y s i c a l p a r t i c l e c r e a t o r s . T h i s t r a n s f o r m a t i o n a l t e r s none of the i n v a r i a n c e p r o p e r t i e s or commutation r e l a t i o n s of the theory. The i n i t i a l work on the d r e s s i n g t r a n s f o r m a t i o n was done by Greenberg and Schweber (1958), who c o n s i d e r e d simple, s o l u b l e t h e o r i e s such as the s c a l a r f i e l d model and the Lee model. In t h i s t h e s i s we have g e n e r a l i z e d the concept of the d r e s s i n g t r a n s f o r m a t i o n i n order to apply i t to more r e a l i s t i c t h e o r i e s . We g i v e a d e t a i l e d p r e s c r i p t i o n f o r determining the d r e s s i n g t r a n s f o r m a t i o n and a l s o f o r determining the Hamiltonian as a f u n c t i o n of the p h y s i c a l p a r t i c l e o p e r a t o r s . Both are c a l c u l a t e d i n a p e r t u r b a t i o n s e r i e s i n the s t r o n g i n t e r a c t i o n c o u p l i n g c o n s t a n t . (A d i f f e r e n t p e r t u r b a t i o n s e r i e s f o r the d r e s s i n g t r a n s f o r m a t i o n has been given by Faddeev (1964)). In S e c t i o n s 4.2 and 4.3, the d r e s s i n g t r a n s f o r m a t i o n i s a p p l i e d to two simple t h e o r i e s - the s c a l a r f i e l d model and the Lee .model. T h i s a p p l i c a t i o n i l l u s t r a t e s many f e a t u r e s of the t r a n s f o r m a t i o n , the p h y s i c a l p a r t i c l e c r e a t o r s , and the dressed Hamiltonian which are present i n more complicated t h e o r i e s . The nucleon-pion t r i l i n e a r i n t e r a c t i o n i s dressed to second order in Chapter 5. We c o n s i d e r the r e s u l t i n g p h y s i c a l nucleon-pion and nucleon-nucleon i n t e r a c t i o n s . Using the Cloudy Bag Model vertex f u n c t i o n , we c a l c u l a t e the second-order nucleon-nucleon p o t e n t i a l . We d i s c o v e r that p r o v i d i n g the two nucleons are not touching, t h i s p o t e n t i a l i s simply a one pion exchange p o t e n t i a l t h a t has been s l i g h t l y m o d i f i e d i n s t r e n g t h . 3 Many of the Appendices provide u s e f u l mathematical formulae and techniques used i n the t h e s i s . For example, they d i s c u s s the r o t a t i o n m a t r i c e s , s p h e r i c a l harmonics, angular momentum c o u p l i n g , and Bessel f u n c t i o n s . Other Appendices p r o v i d e some background to concepts used i n the t h e s i s , d i s c u s s i n g such t h i n g s as two-body p o t e n t i a l s or one pion exchange p o t e n t i a l s . In Appendix H we have g e n e r a l i z e d the t r i l i n e a r i n t e r a c t i o n to i n c l u d e fermions and bosons of a r b i t r a r y s p i n and i s o s p i n . Thus our technique can be a p p l i e d not only to i n t e r a c t i o n s of nucleons and pions, but to i n t e r a c t i o n s of other fermions and bosons as w e l l . F i n a l l y , i n Appendix K, we d i s c u s s a d r e s s i n g t r a n s f o r m a t i o n f o r a Poincare i n v a r i a n t system of i n t e r a c t i n g fermions and bosons. 4 Chapter 2 Fermion and Boson Fundamental Dynamical V a r i a b l e s and  T h e i r P r o p e r t i e s T h i s Chapter w i l l provide the background information necessary f o r an understanding of the r e s t of the t h e s i s . We w i l l i n troduce p a r t i c l e c r e a t o r s and a n n i h i l a t o r s , and we w i l l g i ve t h e i r commutation r e l a t i o n s and space-time t r a n s f o r m a t i o n p r o p e r t i e s . 2.1 The Fundamental Dynamical V a r i a b l e s Our system of fundamental fermions and bosons with a r b i t r a r y s p i n and i s o s p i n i s d e s c r i b e d by a H i l b e r t space which i s a d i r e c t product of fermion Fock space and boson Fock space. The fundamental dynamical v a r i a b l e s , i n terms of which a l l observables and op e r a t o r s can be expressed, are the p a r t i c l e c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s . These are d e f i n e d as f o l l o w s : F ^ T ( x ) , when a c t i n g on the vacuum s t a t e , y i e l d s a one-fermion ket corresponding to an elementary fermion at p o s i t i o n x, having spi n s, z - a x i s p r o j e c t i o n m, i s o s p i n i , and i s o s p i n z-a x i s p r o j e c t i o n fx. The a d j o i n t operator, F ^ ( x ) , when a c t i n g on t h i s one-fermion ket, g i v e s the vacuum s t a t e . E q u i v a l e n t l y , we can d e f i n e the momentum space fermion c r e a t o r s and a n n i h i l a t o r s , F ^ J ^ C r j ) and F ^ J r j ) • These c r e a t e , or dest r o y , an elementary fermion with momentum p_. The two c r e a t o r s are r e l a t e d by a F o u r i e r transform: 5 (2.1.1) Note that ^ o l F ^ T ( P _ ) = ^ F ^ C * ) = O (2.1.2) and S i m i l a r l y , B^^(x_) , when a c t i n g on the vacuum s t a t e , g i v e s a p o s i t i o n ket corresponding to an elementary boson at x having spin s with p r o j e c t i o n m and i s o s p i n i with p r o j e c t i o n JJ. . The op e r a t o r s B ^ ( x ) , B^"^(p_) , and B^(p_) are d e f i n e d analogously to the fermion case. The p o s i t i o n and momentum c r e a t o r s are r e l a t e d by &*u. U> = J d P e <-f> (2.1.4) The f o l l o w i n g commutation r e l a t i o n s are s a t i s f i e d by the fundamental dynamical v a r i a b l e s : i F ^ ^ V F ^ y U l ' ) ] = ^ I - | ' U s 5 ' S ^ S c c ' ^ ' (2.1.5) U^ CI^ B^ CI^ ] = L<+ ,^BSj:.ftlO] = o (2'1-8) IK^UJ, F^U'\I = [ B ^ + C i ^ F ^ f C i ^ ] = o (2.1.9) where [ ] denotes a commutator and { } an anticommutator; i_ r e p r e s e n t s e i t h e r x or £. 6 2.2 Space-Time Transformation P r o p e r t i e s of the Fundamental  Dynamical V a r i a b l e s In t h i s S e c t i o n we d i s c u s s the e f f e c t of displacements, r o t a t i o n s , space i n v e r s i o n and time r e v e r s a l on the p a r t i c l e c r e a t o r s and a n n i h i l a t o r s . These r e s u l t s are a consequence of the requirement of general Poincare i n v a r i a n c e f o r a p h y s i c a l system (Kalyniak (1978)). We w i l l use them i n S e c t i o n 3.1 to determine the form which i n v a r i a n t p a r t i c l e i n t e r a c t i o n s must have. To a s p a t i a l displacement of the system by amount a there corresponds a l i n e a r , u n i t a r y operator D(a) such that Dta^ l F^ f^D fC<0 = F ^ + C £ f a ^ (2.2.1) Bj j + if) D + ta^ = B^M- t l , a )  ( 2 - 2 ' 2 ) The r e s u l t i n g c r e a t i o n operator c r e a t e s a p a r t i c l e at p o s i t i o n x + a rather than x. Using (2.1.1) and (2.1.4) we have Dta^  F ^ C p tf(a) = F ^ ^ e " ' - ' ? m (2.2.3) Di<T> E^^D +(a) = B^t^ e'"'F^ (2.2.4) To a s p a t i a l r o t a t i o n of the system through E u l e r angles d , p ,% there corresponds a l i n e a r , u n i t a r y operator («( £ 7f) , such that s Ktyrt L*)K*uii) = J~ D ^ ^ F w s , ' + c t .^ (2.2.5) m^i) (i) ~- mJs V*,* L<ii) u_ R ) ( 2 - 2 - 6 ) 7 The conventions f o r the E u l e r angles and r o t a t i o n matrices Dm r n' Upv) a r e as. given i n Rose (1957). Note that 2 R = M TL (2.2.7a) = (2.2.7b) where x i s a column v e c t o r and M i s the matrix d e f i n e d i n Rose (1957, p.65). From (2.1.1) and (2.1.4) i t f o l l o w s that »upt^ F ^ + ^ J R ^ T ) = 1 . D J U ^ F ^ V ^ ( 2 . 2 . 8 ) 5 * ^ B l i ^ * + « ^ - £ D „ 5 . ; « P 7 ) 6 ^ c ? . ^ (2.2.9) Both the momentum and s p a t i a l c o o r d i n a t e s , as we l l as the s p i n , are r o t a t e d . To a space i n v e r s i o n t r a n s f o r m a t i o n there corresponds a l i n e a r u n i t a r y operator V such that # F ^ C i ^ 1 " = t F ^ l - r ) (2.2.10) * 6 ^ C ^ = ± B»/» (2.2.11) where the plus s i g n a p p l i e s to p o s i t i v e p a r i t y p a r t i c l e s , and the minus s i g n to negative p a r i t y p a r t i c l e s . %_ represents e i t h e r x or p_. Both the p o s i t i o n and momentum v e c t o r s are i n v e r t e d by t h i s t r a n s f o r m a t i o n . To a time r e v e r s a l t r a n s f o r m a t i o n on the system there corresponds an a n t i l i n e a r , a n t i u n i t a r y operator 3 such that 8 5 3 F ^ f l 0 J + - ^ JT.5[)^^7CO"> F « V f t 2 ^ (2.2.12) s J B ^ C ^ U * = n B J:_S D^mCoTCo) B * y C & } (2.2.13) 3 f d U + = f*C2L) J J f - f * C i ) (2.2.14) where f ( x ) i s an a r b i t r a r y complex f u n c t i o n , and ITF I 2 = 2 = 1 . i s the time r e v e r s a l p a r i t y of the p a r t i c l e . I t i s determined from the space i n v e r s i o n p a r i t y and charge c o n j u g a t i o n p a r i t y by the requirement of o v e r a l l TCP i n v a r i a n c e (see Schweber (1961, p.268)). For nucleons, r £ p = +1; f o r pions, >IB= -1. The time r e v e r s a l t r a n s f o r m a t i o n does not change the p o s i t i o n c o o r d i n a t e s , but does r e v e r s e both the momentum and spin v e c t o r s . Reversing the s p i n i s e q u i v a l e n t to r o t a t i n g the observer through 180 degrees about an a r b i t r a r y a x i s . We have chosen t h i s a x i s to be the y - a x i s , so the s p i n r o t a t i o n has E u l e r angles 0,7T,0. Using LCKO) = d * , m C ^ = C - ^ X ' - w (2.-2.15) as w e l l as (2.1.1) and (2.1.4), i t f o l l o w s that 3 F ^ + l ^ 3 + = TI, rf"" F . : ; + t - ^ (2.2.16) J B^, J+ - n» L-?"° L-p) ( 2 . 2 . , 7) F i n a l l y , to a r o t a t i o n i n i s o s p i n space through E u l e r angles << , |3,Y there corresponds a l i n e a r , u n i t a r y operator $ i Uf>V) such that 9 fixUpi) F^iDKx^fii)* f.. D^upT) F v f a ) (2.2.18) tfrfc^ t i > f f « + U p ^ = £ t - D;/ ^ Ll) (2.2.19) The space-time t r a n s f o r m a t i o n s of F T Y )^( J_ ) and ( A ) are e a s i l y obtained by t a k i n g the a d j o i n t of the t r a n s f o r m a t i o n equations f o r F M / ( |_ ) and B ^ ( f ) given i n t h i s S e c t i o n . 10 Chapter 3_ The T r i l i n e a r Fermion-Boson I n t e r a c t i o n In t h i s Chapter, we in t r o d u c e and d i s c u s s a Hamiltonian f o r s t r o n g l y i n t e r a c t i n g nucleons and pi o n s . (A more general t r i l i n e a r Hamiltonian i n v o l v i n g other fermions and bosons i s c o n s i d e r e d i n Appendix H.) We w i l l show, i n the f i r s t S e c t i o n , that when t h i s i n t e r a c t i o n i s r e q u i r e d to be i n v a r i a n t under c e r t a i n space-time t r a n s f o r m a t i o n s , i t can only then depend on a s i n g l e f u n c t i o n of one v a r i a b l e . In the next S e c t i o n , we s p e c i f y t h i s vertex f u n c t i o n f o r the Cloudy Bag Model. F i n a l l y , we d i s c u s s the concepts of elementary p a r t i c l e and p h y s i c a l p a r t i c l e , w i t h i n the context of the t r i l i n e a r i n t e r a c t i o n . 3.1 R e s t r i c t i o n s Due to C e r t a i n Space-Time I n v a r i a n c e s Using the fundamental dynamical v a r i a b l e s d e f i n e d i n S e c t i o n 2.1, we c o n s t r u c t the f o l l o w i n g Hamiltonian i n v o l v i n g i n t e r a c t i n g s p i n one-half, i s o s p i n one-half, p o s i t i v e p a r i t y fermions, i . e . nucleons, and s p i n zero, i s o s p i n one, negative p a r i t y bosons, i . e . p i o n s : H = Ho t A H i (3.1.1a) (3.1.1b) (3.1.1c) 11 The operator F M^+ (p_) corresponds, i n the n o t a t i o n of S e c t i o n 2.1, to the operator F ^ ^ g ) with S=1/2 and i = l / 2 ; m and jx can take the values ±1/2. We drop the l a b e l s s and i f o r s i m p l i c i t y . S i m i l a r l y , B^ ' C p ) corresponds to B^y(p_) with s=m=0, i= 1 , JLL = ±1,0. A l s o , and (3.1.2) (3.1.3) are the energies of the elementary fermion and boson, r e s p e c t i v e l y . Note that the p a r t i c l e s are t r e a t e d 'semi-r e l a t i v i s t i c a l l y ' by i n c l u d i n g r e l a t i v i s t i c k i n e m a t i c s . Note a l s o that the 'vertex f u n c t i o n ' h ^ ^ * ( g ) i s chosen to be a f u n c t i o n only of g. The t r i l i n e a r i n t e r a c t i o n H, may be p i c t u r e d as f o l l o w s : / F i g . 