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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  AND ITS  TRANSFORMATION  APPLICATION  TO A FERMION-BOSON TRILINEAR INTERACTION by DEBORAH JEAN HEARN B.Sc,  The U n i v e r s i t y  o f S a s k a t c h e w a n , 1979  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  in THE FACULTY OF GRADUATE  STUDIES  DEPARTMENT OF PHYSICS  We a c c e p t to  this  thesis  the required  as conforming standard  THE UNIVERSITY OF BRITISH COLUMBIA July  (c)  1981  D e b o r a h J e a n H e a r n , 1981  In p r e s e n t i n g  this thesis  i n partial  fulfilment of the  r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y of B r i t i s h Columbia, I agree that it  the Library  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 study.  s h a l l make I further  agree that p e r m i s s i o n f o re x t e n s i v e copying o f t h i s  thesis  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 h e a d o f my d e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . understood that for  copying or p u b l i c a t i o n of t h i s  f i n a n c i a l gain  Physics  The U n i v e r s i t y o f B r i t i s h 2075 W e s b r o o k P l a c e V a n c o u v e r , Canada V6T 1W5 Date  DE-6  (2/79)  Ta(vj  thesis  s h a l l n o t be a l l o w e d w i t h o u t my  permission.  Department o f  It i s  oV7 , | 8 I Q  Columbia  written  i i  Abstract In  this  potentials boson  thesis, various are  determined  trilinear Working  vertex  i n Fock  interaction  particles.  We  functions  we  does a  note  not  relations  the  of the theory  perturbatively formulated,  fermion-  fermion-boson  involve  dressing  on t h e f u n d a m e n t a l p a r t i c l e  creators  invariance  the to  into  physical  properties  are preserved.  applied  called  physical  the  determining  and i s  the  explicitly  transformation,  which a c t s  and  interaction  of the b a s i c  that  a n n i h i l a t o r s . They a r e t r a n s f o r m e d  operators,  strong  function.  develop  transformation,  as  space,  trilinear  and  fermion-boson  A precise  dressing  some  and  particle  commutation  technique  transformation  simple  models  in  for is field  theory. The particle  dressing  interactions  interaction. interaction, pion  transformation  When we  applied  find  scattering  implicit  term,  and  i n the second-order  vertex  function derived  discover  coordinate that  between  them  modified  in strength  is  the  the  than given  a  the p h y s i c a l  original  trilinear  nucleon-pion  nucleon-nucleon Hamiltonian.  from t h e C l o u d y  providing  greater  to  dressed  space  distance  in  explicit  a n u c l e o n mass r e n o r m a l i z a t i o n ,  present  nucleon  makes  potential the  twice by  two the  the  be  a  nucleon-  s c a t t e r i n g term Using  Bag M o d e l ,  can  trilinear  the  the nucleon-  calculated.  nucleons are separated bag one  radius, pion  NN7C  the  We by a  potential  exchange p o t e n t i a l  by a f u n c t i o n o f t h e bag r a d i u s .  iii  TABLE OF CONTENTS Abstract  i i  List  of T a b l e s  List  of F i g u r e s  v vi  Acknowledgements  v i i  Chapter  1  Introduction  Chapter  2  F e r m i o n and Boson F u n d a m e n t a l and T h e i r  . Dynamical  Variables  Properties  4  2.1 The F u n d a m e n t a l D y n a m i c a l V a r i a b l e s 2.2 Space-Time  Transformation  Properties  4 of the  Fundamental Dynamical V a r i a b l e s . . . . Chapter  3  The T r i l i n e a r  Fermion-Boson  Interaction  3.1 R e s t r i c t i o n s Due t o C e r t a i n  4  Vertex  Function  i nthe  Bag M o d e l  19  3.3 P h y s i c a l  Bosons  and F e r m i o n s  The D r e s s i n g  Transformation  21  and Some S i m p l e  Applications  22  4.1 The G e n e r a l D r e s s i n g 4.2 The S c a l a r  Field  Transformation  Model  5  Dressing  the T r i l i n e a r  22 26  4.3 The Lee Model Chapter  10  10  3.2 The T r i l i n e a r  Chapter  6  Space-Time  Invariances  Cloudy  1  33 Fermion-Boson  Interaction:  The F e r m i o n - F e r m i o n P o t e n t i a l  40  5.1 D r e s s i n g  40  the T r i l i n e a r  Interaction..  5.2 The S e c o n d - O r d e r N u c l e o n - N u c l e o n P o t e n t i a l . . . 45 5.3 The N u c l e o n - N u c l e o n P o t e n t i a l C l o u d y Bag M o d e l  i n the 50  Chapter  6  Summary  and C o n c l u s i o n s  Bibliography  54  .  57  A  The R o t a t i o n  Appendix  B  The S p h e r i c a l H a r m o n i c s  Appendix  C  I r r e d u c i b l e Tensor Operators Wigner-Eckart  Appendix  D  Angular  Matrices  D^i^ (d p~6 )  Appendix  Y ,(g) and t h e  Theorem  62  Momentum C o u p l i n g .  64 Momenta: The  Clebsch-Gordan C o e f f i c i e n t (b) A d d i t i o n o f T h r e e A n g u l a r  Momenta:  Momenta:  9 j Symbol  68  12j Symbol  Momenta: .  Appendix  E  Two-Particle  Appendix  F  C a l c u l a t i o n of the T r i l i n e a r the  Operators  70  i n Fock Space Vertex  72 Function for  C l o u d y Bag Model  Appendix  G  An E x a c t  Dressing  Appendix  H  A Dressing  76  Operator  Transformation  Fermion-Boson  Trilinear  (a) The G e n e r a l i z e d  f o r the Lee Model  79  for a Generalized  Interaction  Fermion-Boson  82  Trilinear  Interaction (b) D r e s s i n g  64  66  (d) A d d i t i o n o f F i v e A n g u l a r The  and t h e 3 j Symbol.  6 j Symbol  (c) A d d i t i o n o f F o u r A n g u l a r The  60  Arr  (a) A d d i t i o n o f Two A n g u l a r  The  58  82  the Generalized  Interaction....  87  Appendix  I  Some P r o p e r t i e s o f B e s s e l F u n c t i o n s  94  Appendix  J  One P i o n  Exchange  98  Appendix  K  Dressing  a Poincare  P o t e n t i a l s (OPEP) I n v a r i a n t System..  100  V  List  of  Tables  Table  I  Values  of  the Constant  Table  I I  Matrix  E l e m e n t s of t h e  CLM^' Scalar  49 OPEP  ['U'MM^^^XPEP- • 99  vi  List  of F i g u r e s  Figure  1  The T r i l i n e a r  F-B I n t e r a c t i o n . . . .  Figure  2  The S c a l a r F i e l d  Figure  3  The F e r m i o n - F e r m i o n P o t e n t i a l  Figure  4  The F e r m i o n - B o s o n  Figure  5  The Lee Model  Figure  6  The N-V  Figure  7  The V- 6- P o t e n t i a l  Figure  8  The N-6> P o t e n t i a l  Figure  9  The IL -N P o t e n t i a l  Model  11  Interaction i f c ( k , l k / ,K)  30  ( k , k ' ,K)  30  F  ^  Potential  27  P 8  Interaction  Potential  34  ^  38 38 tf  IT™ M.^MM'^"{k,  3  y  l  w  m, fviim  k' ,K) —  —  '>** ( k - k )  The Two-Fermion  Potential  ^sjf/^/k,^  12  The G e n e r a l i z e d  Trilinear  Figure  13  The F e r m i o n - B o s o n  Figure  14  The F e r m i o n - F e r m i o n P o t e n t i a l  Figure  15  The NA.7C T r i l i n e a r  10  The N-N  Figure  11  Figure  P o t e n t i a l lJ~  44  /  ,K)  75  F-B I n t e r a c t i o n  83  Potential  Vertex  44  —  NtJ  Figure  38  8  l/ha ( k k ' ,K)  90  t^pCk-k' )  90  r  93  vii  Acknowledgements I would McMillan, his  first  like  t o thank my  for h i s extensive  constant  encouragement  Secondly,  I  would  Thomas, and D r . G. M i l l e r p a p e r on t h e C l o u d y Bag Lastly,  I  Research C o u n c i l  thank  input  of  s u p e r v i s o r , Dr. ideas  into  J.  this  Malcolm  project  and  interest  throughout  like  t o thank  Serge Theberge, Dr.  for  providing  a  and  i t s progress.  preprint  of  A.W. their  Model. the  for their  Natural financial  Sciences support.  and  Engineering  1  Chapter  j_  The strong  problem  exchange  fermion-boson  form  nucleons  a r e a was t h e c o n c e p t  implying  is a trilinear  a s boson  potentials  terms  of  the  v e r t e x f u n c t i o n . They c a n s e r v e a s  the  phenomenological  on  two  determined  strong  in  interaction  one  In t h i s  for direct  interactions,  These  are  These  f o r the f u n c t i o n a l  fermions.  potentials trilinear  one.  b o s o n - b o s o n , and f e r m i o n - b o s o n  production  of  the u n d e r l y i n g  nucleon-nucleon p o t e n t i a l s .  we d e v e l o p a t e c h n i q u e f o r f i n d i n g  fermion-fermion,  that  provide the b a s i s  of the phenomenological  the  and p i o n s i s an o n g o i n g one.  potentials,  potentials  and a c c u r a t e t h e o r y of  in this  strong interaction  exchange  thesis,  well  of  a useful  one i m p o r t a n t a d v a n c e  boson  boson  of f i n d i n g  interactions  Certainly one  Introduction  as  interaction fermion-boson basis  for  a  H a m i l t o n i a n f o r systems o f  p i o n s and n u c l e o n s a t i n t e r m e d i a t e e n e r g i e s . Using  an a p p r o a c h  variables  of  creators F  T  interaction function that  form  and  B ,  discussed  T  involves  the  this  Cloudy  interaction  Bag  difficulty  interaction  fundamental  is  since  that  Model  F^|0>  2,  i n v o l v e d . We  t o be  does  this  show  invariant  restricts  vertex  the  boson  trilinear  where t h e v e r t e x  i s then used  with i t  Chapter  dynamical  f e r m i o n and  integral  f o r the strong i n t e r a c t i o n  particles  in  transformations greatly  One  the  t h e t h e o r y t o be t h e e l e m e n t a r y  requiring  The  takes  h depends on t h e momenta  space-time h.  which  i n Chapter  under  the vertex  t o determine  a  3  certain function specific  function. fermion-boson  not e x p l i c i t l y  involve  trilinear physical  i s not an e i g e n k e t o f t h e H a m i l t o n i a n .  In  2  Chapter  4  we  formulate  transformation, creators alters  into none  relations  of  of  considered  the  the  simple,  soluble  of  determining the  t h e o r i e s . We  operators.  as  interaction  series  for  the  give  t o two  i n order  detailed  of  coupling constant.  4.2  simple  This  transformation,  who field  to apply  the i t to  prescription  for  and a l s o f o r d e t e r m i n i n g the  physical  (A d i f f e r e n t  transformation  and  4.3,  has  particle in  the  perturbation  been  given  by  application  nucleon-pion  trilinear  nucleon-nucleon function, potential. touching,  we  illustrates  the p h y s i c a l p a r t i c l e  which a r e present  5. We  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  t h e o r i e s - the s c a l a r  Hamiltonian  that  a  function  dressing  Sections  .model.  Chapter  (1958),  (1964)).  In applied  Schweber  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  strong  Faddeev  a  on t h e d r e s s i n g  t h e s i s we have g e n e r a l i z 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  Hamiltonian  commutation  such as the s c a l a r  the d r e s s i n g t r a n s f o r m a t i o n  more r e a l i s t i c  particle  transformation  or  work  and  theories  and t h e Lee m o d e l . In t h i s  concept  Lee  initial  by G r e e n b e r g  dressing  elementary  properties  The  the  creators. This  invariance  done  called  the  particle  theory.  was  technique,  transforming  physical  transformation  model  for  a  consider  calculate  We  discover  this  potential  has been s l i g h t l y  Using  the  physical the Cloudy  a one p i o n  in strength.  dressed  t h e o r i e s . The  nucleon-pion Bag M o d e l  and vertex  nucleon-nucleon  t h a t p r o v i d i n g t h e two  modified  the  t o second order i n  second-order  i s simply  and  c r e a t o r s , and t h e  i s dressed  the r e s u l t i n g  model  many f e a t u r e s of t h e  i n more c o m p l i c a t e d  interaction  interactions.  field  nucleons  exchange  a r e not potential  3  Many o f t h e A p p e n d i c e s p r o v i d e and  t e c h n i q u e s used  rotation  matrices,  spherical  and  background  to concepts  Bessel  harmonics,  functions. used  a s two-body p o t e n t i a l s  in  include  fermions  the  and  momentum  discussing  exchange  not only  but t o i n t e r a c t i o n s  to  such  interaction and i s o s p i n .  interactions  of other  some  potentials.  the t r i l i n e a r  and b o s o n s o f a r b i t r a r y s p i n  pions,  discuss the  angular  thesis,  o r one p i o n  Thus our t e c h n i q u e c a n be a p p l i e d nucleons  they  formulae  Other Appendices p r o v i d e  In A p p e n d i x H we have g e n e r a l i z e d to  mathematical  i n t h e t h e s i s . F o r example,  coupling,  things  useful  of  f e r m i o n s and  bosons as w e l l . Finally, transformation  in  Appendix  for  a  f e r m i o n s and b o s o n s .  Poincare  K,  we  invariant  discuss  a  dressing  system of i n t e r a c t i n g  4  Chapter  2  F e r m i o n and Boson F u n d a m e n t a l D y n a m i c a l V a r i a b l e s and Their  This necessary will give  Properties  Chapter  will  for  understanding  introduce their  an  provide  p a r t i c l e creators  commutation  the  background  of the r e s t  information  o f t h e t h e s i s . We  and a n n i h i l a t o r s , and  r e l a t i o n s and s p a c e - t i m e  we  will  transformation  properties.  2.1  The F u n d a m e n t a l D y n a m i c a l Our  system  arbitrary  spin  of  fundamental  and i s o s p i n  is  a d i r e c t product  The  fundamental dynamical  observables creation  and  and  Variables fermions  i s described  o f f e r m i o n Fock  can  bosons  with  by a H i l b e r t s p a c e  s p a c e and boson Fock  variables,  operators  and  in  terms  of  which space.  which a l l  be e x p r e s s e d , a r e t h e p a r t i c l e  annihilation  operators.  when  on  These  are  defined  as  follows: F^ (x), T  fermion x,  acting  ket c o r r e s p o n d i n g  having  spin  s, z - a x i s  axis  projection  this  one-fermion  t o an e l e m e n t a r y projection  fx. The a d j o i n t ket, gives  Equivalently,  we  and a n n i h i l a t o r s ,  destroy,  an e l e m e n t a r y  m,  define  F^J^Crj)  fermion  y i e l d s a oneat  i , and i s o s p i n z-  F ^ ( x ) , when a c t i n g on  momentum s p a c e  F^Jrj)  transform:  fermion  • These c r e a t e ,  f e r m i o n w i t h momentum p_. The two  r e l a t e d by a F o u r i e r  position  state.  