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

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ON  A  1+1  - DIMENSIONAL  INTERACTING  WITH  S O L I TON-FERMI ON  SYSTEM  SUPERSYMMETRY by  WERNER H. Diplom-Physiker,  Technische  A T H E S I S SUBMITTED THE  KEIL  Universitat  Clausthal,  IN PARTIAL FULFILLMENT  R E Q U I R E M E N T FOR MASTER  OF  THE  DEGREE  OF  SCIENCE  in THE  FACULTY  OF  GRADUATE  Department  We  accept to  THE  this  thesis  the required  UNIVERSITY  Werner  Physics  as  conforming  standard  OF B R I T I S H  Apri1,  ©  of  STUDIES  COLUMBIA  1985  H. K e i l ,  1985  OF  1983  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of  requirements f o r an advanced degree a t the  the  University  o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make it  f r e e l y a v a i l a b l e f o r reference  and  study.  I further  agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may  be granted by the head o f  department or by h i s o r her r e p r e s e n t a t i v e s .  my  It i s  understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain  s h a l l not be allowed without my  permission.  Physics  Department o f The  U n i v e r s i t y of B r i t i s h Columbia  1956  Main M a l l  Vancouver, Canada V6T  1Y3  Date  26. 7.  85  written  i i  Abstract  A space  supersymmetric and  soliton the  one  time  sector  "method  of  of  dimension the  evaluated are that  the  boson  to  to  and  the  for constrained  equations fermion  in a admit  zero-energy  of  the  theory.  solutions theory spatial  are  broken  include  i s doubly  by  translations,  supersymmetry.  of  developed  and  way  to  a  using  equations  is  of  expansion. solutions  for  the  Hamiltonian  motion We  find  for both  The  fermion  consequences  supersymmetry charges  broken.  connected  soliton. the  nontrivial  The  degenerate.  t o be  the  has  topological  i s spontaneously  found  the  the  field.  supersymmetry  state  The  perturbative  soliton  ground  construct  constraints  the  one  generalization  canonical  systems.  the  modified to  with  c o o r d i n a t e s and  order  in  transformation leads  i n the  of  supersymmetry  the  variables  presence  be  using a  canonical  is quantized  first  field  The the  theory  field  in collective  solved  theory  A  "collective"  method  quantum  system  We  theories.  set  Dirac's  is investigated.  coordinates" previously  bosonic  transformed  soliton-fermion  collective  purely of  interacting  It  and  algebra  as  follows  a  that  Finally,  the  with  symmetries  boson  the  has  result the  zero-energy  zero-mode  zero-mode  for  of  the  corresponds  is associated  with  to  iii  Table of C o n t e n t s  Abstract  i i  Acknowledgement  iv  INTRODUCTION 1. Q U A N T I Z A T I O N  1 BY  1.1  Quantization  1.2  Collective  2. Q U A N T I Z A T I O N  around  3.  a classical  static  6  field  6  coordinates OF  S O L I TON-FERMI ON  A  Collective  1+1-DIMENSIONAL  INTERACTING  2.2  Construction  2.3  The  2.4  The H i l b e r t  13  coordinates  for a  soliton-fermion  system  of the Hamiltonian  equations  SUPERSYMMETRY  10  SYSTEM  2.1  field  C O L L E C T I V E - C O O R D I N A T E METHODS  space FOR  A  23  i n g°-order  f o r the s o l i t o n 1+1-DIMENSIONAL  31 sector  38  SOLITON-FERMION  THEORY 3.1  The  40 supersymmetry  generators  f o r the  soliton-fermion  theory 3.2  15  Spontaneous  sector Bibliography  40 supersymmetry  breaking  i n the  soliton 47 53  iv  Acknowledgement  I  would  Semenoff, work.  like  for  His  contributed  to  thank  his help  patient  and  my  research  guidance  explanations  essentially  gratefully  acknowledge  University  of  British  to  the  and  throughout numerous  progress  financial Columbia  supervisor,  of  assistance  Summer  the  Dr.  Gordon  course  suggestions  this in  Graduate  work. form  of  I  of have also  a  Fellowship.  W.  this  1  INTRODUCTION  It  has  been  classical mean  known  field  has  systems  nontrivial the  Skyrme  one  fractional  model  these  properties  time  dimension  Although  this  may  seem  to  physical  s i g n i f i c a n c e , there  Like  1-dimensional  1+1-dimensional  without the  presenting  question  field  of  theory,  i s of  Despite  the  are  no the  many  gauge  the for  shown are  such  as  have best  are  s e e m s .to  be  reasonable  these  closely  various  related  Examples interacting  therein)  attracted a  good  toy  reasons  provides  a  and  the  for  in  assumption  any  studying  i t .  