1 The T r i l i n e a r F-B I n t e r a c t i o n S o l i d l i n e s are fermions; dashed l i n e s are bosons The Hamiltonian (3.1.1) i s t r a n s l a t i o n a l l y i n v a r i a n t and conserves the t o t a l number of fermions. We can see t h i s by n o t i n g t hat the t o t a l momentum operator p (3.1.4) 1 2 and the fermion number operator " - ' ^ t o V ^ ^ ( 3- K 5 ) both commute with H. We now demand that H a l s o be i n v a r i a n t under r o t a t i o n s i n i s o s p i n space, r o t a t i o n s i n o r d i n a r y space, space i n v e r s i o n , and time r e v e r s a l . Ho i s alr e a d y i n v a r i a n t under a l l these t r a n s f o r m a t i o n s , but i n order that H, a l s o be i n v a r i a n t , the form of the f u n c t i o n h ^ ^ ' C g ) appearing i n C3.1.1c) must be r e s t r i c t e d (see eq. (3.1.35)). F i r s t , l e t us c a l c u l a t e the e f f e c t on H, of a r o t a t i o n i n i s o s p i n space. Using the r e s u l t s (2.2.18) and (2.2.19), we see that ' L>' S 'A U ^ F«vJ.' ^  f a d J - (3.1.6) Using (A.1) and (A.8) to combine the three r o t a t i o n matrices i n t o a s i n g l e one, we f i n d where (ji w)t Ijwi^ i s a Clebsch-Gordan c o e f f i c i e n t . (See Appendix D.) In order that (3.1.7) be equal to H,, i t can have no dependence on the angles oL , (•} , Y . T h e r e f o r e , t a k i n g 13 (5=A'=A= 0 and using the prop e r t y (D.12) of the Clebsch-Gordan c o e f f i c i e n t s i n Appendix D, we have • ^ &/*,' J + adj. ( 3* 1' 8> Comparing with (3.1.1c), we see that # r M, = H , (3.1.9) i f • (. "k 1 j^'/A*' I ( * ! A4*/** (3.1.10) From t h i s equation, we note that a l l of the p,,yjzrju3 dependence of h^^^^g) i s contained i n the c o e f f i c i e n t T h e r e f o r e , we may w r i t e ^ ^ S ^ = C i l ^ ^ l V ' ^ - . ^ ^ (3.1.11) Secondly, we determine the e f f e c t on H, of a s p a t i a l r o t a t i o n . From (2.2.8) and (2.2.9), we have • C , U p r t ( & . * U p A F „ ,+ ( ^ t f l d j . (3.1.12) T h i s w i l l equal H, i f i , ^ D V U ^ D * ' ^ , < 3 - ' . ' 3 > 14 We w r i t e ^ " M * ^ = ; C < p (3.1.14) where Y A r n(g) i s a s p h e r i c a l harmonic of order JL , whose p r o p e r t i e s are d e s c r i b e d i n Appendix B. S u b s t i t u t i n g (3.1.14) i n t o (3.1.13), we o b t a i n •D^ „.. D ^ / c ^ r ) < 3 . 1 . 1 5 > Using (B.2), W = D rf .A*p^ W ^ ( 3 . 1 . 1 6 ) we act on both s i d e s of (3.1.15) with J V i ^ C ^ l d c 0 ^ to o b t a i n lV' J b w l P D - - ' u ^ ^ U ( , r ) °""! u ^ 1 3 •' •17' Using (A.8) to combine the f i r s t two r o t a t i o n m a t rices t h i s becomes • D l <*j*T> D J L * " ^ (3.1.18) Now we i n t e g r a t e on both s i d e s with respect to the E u l e r angles <rt , p , ~i . Using (A.9) , we f i n d K™,tn> {f> = i ^nU, L<() C t M a n n I ^  " 0 (i 1 W l V I ^ */> ( 3 ' 1 ' 1 9) T h i s equation determines the m, jin^m dependence of 1 5 (3.1.20) R e f e r r i n g to (3.1.14), we w r i t e l w * l $ > 1 ^ C ^ ( ^ ^ M ! I O K^) (3.1.21) T h i r d l y , we r e q u i r e that H, be i n v a r i a n t under space i n v e r s i o n . From (2.2.10) and (2.2.11) one has • 5,+ C-jf> B^C-^ +adj. (3.1.22) T h i s w i l l equal H, i f k * . . ^ > " k . ^ C - ^ (3.1.23) (One d e a l s with p o s i t i v e p a r i t y bosons i n a s i m i l a r manner. For such bosons we would have i n s t e a d From (3.1.14), we have Krti.w* ("^= 1 i Vl* (-^  h A M W l i n x Co.') = i ^ ^ * ^ ^ k ^ U ) (3.1.24) where the second step f o l l o w s from (B.4). T h e r e f o r e , (3.1.23) can only be s a t i s f i e d i f i . e . , i f i . i s odd. R e f e r r i n g to (3.1.20) f o r h 2 w, r 0 i f f ) i(g l) and to F i g u r e 1, we see that L may be i n t e r p r e t e d as the angular 16 momentum of the boson i n the t r i l i n e a r i n t e r a c t i o n . The Clebsch-Gordan c o e f f i c i e n t il I z*0 i m p l i e s that the only p o s s i b l e odd value of A i s 4 = 1 (3.1.25) That i s , only p-waye pions are allowed. We w r i t e h , l < p = hip (3.1.26) F i n a l l y , we demand that H| be i n v a r i a n t under a time r e v e r s a l t r a n s f o r m a t i o n . From (.2.2.14), (2.2.16) and (2.2.17), we see that •C'-^ ^ ^ B ^ C - g " ) +«dj. <3...27) T h i s equals H, i f , ^ x W1,+ M i . * ^ " < o , « i i ^ a tr> (3.1.28) or, from (3.1.23), k m t = CO k * , . ^ (3.1.29) Now, from (3.1.21) and (3.1.25), we have ^ = -« V L Lp C -w>* m i t W > h*(^ (3.1.30) Using (B.5) and (D.9b), equation (3.1.30) becomes C-**^* i C O W V.^t^Cxl wix-w I 4M , ) K * C ^ (3.1.31) Since m2 i s h a l f odd i n t e g r a l , 1 7 C - Y W ' X - - C - ^ 1 (3. 1 .32) I t f o l l o w s that ^ _ r<l,-rfli +1 II a i I | • ¥ K M i r ) x C ^ » ' £ 0 T ( w > Cii Wxml iMi^V) (3.1.33) Th e r e f o r e , (3.1.29) i s s a t i s f i e d i f M f ) « K*C<p (3.1.34) i . e . , i f h(q) i s a r e a l f u n c t i o n . In summary, the requirements of i n v a r i a n c e under displacements, i s o s p i n r o t a t i o n s , s p a t i a l r o t a t i o n s , space i n v e r s i o n and time r e v e r s a l have determined the vertex f u n c t i o n fo r the t r i l i n e a r i n t e r a c t i o n (3.1.1c) to be Vi^N^> = «' Lk i ^ l ^ H i i ^ M l i ^ Y , , ; ^ ^ (3.1.35) where h(q) i s an a r b i t r a r y , r e a l f u n c t i o n . Thus the i n t e r a c t i o n (3.1.1c) can be w r i t t e n H . = i j ^ J ^ p d ^ ( A l ^ M 3 I r / O C ^ I ^ w l i r v O V , * ( ^ h ( $ V ' C C ^ ^ ^ - ^ + a d J - (3.1.36) At t h i s p o i n t i t i s i l l u s t r a t i v e to show that the above ex p r e s s i o n f o r H( can be put i n t o a m a n i f e s t l y i n v a r i a n t form. F i r s t , we d e f i n e the f o l l o w i n g o p e r a t o r : r T ' M K T^I.WI, • J d * p ^ . ( j O ^ Cj>-^ ( 3 . 1 . 3 7 ) Using the p r o p e r t i e s (2.2.5) and (2.2.18) of the fermion 18 o p e r a t o r s , we f i n d that i •Rl<pi) V ' ^ * 1 , Z , D « V %S> (3-1 -38) and i ttr"f&rt V ^ f t r ^ t ^ ^ y U f i Y ) V ( ^ (3.1.39) i . e . , S ^ ( g ) transforms as a ve c t o r under r o t a t i o n s i n both spin and i s o s p i n space. Indeed, we may i n t e r p r e t t h i s operator as a fermion s p i n - i s o s p i n t r a n s f e r o p e r a t o r . For example, S|i (g) a c t i n g on a one neutron s t a t e with spi n p r o j e c t i o n m=-l/2, turns i t i n t o a one proton s t a t e with spi n up. The spin z - a x i s p r o j e c t i o n i s i n c r e a s e d one u n i t ( A=+1), and the i s o s p i n z-axis p r o j e c t i o n i s a l s o i n c r e a s e d (ja=+1). The t r i l i n e a r i n t e r a c t i o n H, may be expressed i n terms of sA/x(g) a s f o l l o w s . We w r i t e Y,A (g) i n terms of the s p h e r i c a l components of the u n i t v e c t o r g (see Appendix C) as C l f V <U (3'1-40) and s u b s t i t u t e t h i s i n t o (3.1.36). The r e s u l t i s H, = i hty 4 ' S ty ' £ ty + a d J • (3.1.41a) where ity' Bl^S = 5- fc^ i A V l ^ B p t ^ (3.1.41b) Using (A.2), (B.2), and (A.6), we f i n d # U p - 0 | ' S ( $ V 8 ( ^ « ^ ( , i p > * S t ^ ) • B t o ^ ) (3.1.42) and 19 * * u p O %- i t y - B ^ ^ u j s y ) * £ . | < $ V B ^ > (3.1.43) The e x p r e s s i o n (3.1.41) f o r H, i s m a n i f e s t l y i n v a r i a n t under i s o s p i n space and s p a t i a l r o t a t i o n s . Note i t s s i m i l a r i t y to the more c o n v e n t i o n a l CT • | ^  form f o r the nucleon-pion t r i l i n e a r i n t e r a c t i o n , such as the Chew-Low i n t e r a c t i o n d i s c u s s e d i n Schweber (1961, pp.376,377). Indeed, the i n v a r i a n c e requirements considered i n t h i s S e c t i o n determine a l l but the a r b i t r a r y r e a l f u n c t i o n h(q) f o r the nucleon-pion t r i l i n e a r i n t e r a c t i o n . 3.2 The T r i l i n e a r Vertex F u n c t i o n i n the Cloudy Bag Model The Cloudy Bag Model of Theberge, Thomas, and M i l l e r (1980) i n v o l v e s a massive pion f i e l d i n i n t e r a c t i o n with massless quark f i e l d s . The pion f i e l d c ouples to the quark f i e l d s only on a s p h e r i c a l s u r f a c e of r a d i u s R ('the bag'). In t h i s model the bare nucleon and d e l t a p a r t i c l e s are composed of three massless up and down quarks c o n f i n e d to the bag. Using the known, lowest order bag model quark wave f u n c t i o n s , and assuming that the pion f i e l d i s s m a l l , Theberge et a l . have re-expressed the pion-quark i n t e r a c t i o n i n terms of pion-baryon t r i l i n e a r i n t e r a c t i o n s . The r e s u l t i n g Hamiltonian i s a combination of the Lee model Hamiltonian (see Se c t i o n 4.3) and the t r i l i n e a r Hamiltonian d i s c u s s e d i n Se c t i o n 3.1. For f u r t h e r d e t a i l s on the d e r i v a t i o n and consequences of the Cloudy Bag Model, we r e f e r the reader to the paper of Theberge e_t a l . (1980), and r e f e r e n c e s t h e r e i n . 20 In Appendix F we r e l a t e the NN7T pi e c e of the Cloudy Bag Model t r i l i n e a r i n t e r a c t i o n to our equation (3.1.36). We f i n d the t r i l i n e a r vertex f u n c t i o n , which we wr i t e as hcB(^{q) , i s given by f 0 i s the NN7C c o u p l i n g constant and L U l O = — — — (3.2.2) where -j',(qR/-n) i s a s p h e r i c a l B e s s e l f u n c t i o n of order one. The 'form f a c t o r ' U^(q) takes i n t o account the f i n i t e extent of the bare nucleon; i t has the prop e r t y that R^O U n U ^ = 1 (3.2.3) T h e r e f o r e , W < p - h , ( 3 . 2 . 4 ) Hm. In the l i m i t R->0, the Cloudy Bag Model vertex f u n c t i o n becomes e x a c t l y the vertex f u n c t i o n f o r the Chew-Low i n t e r a c t i o n d i s c u s s e d i n Schweber (1961, p.374). 21 3.3 P h y s i c a l Bosons and Fermions The Hamiltonian (3.1.1) has the f o l l o w i n g f e a t u r e : HB/(jf>lo> = f u o C ^ B / C ^ ) 0 > (3.3.1) H F ^ t ^ l O * p„+ C ?^lo> (3.3.2) fo r c V p . ) a n a r b i t r a r y f u n c t i o n . B^~(p_)|0> i s an eigenket of the Hamiltonian and t h e r e f o r e corresponds to a ' p h y s i c a l boson', i . e . , the elementary boson of the theory i s a p h y s i c a l p a r t i c l e with mass ma = m 6 o • However, F m^"(p_)|0> i s not an eigenket of the H a m i l t o n i a n . The elementary fermion of the theory i s not a p h y s i c a l fermion; we say that F T c r e a t e s a 'bare' fermion. The t r i l i n e a r i n t e r a c t i o n i s thus not e x p l i c i t l y a p h y s i c a l p a r t i c l e i n t e r a c t i o n . In the next Chapter we s h a l l develop a technique, the d r e s s i n g t r a n s f o r m a t i o n , f o r o b t a i n i n g p h y s i c a l p a r t i c l e c r e a t o r s and a Hamiltonian which e x p l i c i t l y i n v o l v e s p h y s i c a l " p a r t i c l e i n t e r a c t i o n s . 22 Chapter 4 The D r e s s i n g Transformation and ., Some Simple  A p p l i c a t i o n s As we saw i n the p r e v i o u s Chapter, the t r i l i n e a r i n t e r a c t i o n does not e x p l i c i t l y i n v o l v e p h y s i c a l fermions and bosons. We seek a t r a n s f o r m a t i o n on the fundamental dynamical v a r i a b l e s of the theory which w i l l l e a d to a Hamiltonian expressed i n terms of p h y s i c a l p a r t i c l e o p e r a t o r s . T h i s t r a n s f o r m a t i o n must be u n i t a r y and possess c e r t a i n i n v a r i a n c e p r o p e r t i e s i n order to preserve the b a s i c commutation r e l a t i o n s and t r a n s f o r m a t i o n p r o p e r t i e s of the p a r t i c l e c r e a t o r s and a n n i h i l a t o r s . We w i l l see that the bare p a r t i c l e s a c q u i r e a composite s t r u c t u r e v i a the t r a n s f o r m a t i o n ; thus i t i s c a l l e d a d r e s s i n g t r a n s f o r m a t i o n . In the f o l l o w i n g S e c t i o n s , we w i l l formulate the d r e s s i n g t r a n s f o r m a t i o n e x p l i c i t l y and apply i t to two simple, s o l u b l e models - the s c a l a r f i e l d model and the Lee model. 4.1 The General D r e s s i n g Transformation Consider the u n i t a r y operator U = e D where D1" = - D (4.1.1) and where D i s i n v a r i a n t under t r a n s l a t i o n s , s p a t i a l r o t a t i o n s , space i n v e r s i o n , time r e v e r s a l , and r o t a t i o n s i n i s o s p i n space. The operator D w i l l be s p e c i f i e d f u r t h e r below; i t w i l l be a f u n c t i o n of the fundamental dynamical v a r i a b l e s , so we w r i t e 23 D = D l F . B ^ (4.1.2) Let ff)fj. and where F and B are the fundamental fermion and boson d e s t r u c t i o n o p e r a t o r s , r e s p e c t i v e l y , d e f i n e d in S e c t i o n 2.1. Throughout the r e s t of t h i s S e c t i o n , we w i l l omit the spin and i s o s p i n l a b e l s on the p a r t i c l e o p e r a t o r s , as they only complicate the n o t a t i o n and change none of the r e s u l t s . We use the symbol 0 0 to denote a l l transformed o p e r a t o r s . Because U i s a u n i t a r y o perator, F and B obey the same commutation r e l a t i o n s as do F and B. (See equations (2.1.5) (2.1.9)). Moreover, because of the m v a r i a n c e p r o p e r t i e s of U, F and B w i l l a l s o transform under t r a n s l a t i o n s , s p a t i a l r o t a t i o n s , space i n v e r s i o n , time r e v e r s a l and r o t a t i o n s i n i s o s p i n space a c c o r d i n g to the t r a n s f o r m a t i o n laws i n S e c t i o n 2.2. For any operator A = A(F,B) i n the Fock space, we have OL A (P,B) LL1" = A t ^ B ) (4.1.5) In p a r t i c u l a r , DCF,B ^ U DIF,B) l l T = L H F ^ (4.1.6) and using (4.1.5), (4.1.1), and (4.1.6), we have HlF.BVU/ H l ? , 8 m = e H (F , B ; e. = H (F, B ) (4.1.7) Given D, equation (4.1.7) giv e s the Hamiltonian as some new 24 nj HJ ^, f u n c t i o n H(F,B) of the new, independent fundamental dynamical v a r i a b l e s F and B. Indeed, we can determine the f u n c t i o n a l form of H, expressed below as a f u n c t i o n of the dummy v a r i a b l e s F and B, using -D(F,B> DCF.B) = H t T H . D ] + ^7 C L I 4 , D ] , D 3 + • • • ( 4 . 