the and  state,  isospin  operator,  t h e vacuum  can  creators  are  t h e vacuum  or  creators  5  (2.1.1) Note  that ^olF^ (P_) =  ^ F ^ C * )  T  = O  (2.1.2)  and  Similarly, a position spin  s  B ^ ^ ( x _ ) , when a c t i n g  ket corresponding  with p r o j e c t i o n  operators B ^ ( x ) , the  fermion  on t h e vacuum s t a t e ,  t o an e l e m e n t a r y  m and i s o s p i n  boson a t x  i with projection  B^"^(p_) , and B ^ ( p _ ) a r e d e f i n e d  c a s e . The p o s i t i o n  and momentum  gives having  JJ. . The  analogously to  creators  are related  by  &*u. The  U> =  following  J  commutation  fundamental dynamical  d  P  <-f>  e  relations  are  satisfied  (2.1.4) by  the  variables:  iF^^VF^yUl')]  = ^I-|'Us5'S^Scc'^'  (2.1.5)  U^CI^B^CI^] = L< ^,BSj:.tlO] = o +  IK^UJ, where  [  represents  ]  F^U'\I  denotes  either  = [ B ^  +  a commutator  x or £.  '-  f  C i ^ F ^  f  C i ^ ]  = o  (2 1  (2.1.9)  and { } an a n t i c o m m u t a t o r ;  i_  8)  6  2.2  Space-Time T r a n s f o r m a t i o n Dynamical  Variables  In  Section  this  rotations, creators the  space and  requirement  system  we  inversion  and t i m e  annihilators.  which  the  These r e s u l t s  We  on  invariant  the  particle  a r e a consequence of  invariance  will  Fundamental  of d i s p l a c e m e n t s ,  reversal  of g e n e r a l Poincare  t h e form  of  d i s c u s s the e f f e c t  (Kalyniak (1978)).  determine  Properties  for  use them particle  a  physical  i n S e c t i o n 3.1 t o interactions  must  have. To  a  spatial  corresponds  displacement  a linear,  unitary  Dta^l F^ f ^D C<0 f  position  resulting  Dta^ Di<T> To  d , p ,% such  a  (2.2.1)  +  creation  F ^ C p  that  = F^ C£fa^  +  x + a r a t h e r than  by amount a t h e r e  o p e r a t o r D(a) such  D ta^ = B^M-  B j j + if) The  of t h e system  t  l  , a )  operator  (  creates  (2.1.1)  x. U s i n g  = F^^e"'-'  tf(a)  and  a  spatial  (2.1.4) we  there corresponds  of the system  a linear,  '  2  )  at  have  (2.2.4)  through  unitary  2  (2.2.3)  ? m  F  rotation  -  particle  E^^D (a) = B ^ t ^ e'"'^ +  2  Euler  operator  angles («( £ 7f) ,  that s  Ktyrt  L*)K*uii)  =  m^i)  (i)  ~-  D  J~  J  m  s  ^^F  V*,*  L  <ii)  s w  ,'  +  ct.^ u_ ) R  (2.2.5) ( 2  - 2  6 )  7  The  conventions  D ' Upv)  a  r  e  mrn  f o r the Euler angles  as. g i v e n 2  i n Rose  and r o t a t i o n  ( 1 9 5 7 ) . Note  matrices  that  = M TL  R  (2.2.7a)  = where x i s a column (1957,  (2.2.7b)  vector  and M i s t h e m a t r i x  defined  i n Rose  p.65).  From  (2.1.1) and (2.1.4) i t f o l l o w s t h a t  »upt^ F ^  1. D J U ^ F ^ V ^  ^ J R ^ T ) =  +  (2.2.8)  5  *  ^  B l i ^*  Both spin,  « ^  D„ .;« 7)  - £  (2.2.9)  6^c .^  5  P  t h e momentum a n d s p a t i a l  ?  coordinates,  as well as the  are rotated. To a s p a c e  linear  inversion  unitary operator  #  *  where the  +  F ^ C i ^  6  the plus  minus  either inverted  sign  x  C  To a t i m e  V  such  "  1  ^  ±  t o negative  corresponds  tF ^ l - r )  =  =  there  a  that  (2.2.10)  »/»  (2.2.11)  B  sign applies to p o s i t i v e  o r p_. B o t h  by t h i s  corresponds  ^  transformation  parity  parity  particles.  the position  p a r t i c l e s , and %_  represents  a n d momentum v e c t o r s a r e  transformation. reversal  an a n t i l i n e a r ,  transformation  on  t h e system  a n t i u n i t a r y operator  3  such  there  that  8  5  3 F ^  l 0 J  f  +  -  ^  F « V t 2 ^  JT. ^^ "> [)  (2.2.12)  f  7CO  5  s  J B ^ C ^ U * = nB  3fdU where ITF I  2  It  +  f(x) =  is  -  function,  parity  inversion  by t h e r e q u i r e m e n t  (1961,  (2.2.14)  complex  reversal  space  (2.2.13)  f*Ci)  arbitrary  from t h e  parity  ( s e e Schweber  an  f  i s the time  i s determined  conjugation  D^mCoTCo) B * y C & }  S  f*C2L) J J  =  = 1.  2  J:_  of the parity  of o v e r a l l  particle. and  TCP  r £ = +1;  p.268)). For nucleons,  p  and  charge  invariance for  pions,  >IB= - 1 . The position spin  time  reversal  c o o r d i n a t e s , but does  vectors.  observer chosen Euler  transformation  R e v e r s i n g the s p i n  through this  reverse  180 d e g r e e s a b o u t  axis  to  be  both  not the  change  the  momentum  and  i s equivalent  to r o t a t i n g  an a r b i t r a r y  axis.  the y - a x i s ,  so t h e s p i n  We  the have  rotation  has  a n g l e s 0,7T,0. U s i n g  LCKO) = d * , C ^  = C - ^ X ' - w  m  as w e l l  as  (2.1.1) and  rf""  J B^,  L-?"°  Finally, angles Uf>V)  (2.1.4), i t follows  3 F ^ l ^ 3 = TI, +  $i  does  +  J+ - n»  to a  << , |3,Y such  that  rotation there  that  F.:; t-^ +  (2.2.16)  L-p) in  isospin  corresponds  (2.-2.15)  ( 2 . 2 . , 7) space  a linear,  through  Euler  unitary operator  9  fixUpi)  easily  +  space-time obtained  equations  F  ti>ff« Up^=  tfrfc^ The  f.. D^upT) v a )  F^iDKx^fii)*  by  £ t  transformations taking  the  f o r F M / ( |_ ) and B ^ ( f  D;/  f  Ll)  ^  of F ^( J_ ) and TY)  adjoint ) given  (2.2.18)  (2.2.19) (A )  are  of t h e t r a n s f o r m a t i o n  in this  Section.  10  Chapter  3_  The T r i l i n e a r  In t h i s strongly  Chapter,  Fermion-Boson  we  interacting  i n t r o d u c e and d i s c u s s a H a m i l t o n i a n f o r  nucleons  trilinear  Hamiltonian  considered  i n A p p e n d i x H.) We  that  when  certain single this  this  space-time  involving  pions.  other  will  interaction  function for concepts  of  particle,  w i t h i n the context  fermions  show, i n t h e  In t h e n e x t  Cloudy  Bag  elementary  Section  2.1,  interacting fermions, parity  fundamental  we  spin  i . e . nucleons,  bosons,  and b o s o n s i s Section,  Model.  under  depend on a  S e c t i o n , we  of the t r i l i n e a r  dynamical  then  specify  Finally, and  we  physical  interaction.  Invariances  variables  defined  c o n s t r u c t the f o l l o w i n g Hamiltonian one-half,  general  first  particle  R e s t r i c t i o n s Due t o C e r t a i n Space-Time the  more  i s r e q u i r e d t o be i n v a r i a n t  the  the  Using  (A  t r a n s f o r m a t i o n s , i t can o n l y  discuss  3.1  and  f u n c t i o n o f one v a r i a b l e .  vertex  Interaction  isospin  one-half,  and s p i n  zero,  involving  positive  isospin  in  parity  one, n e g a t i v e  i . e . pions:  H = Ho t A H i  (3.1.1a)  (3.1.1b)  (3.1.1c)  11  The Section  jx  operator 2.1,  can  M  (p_)  corresponds,  t o the o p e r a t o r  take  simplicity.  F ^+  the  values  Similarly,  i= 1 , JLL = ± 1 , 0 .  F^^g) ±1/2.  B^'Cp)  We  in  with  S=1/2  drop  the  corresponds  the  notation  and  i = l/2;  labels  t o B^y(p_)  m  s and with  of and  i for s=m=0,  Also, (3.1.2)  and  (3.1.3)  are  the  energies  respectively.  Note  relativistically' also  t h a t the  function The  by  'vertex  o n l y of  of  the  elementary  that  the  particles  including  fermion are  h^^*(g)  boson,  treated  relativistic  function'  and  'semi-  kinematics.  is  chosen  to  Note be  a  g.  trilinear  interaction  H,  may  be  pictured  as f o l l o w s :  /  Fig.  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 The  conserves noting  Hamiltonian the  total  t h a t the  p  (3.1.1)  number of  total  momentum  lines  are  is translationally fermions.  We  can  bosons invariant see  this  and by  operator  (3.1.4)  1 2  and  the fermion  number  operator  "-'^toV^^ both  commute We  time  demand t h a t H a l s o  space,  rotations  reversal.  Ho  transformations,  is  but  restricted  (see eq.  First, isospin  let  space.  be i n v a r i a n t  i n o r d i n a r y space, already  in  order  of the f u n c t i o n h ^ ^ ' C g )  form  -  K5)  with H.  now  isospin  (3  under  space  invariant  under  t h a t H, a l s o  rotations in  i n v e r s i o n , and a l l  these  be i n v a r i a n t , t h e C3.1.1c)  appearing  in  must  be  the e f f e c t  on H, o f a r o t a t i o n i n  (3.1.35)).  us c a l c u l a t e  Using  the r e s u l t s  (2.2.18) and ( 2 . 2 . 1 9 ) ,  we  see  that  ' Using  L > ' ' A ^ «vJ.' ^ U  (A.1)  and  into a single  where  (ji  Appendix no  F  f  S  (A.8)  one, we  dependence  is  t  In o r d e r on  J-  rotation  (3.1.6) matrices  find  w) Ijwi^  D.)  t o combine t h e t h r e e  a d  the  a  that  Clebsch-Gordan (3.1.7) be e q u a l  angles  oL , (•} , Y .  coefficient.  (See  t o H , , i t can have Therefore,  taking  13  (5=A'=A= 0  and u s i n g t h e p r o p e r t y  coefficients  i n A p p e n d i x D, we have  • Comparing  ^  with  &/*,'  (3.1.1c), #  J + adj.  ( 3  * ' > 1  8  we s e e t h a t  M,  r  (D.12) o f t h e C l e b s c h - G o r d a n  =  H,  (3.1.9)  if  • (. "k 1 From  this  dependence  equation,  of  I  j^'/A*'  we  h^^^^g)  (  note is  *  !  that  A*/**  (3.1.10)  4  a l l of  contained  in  the  the  p,,yj ju zr  3  coefficient  T h e r e f o r e , we may w r i t e  C i l ^ ^ l V ' ^ - . ^ ^  ^^ ^= S  Secondly, rotation.  •C,  From  Uprt  This w i l l  we d e t e r m i n e  the  effect  (2.2.8) a n d ( 2 . 2 . 9 ) ,  on  H,  (3.1.11) of  a  spatial  we have  ( & . * U p A F„,+ ( ^  tfldj.  (3.1.12)  D*'^,  <3-'.'3>  e q u a l H, i f  i,  ^  DVU^  14  We  write  ^  where  Y  A r n  properties  "  M  (g)  *  ^  =  is  a  spherical  harmonic  are d e s c r i b e d i n Appendix  Substituting  (3.1.14)  into  =  a c t on b o t h  D  s i d e s of  V' J b w  l  Using  JL ,  order  whose  (3.1.13),  we  obtain  D^/c^r)  <3.1.15>  (B.2),  W we  of  B.  •D^„.. Using  (3.1.14)  C<p  ;  (A.8) t o combine  D  u  first  to obtain  0  --' ^^  the  (3.1.16)  J Vi^C^ldc ^  (3.1.15) w i t h  P  l  W ^  rf.A*p^  U ( , r )  two  °"" ^ !u  rotation  13  •' • ' 17  matrices  this  becomes  •Dl Now  we  i n t e g r a t e on b o t h  <rt , p , ~i . U s i n g  K™,tn> f > {  This  =i  (A.9) , we  ^ , nU  equation  <*j*T>  D J L * " ^  sides with  (3.1.18)  respect to the E u l e r  angles  find  L<() determines  Ct Mann I ^  the  " 0 (i 1  m, j i n ^ m  W  l  V I ^ */>  dependence  (3  9) ' ' 1  1  of  1 5  (3.1.20) Referring  t o ( 3 . 1 . 1 4 ) , we 1  l w * l $ > Thirdly, inversion.  ^C we  From  require  From  invariant  (2.2.11) one  under s p a c e  has  B^C-^  +adj.  (3.1.22)  "k.^C-^  ^>  parity  would have  ( 3 . 1 . 1 4 ) , we  1  second  bosons i n a s i m i l a r  manner.  For  instead  i V * (-^ h l  = the  (3.1.23)  have  Krti.w* ("^=  Figure  be  (3.1.21)  e q u a l H, i f  s u c h b o s o n s we  i.e.,  H,  C-jf>  (One d e a l s w i t h p o s i t i v e  can o n l y  that  (2.2.10) and  k*..  where  K^)  ( ^ ^ M ! I O  ^  • 5,+ This w i l l  write  step  i ^  ^ *  follows  ^  ^  AMWlinx  k  ^  U  Co.')  )  (3.1.24)  from ( B . 4 ) . T h e r e f o r e ,  (3.1.23)  be s a t i s f i e d i f  i f i . i s odd. R e f e r r i n g 1,  we  see  that  L  t o (3.1.20) f o r h , 2w  may  be i n t e r p r e t e d  r0iff)i  (g ) l  and  to  as the a n g u l a r  16  momentum o f t h e boson  i n the t r i l i n e a r  Gordan c o e f f i c i e n t  il  odd  value  =  implies that the only p o s s i b l e  1  (3.1.25)  i s , o n l y p-waye p i o n s  h,l<p  Finally, reversal  The C l e b s c h -  of A i s 4  That  I z*0  interaction.  =  a r e a l l o w e d . We  write  hip  we demand  (3.1.26)  that  transformation.  H|  From  be  invariant  (.2.2.14),  under  (2.2.16) a n d  a  time  (2.2.17),  we s e e t h a t  •C'-^  ^^B^C-g")  +  «dj.  <3...27)  T h i s e q u a l s H, i f ,  ^  "<o,«ii^  or,  from  x  from  *  ^  (3.1.28)  (3.1.23), = CO  k*,.^  (3.1.21) and ( 3 . 1 . 2 5 ) , ^  Using  .  tr>  a  kmt Now,  W1,+ M i  = -« V L  (B.5) and (D.9b),  Lp  C  equation  we -w>*  (3.1.29)  have mit  W > h*(^  (3.1.30) becomes  C-**^* i C O V.^t^Cxl wix-w I W  Since m  2  i s half  odd  integral,  (3.1.30)  4M,)K*C^  (3.1.31)  1 7  -Y ' W  C  It  follows  -  X  - C - ^  that  ^  K  _ r<l,-rfli +1 M  i  r  Therefore,  )  x  C ^ »  (3.1.29)  summary,  inversion for  isospin  and t i m e  where  requirements rotations,  H.  = i  j  First,  t h i s point H  we d e f i n e  r  T  ' M K  ^  ^  the  rotations,  the vertex  function.  space  function  Thus  (3.1.35)  the i n t e r a c t i o n  I r / O C ^ I ^ w l i r v O V , * (^h($V  -  ^  i t is illustrative (  +  to  a  d  J -  show  the f o l l o w i n g  (3.1.36) that  c a n be p u t i n t o a m a n i f e s t l y  the  above  invariant  form.  operator:  T^I.WI,  • Jd* Using  spatial  under  ( 3 . 1 . 1 c ) t o be  ^ J ^ p d ^ ( A l ^ M 3  for  invariance  written  C  expression  (3.1.33)  i ^ l ^ H i i ^ M l i ^ Y , , ; ^ ^  'C ^ At  of  have d e t e r m i n e d  h ( q ) i s an a r b i t r a r y , r e a l  ( 3 . 1 . 1 c ) c a n be  •¥  iMi^V)  (3.1.34)  interaction  V i ^ N ^ > = «' Lk  I |  Wxml  function.  reversal  the t r i l i n e a r  Cii  (w>  « K*C<p  the  displacements,  T  is satisfied i f  i f h(q) i s a r e a l In  i  II a  ' £0  M f ) i.e.,  ( 3 . 1 .32)  1  properties  p  (2.2.5)  ^ . ( j O ^ and  (2.2.18)  Cj>-^ of  (3.1.37)  the  fermion  18  operators,  we f i n d  that i  •Rl<pi) V ' ^ *  1  , , « Z  V  D  %S>  - -  (3  1  38)  and i  ttr"f&rt i.e.,  isospin  fermion acting  spin-isospin  transfer  on a one n e u t r o n  into  a  one  proton  i s increased  projection  i s also  The A/x(g)  trilinear  a  s  state  with  rotations this  For  We  write  vector  this  hty  spin  operator  as a  example,  projection spin  i n both  m=-l/2,  4  turns  up. The s p i n  z-axis  ( A=+1), and t h e i s o s p i n  z-axis  Y,  terms  of  (g) i n t e r m s o f t h e s p h e r i c a l  A  g  in  ( s e e A p p e n d i x C) a s  <U into  S|i (g)  (ja=+1).  ClfV H, = i  spin  (3.1.39)  i n t e r a c t i o n H, may be e x p r e s s e d  follows.  substitute  ^  interpret  with  one u n i t  components o f t h e u n i t  and  (  operator.  state  increased  V  under  s p a c e . I n d e e d , we may  projection  s  ^yUfiY)  S ^ ( g ) t r a n s f o r m s as a v e c t o r  and  it  V ^ f t r ^ t ^  '-  (3 1  40)  ( 3 . 