nontrivial  in  point  the  for  higher  mechanics, structure Particularly  for  models basic  more  for  every we  will  features  complicated  dimensions.  especially  in  attention.  i n quantum  crucial  valid  theories  without  difficulties.  importance  remain  much  model,  oscillator  theories,  the  admit  that  fields.  simplifications certain to  to  we  considerable  which  references  technical  major  systems,  a  at  theory  expected  solutions  years  have  which  ([7]-[9]).  renormalizability, a  consider. theory  too  be  harmonic  field  ten  by  interacting soliton-fermion  one  the  of  and  baryons  theories  solutions,  which  numbers  ([2]-[6]  for  nonlinear  theories  studies  features  quantum  systems  last  field  and  a  space  unusual  certain  finite-energy  the  Recent  topological  soliton  Among  independent During  that  soliton  in quantized  many  soliton-fermion  time  possess  solutions.  exhibit  include  long  ([1]).  arisen  soliton-like  to  time  equations  interest  a  theories  nontrivial  field  for  This  properties  . of  2  that  depend  In  only  on  addition,  physical  systems.  that  dynamics  the  polymer  i s given  the  global  these  theories  Several of by  structure  authors  electrons a  are  in  of  known  the  to  theory.  describe  ([ 4 ] , [ 1 0 ] - [ 1 4 ]) a  linearly  1+1-dimensional  actual  have  shown  conjugated  diatomic  soliton-fermion  Hamiltonian.  Summarizing 1+1-dimensional methods, is  now  as  this  theories  well  well  as  a  as  model  discussion, a  "proving  for  supersymmetric. many  can  choose  the  Supersymmetric  unusual  properties,  of  divergences  are  now  to  play  a  combines  this  unusual  believed  topology by  (solitons)  several  the  trivial  of  that  topology  new  of ideas  matter  and  systems,  major  ([15]-[20])  a l t e r s the  as  at  in  first  with  a  interest. discuss  conventional  known  the  description  They A  nontrivial  for  is  be  symmetry  physics.  This  how  to  to  order.  particle  symmetry  who  are  systems  boson-fermion  least  role  i s c e r t a i n l y of  these  theories  such to  for  field  confirmed  presence  of  fields  with  topology.  One fact  authors  soliton  for  condensed  action  field  cancellation  that  usefulness  ground"  certain  and  model  the  established.  Furthermore,we  possess  brief  the  problems  standard -  of  theories  perturbation  nontrivial  boundary  containing  theory  cannot  conditions  on  solitons  take the  the  is  the  field  solutions  of  the  3  field  equations  these  systems  ranging  new  on  open  Jackiw, developed  a  way  they  c-number  spaces  for  formalism soliton  fields  a  to  the  method,  thirty  the  years  path  theories  many a u t h o r s the  present  integral The a  set  of  freedom". extended, added.  using  time  index  therein).  that the  approaches  framework  quantization.  system  to  theory as  a  can  the  of  include  the  consistent  classical  then  that  To in a  variables  This  to  "collective  quantize  method  as  very  theory  of  with  be  quantized  via  Dirac  developed  more  developed  the  particle  to  be  special  problems  soliton  system  "internal as  the  quantum  c a l c u l a t i o n s these  by  care  in  is  for  i s favored  transformations  viewed  which  systems  Faddeev  method  ordering  v a r i a b l e s " and can  by  i t requires  canonical for  constrained  elegant  but  operator  transformation  practical  within  techniques  dependent  classical  theory  field  to  ( [ 2 ] , [ 2 6 ] - [ 2 8 ])  viewed  of  developed,  methods  references  quantum  -  been  about  ([29],[30]).  Physically this  For  the  set  integral  due  field  system  new  ([31]).  case,  canonical  new  in  ([23 ] , [ 3 2 ] , [ 3 3 ] ) ,  for  have  a l .  fashion  constrained  ago  et  canonical  a l t e r n a t i v e approach  Feynman  gauge  scalar  and  the  This  An  a  transformed -  and  Tomboulis  solution  constraints.  than  information  techniques  ([21]~[25]  straightforward  canonical  obtain  semiclassical approximation  method  Hamiltonian  To  different  Goldstone,  in a  classical  into account.  several  from  theorems  problem  -  the  in path  ([26]).  