1 . 8 ) Our s t r a t e g y w i l l be to c a l c u l a t e H from t h i s equation; from (4.1.7) we know that the Hamiltonian H(F,B) i s e q u i v a l e n t to A J A> / V A / A J H(F,B) where F and B are the transformed o p e r a t o r s . The t o t a l momentum operator i s P I F . B ^ = E t F , B ; (4.1.9) where P t r , B ) = e P C F . S ) e (4.1.10) Since D(F,B) i s t r a n s l a t i o n a l l y i n v a r i a n t , i . e . , C P , D ] = O (4.1.11) equations (4.1.9) and (4.1.10) imply f C F . t f = E CF.B) (4.1.12) Now suppose that we can w r i t e the Hamiltonian as H = Ho A- X H, (4.1.13) and that D can be expanded i n a p e r t u r b a t i o n s e r i e s i n A: oo \)= X- A° Dn (4.1.14) Equation ( 4 . 1 . 8 ) then becomes the f o l l o w i n g p e r t u r b a t i o n s e r i e s f o r H: 2 5 H = Ho + A { M, + [Mo ,0 , ] } + A* i [ H , , D , ] t Til M . , D , 2 , D . ] + r H o , D ^ l ] + . . . (4.1.15) ^ + , v t We now demand that the transformed operators F and B create physical p a r t i c l e s . That i s , we require F |0> and B |0> to be eigenkets of the Hamiltonian H(F,B), or equivalently of H(F,B). That i s , H l F , ^ F + l ^ l 0 > = <fptj>) F+(^ lo> (4.1.16) Jf ( F , ^ B+ (o)|o> = £ B(jf) B*tf>)|o> (4.1.17) where cV(p) and £ 3 ( 2 ) are some functions of p_. Equations (4.1.16) and (4.1.17) w i l l hold i f , apart from the terms F^"F and B'B, there are no terms in H(F,B), where again F and B are dummy variables, which contain only one fermion or boson annihilator, i . e . , terms of the form F t F B f f F T F B T B t , etc. (4.1.18) H(F,B) may, however, contain terms of the form F^F+FF, B+B1"BB, F rB TFB, F+F+FFB+, etc. (4.1.19) which correspond, respectively, to direct fermion-fermion, boson-boson, and fermion-boson interactions, and boson production on two fermions. These terms appear in the phenomenological Hamiltonian discussed in Hsieh (1978, p.31). We thus choose the D n to eliminate terms of the form (4.1.18) from H(F,B) given by (4.1.15). This may be done order by order in A. For the case that H t i s t r i l i n e a r , we choose Dt such that C H 0 ) D , ] = - H , (4.1.20) 26 The e x p r e s s i o n (4.1.15) then becomes H = Ho 4- Xz [± C H u D . l + E H . ,0*1 ] . (4.1.21) Next, knowing D i , D z i s chosen to e l i m i n a t e terms of the form (4.1.18) to second order i n A , and so on. The operator D c o n s t r u c t e d to s a t i s f y the above c o n d i t i o n s i s c a l l e d a d r e s s i n g o p e r a t o r . The corresponding u n i t a r y t r a n s f o r m a t i o n i s a d r e s s i n g t r a n s f o r m a t i o n ; the transformed o p e r a t o r s F and B , which c r e a t e p h y s i c a l p a r t i c l e s , are c a l l e d d ressed c r e a t o r s . We now go on to i l l u s t r a t e the d r e s s i n g t r a n s f o r m a t i o n with some simple examples. 4.2 The S c a l a r F i e l d Model The s c a l a r f i e l d Hamiltonian i s a t r i l i n e a r one i n v o l v i n g fermions and bosons. I t i s very s i m i l a r to the Hamiltonian d i s c u s s e d i n Chapter 3, except that spi n and i s o s p i n are not i n c l u d e d i n the s c a l a r f i e l d theory. The Hamiltonian i s H = Ho t \ H, M,» J d 3 p d ^ [ F+(^ B ^ J f adj. ] (4.2.1a) (4.2.1b) (4.2.1c) 27 (4.2.2) (4.2.3) where m F o and mg,, are the masses of the elementary fermion and boson, r e s p e c t i v e l y . The o p e r a t o r s F and B obey the commutation r u l e s (2.1.5) (2.1.9), o m i t t i n g a l l re f e r e n c e to spin and i s o s p i n ; h(q) i s chosen to be a r e a l f u n c t i o n independent of the fermion momentum. We may p i c t u r e the t r i l i n e a r i n t e r a c t i o n (4.2.1c) as i n Fig u r e 2: / -A V F i g . 2 The S c a l a r F i e l d Model I n t e r a c t i o n The t o t a l momentum operator f o r the system i s (4.2.4) The fermion number operator i s (4.2.5) Both P and N commute with the Hamiltonian (4.2.1). 28 We note that BT(p_)|0> i s an eigenket of the Hamiltonian, with eigenvalue £B 0(]O) , while F^"(p_)|0> i s not an eigenket. Therefore we may take m % a = mB, the mass of the p h y s i c a l boson. We seek a d r e s s i n g t r a n s f o r m a t i o n as o u t l i n e d i n S e c t i o n 4.1. The f o l l o w i n g Di s a t i s f i e s the r e q u i r e d i n v a r i a n c e p r o p e r t i e s , and equation (4.1.20): D, = S d V d 3 f 4,C^ ,^ [F+(^ F(D-^ B(^  -adj.] (4.2.6a) where and M £ , £ > = C F O ( D - ^ + £ » l ^ - £ f t ( ^ (4.2.6c) Note that t h i s operator has a s t r u c t u r e i d e n t i c a l to H,, the term i t must e l i m i n a t e . We w i l l f i n d t h i s to be a general p r o p e r t y of the d r e s s i n g t r a n s f o r m a t i o n . We now can compute [H,,D|] and f i n d [H,,D,] = - [d'pd'c h S i p F h t ) F t y ) A(p,$> T + a d j . (4.2.7) 29 The l a s t term i n [H,,D (], being of the form (4.1.18), must be e l i m i n a t e d from the dressed Hamiltonian given by (4.1.21). can be c o n s t r u c t e d to accomplish t h i s . The momentum P i s given by (4.1.9) and (4.1.12) to be P = Jd3p £ [ F f ( ^ Ftp + B f(£) B ^ ] (4.2.8) The Hamiltonian, dressed to second order, i s given by (4.1.7) and (4.1.21) as rUF,B") = M CF , B *) - T+ V F F f V P B (4.2.9) T= 5d 3 P E£pt^ FfCj>) t £B B+C^Sl^] (4.2.10) V F r - ^ d 3 f e d V d * K t f P p t ^ f e \ k O ^ (4.2.11) v F 6 = iSd^tffe'd^ I ^ B ^ . k ' , ^ F f [ t k ^ B f ( i ! < - t ) B ^ K - k O F ( . i K f f e ' ) (4.2.12) where = (4.2.13) - £ * t ^ ~ A" Jd ' o ^ (4.2.14) o r w t h , k . ^ B zi^lCiiliM) (4.2.15) l " ^ C t K + k H f e C t ^ - ^ (4.2.16) 30 R e f e r r i n g to the d i s c u s s i o n i n Appendix E (eq. ( E . 7 ) ) , we recognize iK? and l^a as the matrix elements of the fermion-fermion and fermion-boson momentum space p o t e n t i a l s . These f u n c t i o n s are p i c t u r e d i n F i g u r e s 3 and 4: i s - * ' k-k1 F i g . 3 The Fermion-Fermion P o t e n t i a l VhF(k,W^ ,K) IK+fe. K F i g . 4 The Fermion-Boson P o t e n t i a l l/F B(k,k" ,K) Note t h a t , to second order, there are two d i f f e r e n t mechanisms ( F i g . 4) by which FB s c a t t e r i n g can take p l a c e ; these correspond to the two terms i n 1 T F Q . Note a l s o that r U F , ^ F f ( £ ) l o > = M E ) F + C^)!o> ( 4 . 2 . 1 7 ) ULTt%) B f t ^ | o > = E B C ^ B ^ E ^ io> ( 4 . 2 . 1 8 ) and 31 where £ F (p_) i s the 'renormalized' fermion energy, given to second order by (4.2.14). Thus F'(p) now c r e a t e s a p h y s i c a l fermion. From (4.1.3) and (4.1.4) we see F f t ^ = F + C ^ - L > + ( ? ) , D ] + 7T [ L F f C ^ 5 D ] , D ] +... (4.2.19) = r 3 f ( ^ - [ B + C ^ , D ] + ~ [ [ ^ 1 ^ , 0 1 , 0 ] + (4.2.20) s\i 4-C a l c u l a t i o n of these dressed c r e a t o r s shows that F '(p_) i s given by an i n f i n i t e s e r i e s of terms; the f i r s t term i s the bare fermion c r e a t o r F^, the second term i n v o l v e s F^B T, the t h i r d + 4 4-term i n v o l v e s F B B 1, and so on. Thus one says that the p h y s i c a l fermion c o n s i s t s of a bare fermion surrounded by a 'cloud' of bosons. A l s o , we f i n d that B' equals B' p l u s a s e r i e s of terms which i n v o l v e fermion o p e r a t o r s , a l l of which give zero when AJ X 4-a c t i n g on the vacuum s t a t e . Thus B (p_) 10> = B'(p_)|0>, although B (p) 4 B (p_) . The d r e s s i n g t r a n s f o r m a t i o n has not changed the p h y s i c a l one boson s t a t e . If we make the s i m p l i f y i n g assumption that cVo = n1 F o C Z (4.2.21 ) i . e . , the fermion energy i s independent of i t s momentum, we d i s c o v e r s e v e r a l i n t e r e s t i n g r e s u l t s . In t h i s case, (4.2.6c) s i m p l i f i e s to = £ B ($) (4.2.22) and as a r e s u l t the f i n a l two terms i n eq. (4.2.7) v a n i s h . Moreover, now C L W,,D,3 , D , l - O (4.2.23) 32 Thus [H, ,D| ] now c o n t a i n s no terms of the form (4.1.18), so we may take Dz = 0. In turn (4.2.23) i m p l i e s that D 3 = 0. Indeed, = 0 f o r n>1. We may w r i t e the d r e s s i n g t r a n s f o r m a t i o n e x a c t l y as D = D,. The model i s thus s a i d to be s o l u b l e . Note that when (4.2.21) ho l d s , the dressed Hamiltonian no longer c o n t a i n s a fermion-boson s c a t t e r i n g term. I t does, however, have a fermion mass r e n o r m a l i z a t i o n and a fermion-fermion s c a t t e r i n g term. We f i n d equations (4.2.14) and (4.2.15) become, r e s p e c t i v e l y , m Fc* - *rv.c* - A z J H^p" (4.2.24) and V * F F ^ S £ b C o_) (4.2.25) The corresponding c o o r d i n a t e space fermion-fermion p o t e n t i a l may be obtained by from equation (E.11). I t i s fFP(r)= ^ d ^ e'* C ^ l k F (4.2.26) For the s p e c i f i c c h o i ce I (4.2.27) fo r the vertex f u n c t i o n , the i n t e g r a l i n (4.2.26) can be eva l u a t e d . We o b t a i n a c o o r d i n a t e space Yukawa p o t e n t i a l , namely = - ^ A ^ / c e (4.2.28) r The i n t e g r a l (4.2.24) i s mathematically d i v e r g e n t ; equation (4.2.24) i s taken as a d e f i n i t i o n of the bare mass mFo , i n terms of the p h y s i c a l mass mP and the ( i n f i n i t e ) i n t e g r a l . 33 In c o n c l u s i o n , the s c a l a r f i e l d model has i l l u s t r a t e d some i n t e r e s t i n g f e a t u r e s of the t r i l i n e a r i n t e r a c t i o n and the d r e s s i n g t r a n s f o r m a t i o n . The p h y s i c a l p a r t i c l e i n t e r a c t i o n s which were i m p l i c i t i n the t r i l i n e a r Hamiltonian (4.2.1) have been made e x p l i c i t through the d r e s s i n g t r a n s f o r m a t i o n . P h y s i c a l p a r t i c l e p o t e n t i a l s can be determined from the t r i l i n e a r vertex f u n c t i o n s . The p h y s i c a l p a r t i c l e c r e a t o r s can be c a l c u l a t e d and we f i n d that the p h y s i c a l fermion corresponds to a bare fermion surrounded by a cloud of bosons. F i n a l l y , we have shown that the s c a l a r f i e l d model i s s o l u b l e only when the approximation (4.2.21) i s made. For the general case that (4.2.2) holds, the d r e s s i n g operator i s given by an i n f i n i t e s e r i e s of terms and thus can onl y be determined by a p e r t u r b a t i v e procedure such as we have developed. 4.3 The Lee Model The Lee model d e s c r i b e s a t r i l i n e a r i n t e r a c t i o n i n v o l v i n g two d i s t i n g u i s h a b l e fermions and one boson. L i k e the s c a l a r f i e l d model, the spin and i s o s p i n of the p a r t i c l e s i s ne g l e c t e d . The Lee Hamiltonian i s : H = Mo + A 14 (4.3.1a) 2 (4.3.1b) (4.3.1c) 34 The momentum operator i s z p = K P I F ; (4.3.2) F,^ , Fz^, B' are the c r e a t i o n o p e r a t o r s f o r p a r t i c l e s t r a d i t i o n a l l y c a l l e d N, V, and 6\ r e s p e c t i v e l y . k o ty* CpZc2 + ^ C 4 ] V z c ( = l , 2 (4.3.3) i s the energy of the elementary fermion, and > xc x 4- r/W c 4 ] (4.3.4) i s the energy of the elementary boson. h(g) i s chosen to be a r e a l f u n c t i o n . The Lee model i s a l s o s o l v a b l e when h = h(p_,g) but t h i s only complicates the n o t a t i o n and changes none of the e s s e n t i a l r e s u l t s . Note that momentum i s conserved by the i n t e r a c t i o n , as i s the t o t a l number of N and V p a r t i c l e s and the d i f f e r e n c e between the number of N and 6*- p a r t i c l e s . The i n t e r a c t i o n H, may be p i c t u r e d as i n F i g u r e 5: IP / \ \ F i g . 5 The Lee Model I n t e r a c t i o n Thick s o l i d l i n e s are V p a r t i c l e s , s o l i d l i n e s are N p a r t i c l e s , and dashed l i n e s are & p a r t i c l e s . 