1 . 3 6 ) . The r e s u l t i s  ' S ty ' £  ty  +  a  d  J •  (3.1.41a)  where ity' Using #Up-0  and  (A.2),  Bl^S =  5-  fc^  i  ( B . 2 ) , and ( A . 6 ) , we  | ' S ( $ V  8 ( ^ « ^ ( , i p >  A  V l ^ B p t ^  (3.1.41b)  find * S  t ^ ) • B t o ^ )  (3.1.42)  19  * * u p O %- i t y - B ^ ^ u j s y ) The under to  e x p r e s s i o n (3.1.41)  isospin  the  more  trilinear discussed  such  (1961,  function  H,  is  form  for  the  h(q)  for  invariant  i t s similarity  the  nucleon-pion  Chew-Low  interaction  pp.376,377). Indeed, Section  (3.1.43)  manifestly  r o t a t i o n s . Note  as  considered i n this  real  £.|<$VB^>  CT • | ^  conventional  i n Schweber  arbitrary  for  space and s p a t i a l  interaction,  requirements  *  the invariance  determine  a l l but  the nucleon-pion  the  trilinear  interaction.  3.2  The T r i l i n e a r The  Cloudy  Bag M o d e l o f T h e b e r g e ,  i n v o l v e s a massive fields.  up  pion f i e l d  The p i o n f i e l d  spherical bare  Vertex F u n c t i o n i n the Cloudy  surface  bag model quark  field  i s small,  Theberge  resulting  Hamiltonian  Hamiltonian  (see Section i n Section  the  paper  quark  fields  on  only  a r e composed o f t h r e e  wave f u n c t i o n s ,  i n terms  consequences  with massless  a  R ('the b a g ' ) . In t h i s model t h e  particles  interaction  and  (1980)  massless  c o n f i n e d t o t h e b a g . U s i n g t h e known, l o w e s t  order  discussed  in interaction  radius  n u c l e o n and d e l t a and down q u a r k s  Thomas, and M i l l e r  c o u p l e s t o the quark  of  Bag M o d e l  e t a l . have  of p i o n - b a r y o n is  a  4.3)  trilinear  combination and  that  the pion  re-expressed the pion-quark  the  3.1. F o r f u r t h e r  of t h e C l o u d y  of Theberge  and a s s u m i n g  interactions. of  trilinear  details  Bag M o d e l ,  the  Lee  The model  Hamiltonian  on t h e d e r i v a t i o n  we r e f e r  the reader to  e_t a l . ( 1 9 8 0 ) , and r e f e r e n c e s t h e r e i n .  20  In A p p e n d i x F we r e l a t e Model the  trilinear  trilinear  t h e NN7T  interaction  vertex  piece  of  t o our equation  f u n c t i o n , w h i c h we  write  the  Cloudy  Bag  ( 3 . 1 . 3 6 ) . We  find  as  hc (^{q) B  ,  is  g i v e n by  f  0  i s t h e NN7C  c o u p l i n g c o n s t a n t and  LUlO where  =  -j',(qR/-n)  The  'form  the  bare nucleon;  is  factor'  R^O  — — —  (3.2.2)  a spherical Bessel  U^(q) t a k e s  into  account  i t has t h e p r o p e r t y U  n  U  ^  =  f u n c t i o n of order one. the f i n i t e  extent of  that (3.2.3)  1  Therefore, Hm.  In  the l i m i t  W < p  -  h  ,  (  R->0, t h e C l o u d y Bag M o d e l v e r t e x  exactly  the  vertex  discussed  i n Schweber  function (1961,  for  p.374).  the  function  Chew-Low  3. . ) 2  4  becomes  interaction  21  3.3  Physical The  Bosons and F e r m i o n s  Hamiltonian  (3.1.1) h a s t h e f o l l o w i n g  HB/(jf>lo> = H F ^ t ^ l O for  cVp.)  a  n  Hamiltonian i.e.,  and t h e r e f o r e  the elementary 6o  Hamiltonian.  physical trilinear  fermion;  the d r e s s i n g and  In  (3.3.1)  p„+ C ^lo>  (3.3.2)  B^~(p_)|0>  corresponds  • However, F ^"(p_)|0> m  The e l e m e n t a r y we say t h a t  the next  T  a  'physical  an  boson',  eigenket  of the t h e o r y  c r e a t e s a 'bare'  not e x p l i c i t l y  for  of the  i s a physical particle  i s not  fermion  Chapter  Hamiltonian  " particle interactions.  F  i s an e i g e n k e t  to  boson o f t h e t h e o r y  transformation, a  B/C^)0>  ?  i n t e r a c t i o n i s thus  interaction.  creators  *  arbitrary function.  w i t h mass ma = m the  f u o C ^  feature:  we s h a l l  of  i s not a  fermion.  The  a physical particle develop  obtaining  which e x p l i c i t l y  a  technique,  physical involves  particle physical  22  Chapter  4  The  Dressing  Transformation  and  ., Some  Simple  Applications As  we  interaction bosons.  saw does not  We  variables  seek  of  expressed  in  the  theory  transformation  must be  in order  transformation  annihilators. composite dressing In  the  The  explicitly  scalar  General  where D  to  a  dynamical  operators.  the  that  the  particle  relations  creators  bare p a r t i c l e s thus  This  invariance  b a s i c commutation  of  and  Hamiltonian  certain  transformation;  apply  model and  Dressing  the  and  acquire  i t is called  a a  the  the  two  Lee  the  dressing  simple,  soluble  model.  operator  i s i n v a r i a n t under  will  formulate  i t to  D" = - D  where  D  D  will  Transformation  unitary  e  operator of  and  field  i n v e r s i o n , time  function  lead  possess  the  fermions  fundamental  particle  f o l l o w i n g S e c t i o n s , we  U =  space  the  trilinear  transformation.  Consider  and  see  s t r u c t u r e v i a the  models - the  The  to preserve  the  physical  will  u n i t a r y and  will  on  physical  properties  We  transformation  4.1  which  of  Chapter,  involve  a transformation  terms  and  previous  explicitly  in  properties  the  reversal, be  (4.1.1)  1  translations, and  spatial rotations,  rotations in isospin  specified  fundamental dynamical  f u r t h e r below;  space.  it will  v a r i a b l e s , so we  write  be  a  23  D =  DlF.B^  (4.1.2)  Let ff)fj.  and  where F and B a r e t h e f u n d a m e n t a l f e r m i o n and boson operators, rest on  respectively,  of t h i s  the p a r t i c l e  and change all  Section,  Because commutation  operators,  U  is  For  a  relations  also  and  isospin  complicate  use t h e symbol  operator,  a s do F and B. (See  labels  the notation  0 0  to  F and B obey  denote  equations  t h e same  (2.1.5)  because of the m v a r i a n c e p r o p e r t i e s  time  under t r a n s l a t i o n s , reversal  t o the t r a n s f o r m a t i o n any o p e r a t o r  1  spatial  and r o t a t i o n s  laws i n S e c t i o n  A = A(F,B)  OL A (P,B) LL" = In  as they o n l y  unitary  transform  inversion,  according  omit t h e s p i n  2.1. T h r o u g h o u t t h e  operators.  (2.1.9)). Moreover,  space  we w i l l  in Section  none o f t h e r e s u l t s . We  transformed  and B w i l l  defined  destruction  o f U, F  rotations,  in isospin  space  2.2.  i n t h e Fock s p a c e , we  At^B)  have (4.1.5)  particular,  DCF,B ^ and u s i n g  (4.1.5),  U DIF,B) l l  T  = LHF^  ( 4 . 1 . 1 ) , and ( 4 . 1 . 6 ) , we  HlF.BVU/ H l ? , 8 m = e  G i v e n D, e q u a t i o n  H (F , B ; e.  (4.1.7) g i v e s  (4.1.6)  have  = H (F, B )  the Hamiltonian  (4.1.7)  a s some new  24  nj  function  HJ  ^,  H(F,B)  variables  o f t h e new,  below a s a f u n c t i o n  dynamical  o f t h e dummy v a r i a b l e s  form F and  using -D(F,B>  =  Our  strategy  (4.1.7) AJ  fundamental  F a n d B. I n d e e d , we c a n d e t e r m i n e t h e f u n c t i o n a l  of H, e x p r e s s e d B,  independent  A>  we  /V  H(F,B) where The  t  H  will  DCF.B)  T H . D ] +  ^7 C L I 4 , D ] , D 3  be t o c a l c u l a t e  know  that  A/  AJ  H  from  the Hamiltonian  F and B a r e t h e t r a n s f o r m e d  total  + • • •  this H(F,B)  ( 4 . 1 . 8 )  equation;  from  i s equivalent to  operators.  momentum o p e r a t o r i s P IF.B^ = E t F , B ;  (4.1.9)  where P tr,B) S i n c e D(F,B)  =  PCF.S)  e  i s translationally  e  (4.1.10)  invariant, i . e . ,  C P ,D ] = O equations  (4.1.9) and (4.1.10) fCF.tf  Now  (4.1.11)  =  suppose t h a t  imply  E CF.B)  we c a n w r i t e  (4.1.12) the Hamiltonian as  H = Ho A- X H, and t h a t  D c a n be expanded  i n a perturbation  (4.1.13) series  in  A:  oo \)=  Equation for  H:  (4.1.8)  X-  A° Dn  t h e n becomes t h e f o l l o w i n g  (4.1.14)  perturbation  series  25  H  =  Ho +  { M, + [Mo , 0 , ] }  A  + A* i [ H , , D , ] t Til  M.,D,2,D.] + r H o , D ^ l ] + . . .  (4.1.15)  ^ + We  now  demand  that  the  transformed operators F  c r e a t e p h y s i c a l p a r t i c l e s . That i s , we r e q u i r e F |0> to  be  eigenkets  , v  t  and B  and  B |0>  of the Hamiltonian H(F,B), or e q u i v a l e n t l y of  H(F,B). That i s , F l^l0>  HlF,^ Jf  where  ( F , ^B  (o)|o>  +  cV(p) and  <fptj>) F (^  =  +  £ 3 ( 2 ) are  +  =  £ (jf) B  some  lo>  (4.1.16)  B*tf>)|o>  functions  of  (4.1.17)  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 i n H(F,B), where again F and B are dummy variables, i.e.,  which c o n t a i n only one fermion or boson a n n i h i l a t o r ,  terms of the form F  t  F  B  f  F FB B , T  f  T  t  etc.  (4.1.18)  H(F,B) may, however, c o n t a i n terms of the form F^F+FF, which  B B1"BB, +  correspond,  boson-boson, production  r  F+F+FFB+,  T  respectively,  and on  F B FB,  to  fermion-boson  two  fermions.  direct  etc.  (4.1.19)  fermion-fermion,  interactions,  These  terms  and  appear  boson in  the  phenomenological Hamiltonian d i s c u s s e d i n 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). T h i s may be by  order i n A. For the case that H  t  is trilinear,  done  order  we choose D  t  such that CH  0  )  D , ] = - H,  (4.1.20)  26  The  expression H  =  Ho  (4.1.15) t h e n  4- X  CHuD.l +  [±  z  becomes ]  E H . ,0*1  . N e x t , knowing D i , D (4.1.18) t o s e c o n d  i s chosen  order  in  t o e l i m i n a t e terms  A,  and  so  called  a  dressing  transformation operators F  is  and  B  a  operator. dressing  of  the  form  on.  operator D c o n s t r u c t e d to s a t i s f y  The is  z  (4.1.21)  The  t h e above c o n d i t i o n s  corresponding  t r a n s f o r m a t i o n ; the  , which c r e a t e p h y s i c a l  unitary transformed  particles,  are  called  dressed c r e a t o r s . We  now  go  some s i m p l e  4.2  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  examples.  The  Scalar Field  Model  The  scalar  Hamiltonian  field  fermions  and  discussed  i n Chapter  included  with  bosons.  i n the  It  3,  scalar  is  except field  is a trilinear  very  similar  that spin  t h e o r y . The  and  one  t o the  involving Hamiltonian  isospin  Hamiltonian  are  not  is (4.2.1a)  H  =  Ho  t \  H,  (4.2.1b)  (4.2.1c)  M,»  Jd3 d^ p  [ F+(^  B ^ J f adj.  ]  27  (4.2.2)  (4.2.3) where  m  boson,  a n d mg,, a r e t h e masses o f t h e e l e m e n t a r y  Fo  fermion and  respectively.  The  o p e r a t o r s F a n d B obey  (2.1.9), chosen  omitting to  be  a  t h e commutation  a l l reference real  function  to spin  rules  (2.1.5)  and i s o s p i n ; h ( q )  independent  of  the  is  fermion  momentum. We  may  picture  the  trilinear  interaction  (4.2.1c) as i n  F i g u r e 2:  /  -A  V  Fig.  2 The S c a l a r The  total  Field  Model  Interaction  momentum o p e r a t o r  f o r t h e system i s (4.2.4)  The  fermion  number o p e r a t o r i s  (4.2.5)  B o t h P a n d N commute  with  the Hamiltonian  (4.2.1).  28  We n o t e with  that  eigenvalue  B (p_)|0>  i s an e i g e n k e t  £B (]O) ,  while  T  0  Therefore  we may t a k e  We  seek  a  The  f o l l o w i n g Di s a t i s f i e s  and  equation  D, =  m%  = m,  a  B  F^"(p_)|0> the  dressing transformation the  of  the  Hamiltonian,  i s not  an e i g e n k e t .  mass o f t h e  physical  a s o u t l i n e d i n S e c t i o n 4.1.  required  invariance  properties,  (4.1.20):  S V f 4,C^,^[F(^F(-^B(^ -adj.] d  boson.  d 3  +  (4.2.6a)  D  where  and  M£,£> Note the  term  property  that i t must o f the  We now can  =  CFO(D-^  this  operator  +£ » l ^ - £  compute  [H,,D,] = - [ d ' p d ' c  t  ( ^  (4.2.6c)  has a s t r u c t u r e i d e n t i c a l  e l i m i n a t e . We w i l l dressing  f  find  this  to  be  a  t o H,, general  transformation. [H,,D|] a n d f i n d hSip  A(p,$>  Fht)Fty) T  +  adj.  (4.2.7)  29  The  last  be e l i m i n a t e d can  term i n [ H , , D ] , b e i n g (  from t h e d r e s s e d H a m i l t o n i a n  be c o n s t r u c t e d  to accomplish  The momentum P i s g i v e n  P =  Jd p £ 3  The H a m i l t o n i a n , (4.1.7) and  (4.1.9) and  [ F ( ^ Ftp  (4.1.21).  (4.1.12) t o be  + B (£) B ^ ]  f  dressed  g i v e n by  this.  by  (4.2.8)  f  to  second  order,  is  5d  3  T+  E £ p t ^ F Cj>)  t  f  P  V  F F  £B  fV  P  B+C^Sl^]  (4.2.10)  (4.2.11)  P  F 6  iSd^tffe'd^  =  by  (4.2.9)  B  VFr-^d3fedVd*Ktf pt^fe\kO^  v  given  (4.1.21) a s  rUF,B") = M CF , B *) -  T=  of t h e f o r m ( 4 . 1 . 1 8 ) , must  I^B^.k',^  F [tk^B (i!<-t)B^K-kOF(.iK fe')  (4.2.12)  =  (4.2.13)  f  f  f  where  -  o r  w  t h , k . ^  £*t^  ~ A" J d ' o  ^  zi^lCiiliM)  B  (4.2.14)  (4.2.15)  l  "  ^CtK+kHfeCt^-^  (4.2.16)  30  Referring recognize fermion  to  iK?  and  and  functions  the d i s c u s s i o n  i n Appendix  l^a a s t h e m a t r i x  fermion-boson  are pictured  elements of  momentum  in Figures  E  space  (eq. ( E . 7 ) ) , the  we  fermion-  potentials.  These  3 and 4:  k-k  1  i s - * '  Fig.  3 The F e r m i o n - F e r m i o n P o t e n t i a l  Vh (k,W^ F  ,K)  IK+fe.  K  Fig.  4 The F e r m i o n - B o s o n  Note  that,  to  Potential  second  order,  mechanisms  ( F i g . 4 ) by w h i c h FB  correspond  t o t h e two t e r m s  Note a l s o  l / F ( k , k " ,K) B  there  are  two  different  s c a t t e r i n g can t a k e p l a c e ;  these  in 1T Q. F  that  rUF,^  F (£)lo> f  = M E ) F C^)!o> +  (4.2.17)  and  ULT %) t  B t^|o> f  = E B C ^ B ^ E ^ io>  (4.2.18)  31  where  £ (p_)  is  F  second order fermion.  F ' ( p ) now  L> ( ),D]+  +  =  fermion  energy, given t o  creates  a  physical  (4.1.3) a n d (4.1.4) we s e e  F C^-  f  'renormalized'  by ( 4 . 2 . 1 4 ) . Thus  From  F t^=  the  +  ?  7T  [ L F C ^ D ] , D ] +...  (4.2.19)  ~  [[^1^,01,0] +  (4.2.20)  r3 (^-[B C^,D]+ f  +  f  5  s\i 4-  Calculation by  of these dressed  an i n f i n i t e  fermion  s e r i e s of terms;  creator +  term  creators  F^, 4  the  the  second term  term  is  which  involve  we f i n d fermion  fermion  on  that  B' e q u a l s B' p l u s  operators,  a l l of  If  one boson  fermion  several  of  give  zero  when  4-  transformation  has not  although  changed  the  state.  