will  lead  degrees  of  soliton  being  contributions  to  an  are  "quantum f l u c t u a t i o n s "  4  have  to  be  expanded  emphasized,  however,  approximation constrained  In  be  given  to  extend  described  to  include  does  supersymmetry  We  will  In  be  "collective  scalar  the  method  full  quantum  we  of  the  series.  It  involves  no  theory  for  should  be  semiclassical a  classical  with  a  1+1-dimensional  supersymmetry.  Special  problems:  of  method  we  have  just  and the  soliton  affect  the  theory.  as  follows:  chapter  reference  discuss  quantization  coordinate  cft-l'theory,  two  fermions,  the  first  to  will system  presence  proceed  the  following  power  this  soliton-fermion  will  how  a  paper  how  (b)  suitable  that  gives  present  interacting  (a)  but  a  system.  the  emphasis  in  give  a  brief  quantization  [26]  the  we  and  [28].  simplest  review  of  the  method"  by  This  illustrated  theory  is  that  Jackiw  admits  et  a l . with  a  soliton  solutions.  case  In  the  of  a  transform  second  chapter  1+1-dimensional the  old  field  constraints  and  canonical.  Subsequently  in  the  new  show  of  the  will  that  new  generalize  this  method  interacting soliton-fermion  operators  coordinates.  interpretation  we  this we The  to  a  set  of  transformation  evaluate  the  result will  variables  as  new is  to  the  system.  variables  We  with  indeed  energy-momentum confirm  "collective  tensor  the coordinates".  5 Using  the Hamiltonian  equations and  in a  fermion  existence  general of  a  first  operators  solution.  of t h e H i l b e r t  third  and last  a  the ground doublet.  a  spatial  t h e boson  due t o t h e A preliminary  results  are given  with  are quite  f o r the special  of a  whereas  t o an  case  we  will  These  to the  which  have  central  t o be  charges  "N=i" supersymmetry.  show  the c l a s s i c a l fermion  supersymmetric  corresponds  of our theory  the bosonic  transformation.  the  charges"  algebra.  down  state  theory  addresses  or " c e n t r a l  Finally  the existence  transformation,  Our  of the soliton  supersymmetry  supersymmetric  implies  chapter.  chapter  i n t h e supersymmetry  consequence  that  field  system.  "topological"  t h e N=1  the  structure f o r a theory  the calculations  existence  of  space  the second  The presence  in  find  soliton  supersymmetric  degenerate  We  of the c l a s s i c a l  properties.  break  approximation.  and s o l v e  a zero-mode  although  included  construct  contain  concludes  The  order  we  both  discussion solitons  formalism  will  consist  As a  of a  non-perturbatively that soliton  zero-mode zero-mode  solution  via a  always  supersymmetry  i s associated  with  6  1.  1.1  a  method  field  molecular as  theoretic quantum  quantum  equations  the  full  quantum  operator  The  full  quantum  theory  with  a  Consider c6(x t) r  i n two  this  field  equations  for classical  approximations  around  scalar  The  These  order  field  c a n be  viewed  t o the "Born-Oppenheimer" method i n  mechanics.  operators.  static  METHODS  i n the f o l l o w i n g section  analog  zeroth  expansion  COLLECTIVE-COORDINATE  a classical  described  differential  than as  BY  Quantization around  The as  QUANTIZATION  c-number  classical  i s obtained  classical  an  field.  rather  c a n be  expectation  classical  by  are treated  fields  solutions  t o t h e vacuum  describing a  theory  of motion  viewed value  extended  operator  power  object. series  We  will  illustrate  for a  real  scalar  this  ( [26 ] , [ 2 8 ] ) .  the Lagrangian  density  field  dimensions  d) where  U(<p)  power  series  parameter use  from  i s a polynomial expansion  g  like  now  on  we  U((p) = ^  i n <p. assume  2  i s the well  U(q<p). known  Since that The  U  we  are interested  d e p e n d s on standard  a  e x a m p l e we  (2)  The  field  equation  for this  Lagrangian  i s given  by  in a  coupling  c^-self-interaction  of  will  7  (3)  The t-  solutions  independent),  dependent.  For  solutions,  A found:  constant <p  0  For  on  are  which  =  the  we  ±jjj.  to  (3)  static us  can  be  classified  ( t - independent)  only  the  first  two  and  and  as  constant  space-  (x-  and  especially  time-  the  static  is  easily  important.  