35 The c r e a t i o n and d e s t r u c t i o n operators obey the f a m i l i a r fermion and boson commutation r u l e s : \ Rl (J>\ R<^(pOJ = Sip-?') &u' 1 (4.3.5) 5 ^  ( 2 \ Ri'(j»')] = 0 (4.3.6) L & ( £ ) , B F = (4.3.7) CBIE ) , BtpOl = 0 (4.3.8) LF JLCJO, B C E ' ) ] - [ Frfl^, B^CE')] - O (4.3.9) Because WF, ^ ( . E V 0 ^ ~ ty Fv+Cp^|o> (4.3.10) and M B f ( £ n o > = ?Bo B f t p ' ° > (4.3.11) F, T and fit c r e a t e p h y s i c a l p a r t i c l e s . T h e r e f o r e , m|0=m,, the mass of the p h y s i c a l N p a r t i c l e , and m g,0 = mB, the mass of the p h y s i c a l boson. However, Fz^ ~ (p_) | 0> i s not an eigenket of the Hamiltonian H so Fi*" does not c r e a t e p h y s i c a l p a r t i c l e s . We now perform a d r e s s i n g t r a n s f o r m a t i o n a c c o r d i n g to the p r e s c r i p t i o n i n S e c t i o n 4.1. Equation (4.1.20) i s s a t i s f i e d by Di ~ KP d 3 f (P,$) rFj^ C p - ) F, L$ty B(|) - Qdj.] (4.3.12a) where (4.3.12b) * M*ty 36 and A(p,Cp = £. C p - $ W £ B ($0 - f ^ C ^ (4.3.12c) Note that t h i s i s not the same A (p/Q) as i s used in the previous Section. To ensure that A ( £ , 3 ) does not vanish, we require that m2_0<ml+mB, i. e . , the V p a r t i c l e i s stable against decay into N + & . A simple computation now shows that CU,,u,] = - Jd3P d\ J £ £ L ¥+L$\\L*) •K'f 4- M f t h ( < ^ F, + C p - $ ' ) F , t j>- f} j 4- Qdj. (4.3.13) Since [H, , D i ] contains no terms of the form (4.1.18), we take D z - O (4.3.14) A computation of [[H, ,D, ],D, ] indicates that we must choose D 3 = 5d3p d3^ d3(^ ,|) \y±Lp p( ^ _ ^ B ( p - adj.] (4.3.15a) where 37 Th i s e l i m i n a t e s a l l u n s u i t a b l e terms of the form (4.1.18) from the dressed Hamiltonian, to order A . Note that D, and D3 have an i d e n t i c a l t r i l i n e a r s t r u c t u r e . Indeed i t has been p r e v i o u s l y shown (Greenberg and Schweber (1958), Piskunov (1974)) that f o r the Lee model D can be determined e x a c t l y and has a t r i l i n e a r s t r u c t u r e . We give such a c a l c u l a t i o n i n Appendix G, and determine there the p h y s i c a l V p a r t i c l e c r e a t o r Fx1. The dressed Hamiltonian, to second order, i s given by (4.1.21) and (4.3.13): H ( F , §H = T+- V12. + v2i5 +• V1B (4.3.16) T= $d3p I Z Uty F^tjf) R, (jO + £B(^B+(^6ip)] (4.3.17) ^--iW^'d^^uLk^^ (4.3.18) (4.3.19) (4.3.20) where yz = L\f-cx + t V c 4 y Z (4.3.21) k(o) = £zoCp) 4- A X f d 3 o - ^ (4.3.22) * * J T £ z . ( ^ - £ , C ? - ^ - £ B ( f ) = A*h* (k + K') + k ^ k' (4.3.23) 38 (4.3.24) 1^ ,8 U , k ' , ^ = A* h ( ^ - f c ' ) h C H - ^ (4.3.25) C I ITK + I S') +f6(-k!<-fe/) - ( Note that the energy of the V p a r t i c l e has been renormalized (eq. (4.3.22)). The f u n c t i o n s tfz, iXtB, and l^s correspond to F i g u r e s 6, 7, and 8: / F i g . 6 The N-V P o t e n t i a l F i g . 7 The V-fr P o t e n t i a l ifla / F i g . 8 The N-0" P o t e n t i a l There are no N-N or $-& s c a t t e r i n g terms present i n the Hamiltonian to second order. Again we note that the d r e s s i n g t r a n s f o r m a t i o n has made 39 e x p l i c i t the p h y s i c a l p a r t i c l e i n t e r a c t i o n s d e s c r i b e d by the o r i g i n a l t r i l i n e a r i n t e r a c t i o n . M a t r i x elements of the p h y s i c a l p a r t i c l e p o t e n t i a l s , l£ B and 1^a, have been determined i n terms of the t r i l i n e a r vertex f u n c t i o n h ( g ) . The p h y s i c a l p a r t i c l e s (see eq. (G.16)) become composites of elementary ones. 40 Chapter 5 D r e s s i n g the T r i l i n e a r Fermion-Boson I n t e r a c t i o n :  The Fermion-Fermion P o t e n t i a l We saw i n Chapter 3 that the nucleon-pion t r i l i n e a r i n t e r a c t i o n d i s c u s s e d there does not e x p l i c i t l y i n v o l v e p h y s i c a l fermions. In t h i s Chapter, we use the d r e s s i n g t r a n s f o r m a t i o n developed i n S e c t i o n 4.1 to transform the elementary p a r t i c l e c r e a t o r s i n t o p h y s i c a l p a r t i c l e c r e a t o r s . Under t h i s t r a n s f o r m a t i o n , we f i n d that to second order the Hamiltonian c o n t a i n s p h y s i c a l nucleon-nucleon and nucleon-pion i n t e r a c t i o n s . C o n c e n t r a t i n g on the nucleon-nucleon term, we use the vertex f u n c t i o n d e r i v e d from the Cloudy Bag Model to c a l c u l a t e the second-order nucleon-nucleon p o t e n t i a l . 5.1 D r e s s i n g the T r i l i n e a r I n t e r a c t i o n Consider the nucleon-pion Hamiltonian (3.1.1), with vertex f u n c t i o n h given by (3.1.35). B^(p_)|0> i s an eigenket of t h i s Hamiltonian so we take £B<>(2) = £B(E) > t n e energy of the p h y s i c a l p i o n . F ( O^"(p)|0> i s not an eigenket of H. We seek a d r e s s i n g t r a n s f o r m a t i o n as i n S e c t i o n 4.1 which w i l l determine a r*J fsj r^/ dressed Hamiltonian H(F,B) expressed e x p l i c i t l y i n terms of p h y s i c a l p a r t i c l e o p e r a t o r s . A f i r s t - o r d e r d r e s s i n g operator, which s a t i s f i e s the r e q u i r e d i n v a r i a n c e p r o p e r t i e s and (4.1.20) i s : 41 In determining t h i s operator we have taken E K . I ^ - M U . C 1 (5.1.2) We are now able to determine [H, ,D| ]. We f i n d 4- adj . (5.1.3) where r ^ x ^ y ; ' h p l M x ^ ' ' . h ^ y ^ (5.1.5) r pi flu p." - VC£<^ h-r>"*(?o3 ( 5 • ' • 6 , 42 The f i r s t term in [H, ,D| ] can be s i m p l i f i e d u s ing the form (3.1.35) f o r h. We have * hz(<?) •Cz I W X M ' ( x > M » . M " i t w r ) V w ' r^ c^ ) ( 5 . 1 . 8 ) F i r s t , (D.5) i s used to sum over yu*. and ym.", l e a v i n g $p,,/JL-Next, (B.8) i s used to i n t e g r a t e over dcfl.<j, to give Sw ',,»«". F i n a l l y , we use (D.5) to sum over m^  and o b t a i n Sr<\,,tn • Thus * ~lfd% W W ( 5 . 1 . 9 ) The l a s t term i n [H, ,DV ] i s of the form (4 . 1.18) and must t h e r e f o r e be e l i m i n a t e d from the dressed Hamiltonian given by ( 4 . 1 . 2 1 ) . We choose • f v , / ( p ^ F r t v l x ( ? . ^ fy,t^fy,«(jO - a d j . ( 5 . 1 . 1 0 ) so t h a t [HC,D2.] e x a c t l y equals minus one-half of the l a s t term in [H,,D| ] . The r e s u l t i n g Hamiltonian, to order )\~, i s : H(F ,tn = T t VTC^ + VMM ( 5 . 1 . 1 D 1 = ^ ^ P ^VLP * (5.1.12) 43 - ^  (t K+1) ^ „ + ( t K-^ V ^ * ^  ^ L i fc+fe') ( 5 . 1 . 1 3 ) V H M = ^ 21 ( Jd*d*fe'd&X ^ ^ ' ^ ^ ( K - k ' V ^ <AK + |0 F ^ C i k - ^ F ^ t i R - f e O F ^ t ^ ^ O ( 5 . 1 . U ) where m,cx = nv,cz - A 1 U l do _h!i$l_ ( 5 . 1 . 1 5 ) - ( A l ^ x l fc/0(ilwh.* \ £ w O Ctl/x1 j u ^ l t ^ C * ! wkA'llwv)] ( 5 . 1 . 1 6 ) . w x A i t ^ . H ^ i ^vM ,v (* , n , , A , , i^ ( 5 , 1 , 1 7 ) We can see from i t s s t r u c t u r e that V XNJ = V ^ t . using (B . 4 ) and (D . 9 ) , one can e a s i l y show that V^^ = as w e l l . The nucleon mass has been renormalized (eq. ( 5 . 1 . 1 5 ) ) . The f u n c t i o n s ( 5 . 1 . 1 6 ) and ( 5 . 1 . 1 7 ) may be p i c t u r e d as in F i g u r e s 9 and 10: 44 •J )<4-k ClnOtJ^ J|I,) k+k' F i g . 9 The 7C -N P o t e n t i a l iK^^'^fM'p- (k,k',K) k-k 'fyu") F i g . 10 The N-N P o t e n t i a l %^  M x M i (k-k' ) W i W t i n m — — From the F i g u r e s , we see c l e a r l y that the Clebsch-Gordan c o e f f i c i e n t s appearing i n (5.1.16) and (5.1.17) serve to impose angular momentum and i s o s p i n c o n s e r v a t i o n at each v e r t e x . The two terms i n correspond to the two d i f f e r e n t diagrams f o r 7L-N s c a t t e r i n g shown in F i g u r e 9. The p h y s i c a l one-nucleon c r e a t o r may be c a l c u l a t e d from (5.1.18) One f i n d s that the p h y s i c a l nucleon i s a composite p a r t i c l e 45 c o n s i s t i n g of a bare nucleon surrounded by a c l o u d of p i o n s . We have now seen s e v e r a l examples of d r e s s i n g t r a n s f o r m a t i o n s and may i n f e r c e r t a i n p r o p e r t i e s of these t r a n s f o r m a t i o n s . To order n, Dr» i s c o n s t r u c t e d to have the same s t r u c t u r e as the u n s u i t a b l e term which must be c a n c e l l e d . Because of the basic commutation r e l a t i o n s , [Ho,Dn] w i l l always c a n c e l e x a c t l y and only those terms r e q u i r e d . The p h y s i c a l p a r t i c l e s become composites of elementary ones v i a the t r a n s f o r m a t i o n . P h y s i c a l p a r t i c l e i n t e r a c t i o n s are expressed e x p l i c i t l y i n the dressed Hamiltonian. The s e r i e s f o r the d r e s s i n g operator and the Hamiltonian w i l l terminate or be summable only f o r very simple models. 5.2 The Second-Order Nucleon-Nucleon P o t e n t i a l In t h i s S e c t i o n we use the t r i l i n e a r fermion-boson i n t e r a c t i o n of Chapter 3 and r e s u l t s of d r e s s i n g i t to determine a c o o r d i n a t e space nucleon-nucleon p o t e n t i a l i n terms of the vertex f u n c t i o n h ( q ) . I t i s a second-order p o t e n t i a l i n the sense that i t i s obtained from the second-order term V,^ i n H(F,B). F i r s t , we r e w r i t e VMOJ i n eq. ( 5 . 1 . 1 4 ) i n terms of the two-nucleon operators A d e f i n e d by ( E . 1 2 ) to f i n d V. S K S ' M ' 15.2.1a) 46 where (5.2.1b) To a r r i v e at t h i s form f o r V Sti S'M'^Q) w e used (B.6) to combine the two s p h e r i c a l harmonics i n (5.1.17) i n t o a s i n g l e one. Based on the d i s c u s s i o n i n Appendix E, we recognize l T ^ ^ , ( g ) as the matrix elements of the momentum space two-nucleon p o t e n t i a l . Many of the sums over s p i n and i s o s p i n p r o j e c t i o n quantum numbers i n (5.2.1) can be evaluated, as we now show. We apply (D.18) to the f i r s t three i s o s p i n Clebsch-Gordan c o e f f i c i e n t s appearing i n (5.2.1) to f i n d JTTo \sz cr vU (t^^|(J^ (5-2-2) where t Vx cy y r j denotes a 6j symbol (see Appendix D). Using (D.5) to sum over ^ and /JL, we f i n d that the ex p r e s s i o n i n square b r a c k e t s reduces to C 3 = 1 W<SVz] W V (5.2.3) 47 We now c o n s i d e r the f i v e Clebsch-Gordan c o e f f i c i e n t s i n s i d e the c u r l y b r a c k e t s i n (5.2.1b). Rearranging the f o u r t h c o e f f i c i e n t u s ing (D.9d), we f i n d that these f i v e c o e f f i c i e n t s are r e l a t e d e x a c t l y by (D.25). Thus Us+i)(H+r>U)(z) ( S 4 H o ^ l S / H , , ) J_ J l 2. ^ i S ? (5.2.4) / 1 ' X\ where (. Vx S J i s a 9j symbol (see Appendix D). We can combine the r e s u l t s (5.2.3) and (5.2.4) to w r i t e ^ - ? t s ^ i s ^ vul^^sr(?) (5.2.5B) where X X >|2S+I CMOO l i o ) \ ^ I (5.2.5b) Before c o n t i n u i n g l e t us co n s i d e r the v a r i o u s f a c t o r s i n t h i s p o t e n t i a l . F i r s t , we note that the f a c t o r <W ^p' makes the co n s e r v a t i o n of i s o s p i n and i t s z- a x i s p r o j e c t i o n manifest. Secondly, i t can e a s i l y be shown that the requirement of r o t a t i o n a l i n v a r i a n c e alone means that ^ ^ ^ 1 ( 3 . ) must be of the 48 form (5.2.5a). The s p e c i f i c form (5.2.5b) i s a consequence of the t r i l i n e a r i n t e r a c t i o n with which we began. T h i r d l y , note that the Clebsch-Gordan c o e f f i c i e n t ( l l o o l i o ^ ) i s non-zero only f o r I = 0,2, corresponding to the s c a l a r and tensor p a r t s of the strong i n t e r a c t i o n , r e s p e c t i v e l y . The c o n s e r v a t i o n of sp i n i s i m p l i c i t i n the 9j symbol, which i s zero unless the three angular momenta i n any row or column form a t r i a d . Thus, f o r j l = 0,2, S = s' n e c e s s a r i l y . The quantum numbers M and M' are not r e q u i r e d to be equal. Based on the above d i s c u s s i o n , we may w r i t e the nucleon-nucleon i n t e r a c t i o n (5.1.14) as N L i - A* i d ' k d W K J _ , V*Z A ^ + C k , k ) ^ ( 5 . 2 . 6 ) where 1=0,2. . LS± *\ di \St\') ) I ' -* ( (5.2.8) T h i s constant s p e c i f i e s . t h e s p i n and i s o s p i n dependence of the p o t e n t i a l . We have determined i t s numerical values using Messiah (1958, pp. 1065 - 68) to eva l u a t e the 6j and 9j symbols and values are given i n Table I f o r v a r i o u s t o t a l spins and i s o s p i n s . We may. use (E.11) to r e l a t e the momentum space nucleon-49 nucleon p o t e n t i a l ^ M M ' ( g ) to the corresponding c o o r d i n a t e space potent i a l : tfrtK' ( ^ " E' ^  ^ M « ' ^ (5.2 . 9 ) Using (5.2.7), (B.7), and (B.8), we o b t a i n iTgJ. ( O = X- ^ ( r J i r J ^ M (5.2.10) JUo,2. where < f ( r l - H*l< a ^ / < s . 2 . n ) Thus, given a t r i l i n e a r v e r t e x f u n c t i o n h(q) we can compute the corresponding c o o r d i n a t e space second-order N-N p o t e n t i a l by performing the i n t e g r a t i o n i n equation (5.2.11). JU o S = 0 a O S = 1 M^K" SMO I J M I Z I - M V - N I I M O » 'TO <r= \ 3 ^ SM^ 1 Sr\o — CIZMM'-MIIMO S i r Table I Values of the Constant a 50 5.3 The Nucleon-Nucleon P o t e n t i a l i n the Cloudy Bag Model The c o o r d i n a t e space nucleon-nucleon p o t e n t i a l f u n c t i o n s s JL 1}MM! ( r ) , given by (5.2.11), are the matrix elements of the s c a l a r ( 1 = 0 ) and tensor ( 1 = 2 ) p a r t s of the second-order nucleon-nucleon p o t e n t i a l taken between two-nucleon s t a t e s having spi n S and i s o s p i n CT . We now c a l c u l a t e these matrix elements by s u b s t i t u t i n g the Cloudy Bag Model vertex f u n c t i o n given by (3.2.1) i n t o (5.2.11). We have [ ^ * V > l c S H = ^ OS' H^r,*) ( 5 . 3 . , , where CO Nitr.RV f d$ oi+Vc1 I t ^ ^ fts0)2 (5.3.2) o » We show i n Appendix I that f o r the case r > 2R, i . e . , when the two bags do not o v e r l a p , the i n t e g r a l s Nj(.