cVo  discover  'cloud'  a s e r i e s of terms  which  X  we make t h e s i m p l i f y i n g a s s u m p t i o n  the  a  the p h y s i c a l  t h e vacuum s t a t e . Thus B (p_) 10> = B'(p_)|0>,  (p) 4 B (p_) . The d r e s s i n g  physical  bare  T  s u r r o u n d e d by  AJ  acting  the  4-  c o n s i s t s of a bare Also,  i s given  involves F^B , the t h i r d  1  bosons.  i.e.,  first  F '(p_)  i n v o l v e s F B B , and so o n . Thus one s a y s t h a t  fermion  B  shows t h a t  = n1  energy  that (4.2.21 )  Z F oC  is  independent  interesting results.  In  o f i t s momentum, we  this  case,  (4.2.6c)  s i m p l i f i e s to = and  as  a  result  the  £  final  B  ($) two t e r m s i n e q . (4.2.7)  (4.2.22) vanish.  M o r e o v e r , now C  L W,,D,3  , D,l - O  (4.2.23)  32  Thus we may  [H, ,D| ] now c o n t a i n s no t e r m s o f t h e form  take  Indeed, exactly  In  turn  (4.2.23)  implies  that  D  3  so  = 0.  = 0 f o r n>1. We may w r i t e t h e d r e s s i n g t r a n s f o r m a t i o n as D = D,. The model  Note longer  Dz = 0.  (4.1.18),  that  when  contains  however,  have  a a  i s thus  fermion  mass  term.  find  scattering  become,  respectively,  t o be  soluble.  (4.2.21) h o l d s , t h e d r e s s e d H a m i l t o n i a n fermion-boson  fermion  said  We  m c * - *rv.c* - A  scattering  term.  It  r e n o r m a l i z a t i o n and a equations  does,  fermion-  (4.2.14) and  J H^p"  z  F  no  (4.2.15)  (4.2.24)  and V * F F ^ The potential  b C o  corresponding  _)  (4.2.25)  coordinate  may be o b t a i n e d by from  f (r)=  ^ d ^  FP  For  £  S  the s p e c i f i c  e'*  C  ^  space  equation  lk  fermion-fermion  (E.11). I t i s (4.2.26)  F  choice  I  for  the  vertex  function,  (4.2.27) the  integral  e v a l u a t e d . We o b t a i n a c o o r d i n a t e s p a c e  =  - ^ A ^ / c  in  (4.2.26)  c a n be  Yukawa p o t e n t i a l ,  namely  (4.2.28)  e  r The  integral  (4.2.24) i s t a k e n of  the p h y s i c a l  (4.2.24) i s m a t h e m a t i c a l l y as a d e f i n i t i o n  mass m  P  divergent; equation  of the bare  and t h e ( i n f i n i t e )  mass m  integral.  Fo  , i n terms  33  In c o n c l u s i o n , interesting dressing which  features  implicit  been made e x p l i c i t  through  the  trilinear  The  physical  the d r e s s i n g  by a c l o u d model  fermion  thus can o n l y we have  4.3  is  is  soluble  given  and  (4.2.1) have Physical  from t h e t r i l i n e a r  vertex  c a n be c a l c u l a t e d and  c o r r e s p o n d s t o a bare  only case  the  interactions  transformation.  fermion  we have shown t h a t t h e when  that  the approximation  (4.2.2)  by an i n f i n i t e  be d e t e r m i n e d by a p e r t u r b a t i v e  holds,  the  s e r i e s o f terms and procedure  such  as  developed.  Lee  model d e s c r i b e s  distinguishable  field The  Hamiltonian  some  The L e e M o d e l The  two  particle  of bosons. F i n a l l y ,  (4.2.21) i s made. F o r t h e g e n e r a l operator  interaction  p a r t i c l e creators  the p h y s i c a l  field  dressing  model has i l l u s t r a t e d  i n the t r i l i n e a r  The p h y s i c a l  that  surrounded scalar  field  p o t e n t i a l s c a n be d e t e r m i n e d  functions. we f i n d  of  transformation.  were  particle  the s c a l a r  a trilinear  f e r m i o n s and  model, t h e s p i n  one  interaction  boson.  Like  involving  the  scalar  and i s o s p i n o f t h e p a r t i c l e s i s n e g l e c t e d .  Lee H a m i l t o n i a n i s :  H =  Mo + A 14  (4.3.1a)  2  (4.3.1b)  (4.3.1c)  34  The momentum o p e r a t o r i s  p= F,^ ,  K  B'  z  k o  I F ;  P  F ^,  traditionally  is  z  are  called Z  2  creation 6\  and  + ^  C  ]  4  V  operators  x  x  4- r/W  c  4  for particles  respectively. c(=l,2  z  the energy of the elementary f e r m i o n ,  > c  is  the  N, V,  Cp c  ty*  (4.3.2)  (4.3.3)  and  (4.3.4)  ]  the energy of the elementary boson. h(g)  solvable and  i s c h o s e n t o be a r e a l when h = h(p_,g) but t h i s  c h a n g e s none o f t h e e s s e n t i a l Note  that  momentum  the  total  the  number o f N and The  number  function. only  interaction  H,  IP  Lee model  complicates  the  i s also notation  results.  i s conserved  of N and V p a r t i c l e s 6*-  The  by t h e i n t e r a c t i o n , and  the d i f f e r e n c e  particles. may  be p i c t u r e d  as  in Figure  5:  /  \  \  Fig.  5 The Lee M o d e l 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 a r e N p a r t i c l e s , and d a s h e d l i n e s a r e & p a r t i c l e s .  as i s between  35  The c r e a t i o n and d e s t r u c t i o n o p e r a t o r s fermion  a n d boson commutation  Ri'(j»')] =  L&(£), B CBIE)  F  , BtpOl  &u'  E  familiar  (4.3.5)  1  (4.3.6)  0  =  (4.3.7)  = 0  (4.3.8)  - [ Frfl^,  LFJLCJO, B C ' ) ]  the  rules:  \ Rl (J>\ R < ^ ( p O J = Sip-?')  5^( 2 \  obey  B^CE')] - O  (4.3.9)  Because WF, ^ ( . E V ^  ~  MB (£no>  = ?Bo  0  ty  F +Cp^|o>  (4.3.10)  B tp'°>  (4.3.11)  v  and f  F,  and  T  fit  create  physical particles.  mass o f t h e p h y s i c a l N p a r t i c l e , physical  Therefore,  and m g, = m , 0  B  m =m,, t h e |0  t h e mass  of  the  boson.  However,  F ^~ (p_) | 0> i s n o t an e i g e n k e t z  so F i * " d o e s n o t c r e a t e p h y s i c a l dressing  f  transformation  particles.  according  of the H a m i l t o n i a n We  now  to the p r e s c r i p t i o n  perform  H a  in Section  4.1. Equation  Di ~ K P  d 3  (4.1.20)  f  is satisfied  (P,$) rFj^Cp-)  by  F, L$ty B(|) - Qdj.]  (4.3.12a)  where (4.3.12b)  *  M*ty  36  and  A(p,Cp Note  that  this  = £. C p - $ W  S e c t i o n . To ensure that  require  that  m_<m+mB, 2  0  (4.3.12c)  B  i s not the same  previous  decay  £ ($0 - f ^ C ^ A (p/Q)  A(£,3)  does  i . e . , the V p a r t i c l e  l  as i s used i n the not vanish, i s stable  we  against  into N + & . A simple computation  now shows that  CU,,u,] = - Jd P d\ J £ £ L  ¥+L$\\L*)  3  •K'f  4- M f t  h  (  <  ^ F, C - $ ' )  F,tj>-f}j  +  p  4- Qdj. Since  [H,,Di]  contains  no terms of the  form  (4.3.13) (4.1.18),  we  take D A computation  z  - O  of [[H, ,D, ],D, ] i n d i c a t e s that we must  D = 5d p d^ d(^,|) 3  3  where  (4.3.14)  3  3  \y± p L  p  (  ^ _ ^ B ( p - adj.]  choose (4.3.15a)  37  This the  eliminates  a l lunsuitable  dressed Hamiltonian, N o t e t h a t D,  Indeed  it  has  determined  particle  previously  (1974))  exactly  calculation  have an i d e n t i c a l  3  (1958), Piskunov  in  that  3  structure.  ( G r e e n b e r g and Schweber  the  Lee  model  s t r u c t u r e . We  D  can  give  be  such a  the p h y s i c a l  V  F . 1  x  to  second  order,  is  given  by  (4.3.13): H ( F , §H = T+-  T= $d p  for  trilinear  G, and d e t e r m i n e t h e r e  The d r e s s e d H a m i l t o n i a n , (4.1.21) and  shown  and has a t r i l i n e a r  Appendix  creator  A.  to order  and D been  terms o f t h e f o r m (4.1.18) f r o m  I  Z  V12. + v i5 +• V 2  (4.3.16)  1B  F^tjf) R, (jO + £B(^B+(^6ip)]  Uty  ^--iW^'d^^uLk^^  (4.3.17)  (4.3.18)  (4.3.19)  (4.3.20) where = L\f-c  x  k(o)  = £ z o C p ) 4- A f d o X  *  * =  3  J  + t V c  4  y  yz  (4.3.21)  Z  ^  (4.3.22)  T £z.(^ -£,C -^-£ (f) ?  A*h* ( k + K')  B  +  k ^  k'  (4.3.23)  38  (4.3.24)  1^,8 U , k ' , ^ =  (4.3.25)  A* h ( ^ - f c ' ) h C H - ^ C I I T K + I S ' ) +f (-k!<-fe ) ( /  6  Note  that  renormalized The and  the  energy  of  the  V  particle  has  been  (eq. (4.3.22)).  functions  tf , z  iXtB,  and l ^ s c o r r e s p o n d  t o F i g u r e s 6, 7,  8:  /  Fig.  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  /  Fig.  8 The N-0" P o t e n t i a l  T h e r e a r e no N-N o r $-& Hamiltonian Again  t o second  order.  we  that  note  scattering  terms  present  i n the  t h e 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  explicit  the p h y s i c a l  original  trilinear  particle  potentials,  terms  of  particles  the  particle  interaction. Matrix  trilinear  (see eq.  interactions  (G.16))  l £ and B  vertex  1^ , a  described  elements of the  have  function  been  by  the  physical  determined  h(g).  The  in  physical  become c o m p o s i t e s o f e l e m e n t a r y  ones.  40  Chapter  5  D r e s s i n g the T r i l i n e a r The  We  saw  interaction fermions. developed  Fermion-Fermion in  Chapter  this  i n S e c t i o n 4.1 into  transformation,  Concentrating  we  5.1  Dressing  the  that to  p i o n . F ^"(p)|0>  dressing  t r a n s f o r m a t i o n as r*J  physical  particle  A required  fsj  the  t e r m , we Model  to  this  Hamiltonian interactions.  use  the  vertex  calculate  the  Interaction Hamiltonian  B^(p_)|0>  i s not  (O  Hamiltonian  order  particle  Under  nucleon-pion  Bag  £B<>(2)  physical  dressed  and  elementary  potential.  (3.1.35). take  involve physical  creators.  nucleon-nucleon  the n u c l e o n - p i o n  we  the  second  the Cloudy  trilinear  the d r e s s i n g t r a n s f o r m a t i o n  particle  the T r i l i n e a r  so  nucleon-pion  explicitly  to transform  nucleon-nucleon  h g i v e n by  Hamiltonian  the  use  nucleon-nucleon  on  Consider function  find  d e r i v e d from  second-order  we  physical  contains physical  function  that  Chapter,  Interaction:  Potential  d i s c u s s e d t h e r e does not In  creators  3  Fermion-Boson  =  an  (3.1.1),  i s an  £B(E)  >  t  eigenket  i n S e c t i o n 4.1  n  with  vertex  eigenket  of  this  energy  of  the  e  of  H.  which w i l l  We  seek  determine  a a  r^/  H(F,B)  expressed  explicitly  i n terms of  operators.  first-order  dressing  operator,  i n v a r i a n c e p r o p e r t i e s and  which  (4.1.20) i s :  satisfies  the  41  In  determining  this  operator  EK.I^-MU.C  We  a r e now  able  we  have  taken (5.1.2)  1  t o d e t e r m i n e [H, ,D| ] . We  find  4- a d j .  (5.1.3)  where  r ^ x ^ y ; '  r  h  p  l  M  x^''  .  h  ^  y  (5.1.5)  ^  pi flu p."  - VC£<^ h-r>"*( o3 ?  (5  •'•  6,  42  The  first  term  (3.1.35) f o r h. We  i n [H, ,D| ] can  be  simplified  r^c^)  •Cz I W X M ' ( x > M » . M " i t w r ) V ' w  Next,  (D.5)  (B.8)  Finally,  we  i s used  is use  used (D.5)  *  The  last  therefore ( 4 . 1 . 2 1 ) .  term  be We  that  in  [H,,D| ] .  to  integrate  t o sum  o v e r m^  d  i n [H, ,D  resulting  ^  form  from  ym.",  h (<?) z  ( 5 . 1 . 8 )  leaving  $p,,/JL-  dcfl.<j, t o g i v e  obtain  Sr<\,,tn  •  S ',,»«". w  Thus  W  ( 5 . 1 . 9 )  form  (4.1.18)  and  must  the d r e s s e d H a m i l t o n i a n given  by  choose  r  t  v  l  x  ( .^  P  fy,t^fy,«(jO  ?  - adj.  e q u a l s minus o n e - h a l f of t h e  C  ^  and  ] i s of the  V  Hamiltonian, to order  H(F,tn = T  1 =  over  W  ~lf %  [H ,D2.] e x a c t l y  The  o v e r yu*. and  t o sum  eliminated  •fv,/(p^F so  the  have  *  First,  using  ^V P L  t  V^ TC  +  VMM  *  (5.1.10)  last  term  )\~, i s :  ( 5 . 1 . 1 D  (5.1.12)  43  - ^ (t K+1) ^ „ ( t K - ^ V ^ * ^ +  VHM=  21  ^  ^  (  Jd*d*fe'd&X  < A K + |0  ^  Li  (5.1.13)  fc+fe')  ^ ^ ' ^ ^ ( K - k ' V  (5.1.U)  F ^ C i k - ^ F ^ t i R - f e O F ^ t ^ ^ O  where  m,c = nv,c x  -(Al^xl  fc/0(ilwh.*  .  - A  U do _h!i$l_ l  1  \£wO  wxAi We  and  z  Ctl/x  1  (D.9),  one c a n e a s i l y  ^v v * M,  that  and  10:  (5.1.16)  ,n,,A,,i  ^  (5,1,17)  X  show t h a t V ^ ^ =  and ( 5 . 1 . 1 7 )  (  V NJ = V ^ t . u s i n g ( B . 4 )  The n u c l e o n mass h a s been r e n o r m a l i z e d functions  (5.1.16)  j u ^ l t ^ C * ! wkA'llwv)]  t ^ . H ^ i  c a n s e e from i t s s t r u c t u r e  (5.1.15)  as w e l l . (eq. ( 5 . 1 . 1 5 ) ) .  may be p i c t u r e d  The  as i n F i g u r e s  9  44  •J )<4-k  ClnOJ^J|) ,  t  I  k+k'  Fig.  iK^^'^fM'p-  9 The 7C -N P o t e n t i a l  (k,k',K)  k-k'fyu")  Fig.  10 The N-N P o t e n t i a l  %^  (k-k' )  M x M i  WiWtinm  From t h e F i g u r e s , coefficients  appearing  we see c l e a r l y  terms  in  7L-N s c a t t e r i n g The  that  conservation  correspond shown i n F i g u r e  physical  —  the  Clebsch-Gordan  i n (5.1.16) and (5.1.17) s e r v e  a n g u l a r momentum and i s o s p i n two  —  at  each  t o impose  vertex.  The  t o t h e two d i f f e r e n t d i a g r a m s f o r 9.  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  is  a  composite  particle  45  consisting We  of a b a r e n u c l e o n have  now  transformations  structure Because cancel  as  seen  and  transformations.  may  To  the  particles  only  become  explicitly  the  5.2  The  vertex  of Chapter  sense  function that  H(F,B).  two-nucleon  of  must  It  we  operators  use  elementary  The  are  The  will  these same  cancelled. always  physical  ones  interactions  via  the  expressed  series  f o r the  terminate  or  be  Potential  the  results  is  r e w r i t e VMOJ  A  be  [Ho,Dn] w i l l  trilinear  of d r e s s i n g  nucleon-nucleon p o t e n t i a l h(q).  of  models.  we  3 and  dressing  t o have t h e  required.  Hamiltonian  i t i s o b t a i n e d from  First,  terms  pions.  properties  relations,  Nucleon-Nucleon  Section  a c o o r d i n a t e space  of  Hamiltonian.  f o r very simple  this  examples  which  particle  the  Second-Order  interaction  those  dressed  o p e r a t o r and  In  term  composites  Physical  summable o n l y  of  certain  commutation  transformation.  