solution  </> 0  to  We  will  return  static  case  the  impose  the  (3)  with  potential  to  this  solution  field  equation  (2) later  reduces  on.  to  following conditions  (asymptotic  non-triviality)  0  0  Then  (3)  is integrated  to  or +  where  the  prime  and  will  +  denote  from  c o r> if.  now  on  (4)  differentiation  with  8  respect  to  the  explicit  argument.  The  remains constant (at  i s set  least  on  These  to  the  zero,  classical  static An  explicit  The  s o l u t i o n s are 0 (x)  =  L  with  x  a  0  real  the  finite  ±™-  integration is  ensured  level).  example the  tanh  parameter  i f the  physical consistency  s o l u t i o n s with  solitons. static  so  energy  finite  energy  i s provided  well-known  by  are  usually  our  ^-interaction.  "kinks"  m(x-x )  (5)  0  denoting  called  the  center  of  the  soliton.  We  i  note  that  will  be  for  the  We  since 0(g~  of  V= 1  )  U as  well  solutions.  will  construct  now  solutions.  The  operator  field  I <p  0  quantum  of  (*.t)  0(g  _ 1  Hilbert  )  -  r\ .  and  the  values  of  techniques.  This  part  $  quantum  the  (4).  the  The  solution  same  around  constant  is  true  our solution  <j> . 0  (x.t)  usual  be  classical  as  c o n t r i b u t i o n of  0(g ),n>O.  The  n  Fockspace from  multiparticle  can  our  theory  consider  generated  of  i n g,  equation  quantum  i s the  bosonic  Expectation  sector".  *I  i s expanded  space  describe  we  )  _ 1  to  the  First  { | 0 > , | k > . . . | k , . . . k „ > . . .} elements  0(g  due  constant  classical  with  i s of  1  a  states  evaluated theory  vacuum  will  by be  state  with  standard called  |0>  momenta  whose k  .  perturbative the  "vacuum  9 To  construct  include  "quantum  general  state  The  soliton  Jackiw  matrix the  the  theory,  takes  the  "soliton  vacuum  of  that g  given  by  so  to  that  a  | p , k k „ > .  m  a  these  systematic  on.  The  soliton  B  assumptions  is possible.  the  p,  have  < p , 1 , . . . l | . , # | k , . . . k > - 0.  that  later  of  mentioned, soliton  i s however,  there  we  would  sector  the  different left  We part  into  lead  expansion will of  to a  of  return  the  account  as  like  $  =  t o expand  <p  ti  +  two  the to  theory  is called  the  field  analogous to  the  which  b)  how  impose  solution  new  set of  set of theory 0tl  example  of  solutions (1)  the  the boost  <p i s n o t ti  <t>l  ~  theory  parametrized  i s invariant  i s not, but  d e p e n d i n g on  <j>^ s h a l l  under  transforms parameter.  the  field  be  expanded  translational covariance  to these  field  our  since  by  x . 0  Lorentz  into Thus  we  problems  around  The  the  straightforward As  solution  solutions  to  so  whole  while  a)  parameter  a  static  with  not  invariant.  exists  generally,  boosts,  are  especially  momentum  i s now  stable:  shown  structure  with  we  sector.  translationally  More  |p>  solutions  sector".  i n the  This  shown  have  static  space  t o be  existence  already  operator  states"  i n powers  space  f o r the  Hilbert  ([28])  elements  As  soliton  i s assumed  Hilbert  that  theory  i n our  et a l .  consistent  a  problems  variables  dependence  of  0..  i s given  which  enable  and  on  by  a  transformation  us  to control  the  to a  has  10  1 .2  Collective  We  start  level. with the  field  now  I  In  impose  f  field  -  ~ U ( $ )  variables are with the  {*,n,X,P}  t o keep  by  -  0  t o a s e t o f new  defining  (« -  X(t))  ,  XID  £  ,t)  (6a)  , £(t) = Jdx*'tf'  2  t h e number  of independent  variables  constant  we  the constraints  we  modify  a r e now  the  form  A  Given  i  J  dx  £ <j>  J  i x TT ^  d  dealing with  our Poisson  ([29],[30]). in  $ 2  ^  = 4> t o g e t h e r  { TT(-.t) . F ( j . - t ) J  transformation  r  Since  • I  cononical  momentum n  -  a canonical  = Jdx^'  order  - c-number  brackets  («.t) -  0  2  density  The s t a n d a r d  <t> a n d t h e c o n j u g a t e  perform  M  <t>^.  $ ( * . t ) ; £ Lyt)\  variables  where  our Lagrangian  solutions  Poisson I  We  the discussion at the c l a s s i c a l  Consider  static  usual  coordinates  brackets  =  0  (6c)  -. 0  a constrained  according  (6d) system  to the Dirac  we  have t o  procedure  a s e t o f c o n s t r a i n t s on t h e f i e l d  variables  11  =  h {<p,ir) K  we  impose  for  the  f,g.  these  on  Furthermore  we  this  a l l the  from  our  standard  can  simply  by the  system  variable,  variables  now  brackets  procedure  our  f o r any  case  we  two  brackets  functions  obtain  vanish.  