(r,R) are e a s i l y e v a l u a t e d using a r e s u l t given i n Watson (1966), i n v o l v i n g a contour i n t e g r a t i o n over products of B e s s e l f u n c t i o n s . Remarkably, the i n t e g r a l s Nx.(r,R) f a c t o r i n t o the product of a simple f u n c t i o n of the bag r a d i u s R and a f u n c t i o n of the co o r d i n a t e r . Indeed, t a k i n g k = mK, b = R, and a = r i n equations (1.13) and (1.14), we d i s c o v e r C t f ^ o L a * = [iJ-MM'V^OPaP S ( f 0 (5.3.3) where < 3 ^ = C/*RV L (5.3.4) [ /0" M H/ ( r _ ) ] 0 p E P are the matrix elements of the c e n t r a l and 51 tensor one pion exchange p o t e n t i a l , which i s d i s c u s s e d i n Appendix J , and LX= ^ (5.3.5) S p e c i f i c a l l y , the Cloudy Bag Model p o t e n t i a l s are t>5S.°co3CBM - V o o C O i - u f, lk) all'0 e ^ r 9 I R ) r r>2R (5.3.6) and (^r) ' — r 7 2 R (5.3.7) Thus, when the two nucleon bags are not touching (r>2R), the Cloudy Bag Model nucleon-nucleon p o t e n t i a l i s e x a c t l y the same f u n c t i o n of the nucleon s e p a r a t i o n r_ as i s the p o t e n t i a l c a l c u l a t e d from one pion exchange. The s p i n and i s o s p i n dependence of the two p o t e n t i a l s i s a l s o i d e n t i c a l . In l i g h t of the d i s c u s s i o n i n Appendix J , we recognize the constants a ^ i as being p r o p o r t i o n a l to the s p i n - i s o s p i n matrix elements of the nucleon o p e r a t o r s 0",. ( £ = n ) and S l x ( A = 2 ) , which appear i n The two p o t e n t i a l s V C B H and V o p e p d i f f e r i n o v e r a l l s t r e n g t h by the f u n c t i o n g(R). Note that Thus, f o r r > 0, the Cloudy Bag Model p o t e n t i a l becomes e x a c t l y 52 the one pion exchange p o t e n t i a l as R->0. R e c a l l (eq. (3.2.4)) that i n t h i s l i m i t we o b t a i n the Chew-Low i n t e r a c t i o n . We see that f o r t h i s i n t e r a c t i o n , the i n t e g r a l (5.2.12) d i v e r g e s when I =0. T h i s divergence g i v e s r i s e to the £ (r) term i n the one pion exchange p o t e n t i a l . There are no such divergences i n the Cloudy Bag Model p o t e n t i a l . The f u n c t i o n un(q), which accounts f o r the f i n i t e s i z e of the bare nucleon, a c t s as a p h y s i c a l l y meaningful c u t o f f f u n c t i o n which keeps a l l i n t e g r a l s i n the theory f i n i t e . I t i s i n t e r e s t i n g to note that f o r the value R = 0.72 fm. p r e d i c t e d by Theberge ejt a l . we f i n d g (0.7Z f r r O = \.0S (5.3.9) The Cloudy Bag Model p o t e n t i a l d i f f e r s only s l i g h t l y from the one pion exchange p o t e n t i a l , by the f a c t o r 1.05.. So f a r we have c o n s i d e r e d the Cloudy Bag Model p o t e n t i a l f o r the case r > 2R. The i n t e g r a l s Njj(r,R) can a l s o be evaluated f o r the case r < 2R, although t h i s region i s p h y s i c a l l y l e s s c l e a r because i t i m p l i e s an o v e r l a p of the nucleon bags. For completeness, however, we give the r e s u l t f o r 1=0, obtained from a contour i n t e g r a t i o n : - T ( I*" z ^ - ^ R ^ j r^ZR (5.3.10) T h i s s o l u t i o n matches smoothly with (5.3.6) i n the l i m i t r — » 2 R . 53 U n l i k e the s o l u t i o n f o r r > 2R, however, i t does not f a c t o r i n t o a f u n c t i o n of R and a f u n c t i o n of r . F i n a l l y , i n the l i m i t r->0, i t goes to <\ e.~^Z - ( l - ^ R ^ j (5.3.11) which becomes i n f i n i t e f o r R->0. 54 Chapter 6 Summary and C o n c l u s i o n s Our study of s t r o n g l y i n t e r a c t i n g fermions and bosons began with the nucleon-pion t r i l i n e a r i n t e r a c t i o n of Chapter 3. Thi s i n t e r a c t i o n was extended to i n c l u d e fermions and bosons of a r b i t r a r y s p i n and i s o s p i n i n Appendix H. We found that the requirement of i n v a r i a n c e under t r a n s l a t i o n s , s p a t i a l r o t a t i o n s , space i n v e r s i o n , time r e v e r s a l , and r o t a t i o n s i n i s o s p i n space g r e a t l y r e s t r i c t e d the form of the t r i l i n e a r vertex f u n c t i o n . We were able to use the Cloudy Bag Model to o b t a i n a s p e c i f i c e x p r e s s i o n f o r the NNJt t r i l i n e a r vertex f u n c t i o n . The fermion-boson t r i l i n e a r i n t e r a c t i o n does not e x p l i c i t l y i n v o l v e p h y s i c a l p a r t i c l e s , because F^"|0> i s not an eigenket of the Hamiltonian. In Chapter 4, we developed a d r e s s i n g t r a n s f o r m a t i o n which acted on the bare p a r t i c l e c r e a t o r s and a n n i h i l a t o r s to transform them i n t o p h y s i c a l p a r t i c l e o p e r a t o r s . We d e r i v e d an e x p r e s s i o n (eq. (4.1.20)) f o r the Hamiltonian as a f u n c t i o n of the p h y s i c a l p a r t i c l e c r e a t o r s and a n n i h i l a t o r s , and gave a p r e c i s e method f o r determining the d r e s s i n g t r a n s f o r m a t i o n to any d e s i r e d order i n p e r t u r b a t i o n theory. A p p l i c a t i o n of t h i s d r e s s i n g t r a n s f o r m a t i o n to the s c a l a r f i e l d model, the Lee model, the nucleon-pion t r i l i n e a r i n t e r a c t i o n , and to the g e n e r a l i z e d fermion-boson t r i l i n e a r i n t e r a c t i o n i l l u s t r a t e d s e v e r a l common f e a t u r e s of the t r a n s f o r m a t i o n . F i r s t , we d i s c o v e r e d that the p e r t u r b a t i o n s e r i e s f o r the d r e s s i n g operator only terminates f o r very simple i n t e r a c t i o n s , and that f o r r e a l i s t i c t h e o r i e s a technique such as we have developed must be used to f i n d the operator D. Secondly, to any 55 given order i n p e r t u r b a t i o n theory, D n i s c o n s t r u c t e d to have the same s t r u c t u r e as the u n s u i t a b l e term of the form (4.1.18) which must be e l i m i n a t e d from the dressed Hamiltonian. [H 0 ,D n] w i l l then c a n c e l e x a c t l y and only the term r e q u i r e d . T h i r d l y , the d r e s s i n g t r a n s f o r m a t i o n g i v e s a s p e c i f i c e x p r e s s i o n f o r the p h y s i c a l p a r t i c l e c r e a t o r s as composites of the elementary p a r t i c l e c r e a t o r s . We found a fermion mass r e n o r m a l i z a t i o n i n the second-order c a l c u l a t i o n s ; to higher orders we expect an analogous v e r t e x r e n o r m a l i z a t i o n . F i n a l l y , the p h y s i c a l p a r t i c l e i n t e r a c t i o n s i m p l i c i t i n the t r i l i n e a r Hamiltonian are made e x p l i c i t by the d r e s s i n g t r a n s f o r m a t i o n , l e a d i n g to i n t e r a c t i o n p o t e n t i a l s t h at can be c a l c u l a t e d from the o r i g i n a l t r i l i n e a r vertex f u n c t i o n . A second-order nucleon-nucleon p o t e n t i a l f o r the Cloudy Bag Model was determined in Chapter 5. We found that when the two nucleon bags were not touching, t h i s p o t e n t i a l was simply the one pion exchange p o t e n t i a l m u l t i p l i e d by a f u n c t i o n of the bag r a d i u s . As the bag r a d i u s goes to zero, the Cloudy Bag Model N-N p o t e n t i a l goes to the one pion exchange p o t e n t i a l . The nucleon-pion Hamiltonian, when dressed to second order, was found to c o n t a i n not only a nucleon-nucleon i n t e r a c t i o n but a l s o a d i r e c t nucleon-pion i n t e r a c t i o n . When the theory i s extended to i n c l u d e the A , we o b t a i n N- A / A ~ A , and A - % i n t e r a c t i o n s as w e l l . P o t e n t i a l s f o r these i n t e r a c t i o n s can be c a l c u l a t e d analogously to our dete r m i n a t i o n of the nucleon-nucleon p o t e n t i a l . When the fermion-boson t r i l i n e a r Hamiltonian i s dressed to t h i r d order, we f i n d a term d e s c r i b i n g boson p r o d u c t i o n on two fermions, i . e . $ i r F^F^FF^B^". The d r e s s i n g 56 t r a n s f o r m a t i o n approach allows us to c a l c u l a t e if i n terms of an i n t e g r a l i n v o l v i n g the b a s i c t r i l i n e a r v ertex f u n c t i o n . T h i s c o n t r a s t s with the Hamiltonian c o n s i d e r e d by Hsieh (1978), where t h i s f u n c t i o n l)~ i s determined only phenomenologically. Thus our work on the d r e s s i n g t r a n s f o r m a t i o n can be used to f i n d such i n t e r a c t i o n p o t e n t i a l s . In t u r n , they can serve as a b a s i s f o r the f u n c t i o n a l form of a long-range, strong i n t e r a c t i o n Hamiltonian f o r systems of pions and nucleons at intermediate e n e r g i e s . 57 B i b l i o g r a p h y Abramowitz, M. and Stegun, I., 1965, Handbook of Mathematical  F u n c t i o n s , Dover, New York. D i r a c , P.A.M., 1949, Rev. Mod. Phys. 2J_, 392. Edmonds, A.R., 1960, Angular Momentum i n Quantum Mechanics, P r i n c e t o n U n i v e r s i t y Press, P r i n c e t o n . Faddeev, L.D., 1964, S o v i e t P h y s i c s Doklady 8, 881. G l o c k l e , W. and M u l l e r , L. , 1981 , Phys. Rev. C 2_3, 1183. Greenberg, O.W. and Schweber, S.S., 1958, Nuovo Cim. 8, 378. Hsieh, W.W., 1978, M.Sc. T h e s i s , U n i v e r s i t y of B r i t i s h Columbia. Jahn, H.A. and Hope, J . , 1954, Phys. Rev. 93, 318. Kal y n i a k , P., 1978, M.Sc. T h e s i s , U n i v e r s i t y of B r i t i s h Columbia. Messiah, A., 1958, Quantum Mechanics v.2, Wiley, New York. Moravcsik, M.J., 1963, The Two-Nucleon I n t e r a c t i o n , Clarendon, Oxford. Ord-Smith, R.J. , 1954, Phys. Rev. 9_4, 1227. Piskunov, V.N., 1974, Theor. Math. Phys. J_5, 546. Rose, M.E., 1957, Elementary Theory of Angular Momentum, Wiley, New York. Schweber, S.S., 1961, An I n t r o d u c t i o n to R e l a t i v i s t i c Quantum  F i e l d Theory, Harper and Row, New York. Theberge, S., Thomas, A.W. and M i l l e r , G., 1980, Phys. Rev. D 22, 2838. Watson, G.N., 1966, Theory of Be s s e l F u n c t i o n s , Cambridge U n i v e r s i t y Press, Cambridge. 58 4 Appendix A The Rota t i o n M a t r i c e s D^i m (A g>t ) T h i s Appendix l i s t s some u s e f u l p r o p e r t i e s and formulae i n v o l v i n g the D m w' (<Ap~f ) , the i r r e d u c i b l e matrix r e p r e s e n t a t i o n s of the r o t a t i o n group. We take the conventions f o r these matrices to be those used i n Rose (1957) and Messiah (1958). Note that these d i f f e r from the conventions used i n Edmonds (1960). From Rose (1957, p.54), we have Upr) = £ W C-tf .-p,-^ (A.2) When A. = 0, Da°0 U S T ) = i <A-3> When o( =Y= 0, the r e a l m a t r i c e s d^Vn are d e f i n e d by <£>mW " ^ C o f > o ) (A.4) Note t h a t The o r t h o g o n a l i t y p r o p e r t i e s of the r o t a t i o n m a t r i c e s a r e , from Rose (1957, p.73): ^ D J L * $**m U p ) = $„,.„,<• (A.6) X D^* Wfctf 0™« (^pTT) » oW ffjno' ^ p o ^ '"mm * v... ... (A. 7) 59 The r u l e f o r combining two r o t a t i o n m a t rices i n t o one i s (Rose (1956, p.58)): F i n a l l y , from Rose (1957, p.75), we have Z7C T 2A. (A.8) = S*».W Sm,,*,. Sj,j^ C ^ T , ) (A. 9 ) 60 Appendix B The S p h e r i c a l Harmonics (q) T h i s Appendix l i s t s some u s e f u l p r o p e r t i e s and formulae i n v o l v i n g the s p h e r i c a l harmonics. We w r i t e y w ty * ie,**) ( B , 1 ) where B and ^ are the p o l a r and azimuthal angles, r e s p e c t i v e l y , used to s p e c i f y the d i r e c t i o n of the v e c t o r g. Yjtm transforms under r o t a t i o n s a c c o r d i n g to (Rose ( 1957, p.60)) : il V A m - 2- DmWi y l w i, ( B > 2 ) where - ^ % (B.3) In (B.3), g i s a column v e c t o r i n i t s three C a r t e s i a n c o o r d i n a t e s and M i s the matrix given i n Rose (1957, p.65). From Rose (1957, p.61) we have Urn t-£> = 0 A ^ (B.4) C C | ) * CO" ( B . 5 , where ( j i j z M . r / K I j f O i s a Clebsch-Gordan c o e f f i c i e n t (see Appendix D). From Rose (1957, p.81), ^ - H - J . J u i ' l i W O W ^ k l r i ( B. 7 l 61 A l s o , from Rose (1957, p.75), we have J dG/H timty ft*, ty = j I ^ (S,(P) Y l W (a,^ * n * d G dCD o o 62 Appendix C I r r e d u c i b l e Tensor Operators and the Wigner-Eckart  Theorem An i r r e d u c i b l e tensor operator of rank L i s a set of 2L+1 q u a n t i t i e s T l H (-L < M < L) which transform under r o t a t i o n s a c c o r d i n g to (Rose (1957, p.77)): i -LM' (C.1) (Note that the above operator R i s not the Fock space operator # u p " ) ) . Any vec t o r k = (k*, ky, k z) i s an i r r e d u c i b l e tensor of rank one, with s p h e r i c a l components A l t e r n a t i v e l y , the s p h e r i c a l components of a vec t o r can be w r i t t e n i n terms of the s p h e r i c a l harmonics: k, - ^ . |S" V w t ^ . , , ± , , 0 ( C 3 ) The r o t a t i o n a l l y i n v a r i a n t s c a l a r product of two v e c t o r s can be w r i t t e n i n terms of t h e i r s p h e r i c a l components: = X. t-v4 k* a , (c.4) Consider the matrix element of the i r r e d u c i b l e tensor operator T u <^ between angular momentum s t a t e s . The Wigner-Eckart theorem gi v e s the dependence of t h i s matrix element on the p r o j e c t i o n quantum numbers. I t s t a t e s (Rose (1957, p.85)): 6 3 where \\ T\ ll j V i s c a l l e d the reduced matrix element of the set of tensor o p e r a t o r s T L M . For the case that the T L ^ are the angular momentum operator s J , t h i s reduced matrix element i s (Rose (1957, p.89)): 5lVj>= vj iCj + .) Sjj' ( C . 