dressing  infer  unsuitable  and  in  several  by a c l o u d  o r d e r n, Dr» i s c o n s t r u c t e d  of t h e b a s i c exactly  surrounded  the  V.S K S ' M '  by  (5.1.14) (E.12)  to  determine  terms  of  potential  second-order  i n eq.  defined  i t to  in  a second-order  fermion-boson  term  i n the V,^  i n terms  the  of  in the  find  15.2.1a)  46  where  (5.2.1b)  To a r r i v e a t t h i s combine  the  two  f o r V ti S'M'^Q)  form  w  used  e  S  spherical  harmonics  (B.6)  i n (5.1.17) i n t o a  to  single  one. Based  on  lT^^,(g) nucleon  as  the  discussion matrix  p o t e n t i a l . Many  projection now  the  quantum  in  Appendix  elements  of  the  numbers  E,  we  recognize  o f t h e momentum s p a c e two-  sums  over  i n (5.2.1)  spin  and  isospin  c a n be e v a l u a t e d , as we  show. We a p p l y (D.18) t o t h e f i r s t  coefficients  a p p e a r i n g i n (5.2.1)  (D.5)  ^  and  Clebsch-Gordan  to find  vU ^^ ^ -|(J  z  cr  a  6 j symbol  /JL, we  find  t Vx cy y r j d e n o t e s t o sum o v e r  isospin  (t  \s  JTTo where  three  (5 2 2)  (see Appendix  that  the  D). Using  expression  in  square b r a c k e t s reduces t o  C  3  =  1  W<SVz]  W  V  (5.2.3)  47  We now c o n s i d e r the  curly  brackets  coefficient are  the f i v e  using  in  Clebsch-Gordan (5.2.1b).  (D.9d), we f i n d  r e l a t e d e x a c t l y by ( D . 2 5 ) .  that  coefficients  Rearranging these  five  inside  the  fourth  coefficients  Thus  J_  Us+i)(H+r>U)(z)  (S4Ho^lS H /  , ,  )  Jl  2. i  S ?  ^ (5.2.4)  \  X / (.  1  where  ' Vx S J i s a 9 j symbol  We c a n combine  ^  -  ?  the r e s u l t s  t  s  ^  i  s  (see Appendix D ) . (5.2.3) a n d (5.2.4) t o w r i t e  ..  v ^^ r )  ^  ul  s  (5 2 5B)  (?  where  >|2S+I  Before this  continuing  potential.  conservation  lio)  CMOO  First,  of i s o s p i n  \  X  X  ^  I  l e t us c o n s i d e r we n o t e and  Secondly,  i t can  easily  rotational  i n v a r i a n c e alone  that  (5.2.5b)  the various  t h e f a c t o r <W ^ p '  i t s z-axis  be  shown  means t h a t  that  projection the  ^^^1(3.)  factors in makes t h e manifest.  requirement of must be o f t h e  48  form the  (5.2.5a). trilinear  that  the  only  for I  of  the  spin  form  angular  we  coefficient  = 0,2, c o r r e s p o n d i n g  strong  (5.2.5b) i s a c o n s e q u e n c e o f  w i t h which  Clebsch-Gordan  interaction,  began.  Thirdly,  ( l l oolio^)  to the s c a l a r  note  i s non-zero  and t e n s o r  parts  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 o f  i n t h e 9 j symbol,  which  is  momenta i n any row o r column  j l = 0,2, S = s'  for  specific  interaction  i s implicit  three  are  The  zero  unless  form a t r i a d .  n e c e s s a r i l y . The quantum numbers M  the Thus,  and  M'  n o t r e q u i r e d t o be e q u a l . Based  nucleon  on  the  interaction  above d i s c u s s i o n , we may w r i t e t h e n u c l e o n (5.1.14) a s  NLi - A* i d ' k d W K  A^ Ck,k)  J _ , V*Z  ^ ( 5 . 2 . 6 )  +  where  1=0,2.  . LS± *\ di \St\')  This constant the  potential.  Messiah and  specifies.the  We  have  ) I  are  given  determined  in  -* (  s p i n and i s o s p i n  Table  (5.2.8)  dependence  i t s numerical  (1958, pp. 1065 - 68) t o e v a l u a t e  values  '  values  t h e 6 j and 9 j  I for various total  of  using  symbols  s p i n s and  isospins. We may. u s e (E.11) t o r e l a t e  the  momentum  space  nucleon-  49  nucleon  potential  potent  M M  '(g)  to the corresponding  coordinate  space  ial:  tfrtK'  Using  ^  (5.2.7),  ^  (  E  "  ( B . 7 ) , and  iTgJ. ( O =  '^  ^M«'  ( B . 8 ) , we  (5.2.9)  ^  obtain  ^ ( r J i r J ^ M  X-  (5.2.10)  JUo,2.  where <  f  (rl -  H*l< a ^ /  Thus, g i v e n a t r i l i n e a r the c o r r e s p o n d i n g performing  coordinate  the i n t e g r a t i o n  < s  vertex space  0  Table  I  3 ^  Values  (5.2.11).  IJMIZI-MV-NIIMO » 'TO  — CIZMM'-MIIMO  SM^ Sr\o 1  of the Constant  potential  1  ^MK" S M O  <r= \  N-N  a O S=  Sir  a  2  n )  h ( q ) we c a n compute  second-order  in equation  JU o S=  function  . .  by  50  5.3  The N u c l e o n - N u c l e o n  The  s  Potential  c o o r d i n a t e space  i n the Cloudy  nucleon-nucleon  Bag M o d e l  potential  functions  JL  1} !  ( r ) , given  MM  (1=0)  scalar  by  and  nucleon-nucleon having  spin  elements given  potential  [^*V>lcSH where  f$  CO  Nitr.RV  d  o  We the  show  two  =  now  the Cloudy  oi+Vc 1  the  second-order  two-nucleon  calculate  states  these  Bag M o d e l v e r t e x  matrix function  have  OS'  i n Appendix  H^r,*)  I t ^ ^ I that  integration  given over  the i n t e g r a l s  function r.  f o r the case  do n o t o v e r l a p , t h e i n t e g r a l s  contour  equations  We  ^  using a result  coordinate  between  ( 5 . 2 . 1 1 ) . We  evaluated  simple  CT .  of  of the  (5.3.,,  ft 2  (5.3.2)  s0)  »  bags  Remarkably,  parts  taken  substituting  (3.2.1) i n t o  a r e the matrix elements  (1=2)  tensor  S and i s o s p i n  by  by  (5.2.11),  of  bag  taking  (1.13) and ( 1 . 1 4 ) , we  Ctf^oLa*  Watson  products  =  radius  into R  k = m, K  i.e.,  when  Nj(.(r,R) a r e e a s i l y  (1966), of  Nx.(r,R) f a c t o r  the  Indeed,  in  r > 2R,  involving  Bessel  functions.  the product  and a f u n c t i o n b = R,  and  a  of  a  of the  a = r  in  discover  [iJ-MM'V^OPaP  S  (f0  (5.3.3)  where  <3^ [ 0" / /  M H  =  C/*RV  (r_)] p 0  E P  (5.3.4)  L  are the matrix  elements  o f t h e c e n t r a l and  51  tensor  one  pion  exchange  potential,  which  is  discussed  in  A p p e n d i x J , and  LX= ^  Specifically,  t>5S.°co3  CBM  (5.3.5)  t h e C l o u d y Bag M o d e l p o t e n t i a l s a r e -  i - u f , k ) all' l  VooCO  0  e ^  r  r  9  IR)  r>2R  (5.3.6)  and  (^r)  Thus, the  when  the  '  —  r  function  of  the nucleon  from  dependence  o f t h e two p o t e n t i a l s  as  discussion  operators  The strength  Thus,  two  exchange.  R  i s also  potentials  by t h e f u n c t i o n  ( £ = n ) and S  V  C B H  and  g ( R ) . Note  V  l x  (r>2R),  exactly  spin  and  identical.  to the s p i n - i s o s p i n 0",.  is  (5.3.7)  the  r_ a s i s t h e p o t e n t i a l  The  i n A p p e n d i x J , we r e c o g n i z e  being p r o p o r t i o n a l  nucleon  pion  potential  separation  calculated  the  one  2  two n u c l e o n bags a r e n o t t o u c h i n g  C l o u d y Bag Model n u c l e o n - n u c l e o n  same  7  the matrix  isospin  In l i g h t o f  constants  a^i  elements of the  ( A = 2 ) , which appear i n  o p e p  differ  in  overall  that  f o r r > 0, t h e C l o u d y Bag Model p o t e n t i a l  becomes  exactly  52  the  one p i o n  that  in  that  this  for this  I =0. pion  This  limit  interaction, divergence  Bag  theory  size  cutoff  gives  Cloudy pion So  t o note  Bag  (0.7Z  frrO =  acts  in  which  as  a  when  i n t h e one the  accounts  physically  a l l integrals  the  i n the  because  from a c o n t o u r  r < 2R, i t implies however,  \.0S differs  only  this  an o v e r l a p o f give  slightly  from t h e  1.05.. Bag  Njj(r,R)  Model  can a l s o  region the  the r e s u l t  potential  be e v a l u a t e d  i s physically nucleon  bags.  f o r 1=0,  less For  obtained  integration:  - T ( I*" z ^ - ^ R ^ solution  fm.  (5.3.9)  the Cloudy  although  we  R = 0.72  find  r > 2R. The i n t e g r a l s  case  f o r the v a l u e  by t h e f a c t o r  f a r we have c o n s i d e r e d  completeness,  This  un(q),  keeps  that  Model p o t e n t i a l  exchange p o t e n t i a l ,  the case  clear  which  term  divergences  The f u n c t i o n  function  diverges  t o the £ (r)  nucleon,  (eq. (3.2.4))  i n t e r a c t i o n . We see  (5.2.12)  T h e r e a r e no s u c h  by T h e b e r g e ejt a l . we  g  for  Recall  t h e Chew-Low  rise  of the bare  i s interesting  predicted  for  R->0.  finite.  It  The  as  the i n t e g r a l  Model p o t e n t i a l .  the f i n i t e  meaningful  one  we o b t a i n  exchange p o t e n t i a l .  Cloudy for  exchange p o t e n t i a l  matches s m o o t h l y  with  j  r^ZR  (5.3.10)  (5.3.6) i n t h e l i m i t  r—»2R.  53  Unlike a it  the s o l u t i o n  function  f o r r > 2R,  of R and a f u n c t i o n  however,  i t does n o t f a c t o r  of r . F i n a l l y ,  i n the l i m i t  into r->0,  goes t o  <\  e.~^  w h i c h becomes i n f i n i t e  Z  for  -  ( l - ^ R ^ j  R->0.  (5.3.11)  54  Chapter  with  6  Summary and  Our  study  the  nucleon-pion  interaction arbitrary  s p i n and of  t o use  expression The  function  of  a  Bag  the  to  illustrated we  dressing that  developed  to  any  this  Lee  the  acted  discovered only  for r e a l i s t i c be  used  that  function.  we  a  does not an  developed  c r e a t o r s and  eigenket  the  creators  find  the  and  operators.  the  as  a  and  dressing  in perturbation  theory.  scalar  field  trilinear  interaction,  trilinear  interaction  of  the  perturbation  theories a  of  dressing  annihilators,  to the  We  explicitly  Hamiltonian  determining  nucleon-pion  space  specific  a  bare p a r t i c l e  order  the  function.  obtain  (4.1.20)) f o r the  features  b o s o n s of that  into physical p a r t i c l e  terminates  to  vertex  vertex  fermion-boson  common  This  spatial rotations,  to  the  3.  rotations in isospin  dressing transformation  several  must  on  desired  generalized  found  Model  for  model, t h e  operator  We  trilinear  4,  them  method  H.  b o s o n s began  and  b e c a u s e F^"|0> i s not  Chapter  (eq.  fermions  interaction  physical particle  precise  of  the  trilinear  expression  the  Application model,  In  and  and  of C h a p t e r  translations,  trilinear  which  transformation  and  Cloudy  to transform  d e r i v e d an  First,  form of  physical particles,  annihilators  and  the  Hamiltonian.  gave  i n Appendix  reversal,  f o r t h e NNJt  transformation  We  the  fermion-boson  involve  include  i n v a r i a n c e under time  fermions  interaction  to  isospin  restricted  were a b l e  trilinear  extended  inversion,  greatly  the  strongly interacting  was  requirement space  of  Conclusions  for very technique operator  transformation. series  simple such D.  for  the  interactions, as  we  Secondly,  have to  any  55  given the  order same  i n perturbation theory, D  constructed  s t r u c t u r e a s t h e u n s u i t a b l e term  w h i c h must be e l i m i n a t e d from will  is  n  then  cancel  the dressed  e x a c t l y and o n l y  particle  particle the  creators  c r e a t o r s . We  second-order  analogous  vertex  interactions explicit  (4.1.18)  Hamiltonian.  [ H ,D ]  t h e term  calculations;  implicit  in  the  expression of  the  renormalization  in  higher  o r d e r s we e x p e c t  an  trilinear  t h a t c a n be c a l c u l a t e d  was  nucleon  the p h y s i c a l Hamiltonian  from  the  particle are  original  nucleon-nucleon  determined  i n Chapter  bags were n o t t o u c h i n g ,  p i o n exchange p o t e n t i a l  potential  made  trilinear  5. We  this  f o r the Cloudy  found  potential  multiplied  was  potential  potential.  The  nucleon-pion  was f o u n d a  extended  when d r e s s e d  t o c o n t a i n not only a nucleon-nucleon direct  nucleon-pion  t o i n c l u d e t h e A , we  interactions calculated nucleon  Hamiltonian,  as  well.  analogously  interaction.  Potentials to  our  f o r these  determination  p o t e n t i a l . When t h e f e r m i o n - b o s o n  dressed  production  to third on  two  order,  we  fermions,  find  a  the  Bag M o d e l  t o second  and  interactions the  trilinear term  but  the theory i s  A~A ,  of  N-N  order,  interaction  When  N- A /  obtain  simply  by a f u n c t i o n o f t h e bag the Cloudy  goes t o t h e one p i o n e x c h a n g e  Bag  t h a t when t h e two  r a d i u s . As t h e bag r a d i u s goes t o z e r o ,  is  elementary  function.  Model  also  f o r the  mass  renormalization. Finally,  A second-order  one  to  n  required. Thirdly,  composites  a fermion  0  by t h e 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 t o i n t e r a c t i o n  potentials vertex  found  as  have  o f t h e form  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 physical  to  A -% c a n be  nucleon-  Hamiltonian  describing  boson  i . e . $ i r F^F^FF^B^". The d r e s s i n g  56  transformation integral  involving  c o n t r a s t s with this work  the b a s i c  on  the  the H a m i l t o n i a n  dressing  potentials.  functional  Hamiltonian energies.  form  to c a l c u l a t e  trilinear  f u n c t i o n l)~ i s d e t e r m i n e d  interaction the  a p p r o a c h a l l o w s us  vertex  function. (1978),  only phenomenologically.  In t u r n , t h e y a  i n t e r m s of  c o n s i d e r e d by H s i e h  t r a n s f o r m a t i o n can  of  if  can  used  s e r v e as a  long-range,  f o r s y s t e m s of p i o n s and  be  strong  nucleons  at  an  This where  Thus  our  to f i n d  such  basis  for  interaction intermediate  57  Bibliography Abramowitz, M. and S t e g u n , I . , 1965, Handbook o f M a t h e m a t i c a l F u n c t i o n s , D o v e r , New Y o r k . Dirac,  P.A.M., 1949, Rev. Mod. P h y s .  2J_,  392.  Edmonds, A.R., 1960, A n g u l a r Momentum i n Quantum Princeton U n i v e r s i t y Press, Princeton. Faddeev,  L.D., 1964, S o v i e t  Glockle,  W. and M u l l e r ,  Greenberg,  O.W.  H s i e h , W.W.,  Physics  Doklady  8, 881.  L. , 1981 , P h y s . Rev. C 2_3, 1183.  and Schweber, S.S., 1958, Nuovo Cim. 8, 378.  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  J a h n , H.A. a n d Hope, J . , 1954, P h y s . Rev. 93, Kalyniak,  P., 1978, Columbia.  M.Sc.  Thesis,  M e s s i a h , A., 1958, Quantum M e c h a n i c s  R . J . , 1954, P h y s .  Columbia.  318.  University  v . 2 , W i l e y , New  M o r a v c s i k , M.J., 1963, The Two-Nucleon Oxford. Ord-Smith,  Mechanics,  Interaction,  of  British  York. Clarendon,  Rev. 9_4, 1227.  P i s k u n o v , V.N., 1974, T h e o r . Math. P h y s . J_5, 546. Rose,  M.E., 1957, E l e m e n t a r y T h e o r y New Y o r k .  o f A n g u l a r Momentum, W i l e y ,  Schweber, S.S., 1961, An I n t r o d u c t i o n to Relativistic F i e l d T h e o r y , H a r p e r a n d Row, New Y o r k . Theberge, Watson,  S., Thomas, 22, 2838.  