6 ) 64 Appendix D Angular Momentum Coupling In t h i s Appendix, we c o n s i d e r the a d d i t i o n of v a r i o u s numbers of angular momenta to form a t o t a l angular momentum j(m). Because such a c o u p l i n g can take p l a c e v i a d i f f e r e n t intermediate r e p r e s e n t a t i o n s , c o e f f i c i e n t s to connect these r e p r e s e n t a t i o n s must be d e f i n e d . These are the Clebsch-Gordan c o e f f i c i e n t s , 3 j , 6 j , 9j and 12j symbols. A f t e r d e f i n i n g these c o u p l i n g c o e f f i c i e n t s , we l i s t some u s e f u l formulae i n v o l v i n g them. (a) A d d i t i o n of Two Angular Momenta: The Clebsch-Gordan  Coef f i c i e n t and the 3j Symbol Consider a system i n v o l v i n g two angular momenta J, and J t coupled to form a t o t a l angular momentum J : 5, + 5* » £ (D.1) T h i s system can have four simultaneously d i a g o n a l i z a b l e o p e r a t o r s . Let Ij, ^  m.tYi*.^ denote the b a s i s s t a t e i n which J , , iZz / / a r e d i a g o n a l , and ' j i j i j t ^ the s t a t e in which J , , J Z , J , JJ;. are d i a g o n a l . These two r e p r e s e n t a t i o n s are r e l a t 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 : and I j i j z r n , ^ ^ = 2_ I j, j \ jm >^0'v}*-*)iMz IjijxjfYi') (D.3) The elements of t h i s t r a n s f o r m a t i o n , j no )> , are the Clebsch-Gordan c o e f f i c i e n t s , which we denote by 65 ^jijitfhwh. lj * I f c P o s s i b l e t o d e f i n e a phase convention f o r these c o e f f i c i e n t s such that they are r e a l : T h e i r o r t h o g o n a l i t y p r o p e r t i e s are d e r i v e d from the requirement that the two r e p r e s e n t a t i o n s be normalized: (D.5) 2- t j i i z J j m ) ( J . J , tf,. ^ - ^ i W | i l g 0 The c o e f f i c i e n t s have the prop e r t y that u n l e s s m, + m^ = m, and ( j , , j z , j ) form a t r i a d , i . e . , (D.6) (D.7) (D.8) The symmetry r e l a t i o n s of the Clebsch-Gordan c o e f f i c i e n t s are given i n Rose (1957, pp. 38-39): Cj '. j iW.iYij.lJatfiO =- ( - ^ , + " J l - " j 3 m x M ( | j 3 W 1 i ) (D .9a) (D.9b) (j« Ja V1'-Wslji-Wz.) (D.9c) J l + M 2 I 2 i +. I = t-0 vJ^V" ( ^ 0 * - W 3 ^ l o . - M . ) (D. 9d) (r^ ' ^ 'J^ j'xVl ^3 i 1 "1* -»^ < 1 via «0 (D.9e) = ^ " " l ^ r - ^ i s ^ ^ l j ^ ^ ( D- 9 f ) 66 From Rose (1957, p.42) and (D.9), we have ( j , O O j j a l Y O = Sj.jj SrtJ.y^  (D.\0) (O J z O ^ l j j ^ ^ S j x i ' 3 S K ) , M 3 ( D , 1 1 ) ' J ' ^ * ' 0 0 ^ ^ ^ ^ ( D - , 2 ) One can d e f i n e a 3j symbol which i s r e l a t e d to the Clebsch-Gordan c o e f f i c i e n t s by (Edmonds (1960, p.46)): UtfK»*J~ r,- ^ . O x M . ^ I ^ - ^ (D.13) (b) A d d i t i o n of Three Angular Momenta: The 6j Symbol Consider the f o l l o w i n g two schemes for combining three angular momenta J t , J z , J 3 to form a t o t a l J : J, + O x = J , z J,^ J 3 = J (D.14) and Ji + - Iz» • J, +• £xs = 2 (D.15) In the f i r s t case, the o p e r a t o r s J, , Jz, J 3 , J l 2_ , J can be made d i a g o n a l ; we denote the corresponding s t a t e by ^j>iOjn •> i j , jy • I n t ^ i e second case, the operators J, , J 3 , J 2 , , J can be d i a g o n a l i z e d ; the s t a t e i s I j ( , Cjxja") j i x t j y . The 6j symbol, which r e l a t e s these two independent r e p r e s e n t a t i o n s i s d e f i n e d by (Edmonds (i960, p.92)): 67 = ( = = 1 ){ \ W>\ (D-16) The two d i f f e r e n t r e p r e s e n t a t i o n s (D.14) and (D.15) can be b u i l t up from Clebsch-Gordan c o e f f i c i e n t s . For example, • I j . r f O lja.wU> l j i M * > (D.17) S u b s t i t u t i n g such expansions i n t o (D.16) and using (D.5), one f i n d s The 6j symbol i s i n v a r i a n t under a permutation of i t s columns and under interchange of the upper and lower v a r i a b l e s i n each of any two columns. In c e r t a i n cases the 6j symbol can e a s i l y be evaluated n u m e r i c a l l y . From Edmonds (1960, p.95): U J 5 i x \ - & [(!*•,, ( D . 1 9 ) LVx Jj'Yx jxtVi J L J . N . u - - v — I (D.20) (2jx+i ) (2jx+OC2jV) Uj3f0 J i Jx J3 7 J1+J2.4-J3 f" /*."'.'. ."N *. .* N ~7 ^ 2-2jz czjx+o U jOUjs+O ( D . 2 1 ) 68 (c) A d d i t i o n of Four Angular Momenta: The 9 j Symbol Consider the f o l l o w i n g two schemes f o r combining four angular momenta to make a t o t a l J : = In. Ii + I * = J 3 4 I n f ^ = J ( D ' 2 2 ) and The 9j symbol, which r e l a t e s these two independent r e p r e s e n t a t i o n s , i s d e f i n e d by (Edmonds (1960, p.101)): C J i J»- J I T . ) • r i w « M 2 i « 4 . v w , i ( w j h ' j i 0 (d-24> Expanding the s t a t e s i n (D.24) as in (D.17) and using (D.5) twice, one f i n d s * W) a in1if I j x M r/l,* r / l x H | j no) J I J x 0>z OH 03*1 (D.25) A c t i n g on both s i d e s of (D.25) with (j a „VWj,» Ij'^ ') and u s i n g (D.5) g i v e s w b 4 69 m,x r/ko ^ ' ^ « o 3 >nn,Wl» Iju mO.O CjajM rO*Wk mOiO* • mArt \jmr>") ( 0«- W\x r f l 3 v [ j m ^ (D.26) From (D.26) and the p r o p e r t i e s of the Clebsch-Gordan c o e f f i c i e n t s , we see that the 9j symbol i s zero unless the three angular momenta i n any row or column form a t r i a d . The 9j symbol i s i n v a r i a n t under a t r a n s p o s i t i o n , or an even permutation of rows or columns ( i . e . , an even number of exchanges of adjacent rows or columns). Under an odd permutation, the phase changes by In c e r t a i n cases, the 9j symbol can be e a s i l y e v a l u a t e d n u m e r i c a l l y . From (Edmonds (1960, pp. 101,105,106)): CO C a. b e (D.27) K I um i * J OH JzH ") C ) r t 0 3 4 j J ( D < 2 8 ) 70 (d) A d d i t i o n of F i v e Angular Momenta: The 12j Symbol Consider the f o l l o w i n g two schemes f o r combining f i v e angular momenta to make a t o t a l J : and I , + CTa = 3"t» 3s t 3 « - 2" Ix 3H -- Iz* I" f J 2 M = I (D.30) The 12j symbol, which r e l a t e s these two independent r e p r e s e n t a t i o n s , i s d e f i n e d by (Ord-Smith (1954)): = .1 l l J 1 0 i is ot 0i* 05 • fl (D.31 ) Expanding the s t a t e s i n (D.31) i n terms of Clebsch-Gordan c o e f f i c i e n t s , and using (D.5) three times, we f i n d w^ rti* tiflci W „ W » M M " • ( j y ^ ^ s ^ ^ ( j \ ^  w^yvu yY]^)' • (05 jiz ^ $ »0ia Ij'w') C 0" j x s rfl" vY)^ | j to ) ' ( j ' v)W m ' *>3M J'M j i u o" / (D.32) j . J l _ < • II 0 Oft o i 4 71 The symmetry r e l a t i o n s of the 12j symbol are given i n Ord-Smith (1954). When one element i s zero, the 12J symbol s i m p l i f i e s to a b e c } ^ C d f 9 ? = [UevOOq^J^ ^ d f 7 (D.33) g h o 5 J 72 Appendix E T w o - P a r t i c l e Operators i n Fock Space T h i s Appendix pro v i d e s some background to the v a r i o u s two-body p o t e n t i a l s and two-fermion oper a t o r s that are used i n the t h e s i s . T w o - p a r t i c l e operators i n Fock space are c o n s t r u c t e d from the corresponding two-body op e r a t o r s i n the H i l b e r t space of n p a r t i c l e s . In t h i s Appendix we d e a l only with fermion Fock space; the extension to i n c l u d e bosons i s e a s i l y made. For a complete d e r i v a t i o n of the form of t w o - p a r t i c l e o p e r a t o r s i n Fock space see Schweber (1961,pp.140-2). Suppose that the t o t a l p o t e n t i a l Vn of a system of n i d e n t i c a l p h y s i c a l fermions i s due to t w o - p a r t i c l e i n t e r a c t i o n s . Then i n the n-fermion H i l b e r t space, we have p r o v i d i n g t h at v ( ? * , = v e t , , o (E.2) t«i denotes the c o o r d i n a t e s , momentum, s p i n , and i s o t o p i c s p i n of fermion oi. V ( f A , fp) i s the p o t e n t i a l between fermions o{ and The corresponding Hermitian operator i n Fock space i s : ( E . 3 ) 73 where the index j runs over 1,2,3,4. F 1 i s the c r e a t i o n operator f o r a p h y s i c a l fermion and That i s , the xrj- ^ (x,y,x',y') are the c o o r d i n a t e space matrix elements of the two-fermion p o t e n t i a l V(£.,,^p) taken between two-fermion s t a t e s . Making the change of c o o r d i n a t e s 1 ' = R'+yzr' K ' - / 2 r ' (E.5) to center of mass and r e l a t i v e c o o r d i n a t e s , and imposing the c o n d i t i o n of displacement i n v a r i a n c e , t h i s t w o - p a r t i c l e operator becomes V- I jd»r d V d 3 R d V (r.r 1, R-R') -(E.6) Note that V~ can only depend on the d i f f e r e n c e R-R' , i f V i s to be t r a n s l a t i o n a l l y i n v a r i a n t . Using (2.1.1), V can a l s o be expressed as where 74 e e e. v E - s ; For the s p e c i a l case that we f i n d that the i n v e r s e of equation (E.8) i s where i < X W = K<r ^ ^ ( E. n ) We now d e f i n e a two-fermion operator as f o l l o w s : A * a + 0 K ) = 7=- ( s l s i r f l , ^ I S M ) ( i , L i u , a 2 ( ^ ) • ~ S(SXL,U ^ z Mifri*.|i,yu-z • F ^ , + I t t t ^ F ^ t i t - f c ' ) ( E .,2) As a f u n c t i o n of the c o o r d i n a t e s £ and R t h i s becomes The two operat o r s A j ^ ^ " (fc,K) and A ^ ( r , R ) , when a c t i n g on the vacuum s t a t e , c r e a t e a two-fermion s t a t e having t o t a l s p i n s, with p r o j e c t i o n m, and t o t a l i s o s p i n CT , with p r o j e c t i o n LL . They are r e l a t e d , using (2.1.1), by 75 Using the p r o p e r t i e s of the Clebsch-Gordan c o e f f i c i e n t s i n Appendix D, the t w o - p a r t i c l e i n t e r a c t i o n V can be w r i t t e n as i ^ d ' k ' d'K c?;:^ (s,^^ A ^ V ^ A $ ( E . 1 5 ) S<5 S '0 ' where 5 j 1J ^iMj C L H c 3 / ^ y a 3 l a y ) i r j ^ CK.K'.K) (E.16) The p o t e n t i a l f u n c t i o n s IT^- t j W 7y,(k,k/ ,K) are the momentum space matrix elements of the two-fermion p o t e n t i a l V ( f r f , f ^ ) between two-fermion s t a t e s of t o t a l s p i n s, i s o s p i n CT and t o t a l s p i n s' , i s o s p i n C . These f u n c t i o n s may be p i c t u r e d as i n F i g u r e 11: 2 K + is. F i g . 11 The Two-Fermion P o t e n t i a l i T - . - ^ . (k,k ,K) 5J LJMJPJ 76 Appendix F C a l c u l a t i o n of the T r i l i n e a r Vertex F u n c t i o n f o r the  Cloudy Bag Model In t h i s Appendix, we determine the NN/i t r i l i n e a r vertex f u n c t i o n h(q) f o r the Cloudy Bag Model. The one-nucleon matrix elements of the nucleon-pion i n t e r a c t i o n i n the Cloudy Bag Model are (Theberge e_t a l • (1980, eq. ( 2 . 2 5 ) ) ) : <rt iH5SN7« ^ 1 5^S 3fc V V B ^ f Q d j . ( F . 1 a ) ir/"ck)* t ^ r ^ unco < b * . , - k y O t M \ k ^ , T ^ y < F . U » where f<, i s the NN7T c o u p l i n g constant, m^  i s the pion mass, and c J C V , = ( ^ ^ l ) Y l (F.2) U J O O = 3a",(.hR)/ hR ( F . 3 ) where ^ ((kR) i s a s p h e r i c a l Bessel f u n c t i o n of order one and R i s the bag r a d i u s . F u r t h e r , I t i o , ija> i s the s p i n - i s o s p i n s t a t e d e s c r i b i n g a nucleon with spi n p r o j e c t i o n m and i s o s p i n p r o j e c t i o n JUL ; S = £[/z. i s the nucleon s p i n o perator, where <5 has as i t s components the P a u l i s p i n m a t r i c e s ; 1^= t^lz are the s p h e r i c a l components of the nucleon i s o s p i n o p e r a t o r . The above i s w r i t t e n i n u n i t s where fi = c = 1 . Note the s i m i l a r i t y between the i n t e r a c t i o n (F.1) and the form (3.1.41) for the nucleon-pion t r i l i n e a r i n t e r a c t i o n . We determine the vertex f u n c t i o n h(q) f o r the Cloudy Bag Model by 7 7 e v a l u a t i n g the s p i n - i s o s p i n matrix elements i n (F.1b) and comparing the r e s u l t with the one-nucleon matrix elements of H( given by (3.1.36). F i r s t , we separate the s p i n and i s o s p i n p a r t s of the matrix element. Expanding S. • fe i n s p h e r i c a l components using (C.4), we have = ^ t-V* k+ iSdL ^ - k u , ( x ^ | - k M l > (F.4) The Wigner-Eckart theorem (C.5) g i v e s < k rf\, I ] t r Y l i > = t z I M i c H l rfO S ||:£> (F.5) <"z/A, H M l - I / O = Cx» J ^ l t y O ^ II I I-L > (F.6) where i = ^11 X H-k> = J i / z (F.7) S u b s t i t u t i n g (F.5), (F.6), and (F.7) i n t o (F.1), and using (C.3) to express q.