A.W.  and M i l l e r ,  G.N., 1966, T h e o r y of Bessel U n i v e r s i t y Press, Cambridge.  G.,  Quantum  1980, P h y s .  Functions,  Rev. D  Cambridge  58  Appendix This involving of  4 The R o t a t i o n M a t r i c e s D ^ i ( A g>t )  A  m  Appendix the D  m w  the r o t a t i o n We t a k e  i n Rose  lists  some  properties  ' (<Ap~f ) , t h e i r r e d u c i b l e  and f o r m u l a e  matrix representations  group.  the conventions  (1957) and M e s s i a h  t h e c o n v e n t i o n s used From Rose  useful  f o r t h e s e m a t r i c e s t o be t h o s e  ( 1 9 5 8 ) . Note t h a t  i n Edmonds  (1957, p . 5 4 ) ,  these  differ  used from  (1960).  we have  Upr) = £ W  C-tf.-p,-^  (A.2)  When A. = 0,  D° UST) =i a  When  o( =Y=  0, t h e r e a l  <£> W Note  that  The  orthogonality (1957,  ^  X  D  A  3  m a t r i c e s d^Vn a r e d e f i n e d by  "  m  from Rose  <->  0  ^  Cof>o)  properties  of the r o t a t i o n  (A.4)  matrices are,  p.73):  JL*  D^* Wfctf ffjno' ^ po^  $**  m  0™« '"mm  U p ) = $„,.„,<•  (^pTT) » v... oW * ...  (A.6)  (A. 7)  59  The (Rose  rule  (1956,  f o r combining  two  rotation  matrices  into  one  is  p.58)):  (A.8) Finally, Z7C  T  from Rose  (1957,  p . 7 5 ) , we  have  2A.  =  S*».W  Sm,,*,. Sj,j^ C ^ T , )  (A. 9 )  60  Appendix  B  The  This  S p h e r i c a l Harmonics  Appendix  involving  B  some  useful properties  t h e s p h e r i c a l h a r m o n i c s . We y  where  lists  and  ^  respectively, Yjtm  w  ty  are  *  (q)  and  formulae  write  ie,**)  the  polar  (  and  azimuthal  under  rotations according  ,  1  )  angles,  u s e d t o s p e c i f y t h e d i r e c t i o n of t h e v e c t o r  transforms  B  t o (Rose  g. ( 1957,  p.60)) : il  V  -  A m  2- D  y ,  mWi  lwi  ( B > 2 )  where  - ^% In  (B.3),  coordinates  g  and M  From Rose  where  is  column  vector  i s the matrix  (1957, p.61)  we  Urn  t-£>  =  C  C|)  * CO"  (jijzM.r/K  Appendix  a  (B.3)  IjfO  0  is  a  A  given  in  its  i n Rose  three  Cartesian  (1957, p . 6 5 ) .  have  ^  (B.4)  ( B  Clebsch-Gordan  coefficient  .5,  (see  D).  From Rose  (1957, p . 8 1 ) ,  ^ - H - J .  Jui'liWOW^klri  (B  .  7l  61  Also,  J  f r o m Rose  (1957,  H timty ft*, ty =  dG/  j  I o o  p.75),  ^  we  (S,(P)  Y  have  l W  (a,^  * n * dG  dCD  62  Appendix  C  Irreducible  T e n s o r O p e r a t o r s and  the  Wigner-Eckart  Theorem  An  irreducible T  quantities according  l  (-L  H  t o (Rose  tensor  operator  < M < L) (1957,  which  of rank L i s a s e t o f transform  under  2L+1  rotations  p.77)):  i-  (C.1)  LM'  (Note  that  t h e above  operator  R i s not t h e Fock  space  operator  # up")). Any  vector  rank one,  with  k =  ( k * , ky, k )  spherical  Alternatively, written  can  -  the s p h e r i c a l  ^  .  rotationally  be w r i t t e n  |S"  operator theorem  T ^ u<  gives  projection  irreducible  components  of  .,,±,,0  w t  scalar  product  spherical  = X. t-v  k* a  ,  the  element  matrix  angular  t h e dependence  be  two  vectors  (c.4) of  this  quantum numbers. I t s t a t e s  of  (C3)  components:  the  irreducible  momentum s t a t e s . of  can  harmonics:  i n terms of t h e i r  between  tensor  of a v e c t o r  V ^  invariant  4  Consider  an  components  i n terms of the s p h e r i c a l  k, The  is  z  matrix (Rose  tensor  The W i g n e r - E c k a r t element  (1957,  on  p.85)):  the  63  \\ T \ ll j V  where set  is called  of t e n s o r o p e r a t o r s For  the  case  operators J , this  T  that  reduced  L M  the reduced  matrix  element  of t h e  . the  T ^  matrix  5lVj>= vj iCj + .)  L  are  element  Sjj'  the  angular  i s (Rose  (1957,  momentum p.89)):  ( C . 6 )  64  Appendix D  In  A n g u l a r Momentum  this  Appendix,  numbers o f a n g u l a r j(m).  Because  intermediate  we  consider  momenta t o  such  a  form  coupling  representations,  representations coefficients, coupling  Coupling  must  be  the  a  total  can  take  coefficients  defined.  a d d i t i o n of v a r i o u s angular place to  we  list  via different connect  these  These a r e the Clebsch-Gordan  3 j , 6 j , 9 j and 12j s y m b o l s . A f t e r  coefficients,  momentum  defining  these  some u s e f u l f o r m u l a e i n v o l v i n g  them.  (a) A d d i t i o n  o f Two A n g u l a r Momenta:  Coef f i c i e n t  Consider coupled  and t h e 3 j Symbol  a system  t o form a t o t a l  i n v o l v i n g two a n g u l a r angular  5, + 5 * This  system  operators. iZz / J , Z  a  /  can  Ij, ^  Let a  r  e  diagonal,  momenta J ,  and  t  (D.1) four  denote and  These  J  momentum J :  » £  have  m.tYi*.^  J , JJ;. a r e d i a g o n a l .  unitary  The C l e b s c h - G o r d a n  simultaneously the basis  'jijijt^  diagonalizable  state  in  the state  two r e p r e s e n t a t i o n s  which  J, ,  i n which J , ,  a r e r e l a t e d by  transformation:  and Ijijzrn,^^ =  The are  2_  I j, j \ jm  elements of t h i s  the  Clebsch-Gordan  >^0'v}*-*)iMz IjijxjfYi')  transformation, coefficients,  (D.3)  j no )> which  we  denote  , by  65  ^jijitfhwh. lj  *  these c o e f f i c i e n t s  Their  P  I f c  o  s  i b l  s  e  such that  that  o  define  a phase c o n v e n t i o n f o r  they a r e r e a l :  orthogonality  requirement  t  properties  are derived  t h e two r e p r e s e n t a t i o n s  from t h e  be n o r m a l i z e d : (D.5)  20  tjiiz  J j  The c o e f f i c i e n t s  m  ) ( J . J ,tf,.^  -^  have t h e p r o p e r t y  i W | i l  g  (D.6)  that (D.7)  unless  m, + m^ = m, a n d ( j , , j , j ) form a t r i a d , i . e . , z  (D.8) The are  given  symmetry i n Rose  Cj'.jiW.iYij.lJatfiO  relations  of the Clebsch-Gordan  coefficients  (1957, pp. 3 8 - 3 9 ) : =- ( - ^  , +  "  J l  -"  j 3  m x M (  |  j 3 W 1 i  )  (D.9a)  (D.9b)  (j« Ja V '-Wslji-Wz.) 1  (D.9c)  J l + 2 I 2 i +. I M  t-0  =  vJ^V"  9d) (  ^ 0 * -  (r^' ^'J^j'xVl ^ i 3  =  1  W  3 ^ l o . - M . )  "1* -»^< 1  via  «0  ^""l^r-^is^^lj^^  (D.  (D.9e)  (D  -  9f)  66  From Rose  (1957, p.42)  (j, O  jj  O  lYO  a  and  One  J  can  '  ^  *  '  Sj ' x i  0  by  (Edmonds  ^  the  momenta J  following  Momenta: two  , J , J 3 t o form  t  z  J, + O  = J,  x  3  (D.\0)  SK),M  ^  (  3  ^  (  t o the  z  The  6j  a total J  J,  +• £xs  3  -  ,  2  )  p.46)):  schemes  J,^  D  Clebsch-  ^.OxM.^I^-^  (b) A d d i t i o n of T h r e e A n g u l a r Consider  have  is related  (1960,  r,-  UtfK»*J~  angular  ^  0  d e f i n e a 3j symbol w h i c h  Gordan c o e f f i c i e n t s  we  Sj.jj SrtJ.y^  =  (O J z O ^ l j j ^ ^  '  (D.9),  (D.13)  Symbol  f o r combining  three  J:  = J  (D.14)  = 2  (D.15)  and Ji In made  the  +  first  2  case,  diagonal;  ^ j > i O j n •> i j , jy J , ,  - Iz»  J can  be  6j  symbol,  is  d e f i n e d by  the  we •  I  n  t  ^  •  i e  o p e r a t o r s J, , J ,  denote  the  second  case,  d i a g o n a l i z e d ; the  which  relates  (Edmonds  these  (i960,  J , J _ , J can  z  be  l2  corresponding  state  by  the o p e r a t o r s J, ,  state  two  3  is  I j , Cjxja") j (  independent  p.92)):  i x  J , 3  t  j y  .  The  representations  D  ,  1  1  67  =  ( = = 1  The built  two  different  representations  Substituting  -  (D  16)  (D.15) can be  example,  ljiM*>  (D.17)  (D.16) and  using  (D.5),  finds  The columns in  For  lja.wU>  such expansions i n t o  \ W>\  (D.14) and  up f r o m C l e b s c h - G o r d a n c o e f f i c i e n t s .  •Ij.rfO  one  ){  6j  symbol  and under  certain  numerically.  U LVx  J5 ix  From  under  a p e r m u t a t i o n of i t s lower  cases Edmonds  the  6j  symbol  can e a s i l y  Jx  J3  7  be  evaluated  (1960, p . 9 5 ) :  [(!*•,, L  J  .N  .  (D 19)  .  —I  - -v  (2jx+i ) (2j +OC2jV) Uj f0 u  x  Ji  variables  columns.  \- &  Jj'Yx jxtVi J  invariant  i n t e r c h a n g e o f t h e upper and  e a c h o f any two In  is  J1+J2.4-J3 f" / * . " ' . ' . ."N  2jz  czjx+o  (D.20)  3  *.  .* N  UjOUjs+O  ~7 ^2-  (D.21)  68  (c) A d d i t i o n  of F o u r A n g u l a r Momenta:  Consider angular  the  following  two  momenta t o make a t o t a l  = In.  Ii + I * = J  The  9 j Symbol  schemes  for  combining  four  J:  In f ^ = J  3 4  ( D  '  2 2 )  and  The  9j  symbol,  representations,  which  i s defined  by  relates  these  (Edmonds  (1960,  two  p.101)):  J»-  CJ i  JIT.  •r i w « M 2 i « 4 . v w i ( w j ' ,  Expanding twice,  one  the s t a t e s  )  ji0  h  i n (D.24) as  independent  i n (D.17) and  (d  using  -  24>  (D.5)  finds  IjxM  W) in1if  *  a  JI  r/l,* r / l x  J x  0>z  OH  03*1  | j  H  no)  (D.25) Acting and  using  on b o t h  (D.5)  s i d e s of  gives  (j  (D.25) w i t h w  b  4  a  „VWj,»  Ij'^')  69  m,  r/ko  x  ^  ' ^«o3 •  From  we  a n g u l a r momenta is  invariant  rows  or  and  the  see t h a t i n any  rows o r c o l u m n s ) . Under  In c e r t a i n  cases,  From  W\  or column  an even  rfl  x  of  t h e 9 j symbol  row  (i.e.,  CjajM rO*Wk  properties  under a t r a n s p o s i t i o n ,  columns  numerically.  mO.O  \jmr>") ( 0«-  mArt  (D.26)  coefficients,  >nn,Wl» Iju  mOiO*  [jm^  3 v  the  Clebsch-Gordan  i s zero unless  form a t r i a d . or an  (D.26)  even  The  the 9j  three symbol  permutation  number o f e x c h a n g e s  of  adjacent  an odd p e r m u t a t i o n , t h e phase c h a n g e s  t h e 9 j symbol  (Edmonds  can  (1960, pp.  CO  be  easily  by  evaluated  101,105,106)):  C a. b  I um  K  OH JzH ")  of  C)  rt  034  e  i  j J  (D.27)  *  J  (D<28)  70  (d) A d d i t i o n  o f F i v e A n g u l a r Momenta:  Consider  the  a n g u l a r momenta  following  two  The 1 2 j Symbol  schemes  for  combining  five  t o make a t o t a l J :  and I , + CTa = 3"t» The  3s  12j  t 3 « - 2"  symbol,  representations,  = .1  which  i s defined  Expanding coefficients,  W „ W »  3H -- Iz*  relates  I"  these  by ( O r d - S m i t h  f J M =I 2  two  (D.30)  independent  (1954)):  ll  J1  0i  is  ot  • fl  (D.31 ) 05  0i*  w^rti*  Ix  the  states  and u s i n g  i n (D.31) i n t e r m s  (D.5) t h r e e  t i m e s , we  of Clebsch-Gordan find  tiflci  M  M  "  • ( j  y  ^  ^  s  ^  ^  • (05 j i z ^$ »0ia Ij'w')  (j\ ^  yY]^)'  C " j s rfl" vY)^ | j to ) 0  x  j. ' ( j ' v)W ' *>3M  w^yvu  _  m  Oft  J l  J'M o  i 4  <  • II  j i u o" 0 /  (D.32)  71  The symmetry Smith  (1954).  simplifies  relations  When  one  of t h e 12j symbol element  is  are given  zero,  the  12J  in  Ordsymbol  to  a  b  e  C  d  f  g  h  o  c } ^ 9 ? = [UevOOq^J^ ^ 5 J  d  f  7  (D.33)  72  Appendix  E  Two-Particle Operators  T h i s Appendix p r o v i d e s body  potentials  i n Fock  Space  some b a c k g r o u n d  to the v a r i o u s  and t w o - f e r m i o n o p e r a t o r s t h a t a r e u s e d  twoi nthe  thesis. Two-particle the  operators  corresponding  particles. space; complete Fock  extension  identical  that  we  of  total  only  with  space  of n  fermion  i s easily  two-particle  potential  space,  Fock  made. F o r a  operators  in  V  n  of  a system of n  we have  that  =  v e t , ,  o  t«i d e n o t e s t h e c o o r d i n a t e s , momentum, fermion  The  i n the H i l b e r t  from  i s due t o t w o - p a r t i c l e i n t e r a c t i o n s .  Hilbert  v ( ? * ,  of  constructed  (1961,pp.140-2).  fermions  i n the n-fermion  deal  are  i n c l u d e bosons  o f t h e form  the  physical  providing  to  s e e Schweber  Suppose  Then  Appendix  derivation  space  space  two-body o p e r a t o r s  In t h i s  the  i n Fock  oi.  V(f ,fp) A  corresponding  (E.2) spin,  i s the p o t e n t i a l  Hermitian  operator  and  between  i n Fock  isotopic fermions  spin  o{ and  space i s :  (E.3)  73  where t h e i n d e x for  a physical  That  i s , the  j  runs  over  1,2,3,4. F  operator  f e r m i o n and  xrj- ^(x,y,x',y')  elements of the two-fermion two-fermion  i s the c r e a t i o n  1  a r e the c o o r d i n a t e space  potential  V(£.,,^p)  taken  matrix between  states.  M a k i n g t h e change o f c o o r d i n a t e s  to  center  condition  1'=  R'+yzr'  of  mass  K'-/2r'  and r e l a t i v e  of displacement  (E.5)  c o o r d i n a t e s , and i m p o s i n g  invariance, this  the  two-particle operator  becomes  V- I  jd»r d V d R d V 3  ( r . r , R-R') 1  (E.6) Note t h a t be  V~ c a n o n l y depend on t h e d i f f e r e n c e  translationally Using  where  (2.1.1),  invariant. V can a l s o  be e x p r e s s e d  as  R-R' , i f V i s  to  74  e For  we  find  e  the s p e c i a l case  that  the i n v e r s e  e.  v  E  -  s  ;  that  of e q u a t i o n  (E.8) i s  where  < X We A * a  now  +  W  K<r  =  define  ~  ^  a two-fermion  K) =  0  i  7=S(S L,U  ^  X  (s s l  i r  ^  operator  fl,^  as  I S M ) (i,  ( E  .  n )  follows:  L i  u,a ( ^ ) • 2  z  Mifri*.|i,yu-z  • F ^ ,  As  The the s,  a function  +  Ittt^ F^tit-fc')  of the c o o r d i n a t e s  two o p e r a t o r s A j ^ ^ " (fc,K)  vacuum  state,  with p r o j e c t i o n  create m,  and A ^ ( r , R ) ,  a two-fermion  and t o t a l  They a r e r e l a t e d , u s i n g  £ and R t h i s  (2.1.1),  isospin by  state  (E.,2) becomes  when a c t i n g  having  on  total  spin  CT , w i t h p r o j e c t i o n  LL .  75  Using Appendix  the properties  of the Clebsch-Gordan  D, t h e t w o - p a r t i c l e  interaction  i ^ d ' k ' d ' K c?;:^  (s,^^  coefficients in  V c a n be w r i t t e n  A ^ V ^  A  $  as  (  E  .  1  5  )  S<5 S ' 0 ' where  5  j J ^iMj 1  CL c H  The space  potential  matrix  elements  s' , i s o s p i n  Figure  ^ya  l a y )  3  functions  between t w o - f e r m i o n spin  3 /  C .  of  states These  IT^the  tjW7  i r j ^  CK.K'.K)  y,(k,k/ ,K)  two-fermion  of t o t a l  spin  functions  are  the  momentum  potential  V(f ,f^)  s, i s o s p i n CT  may  be  11:  11 The Two-Fermion P o t e n t i a l  i T - . - ^ . ( k , k ,K) J J M JP J  5 L  r f  and t o t a l  pictured  2 K + is.  Fig.  (E.16)  as  in  76  Appendix F  Calculation C l o u d y Bag  In  this  function  i n the  we  Trilinear  Vertex  determine the  Cloudy  one-nucleon  interaction eq.  f o r the  the  Function  for  the  Model  Appendix,  h(q)  The  of  Bag  matrix  Cloudy  Bag  NN/i  trilinear  vertex  Model.  elements  of  the  nucleon-pion  Model a r e  ( T h e b e r g e e_t a l • (1980,  (2.25))):  <rtiH5SN7« ^  t^r  ir/"ck)*  where f<, i s t h e  3  unco  ^  coupling = ( ^ ^  NN7T  c J C V ,  UJOO where is  ^ (kR) i s a (  the  bag  projection has  as  =  < b * . , - k y O t l  c o n s t a n t , m^ ) Y  i s the Pauli  i n u n i t s where fi = c =  (3.1.41)  determine the  for vertex  the  the  nucleon  pion  of  order  i s the  projection  a  )  <F.U»  mass,  spin  m  one  and (F.2)  R  spin-isospin and  operator, 1^=  and  t^lz  isospin where <5 are  i s o s p i n o p e r a t o r . The  interaction  nucleon-pion h(q)  1  the above  1 .  between the  function  ija>  spin matrices;  is written  form  function  nucleon  components of  similarity  i s the  .  F  (F.3)  Itio,  spherical  Note t h e  \ k ^ , T ^ y  a  a nucleon with spin  i t s components t h e  M  (  3 ",(.hR)/ h R  Further,  JUL ; S = £[/z.  fQdj.  l  s p h e r i c a l Bessel  radius.  state describing  VVB^  5^S fc  1  trilinear  f o r the  (F.1)  and  interaction.  C l o u d y Bag  Model  the We by  77  evaluating  the  spin-isospin  comparing  the  given  (3.1.36).  by  First, element. we  result  we  matrix  elements  with the one-nucleon  separate  the s p i n  and  matrix  isospin  S. • fe i n s p h e r i c a l  Expanding  in  (F.1b)  elements  and  of  H  (  p a r t s of the  matrix  components u s i n g  (C.4),  have  = The  t-V* k+  ^  Wigner-Eckart < k  rf\,  <"z/A,  I  ]t  H  M  ^-ku, (x^|-k  iSdL theorem  rYli>  =  l-I/O =  (C.5)  >  M l  (F.4)  gives  t z IMicHl  rfO  Cx» J ^ l t y O  ^  S ||:£>  (F.5)  I-L >  (F.6)  II I  where  =  i Substituting (C.3)  ^11  X  H-k>  ( F . 5 ) , ( F . 6 ) , and  to express  q.^  = Ji/z (F.7)  i n t e r m s of q and  (F.7)  into  ( F . 1 ) , and  a spherical  using  harmonic,  we  find  • ( x I /^i-yj- "ai / O 1  Now  C i l rf^i l-knn^ j ^ C ^ )  c o n s i d e r the m a t r i x  (3.1.36)  between  relations  f o r the  one  elements  nucleon  fermion  of t h e  states.  o p e r a t o r s , we  + ad  interaction  Using find  (F.8)  y  the  H,  in  commutation  78  (F.9) Comparing required  to  (F.8) give  w h i c h we w r i t e  and ( F . 9 ) ,  and  putting  i n ft a n d  the c o r r e c t dimensions t o the vertex  as h  C B h l  (q),  c  as  function  we s e e t h a t  (F.10)  where  6^ (q)  formalisms vertex  i s given can  Rotational  interactions  will  pion  be  character  two  by t h e c h o i c e  t o do t h i s  rotations,  reversal.  invariance  The  because  space  invariance  different  (F.10) f o r t h e both  ( F . 1 ) and ( 3 . 1 . 3 6 ) , a r e r e q u i r e d  translations,  projection  (3.1.3).  made i d e n t i c a l  f u n c t i o n . We a r e a b l e  interactions, under  be  by e q u a t i o n  trilinear  t o be i n v a r i a n t  inversion, determines  have t h e same d e p e n d e n c e on s p i n  and that and  quantum numbers and on t h e a n g l e s g . Space requires one,  and  that this  of the operator  the o r b i t a l is  angular  manifested  S (eq. (F.5)).  momentum  time both  isospin inversion of  the  i n ( F . 1 ) by t h e v e c t o r  79  Appendix  In  G  An E x a c t  this  Appendix  determining discussed h  an  equations  we f o l l o w t h e work o f  exact  dressing  Piskunov  operator  is  analogous  to  the  for  procedure  (4.3.12a) and (4.3.15a) s u g g e s t ,  the  ?  attempt  determining  3  to  find  the  creator  that Fi (p_)|0>  requiring  T  Recall  Fjty  from  Fx  (p.)  given  d(p_,g).  i - terms n  be an e i g e n k e t  for  in  model  the  case  here.)  As  (G.D  We  do  this  by  of d(p_,g) and t h e n  of the Hamiltonian.  (4.1.3) t h a t  = e° F, = K\^-  Using  function  Lee  adj.]  ?  the  (1974)  we w r i t e  F,( -^B(£)-  $ d p d o_ c k f , ^ j  D=  we  f o r the Lee Model  i n S e c t i o n 4.3. (The method o f s o l u t i o n  = h(p_,g)  and  Dressing Operator  f  e~  D  trSip,*]*  (G.1) f o r D, a n d t h e  ir £ £ F ^ > , D J * , D > . . . ?  commutators  (4.3.5) -  (  G  -  2  )  (4.3.9),  find  UFa^.D]  = J d ^ d ( , ^ r, C D , | ) B ^ ) ?  +  +  (G.S)  and  where  The  series  i n (G.2) t h u s  separates  w h i c h c a n be summed e x a c t l y . We  find  into  two s e r i e s ,  each of  80  F (^ = c o s d ( ^ +  fc ^ F,+c -f>Bty  - Jd'f  z  ?1  (  ?  G  -  6  )  where  This  equation  substitute  F  T X  (p)  gives  into  in  T  -  (G.8)  +  V-particle  t h e commutators ( 4 . 3 . 5 ) - ( 4 . 3 . 9 )  (4.3.1) f o r H, e q u a t i o n  [£  [cosdC^  2 0  - A J d ^ hC|Uf  c*V  +  the  scalar  < 0 | F ( p ) and t h e n /  i  and  the  state. expression  (G.8) becomes  - F , Cf>-^ b C p Taking  o f d(£,cj). We now  Fz ljrtlo>  E ^ g ) i s t h e e n e r g y of t h e p h y s i c a l Using  with  terms  the equation  HF^l^lo) where  F,_ (p)  lo>  f  product with  J  ? l  ^]  FA^|o>  = O  of t h i s equation  (G.9) from t h e l e f t  <0|F, ( p / ^ )B(g/ ) g i v e s -  cosd ( ' ) ?  (G. 1 1 )  cos <  Substituting  (G.I 1) i n t o  f*(if> * ?zc CP') + A * f d o 3  (G.10) g i v e s t h e r e s u l t  ;  (G.12)  81  We into  now  solve  (G.11),  both  sides  f o r d(p_,g) by  squaring  with  the  respect  s u b s t i t u t i n g f(p_,g)  r e s u l t i n g equation,  t o g.  We  from  and  (G.7)  integrating  find  = X*  W^C;^  (G.13)  where  (G.14)  Finally,  from  (G.11),  d(£,Q)=  "arc-fan  dressing by  one  first,  Indeed, obtains  second,  (4.3.12),  iteratively  is  that  given  Using particle  exactly  and  third  solve by  and  (G.15),  (G.7)  the  find  that  as  in  one  (G.1), in a  same d r e s s i n g  orders  as  is  can  >  £z(p_) >  power  1  5  )  the given  series  in  transformation  to  given  the  write  w i t h d(p_,g)  (4.3.15). Furthermore,  by we  equations  note  s o l u t i o n to  that  order  if X" 7  (4.3.22). and  Cosq^fcV-ACdSo  physical V-particle state  containing one  G  (G.6)  we  obtain  the  physical  V-  creator:  FzV=  N and  (  i s expanded  (G.12) f o r  equation  have  £,CJ>-$WBC$\I  exactly, D  ( G . 1 3 ) , we  3  m o d e l , we  if this  (4.3.14),  we  The  Lee  transformation  (G.15).  A ,  f o r the  and  (A C^;  r^(^)-  g(E) Thus,  (G.7),  an tJ  is  ^F.*f?-?>BV a  elementary V p a r t i c l e  particle.  superposition with a  state  7  of  (G.16)  a  state  containing  one  82  Appendix H  A-Dressing  Transformation  Boson T r i l i n e a r  In t h i s Chapter  Appendix  In  doing  interactions other  of nucleons  types  of  placed  requirement  that  transformations.  on it  not  only  and p i o n s , and  this  are  but  bosons.  able  also  invariant  under  Secondly,  we p e r f o r m  a dressing  dynamical  Finally,  we c o n s i d e r  the  Hamiltonian,  variables  various  particularly  to  terms  in  of  we examine t h e  certain  as  and  consider  interaction  be  fundamental  spin  interactions  First,  generalized  the  dressed  i n t e r a c t i o n of  and b o s o n s o f a r b i t r a r y  we  fermions  restrictions  on  we g e n e r a l i z e t h e t r i l i n e a r  so,  Fermion-  Interaction  3 to include fermions  isospin.  For A Generalized  by  the  space-time  transformation in  Section  the  5.1.  second-order  the fermion-fermion  scattering  term. Although generalized  formulae  become  interaction,  transformation interaction  the  as  can to  the  be the  theory  applied simpler  more c o m p l i c a t e d  as  of  with  the  dressing  successfully  theories  we  have  this  to  this  previously  studied.  (a) The G e n e r a l i z e d F e r m i o n - B o s o n T r i l i n e a r Consider constructed Section  the  following  Interaction  generalized t r i l i n e a r  Hamiltonian  from t h e f u n d a m e n t a l 'dynamical v a r i a b l e s d e f i n e d  in  2.1: H  =  H  0  + X H,  (H.1a)  83  H»» 7Stmu  J*pffi£V'£V  £yVO?>J  *  (H  '  S'i'mV  (H.1c)  +. a d j .  m.itiairtjju.ya^j  where  A ; ^= [pVt^ S  is  the energy of the elementary  isospin The and  i and i s o s p i n z - a x i s  j  fermion  projection  f e r m i o n s and bosons a r e t r e a t e d  the vertex The  H c  function  i s chosen  (H.2)  y i  o r boson h a v i n g ju. .  'semi-relativistically'  t o be a f u n c t i o n  i n t e r a c t i o n H, may be p i c t u r e d  12 The G e n e r a l i z e d The  total  Trilinear  momentum o p e r a t o r  F-B  of g  as i n F i g u r e  1 £ UM.I.JU)  Fig.  s p i n s,  Interaction  f o r the system i s  12:  only.  -  ,b)  84  The H a m i l t o n i a n H i s require  that  spatial  rotations,  already  satisfies  further  i t be  translationally  invariant space  under r o t a t i o n s  inversion,  these  invariant.  and  We  in isospin  time  r e q u i r e m e n t s , b u t H,  also space, Ho  reversal.  does n o t w i t h o u t  restriction.  First,  we  consider a rotation  in  isospin  space.  We  must  have  tf? US*") x  Using  (2.2.18)  (3.1.6) t h r o u g h invariance  and  H, # i «(J7f) = M,  (H.4)  +  (2.2.19),  ( 3 . 1 . 1 0 ) . We  we  proceed  discover  that  similarly isospin  to steps  rotational  implies  Secondly,  we  consider a spatial  rotation,  demanding  flup-rt H, ft up*) = H, f  Analogously  t o (3.1.13),  • c£,; As  i n (3.1.14),  ^  ( 2 . 2 . 9 ) , we  C;*  -  6 )  find  (H.  71  we w r i t e  Substituting expression  u s i n g (2.2.8) and  ( H  this  expression  comparable t o (3.1.17),  into namely  (H.7),  we  obtain  an  85  D ;„.  Using matrices we  (A.1)  p r o c e e d as  ma.,  m, 3  in  M  (A.8)  and  i n t o one,  and  from  -  1  • (5 S t  Thirdly,  we  the  vertex  (H.8),  £  W < p  we  can  h  that  H  of  second p a i r  9)  rotation into  that  i s given  -  one,  the  m,,  by  write  c^V  A s  M  be  v  <H  pair  discover  function  l s « 0 ( ^ S, rn  s  require  first  We  ( 3 . 1 . 1 9 ) .  and  s  the  t o combine t h e  and  ( 3 . 1 . 1 8 )  (B.5)  ty  W  t o combine  then  M d e p e n d e n c e of  Therefore,  Wrt D » V ^  5  • VL\*?T)  l  lS«0  (H.11)  invariant  under  space  inversion: {P H, This  for  implies,  positive  Substituting this  as  in  <P  the  (H.12)  fermions  vertex  (3.1.23),  and  function  that  negative from  (H.11),  parity we  bosons. see  that  implies C-)  That  H,  (3.1.22) and  parity  for  =  F  is,  &  must  be  X  = - l odd  (H.14) in  order  t o have s p a c e  inversion  86  invariance. Finally,  we c o n s i d e r a  time  reversal  transformation.  We  require  J Using  (2.2.14),  K  to  we have  P  =  C  7 l = 1 and  =  F^ ;.  f  1  i s o d d and h *jf s,i  l,l  s  coupling  5,  7  fermion  t , t , t j  (sz.,ii).  angular  momentum.  For we  the  (q)  spin  a n g u l a r momentum of fermion  similar  wils«<V  C  ?  *' (q) 3  T  - ^  h  lrv>  A  3  <-<fV  s  B £ £ O p 4- Odj.  i s a real  the  as s  z  the  boson  We s e e t h a t  particular  total  t o boson  case  5  spin  spin  (s , i  3  s . 3  ) r  recover the i n t e r a c t i o n  Referring t h e quantum  resulting  from  JL i s t h e o r b i t a l  with  £ + s = s,  that  (H. 18)  function.  a n d F i g u r e s 12, 13, a n d 14, we i n t e r p r e t h * js  1 6 )  (H.1c) c a n be w r i t t e n  s  s in  '  ( 3 . 1 . 3 4 ) , we c o n c l u d e  3  (H.18)  ij=1,  « Following steps  3  &  number  - 1  B  ( H  . • Sdp cl <jL Cut's/xru» l^u,") Csis  •  to  leads to  ^  77_  p  • CA S, miryj, l S - 0  where  this  h  the i n t e r a c t i o n  H ,*i  (H.15)  ^  taken  (3.1.29) t h r o u g h  Thus  = Hi  f  (2.2.16) a n d ( 2 . 2 . 1 7 ) ,  C<  where  H, J  respect  t o the  i n order to conserve  s, = S i = i , = iz = 1 /2 ,  ( 3 . 1 . 3 6 ) . From  s =m =0 , s  3  the properties  87  of  the  Clebsch-Gordan  s = 1/2 and  coefficients  i = 1 for this  case.  • V Noting Y,^  and  the  h(q)  (H.19) w i t h  is  (b) D r e s s i n g Just  while  see  that  Therefore  h ^ J ^ V  1M  I I k "\ M i l a i / O = fc-V Y^  ty  nucleon-pion  boson  we  (H.18),  (H.19)  that  comparing  where  in  B^^(p_)|0>  3 for  Interaction  f o r the Hamiltonian  i s an e i g e n k e t ket  transformation  order, a suitably  we see  f u n c t i o n i n t r o d u c e d i n Chapter  the G e n e r a l i z e d  t h e one f e r m i o n  dressing  (H.20)  interaction.  as we have n o t e d  ket  x  (3.1.35),  the vertex  -m I -5 m,')  C -5 1 m  as  invariant  F^jJ (p_) -  the  of t h e H a m i l t o n i a n  | 0>  developed dressing  (3.1.1),  is  not.  in  Chapter  operator  We  that  one  (H.1),  require 4. To  first  satisfies  (4.1.20) i s  m . r r t z i m j y a ^ x ^ j  £ &  ^  "tyij  where we have  taken  ^  ~  Q  J'  (H.22)  a  88  (H.23) The contain be a  an u n s u i t a b l e  eliminated suitable  will  [ H , , D i ] c a n now  commutator  from  term  o f t h e form F^FBB + a d j . ,  the second-order  c h o i c e of the d r e s s i n g  be  similar  resulting  D  to  second-order  B( F , B )  T-Z- K IV; c  1  S,S S A  %  t|t l t  which  dressed Hamiltonian  operator  g i v e n by  2  B  z  .  This  ( 5 . 1 . 1 0 ) . From  can  through operator  (4.1.21), the  Hamiltonian i s :  =  T  F:;%T  5d k 3  Z  be computed. I t i s f o u n d t o  + \ipB t  (H.24)  Vpp  §$VK£<0]  ^  d V d^K l T  P B  C  <- > H  25  k,k',kV  3  T<H' H*.' +  S.'Sx' i  adj.  (H.26)  C't'i'  i  . . i  F:;".*"  F i j i ( * K- ^ F;# I**- ^ ftt * * ^ f  c  (H.27) where 2.  (H.28)  89  • U s , •*>«>, ls«f) Cs^'s'  m,' IsV^ U ' i .  3  • U ^ ' m ^ ' l i O Cs.s*' «m,^»' I s'^')  • ( i S , <yiWl, 1 5 < 0 ( S i ' S  Is^'V  U's hrt'wii IsuO • z  K»i I s'-iO ( X ' - s / - i ^ ' m / I s V V  3  (H.30) Equation order  t h e f e r m i o n mass r e n o r m a l i z a t i o n ,  to  A*. The  14:  (H.28) g i v e s  functions  'ZTpg  and  op  F  are pictured in Figures  13 and  90  ts*w«.V/.,')  Jt'  /  •4  tSs'-ViV/i,') /  2.  tutrixiVyuO  Fig.  13 The F e r m i o n - B o s o n  Potential  l/^ (_k,k  i  Fig.  14 The F e r m i o n - F e r m i o n  The  fermion-fermion  evaluating (H.30).  