^ i n terms of q and a s p h e r i c a l harmonic, we f i n d • ( x I /^ i -yj- 1 "ai /O C i l rf^i l-knn^ j^C^) + ady (F.8) Now c o n s i d e r the matrix elements of the i n t e r a c t i o n H, i n (3.1.36) between one nucleon s t a t e s . Using the commutation r e l a t i o n s f o r the fermion o p e r a t o r s , we f i n d 78 (F.9) Comparing (F.8) and (F.9), and p u t t i n g i n ft and c as r e q u i r e d to give the c o r r e c t dimensions to the vertex f u n c t i o n which we w r i t e as h C B h l ( q ) , we see that where 6^  (q) i s given by equation (3.1.3). The two d i f f e r e n t formalisms can be made i d e n t i c a l by the ch o i c e (F.10) f o r the vertex f u n c t i o n . We are able to do t h i s because both t r i l i n e a r i n t e r a c t i o n s , (F.1) and (3.1.36), are r e q u i r e d to be i n v a r i a n t under t r a n s l a t i o n s , r o t a t i o n s , space i n v e r s i o n , and time r e v e r s a l . R o t a t i o n a l i n v a r i a n c e determines that both i n t e r a c t i o n s w i l l have the same dependence on spin and i s o s p i n p r o j e c t i o n quantum numbers and on the angles g. Space i n v e r s i o n i n v a r i a n c e r e q u i r e s that the o r b i t a l angular momentum of the pion be one, and t h i s i s manifested i n (F.1) by the ve c t o r c h a r a c t e r of the operator S (eq. ( F . 5 ) ) . (F.10) 79 Appendix G An Exact D r e s s i n g Operator f o r the Lee Model In t h i s Appendix we f o l l o w the work of Piskunov (1974) i n determining an exact d r e s s i n g operator f o r the Lee model d i s c u s s e d i n S e c t i o n 4.3. (The method of s o l u t i o n f o r the case h = h(p_,g) i s analogous to the procedure given here.) As equations (4.3.12a) and (4.3.15a) suggest, we w r i t e D = $ d ? p d3o_ c k f , ^ j F , ( ? - ^ B ( £ ) - a d j . ] ( G . D and attempt to f i n d the f u n c t i o n d(p_,g). We do t h i s by determining the c r e a t o r Fx (p.) i - n terms of d(p_,g) and then r e q u i r i n g t h a t F i T(p_)|0> be an eigenket of the Hamiltonian. R e c a l l from (4.1.3) that Fjty = e° F,f e ~ D = K \ ^ - trSip,*]* ir £ £ F ^ ? > , D J * , D > . . . ( G - 2 ) Using (G.1) f o r D, and the commutators (4.3.5) - (4.3.9), we f i n d U F a ^ . D ] = J d ^ d ( ? , ^ r,+ CD , | )B + ^ ) (G.S) and where The s e r i e s i n (G.2) thus separates i n t o two s e r i e s , each of which can be summed e x a c t l y . We f i n d 80 Fz+(^ = cosd(^ - Jd'f fc ? 1^ F,+c?-f>Bty ( G - 6 ) where T h i s equation g i v e s F,_T (p) i n terms of d(£,cj). We now s u b s t i t u t e F X T (p) i n t o the equation H F ^ l ^ l o ) - Fz + l jrt lo> (G.8) where E ^ g ) i s the energy of the p h y s i c a l V - p a r t i c l e s t a t e . Using the commutators (4.3.5)-(4.3.9) and the ex p r e s s i o n (4.3.1) f o r H, equation (G.8) becomes [cosdC^ [ £ 2 0 c * V - A J d ^ h C | U f ? l ^ ] FA^|o> - F , + Cf>-^ b f C p lo> = O (G.9) Taking the s c a l a r product of t h i s equation from the l e f t with <0|F i(p /) and then with <0|F, ( p / - ^ )B(g/ ) g i v e s J c o s d ( ? ' ) cos < (G. 1 1 ) S u b s t i t u t i n g (G.I 1) i n t o (G.10) gives the r e s u l t f*(if> * ?zc CP') + A* f d 3 o ; (G.12) 81 We now s o l v e f o r d(p_,g) by s u b s t i t u t i n g f(p_,g) from (G.7) i n t o (G.11), squaring the r e s u l t i n g equation, and i n t e g r a t i n g both s i d e s with respect to g. We f i n d W ^ C ; ^ = X* (G.13) where (G.14) F i n a l l y , from (G.11), (G.7), and (G.13), we have d ( £ , Q ) = "arc-fan ( A 3 C ^ ; ( G > 1 5 ) g(E) r^(^)- £ , C J > - $ W B C $ \ I Thus, f o r the Lee model, we f i n d that one can w r i t e the d r e s s i n g t r a n s f o r m a t i o n e x a c t l y , as in (G.1), with d(p_,g) given by (G.15). Indeed, i f t h i s D i s expanded i n a power s e r i e s i n A , one o b t a i n s e x a c t l y the same d r e s s i n g t r a n s f o r m a t i o n to f i r s t , second, and t h i r d o r d e r s as i s given by equations (4.3.12), (4.3.14), and (4.3.15). Furthermore, we note that i f we i t e r a t i v e l y solve (G.12) f o r £z(p_) > the s o l u t i o n to order X7" i s t h a t given by equation (4.3.22). Using (G.15), (G.7) and (G.6) we o b t a i n the p h y s i c a l V-p a r t i c l e c r e a t o r : FzV= C o s q ^ f c V - A C d S o ^F.*f?-?>BV 7 (G.16) The p h y s i c a l V - p a r t i c l e s t a t e i s a s u p e r p o s i t i o n of a s t a t e c o n t a i n i n g an elementary V p a r t i c l e with a s t a t e c o n t a i n i n g one N and one tJ p a r t i c l e . 82 Appendix H A-Dressing Transformation For A G e n e r a l i z e d Fermion- Boson T r i l i n e a r I n t e r a c t i o n In t h i s Appendix we g e n e r a l i z e the t r i l i n e a r i n t e r a c t i o n of Chapter 3 to i n c l u d e fermions and bosons of a r b i t r a r y s p i n and i s o s p i n . In doing so, we not only are ab l e to c o n s i d e r i n t e r a c t i o n s of nucleons and p i o n s , but a l s o i n t e r a c t i o n s of other types of fermions and bosons. F i r s t , we examine the r e s t r i c t i o n s p l a c e d on t h i s g e n e r a l i z e d i n t e r a c t i o n by the requirement that i t be i n v a r i a n t under c e r t a i n space-time t r a n s f o r m a t i o n s . Secondly, we perform a d r e s s i n g t r a n s f o r m a t i o n on the fundamental dynamical v a r i a b l e s as i n S e c t i o n 5.1. F i n a l l y , we c o n s i d e r the v a r i o u s terms i n the second-order dressed Hamiltonian, p a r t i c u l a r l y the fermion-fermion s c a t t e r i n g term. Although the formulae become more complicated with t h i s g e n e r a l i z e d i n t e r a c t i o n , the theory of the d r e s s i n g t r a n s f o r m a t i o n can be a p p l i e d as s u c c e s s f u l l y to t h i s i n t e r a c t i o n as to the simpler t h e o r i e s we have p r e v i o u s l y s t u d i e d . (a) The G e n e r a l i z e d Fermion-Boson T r i l i n e a r I n t e r a c t i o n Consider the f o l l o w i n g g e n e r a l i z e d t r i l i n e a r Hamiltonian c o n s t r u c t e d from the fundamental 'dynamical v a r i a b l e s d e f i n e d i n S e c t i o n 2.1: H = H0 + X H, (H.1a) 83 H»» 7- J*pffi£V'£V * £yVO?>J (H-,b) S t m u ' S'i'mV m.itiairtjju.ya^ j +. adj. (H.1c) where A S ; ^ = [ p V t ^ c H j y i (H.2) i s the energy of the elementary fermion or boson having spi n s, i s o s p i n i and i s o s p i n z-axis p r o j e c t i o n ju. . The fermions and bosons are t r e a t e d ' s e m i - r e l a t i v i s t i c a l l y ' and the vertex f u n c t i o n i s chosen to be a f u n c t i o n of g only. The i n t e r a c t i o n H, may be p i c t u r e d as i n F i g u r e 12: 1 £ UM.I.JU) F i g . 12 The G e n e r a l i z e d T r i l i n e a r F-B I n t e r a c t i o n The t o t a l momentum operator f o r the system i s 84 The Hamiltonian H i s t r a n s l a t i o n a l l y i n v a r i a n t . We a l s o r e q u i r e that i t be i n v a r i a n t under r o t a t i o n s i n i s o s p i n space, s p a t i a l r o t a t i o n s , space i n v e r s i o n , and time r e v e r s a l . Ho a l r e a d y s a t i s f i e s these requirements, but H, does not without f u r t h e r r e s t r i c t i o n . F i r s t , we con s i d e r a r o t a t i o n i n i s o s p i n space. We must have tf?x US*") H, # i + «(J7f) = M, (H.4) Using (2.2.18) and (2.2.19), we proceed s i m i l a r l y to steps (3.1.6) through (3.1.10). We d i s c o v e r that i s o s p i n r o t a t i o n a l i n v a r i a n c e i m p l i e s Secondly, we c o n s i d e r a s p a t i a l r o t a t i o n , demanding flup-rt H, ftf up*) = H, ( H - 6 ) Analogously to (3.1.13), u s i n g (2.2.8) and (2.2.9), we f i n d • c£, ; ^ C ; * ( H . 7 1 As i n (3.1.14), we w r i t e S u b s t i t u t i n g t h i s e x p r e s s i o n i n t o (H.7), we o b t a i n an ex p r e s s i o n comparable to (3.1.17), namely 85 • VL\*?T) DM5;„. Wrt D » V ^ < H- 9 ) Using ( A . 1 ) and ( A . 8 ) to combine the f i r s t p a i r of r o t a t i o n m a t r ices i n t o one, and then to combine the second p a i r i n t o one, we proceed as i n ( 3 . 1 . 1 8 ) and ( 3 . 1 . 1 9 ) . We d i s c o v e r that the m,, ma., m3, M dependence of the vertex f u n c t i o n i s given by T h e r e f o r e , from (B.5) and ( H . 8 ) , we can w r ite W ty - 1 £s  W < p h A s c ^ V • ( 5 t S s l s « 0 ( ^ S, rn M l l S « 0 (H.11) T h i r d l y , we r e q u i r e that Hv be i n v a r i a n t under space i n v e r s i o n : {P H, <PF = H, (H.12) T h i s i m p l i e s , as in (3.1.22) and (3.1.23), that f o r p o s i t i v e p a r i t y fermions and negative p a r i t y bosons. S u b s t i t u t i n g f o r the vertex f u n c t i o n from (H.11), we see that t h i s i m p l i e s C - ) X = - l (H.14) That i s , & must be odd i n order to have space i n v e r s i o n 86 i n v a r i a n c e . F i n a l l y , we consid e r a time r e v e r s a l t r a n s f o r m a t i o n . We r e q u i r e J H, J f = Hi (H.15) Using (2.2.14), (2.2.16) and (2.2.17), t h i s leads to K C <P = C ^ h ^ ( H ' 1 6 ) where we have taken 7l p = 1 and 77_B = - 1 « F o l l o w i n g steps s i m i l a r to (3.1.29) through (3.1.34), we conclude Thus the i n t e r a c t i o n (H.1c) can be w r i t t e n H ,*i . • Sd3p cl3<jL Cut's/xru» l^ u,") Csiss wi3ls«<V • CA S, miryj, l S - 0 T l r v > h A s <-<fV • f F^ 1 ; . C ? - ^ B £ £ O p 4- Odj. (H. 18) where & i s odd and hs,i*jfs l , l*' 3(q) i s a r e a l f u n c t i o n . R e f e r r i n g to (H.18) and F i g u r e s 12, 13, and 14, we i n t e r p r e t the quantum number s i n h 5 ,* 7js t , t , t j ( q ) as the t o t a l s p i n r e s u l t i n g from c o u p l i n g fermion spi n s z to boson spin s 3 . JL i s the o r b i t a l angular momentum of the boson ( s 5 , i 3 ) with res p e c t t o the fermion ( s z . , i i ) . We see that £ + s = s, r i n order to conserve angular momentum. For the p a r t i c u l a r case that s, = S i = i , = iz = 1 /2 , s s =m3 =0 , ij=1 , we recover the i n t e r a c t i o n (3.1.36). From the p r o p e r t i e s 87 of the Clebsch-Gordan c o e f f i c i e n t s i n ( H . 1 8 ) , we see that s = 1/2 and i = 1 f o r t h i s case. T h e r e f o r e • V1M h ^ J ^ V (H.19) Noting that Y,^  ty I I k "\ M i l a i / O = fc-V Y^ C -5 1 mx -m I -5 m,') (H.20) and comparing (H.19) with (3.1.35), we see where h(q) i s the vertex f u n c t i o n introduced i n Chapter 3 for the nucleon-pion i n t e r a c t i o n . (b) D r e s s i n g the G e n e r a l i z e d I n t e r a c t i o n Just as we have noted f o r the Hamiltonian (3.1.1), the one boson ket B^^(p_)|0> i s an eigenket of the Hamiltonian (H.1), while the one fermion ket F^jJ-(p_) | 0> i s not. We r e q u i r e a d r e s s i n g t r a n s f o r m a t i o n as developed i n Chapter 4. To f i r s t o r der, a s u i t a b l y i n v a r i a n t d r e s s i n g operator that s a t i s f i e s (4.1.20) i s m . r r t z i m j y a ^ x ^ j £ & ^ "tyi j ^ ~ Q J' (H.22) where we have taken 88 (H.23) The commutator [ H , , D i ] can now be computed. I t i s found to c o n t a i n an u n s u i t a b l e term of the form F^FBB + a d j . , which can be e l i m i n a t e d from the second-order dressed Hamiltonian through a s u i t a b l e c h o i c e of the d r e s s i n g operator Bz . T h i s operator w i l l be s i m i l a r to D 2 given by (5.1.10). From (4.1.21), the r e s u l t i n g second-order Hamiltonian i s : T-Z- K IV; c1 F : ; % T ^ §$VK£<0] <H-25> B ( F , B ) = T + \ipB t Vpp (H.24) Z S , S A S % t | t t l 3 T<H' H*.' 5d 3 k d V d^ K l T P B C k , k ' , k V + adj. (H.26) S . ' S x ' C ' t ' i ' i i . . i F:;".*" F i j i f ( * K- ^ F;# I**- ^  fttc * * ^ (H.27) where 2. (H.28) 89 • Us, •*>«>, ls«f) Cs^ 's3' m,' IsV^ U ' i . Is^'V • U ^ ' m ^ ' l i O Cs.s*' «m,^ »' I s'^') U's z hrt'wii IsuO • • ( i S , <yiWl, 1 5< 0 ( S i ' S 3 K»i I s'-iO (X'-s/-i^'m/ I s V V (H.30) Equation (H.28) g i v e s the fermion mass r e n o r m a l i z a t i o n , to order A*. The f u n c t i o n s 'ZTpg and opF are p i c t u r e d i n F i g u r e s 13 and 14: 90 Jt' tSs ' -ViV/ i , ' ) tutrixiVyuO ts*w«.V/.,') • 4 / / 2. K +• k' ( S t v/i, i iya F i g . 13 The Fermion-Boson P o t e n t i a l l/^ Q(_k,k / ,K) i K +•)}1 F i g . 14 The Fermion-Fermion P o t e n t i a l tJpp(k-k/ ) The fermion-fermion p o t e n t i a l can be s i m p l i f i e d by e v a l u a t i n g many of the sums over p r o j e c t i o n quantum numbers in (H.30). F i r s t , we r e w r i t e the fermion-fermion i n t e r a c t i o n i n 91 terms of the two-fermion o p e r a t o r s d e f i n e d i n Appendix E: ' +-Qdj. (H.31) where - 2 1 . . H J U K i v ^ s , S i S s t , i » i 3 : 1  * Hrc (zt+O r • |(-,> ( S . S X ' M , ^ ' | S U M 5 a . S , ' I s ' M ' U l ' * "l'm I LA) ( . S x S j wix n f) 8l •US|W)*I.U«0 CSz'SsVyia'no, U V ) (1's,' U V ) } (H.32) As i n (5.2.3), we use (D.18) and (D.5) to show Z C ] = L-) ^f,)LUM t- 6 l " ( H.33) Next, we rearrange c e r t a i n of the c o e f f i c i e n t s i n the c u r l y b rackets i n equation (H.32) using (D.9), and then apply (D.32) to o b t a i n 21 ^ } - 2 1 (_^ 5'' + S l t s s + 2,Si - 2.S + s' - A±A' +& + JL' +rt m,Mirnr* w i.W m L-^A) _ .s,' + Sx-Sj-s+s'-A-A'4-UV f s j V s*' ' t' ) CSLMA U ' H ' ) j ^ s / s' | ( A JL' 5, L v • [t2SzVlM2A+lMzil+l)(2*+l')(2L+0 U S + O j V * (H.34) 92 5» A' S*' S , Sa s,' s' X . where (_ x ^ s L J i s a 12] symbol, whose p r o p e r t i e s are d e s c r i b e d i n Appendix D. Thus we may w r i t e the fermion-fermion p o t e n t i a l as U S + I ) (2JH-Q Czg.'