many  First,  Potential  potential  o f t h e sums o v e r  we r e w r i t e  the  /  Q  K +• k ' ( S v/i, i iya t  ,K)  K +•)}  1  tJpp(k-k/ )  can  be  projection  simplified  by  quantum numbers i n  fermion-fermion  interaction  in  91  t e r m s of t h e t w o - f e r m i o n  operators  d e f i n e d i n Appendix  '  +-Qdj.  E:  (H.31)  where -  2 1 .  .  iv  K  J U  H  s,SiS t,i»i3  :  s  ^  1  * Hrc (zt+O  • |(- >  (S.SX'M,^' | S U M 5 a . S , ' I s ' M ' U l ' *  ,  r  "l'm I LA) ( . S x S j wix nf) l 8  •US|W)*I.U«0 CSz'SsVyia'no, U V ) (1's,' As  i n ( 5 . 2 . 3 ) , we  Z  C  ]  =  N e x t , we brackets to  use (D.18) and  rearrange c e r t a i n  in  equation  (H.32)  (D.5) t o show  ^f,)LUM  L-)  UV)}  -  t  6  l  "  ( H  of the c o e f f i c i e n t s  (H.32) u s i n g  ( D . 9 ) , and t h e n  .33)  i n the c u r l y apply  (D.32)  obtain 21  ^  m, rnr* Mi  }  -  21  (_^'' 5  + S  l  t s  s  +  2,Si - 2.S + s' - A±A' +& + JL' +rt  wi.W m  L-^A) _  f s j V  .s,' + S x - S j - s + s ' - A - A ' 4 - U V  ' ' t  CSLMA U ' H ' )  )  j ^ ( A  •  [t2SzVlM2A+lMzi +l)(2*+l')(2L+0 l  US+Oj * V  s*'  s'  s /  JL'  5,  | L  v  (H.34)  92  A'  5»  S*'  S  s'  X  , Sa s,'  where  ( _  ^  x  s  L  .  J  is  a  12]  symbol,  whose  d e s c r i b e d i n A p p e n d i x D. Thus we may w r i t e t h e f e r m i o n - f e r m i o n  properties  p o t e n t i a l as  U S  .(SLMAIS-MO The nucleon  form o f t h i s  S I  and  z-axis projection i s manifest.  we  Si = S I = S I ' = S i ' - i i  are  considering  and i n b o t h  (5.2.6),  using  Equation  nucleons F  F  3  interacting  nucleon-delta  potential,  function. If for  we  wished  example,  p i e c e o f t h e C l o u d y Bag M o d e l H a m i l t o n i a n , obtain  the s  case  =1 , i.e.,  through  c a n be shown t o e q u a l  (H.35) g i v e s a f e r m i o n - f e r m i o n vertex  to  of i s o s p i n  in  3  by  pion  VMN g i v e n by  (H.21) and (D.33)..  of a t r i l i n e a r  15,  Indeed,  s =m = 0 , and i  a  exchange, t h e i n t e r a c t i o n V  required  the conservation  = i t = i ,' = i ' = l / 2 , two  i)  to the nucleon-  B o t h have t h e s t r u c t u r e  invariance,  that  Czg.'+  { H 3 5 )  similar  rotational isospin  (2JH-Q  5  p o t e n t i a l i s very  p o t e n t i a l (5.2.5).  + I)  '' ' * .  S  are  h  x*/i.  '3' '  i  n  t  n  e  we  p o t e n t i a l i n terms to  calculate  a  c o u l d use t h e NATC pictured  same  in  way as we  Figure found  93  h  C  B  H  (q)  i n Appendix  vertex  function  desired  N-A  F. We  to  would  then  integrate  as  be a b l e in  t o use  Section  5.3,  potential.  SiSxS, 1  is  l,ulj  (<p  N  1  \ \  V  \  \ \  Fig.  15 The N A T 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 .  this giving  N&7C the  94  Appendix In  I_  this  functions consider For the  Some P r o p e r t i e s o f B e s s e l Appendix,  encountered  in  and  the d i s c u s s i o n  a c o m p l e t e d i s c u s s i o n of B e s s e l  Ji/(z)  list  of  denotes order  an  V  ordinary  and  complex  integral  are c a l l e d  order  spherical  Bessel  ordinary  some  Bessel  i n C h a p t e r 5. We  then  functions.  f u n c t i o n s , we  refer  to  (1966).  unrestricted,  the  introduce  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 Bessel  book by Watson  kind,  we  Functions  Bessel  argument  z.  v a r i a b l e s . The  Both Bessel  s p h e r i c a l Bessel  f u n c t i o n of the f i r s t  Bessel  f u n c t i o n of the  functions  y  and  first  z  are  f u n c t i o n s of  half-  functions. j n ^ ^ is a z  kind.  It  is  related  to  by  (1.1 ) The  s p h e r i c a l Bessel  Abramowitz  and S t e g u n  f u n c t i o n s of t h e s e c o n d k i n d a r e , from  (1965,  p.433):  n=o,± I, We  will  also  require  f u n c t i o n s of t h e f i r s t , i_ (z),  and  of  spherical  n  the  Stegun  the  s e c o n d , and  modified third  k ( z ) , f o r n = 0,±1,.... n  (1965, pp. YlT-VZ CO  Bessel 443-4), we =  = e  functions.  spherical  kinds,  They  Bessel  namely  are d e f i n e d  From  (1.2)  • •  i (z), n  i n terms  Abramowitz  and  have  1» (&  (-E^artjZ ^ K)  (1.3)  95  SCn+Oxi/z  1  y oO  = e.  (f^QrgZ  p  (1.4)  (1.5) We  now  list  from A b r a m o w i t z  1a(*>  some  Bessel  and S t e g u n  f u n c t i o n s of s m a l l  (1965, pp.  i  Sin  433-4,  2  cos  •jo C O  =  ClO = ~  -U  4-a  *  C  o  S  taken  443-4). C05  Sin 2 _ x  order,  £  (1.6)  *  cost  _  Sir> 2  (1.7)  2: cosVi 2 2: ~  3  ; f -jr )  Sinn £  -  —  C  0  5  h  ^  (1.8)  22  b,^  = JL  22  e~* (/ + ~ )  2  X- > s  (1.9)  96  The  small argument l i m i t  of the s p h e r i c a l B e s s e l  functions  is: 2^0  CZn + 0 ! I  -C2n-0'. ! ( i ) n + l  %(Z)^=LJZ  (1.10)  where (2n+1)!! = (2n+1)(2n-1 )...(3)(1 ) We  now  consider  s p h e r i c a l Bessel choosing  p=3/2,  certain  contour  integrations involving  f u n c t i o n s . From Watson (1966, eq. ( 9 ) , yu,=3/2=yq , 2  b,=b =b, 2  and V = l+l/2  p.430),  f o r X an  even i n t e g e r , we have oo  x +k 2  2  = - [Xj/iCb^l" 2  K^Lak)  Cn)  p r o v i d i n g a>2b and 1<4. W r i t i n g the o r d i n a r y B e s s e l ones using co  (1.1),  f u n c t i o n s i n terms of s p h e r i c a l  (1.3), and (1.5), t h i s becomes  ,  l + i/z  97  When  £ = 0 , using  ( 1 . 8 ) and ( 1 . 9 ) ,  we  find  a>2b  When  £ = 2 , using  ( 1 . 8 ) and ( 1 . 9 ) ,  the i n t e g r a l  (1.13)  becomes  CO  -0>  2- db^fe  ak (o^  bfc s,w a >2b  (  I  '  1  4  )  98  Appendix  In of  J  One  this  Pion  Exchange P o t e n t i a l s  Appendix,  nucleon-nucleon  potentials for  are  example,  the  and  book by with  a  i n most  the  r e s u l t s of  from  one  texts  on  pion  calculations  exchange.  These  physics.  See,  nuclear  Moravcsik  (1963).  suitable  non-relativistic limit  of  the  interaction  considering  following  summarize  potentials  discussed  Beginning Lagrangian  we  (OPEP)  one  pion  second-order  exchange  in a perturbative  nucleon-nucleon  potential  theory,  the  i s obtained:  • Mrcr  r  s He  where  5  G", of  and  nucleons  isospin The the  Cix  are 1 and  2,  proportional  Pauli  to  spin  operators  respectively;  separation above  this  - G \ ' 0 * .  S C . - r ^ . r  f i s the  mass. The  Note t h a t  operator  the  operators;  relative pion  =  a  NN7T of  Z,  the  interaction and  T  a tensor  a  are  is r = r  in units  has  z  Hilbert the  constant  nucleons  i s written  in the  and  coupling  (J.2b)  =  2  .08).  - r_ ; m (  where -ft = c =  scalar part,  analogous  (f z  spaces  part  K  1.  which  proportional  is  to  is the  S , . IZ  Consider  the  s c a l a r OPEP, f o r  r^O.  Taking  matrix  elements  99  between  two-nucleon  I  s t a t e s of s p i n  S and i s o s p i n  <o , we o b t a i n  1  (J.3) A simple ^SM  calculation  ICJ, - q  z  IS'M'>  shows Sss' < W  =  ^ "3  5=S'^0 s =S = I  (J.4)  <5 - C3'  (J.5)  Similarly - 5 l  The  resulting  matrix  e l e m e n t s of t h e p o t e n t i a l  = I  are given i n  Table I I .  X= o S= o  S= I  r  r  Table  II  Matrix  Elements  of  -•9 $MM< r  r  t h e S c a l a r OPEP  L. V  H  , (.£U  100  Appendix K  Dressing a Poincare  In t h i s dressing  A p p e n d i x we c o n s i d e r t h e c o n s e q u e n c e s o f a p p l y i n g a  t r a n s f o r m a t i o n t o a system which  Poincare under  transformation,  translations,  reversal, To  rotations,  ten H e r m i t i a n  space  operators P , J J  algebra  , H, K  J  and  time  boosts. invariant,  variables  (j=l,2,3)  J  a  invariant  inversion  i s Poincare  the fundamental dynamical  under  i s not only  under homogeneous L o r e n t z  d e s c r i b e a system which from  i s invariant  i . e . , one w h i c h  spatial  but a l s o  construct  Poincare  I n v a r i a n t System  ( s e e , f o r example, K a l y n i a k  one must  of the system  satisfying  (1978,  the  p.23):  (K.1a)-(K.Ic)  f>,H>o Ck*,  P ]  -  h  [>*,  CT^H] - o S  [k>>H] = - c f c P  H/c  j b  -if,  |<  A  j w  (K.2a)-(R.2d)  2  C^,K ]=-'^^kJ- J / c l  k  J  e  l  (K.3a)-(R.3b) j.h.JL * 1»^»3  The  momentum  translations; rotations;  For operator equations  a  the operator  P  of  P  H  as  and  B^B  operators  corresponding  the  generator  is  the  free  fermions  in  (4.2.4)  and  (4.2.1b).  J  terms  and  expressions  n-particle  space  can  operators.  of s p a t i a l of  time boosts.  and b o s o n s t h e momentum  are given  Fock  spatial  of L o r e n t z  H  in  of  generator  K i s the generator  and the H a m i l t o n i a n such  is  momentum J i s t h e g e n e r a t o r  Hamiltonian  system  i n v o l v e F^F particle  the angular  the  translations;  operator  be For  D  Fock and K for  space 0  will  these  obtained  by also free  from t h e  example,  see  101  Kalyniak and  (1978, pp.76,77)  Lorentz To  boost  Poincare  invariant  a n d b o s o n s , we i n t r o d u c e  form o f D i r a c  n-particle  angular  momentum  operators.  describe a  fermions  f o r the  (1949),  system  of  interacting  interactions using  the i n s t a n t  i . e . , we l e t  I  -  J o  M = Wo t A (41 k Thus  we  have  generators (K.1)  -  as  P * J<> » 0  as  depend  IS  which  the  boson  satisfy  relations  momentum  the  only  the vertex  previously  Chapter  on g . T a k i n g f u n c t i o n on 2  algebra,  dynamical  just  Poincare  with  algebra  in we  this  chose  h = h(p_,g), a n <  3 g  order  the  to s a t i s f y the  systems  which  we  vertex  function  to  t h e f u n c t i o n a l dependence be  restricted  by  the  requirements.  a dressing transformation  on  the  fundamental  s y s t e m . The t r a n s f o r m a t i o n  (4.1.1) - ( 4 . 1 . 4 ) .  Since /\J  transformation  p_  t h e s i s c o u l d n o t be made  will  invariance  v a r i a b l e s of t h i s  equations  in  momentum  a s i t s d e p e n d e n c e on s p i n was d e t e r m i n e d  3 by s p a c e - t i m e  Now c o n s i d e r  g  i n v o l v i n g K a n d H. The  i n v a r i a n t because  Poincare  by  '  a n d bosons  t h e i n t e r a c t i o n s H, a n d K, t o be t r i l i n e a r . The  considered  Poincare  in  H  fermions  f u n c t i o n h must be a f u n c t i o n o f t h e f e r m i o n  commutation  of  (K.4)  t  a system of i n t e r a c t i n g  choose  well  have  +A k  c  (K.3).  We vertex  = K  i s u n i t a r y , the operators  the **~>  P , J 0  e  /v  i s given dressing  f\J  , H, K must  still  1 02  obey  the  Poincare  algebra. 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  rotations, i.e.,  CP so  the  tlo, D] * O  -  f u n c t i o n a l form of  dressing  As  D]  0 )  P  0  (K.5)  and  J  0  will  not  be  c h a n g e d by  the  transformation:  i s Section  4.1  we  write co  D=  The  dressed  given  by  Hamiltonian  a similar  We  >  X.  now  x  (  be  given  by  (4.1.15); K  will  K  -  8  )  be  expression.  choose D  + A  0  D n  H will  to eliminate unsuitable  h  (4.1.18) from K t o o r d e r  K = k  n  \ \  I  n.  k, , 0 . 1  t e r m s of  the  form  Thus  + CKo ,  D»l  (K.9)  ] + . •'•  That i s , K where K  n  =  k  contains  annihilator  (other  no  dressed  terms  than  (4.1.20) f o r H h o l d s , The  A  4-  0  as  K  (K.10)  n  with  a  F^F  and  B^B).  we  now  show.  operators  single This  satisfy  fermion  implies that  the  Poincare  or  boson  equation  algebra;  therefore  (K.1 1 )  103  We a l s o h a v e , from  Taking  (K.2d),  -i^fe  H./c*  -•'^Sife  H./c  t h e commutator  *  Cko , P  e  ]  (K.12)  [W, , ? o ] J  =  1  k  J  of e q u a t i o n  4- I T W  (K.1.3)  k  (K.9) w i t h  ,0*3, P6 ]| (K.13)  find  + •-.  k  Note t h a t we c a n use ( K . 5 ) ,  , we  and  a  (K.14)  Jacobi  identity  to  show, f o r example,  0  = -6K/CTherefore,  using  (K.11),  S  j k  rH,,0,]  (K.12),  (K. 1 5)  (K.13)  and  (K.5) e q u a t i o n  (K.14) becomes + • A* ^ t H . , D, ] + F u  H =  or  0  jDil}  +•• •  (K. 1:6)  CO  %  where H  0  =  l\  0  +  T > n-2.  n  H  (K.17)  n  a l s o c o n t a i n s no u n s u i t a b l e terms o f t h e form  (4.1.18).  A/  This  i s t h e same s e r i e s  (4.1.20).  Thus  for  H  that  we  obtained  in  equation  i f we c h o o s e D t o 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 s u c h  terms  from  H  as  well. Is satisfied  the order  Poincare by  invariance  order  in A ?  of  the  Certainly  resulting (K.1)  is  theory still  1 04  satisfied  by P  and J  c  Z?o , H n ] From  (K.11),  0  . Also,  =  T3o  in light  ,Hn]  (K.12) and ( K . 5 ) , [ L  .PoM  o f ( K . 5 ) , we have  = O  (K.18)  i t must  Sjfe  »  Qn  hold that (K.19)  f e  and  [k. Thus  (K.2)  Consider  is  ,X > * ] =  J  satisfied  i n £ JL k * jfe  (K.20)  n  by t h e d r e s s e d  generators  t o o r d e r n.  (K.3a): Li^  , ft] = - i * P o *  (K.21)  Since tKo this  J  ,HoI  - -.^Po  (K.22)  J  implies  +  Thus series  A " " rk„ , i L ] = 0  £  J  (K.3) c a n n o t  be s a t i s f i e d  f o r K and H a r e t r u n c a t e d ,  Poincare In dressing  i n v a r i a n c e of the theory summary,  we  the  to  dressing  u n s u i t a b l e t e r m s o f t h e form  a  by o r d e r  (K.23) w i l l  i n A . I f the  n o t .hold, and  the  i s destroyed.  have shown t h a t  transformation  constructing  order  (K.23)  i t i s p o s s i b l e to apply a  Poincare operator  (4.1.18) from  invariant Dn  to  system  eliminate  K, t h e g e n e r a t o r  by the for  105  Lorentz can  be  boosts.  Procedures s i m i l a r  u s e d t o c a r r y out  consequence, t e r m s and truncate An Poincare  the  thus the  F  this  dressed and  series  B  T  dressing  Hamiltonian  perturbative  i n v a r i a n t system  given  i n Chapter  transformation. contains  no  create physical p a r t i c l e s .  f o r H o r K and  alternative  to those  i s given  maintain  approach  to  i n Glb'ckle and  a  unsuitable One  Poincare  As  4  cannot  invariance.  describing Mu'ller  a  (1981).  

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