+ i ) .(SLMAIS-MO S I S'' 5' * . { H 3 5 ) The form of t h i s p o t e n t i a l i s very s i m i l a r to the nucleon-nucleon p o t e n t i a l (5.2.5). Both have the s t r u c t u r e r e q u i r e d by r o t a t i o n a l i n v a r i a n c e , and i n both the c o n s e r v a t i o n of i s o s p i n and i s o s p i n z - a x i s p r o j e c t i o n i s man i f e s t . Indeed, i n the case that Si =SI = SI' = S i ' - i i = i t = i ,' = i a ' = l / 2 , s 3 =m3 = 0 , and i s = 1 , i . e . , we are c o n s i d e r i n g two nucleons i n t e r a c t i n g through pion exchange, the i n t e r a c t i o n V F F can be shown to equal VMN given by (5.2.6), using (H.21) and (D.33).. Equation (H.35) g i v e s a fermion-fermion p o t e n t i a l i n terms of a t r i l i n e a r vertex f u n c t i o n . I f we wished to c a l c u l a t e a n u c l e o n - d e l t a p o t e n t i a l , f o r example, we c o u l d use the NATC p i e c e of the Cloudy Bag Model Hamiltonian, p i c t u r e d i n F i g u r e 15, to o b t a i n h x*/i. '3' ' i n t n e same way as we found 93 h C B H ( q ) i n Appendix F. We would then be able to use t h i s N&7C ver t e x f u n c t i o n to i n t e g r a t e as i n S e c t i o n 5.3, g i v i n g the d e s i r e d N-A p o t e n t i a l . S i S x S , l , u l j 1 is (<p N 1 \ \ V \ \ \ F i g . 15 The NAT T r i l i n e a r Vertex Thick s o l i d l i n e s are A p a r t i c l e s , s o l i d l i n e s are nucleons, dashed l i n e s are p i o n s . 94 Appendix I_ Some P r o p e r t i e s of B e s s e l Functions In t h i s Appendix, we in t r o d u c e and l i s t some Be s s e l f u n c t i o n s encountered i n the d i s c u s s i o n i n Chapter 5. We then c o n s i d e r c e r t a i n i n t e g r a l s i n v o l v i n g these Bes s e l f u n c t i o n s . For a complete d i s c u s s i o n of B e s s e l f u n c t i o n s , we r e f e r to the book by Watson (1966). Ji/(z) denotes an o r d i n a r y B e s s e l f u n c t i o n of the f i r s t k i nd, of order V and argument z. Both y and z are u n r e s t r i c t e d , complex v a r i a b l e s . The B e s s e l f u n c t i o n s of h a l f -i n t e g r a l order are c a l l e d s p h e r i c a l B e s s e l f u n c t i o n s . j n ^ z ^ i s a s p h e r i c a l B e s s e l f u n c t i o n of the f i r s t k i n d . I t i s r e l a t e d to the o r d i n a r y B e s s e l f u n c t i o n s by The s p h e r i c a l Bessel f u n c t i o n s of the second kind are, from Abramowitz and Stegun (1965, p.433): We w i l l a l s o r e q u i r e the m o d i f i e d s p h e r i c a l Bessel f u n c t i o n s of the f i r s t , second, and t h i r d kinds, namely i n ( z ) , i _ n ( z ) , and k n ( z ) , f o r n = 0,±1,.... They are d e f i n e d i n terms of the s p h e r i c a l B e s s e l f u n c t i o n s . From Abramowitz and Stegun (1965, pp. 443-4), we have (1.1 ) n=o,± I, • • (1.2) YlT-VZ CO = 1» (& (-E^artjZ ^ K) = e (1.3) 95 1 SCn+Oxi/z = e. y p o O ( f^QrgZ ( 1 . 4 ) ( 1 . 5 ) We now l i s t some Bess e l f u n c t i o n s of small order, taken from Abramowitz and Stegun (1965, pp. 433-4, 443-4). 1a(*> S i n i Sin 2 _ x cos 2 C05 £ ( 1 . 6 ) •jo C O = -U C l O = ~ C o S * 4-a * cost _ Sir> 2 2: cosVi 2 2: 3 ~ ; f -jr ) Sinn £ - — C 0 5 h ^ 22 b , ^ = JL e~* (/ + ~ ) 2 2 2 Xs- > ( 1 . 7 ) (1 . 8 ) ( 1 . 9 ) 96 The small argument l i m i t of the spherical Bessel functions i s : 2^0 CZn + 0 ! I %(Z)^=LJZ -C2n-0'. ! ( i ) n + l (1.10) where (2n+1)!! = (2n+1)(2n-1 )...(3)(1 ) We now consider certain contour integrations involving spherical Bessel functions. From Watson (1966, eq. (9), p.430), choosing p=3/2, yu,=3/2=yq2, b,=b2=b, and V = l+l/2 for X an even integer, we have oo x2 + k2 = - [Xj/iCb^ l2" K^Lak) Cn) providing a>2b and 1<4. Writing the ordinary Bessel functions in terms of spherical ones using (1.1), (1.3), and (1.5), this becomes co , l + i / z 97 When £ = 0 , using (1.8) and ( 1.9), we f i n d a>2b ( 1 .13 ) When £=2, using (1.8) and ( 1.9), the i n t e g r a l becomes CO - 0 > 2- db^ fe ak (o^ bfc s,w ( I ' 1 4 ) a >2b 98 Appendix J One Pion Exchange P o t e n t i a l s (OPEP) In t h i s Appendix, we summarize the r e s u l t s of c a l c u l a t i o n s of nucleon-nucleon p o t e n t i a l s from one pion exchange. These p o t e n t i a l s are d i s c u s s e d i n most t e x t s on nuclear p h y s i c s . See, f o r example, the book by Moravcsik (1963). Beginning with a s u i t a b l e n o n - r e l a t i v i s t i c l i m i t of the Lagrangian i n t e r a c t i o n and c o n s i d e r i n g one pion exchange in a p e r t u r b a t i v e theory, the f o l l o w i n g second-order nucleon-nucleon p o t e n t i a l i s obtained: • M r c r r s He where 5 a = S C . - r ^ . r - G \ ' 0 * . ( J.2b) G", and C i x are the P a u l i s p i n o p e r a t o r s in the H i l b e r t spaces of nucleons 1 and 2, r e s p e c t i v e l y ; Z, and T z are the analogous i s o s p i n o p e r a t o r s ; f i s the NN7T c o u p l i n g constant ( f 2 = .08). The r e l a t i v e s e p a r a t i o n of the nucleons i s r = r z - r_( ; mK i s the pion mass. The above i s w r i t t e n i n u n i t s where -ft = c = 1. Note that t h i s i n t e r a c t i o n has a s c a l a r p a r t which i s p r o p o r t i o n a l to and a tensor p a r t , p r o p o r t i o n a l to the operator SIZ, . Consider the s c a l a r OPEP, f o r r^O. Taking matrix elements 99 between two-nucleon s t a t e s of spin S and i s o s p i n <o , we o b t a i n I 1 A simple c a l c u l a t i o n shows ^SM ICJ, - q z IS'M'> = Sss' < W ^ " 3 5=S'^0 S i m i l a r l y - 5 l s = S = I <5 - C3' = I (J.3) (J.4) (J.5) The r e s u l t i n g matrix elements of the p o t e n t i a l are given i n Table I I . X= o S = o S= I r -•9 $MM< r r r Table II Matrix Elements of the S c a l a r OPEP L. V H , (.£U 100 Appendix K Dressing a Poincare I n v a r i a n t System In t h i s Appendix we c o n s i d e r the consequences of a p p l y i n g a d r e s s i n g t r a n s f o r m a t i o n to a system which i s i n v a r i a n t under a Poincare t r a n s f o r m a t i o n , i . e . , one which i s not only i n v a r i a n t under t r a n s l a t i o n s , s p a t i a l r o t a t i o n s , space i n v e r s i o n and time r e v e r s a l , but a l s o under homogeneous Lorentz boosts. To d e s c r i b e a system which i s Poincare i n v a r i a n t , one must c o n s t r u c t from the fundamental dynamical v a r i a b l e s of the system ten Hermitian operators P J, J J , H, KJ ( j = l , 2 , 3 ) s a t i s f y i n g the Poincare a l g e b r a (see, f o r example, Kalyniak (1978, p.23): (K.1a)-(K.Ic) f > , H > o C T ^ H ] - o [>*, -if, e j w |<A Ck*, P h] - S j b H / c 2 (K.2a)-(R.2d) [k>>H] = - c f c P J C ^ , K k ] = - ' ^ ^ k J - J l / c l (K.3a)-(R.3b) j.h.JL * 1»^»3 The momentum operator P i s the generator of s p a t i a l t r a n s l a t i o n s ; the angular momentum J i s the generator of s p a t i a l r o t a t i o n s ; the Hamiltonian H i s the generator of time t r a n s l a t i o n s ; the operator K i s the generator of Lorentz boosts. For a system of f r e e fermions and bosons the momentum operator P and the Hamiltonian H are given i n Fock space by equations such as (4.2.4) and (4.2.1b). J D and K 0 w i l l a l s o i n v o l v e F^F and B^B terms and expressions f o r these f r e e p a r t i c l e o p e r a t o r s i n Fock space can be obtained from the corresponding n - p a r t i c l e o p e r a t o r s . For example, see 101 Kalyniak (1978, pp.76,77) f o r the n - p a r t i c l e angular momentum and Lorentz boost o p e r a t o r s . To d e s c r i b e a Poincare i n v a r i a n t system of i n t e r a c t i n g fermions and bosons, we introduce i n t e r a c t i o n s u s i n g the i n s t a n t form of D i r a c (1949), i . e . , we l e t I - J o M = Wo t A (41 k = K c + A k t (K.4) Thus we have a system of i n t e r a c t i n g fermions and bosons with generators P 0 * J<> » H ' IS which s a t i s f y the Poincare a l g e b r a (K.1) - (K.3). We choose the i n t e r a c t i o n s H, and K, to be t r i l i n e a r . The vertex f u n c t i o n h must be a f u n c t i o n of the fermion momentum p_ as w e l l as the boson momentum g i n order to s a t i s f y the commutation r e l a t i o n s i n v o l v i n g K and H. The systems which we have c o n s i d e r e d p r e v i o u s l y i n t h i s t h e s i s c o u l d not be made Poincare i n v a r i a n t because we chose the vertex f u n c t i o n to depend only on g. Taking h = h(p_,g), the f u n c t i o n a l dependence of the verte x f u n c t i o n on 2 a n <3 g w i l l be r e s t r i c t e d by the Poincare a l g e b r a , j u s t as i t s dependence on spin was determined in Chapter 3 by space-time i n v a r i a n c e requirements. Now c o n s i d e r a d r e s s i n g t r a n s f o r m a t i o n on the fundamental dynamical v a r i a b l e s of t h i s system. The tr a n s f o r m a t i o n i s given by equations (4.1.1) - (4.1.4). Since the d r e s s i n g /\J **~> /v f\J t r a n s f o r m a t i o n i s u n i t a r y , the op e r a t o r s P 0 , J e , H, K must s t i l l 1 02 obey the Poincare a l g e b r a . D i s i n v a r i a n t under t r a n s l a t i o n s and r o t a t i o n s , i . e . , C P 0 ) D] - tlo, D] * O (K.5) so the f u n c t i o n a l form of P 0 and J 0 w i l l not be changed by the d r e s s i n g t r a n s f o r m a t i o n : As i s S e c t i o n 4.1 we w r i t e co D = X. > n D n ( K - 8 ) The dressed Hamiltonian H w i l l be given by (4.1.15); K w i l l be given by a s i m i l a r e x p r e s s i o n . We now choose D h to e l i m i n a t e u n s u i t a b l e terms of the form (4.1.18) from K to order n. Thus K = k 0 + A x \ \ I k, ,0.1 + CKo , D» l ] + . •'• (K .9) That i s , K = k 0 4- A K n (K.10) where K n c o n t a i n s no terms with a s i n g l e fermion or boson a n n i h i l a t o r (other than F^F and B^B). T h i s i m p l i e s that equation (4.1.20) f o r H holds, as we now show. The dressed o p e r a t o r s s a t i s f y the Poincare a l g e b r a ; t h e r e f o r e (K.1 1 ) 103 We a l s o have, from (K.2d), - i ^ f e H./c* * C k o J , P e k ] (K.12) -•'^ Sife H./c 1 = [W , J , ? o k ] (K.1.3) Taking the commutator of equation (K.9) with , we f i n d 4- ITW ,0*3, P6k]| + • - . (K.14) Note that we can use (K.5), (K.13) and a J a c o b i i d e n t i t y to show, f o r example, 0 = -6K/C- S j k rH,,0,] (K. 1 5) The r e f o r e , u s i n g (K.11), (K.12), (K.13) and (K.5) equation (K.14) becomes H = + • A* ^ t H . , D, ] + Fu 0 j D i l } + • • • (K. 1:6) or CO % = l\0 + T > n H n (K.17) n-2. where H 0 a l s o c o n t a i n s no u n s u i t a b l e terms of the form (4.1.18). A / T h i s i s the same s e r i e s f o r H that we obtained i n equation (4.1.20). Thus i f we choose D to e l i m i n a t e a l l u n s u i t a b l e terms from K , we a u t o m a t i c a l l y e l i m i n a t e a l l such terms from H as w e l l . Is the Poincare i n v a r i a n c e of the r e s u l t i n g theory s a t i s f i e d order by order i n A ? C e r t a i n l y (K.1) i s s t i l l 1 04 s a t i s f i e d by P c and J 0 . A l s o , i n l i g h t of (K.5), we have Z?o , H n ] = T3o , H n ] = O (K.18) From (K.11), (K.12) and (K.5), i t must hol d that [ L . P o M » Sjfe Qn f e (K.19) and [ k . J , X > * ] = in £jfeJL k n * (K.20) Thus (K.2) i s s a t i s f i e d by the dressed generators to order n. Consider (K.3a): L i^ , ft] = - i * P o * (K.21) Since t K o J ,HoI - -.^Po J (K.22) t h i s i m p l i e s + £ A " " rk„ J , i L ] = 0 (K.23) Thus (K.3) cannot be s a t i s f i e d order by order i n A . If the s e r i e s f o r K and H are t r u n c a t e d , (K.23) w i l l not .hold, and the Poincare i n v a r i a n c e of the theory i s destroyed. In summary, we have shown that i t i s p o s s i b l e to apply a d r e s s i n g t r a n s f o r m a t i o n to a Poincare i n v a r i a n t system by c o n s t r u c t i n g the d r e s s i n g operator Dn to e l i m i n a t e the u n s u i t a b l e terms of the form (4.1.18) from K, the generator f o r 105 Lorentz boosts. Procedures s i m i l a r to those given i n Chapter 4 can be used to c a r r y out t h i s d r e s s i n g t r a n s f o r m a t i o n . As a consequence, the dressed Hamiltonian c o n t a i n s no u n s u i t a b l e terms and thus F and B T c r e a t e p h y s i c a l p a r t i c l e s . One cannot t r u n c a t e the s e r i e s f o r H or K and maintain Poincare i n v a r i a n c e . An a l t e r n a t i v e p e r t u r b a t i v e approach to d e s c r i b i n g a Poincare i n v a r i a n t system i s given i n Glb'ckle and Mu'ller (1981). 

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