defines  to  a  set  construct the  the  equations  them  We  here.  turns be the  a of  canonical new  transformation  "collective  constraints.  and  of  {f,g}  Dirac  (7b)  calculation  can  coordinate  The  for  energy-momentum of  motion  s u b s t i t u t i n g (6a)-(7b)  reproduce the  s u b s t i t u t i n g the  bracket  procedure  variable  with  and  by  1  described  generators  Since  =  The  We  Poisson  remaining  coordinates"  system  take  {X,P} and  our  conventional  Adopting  a=1 i • • • t N  0  out  to  into  results note  be  are  however,  just  P.  for the  our old  rather that  Thus  tensor,  X,  p h y s i c a l l y i n t e r p r e t e d as  new  ([26]).  lengthy  its  Lorentz  variables  theory  the  the  we  total  will  momentum  conjugate  the  center-of-mass  new  collective  soliton.  quantization  of  our  is straightforward.  theory We  in  the  simply  follow  the  not  canonical  of  12  quantization Hilbert see  space,  from  field  scheme:  quantum  been  care  To  maintain  change  theory general  precisely by  part  o f by  the  c o n s i s t e n c y we  will  c-number  translational  to  and  part  in a  As  desired result.  i n t o d u c t i o n of a had  operators  t o commutators.  our  the s t a t i c The  a r e now  #  cl  we  plus  noninvariance  "position  omit  a detailed  are going  o f an  an  has  operator"  i n t r o d u c e c o n s t r a i n t s and the commutation  d i s c u s s i o n of  to extend  interacting  can  The  relations  in a  way.  s i n c e we case  variables  correspond  t h e c a n o n i c a l momenta  nontrivial  We  gives  are given  additional taken  field  the brackets  (6a-f) t h i s  operators  the  these  the p u r e l y methods  fermion-boson  bosonic  t o the  theory.  more  X.  13  2.  Q U A N T I Z A T I O N OF  A  1+1-DIMENSIONAL  INTERACTING SOLITON-FERMION  SYSTEM.  We by  will  <(> a  real  2-spinor  boson  We  not  however,  but  imposes  that  condition.  results  only  technical  brackets  for Dirac will  fermions  will  classical  we  fermion  remain  and l e a v e s  now  chapter part  examine  a n d some  how  1  with  of the theory  the Majorana  o f them  are  i n polymers. degree  First  remains  method,  anticommutators.  simplest  (8).  I t does  that our  familiar The  of freedom i n  possible spinor.  to generalize our r e s u l t s  to the Lagrangian  0(g~ )  i n .  the following .  the charge  us w i t h . t h e  charge  chapter..  i n t h e sense  standard  fractionization removes  the  of our q u a n t i z a t i o n  with  unchanged  of charge  £  provides the  are interested  t o be c o m p a t i b l e  fermions  (a complex  i n the previous  restrictions  have  described  t r a n s p o s i t i o n ) a n d V(</>)  of a Majorana  the generality  c o n d i t i o n merely  previous  discussed  the choice  affect  the context  We  matrix  I t i s s t r a i g h t f o r w a r d to repeat  construction  Dirac  term  2-spinor T  denoting  T  theory  c o n d i t i o n <// = C i / / ,  of the Lagrangian,  anticommutator  Majorana  and  interaction  note  \p a M a j o r a n a  field,  field  density  t o the Majorana  matrix  supersymmetry  in  Lagrangian  scalar  subject  conjugation  The  the 1+1-dimensional  the supersymmetric  with  the  consider  we  note  unchanged  from  the  that the i f we  take  14  the V  interaction i s at  least  fulfills  this  operator  field  In fermion The so  of  0(g°).  condition.  the  lowest -  to  between  0(g).  the  unchanged.  On  to  between  quantum  since  the  the  the  We and  can  shift  </>'-model,  and  for  (8)  leads  set  the  that  example  to  the  only  introduce  level  the  operator  the  field  0(q°)  will  to  the  V'(0  t L  occur  theory. will  suffices  to  0(g°).  collective the  occur  be  zero.  be  at 1  )  least remain are  Interactions  in higher  orders.  is  trivial  the  soliton  due  position  t  fermions  ).  In  V'(0 )  the  to  in general  classical  to  i n 0(g~  motion  It  by  -  <9(g°)-term  i s seen  potential  effects  up  of  field  Furthermore  fermions  free  t V  to  classical  0(g°)-approximation  the  V'(<£ ).  spinor  0(g°).  fermions  nontrivial  boson  of  quantum  a  can  fermions  equations  sector  operators  the  the  and  to  we  Grassmann  c-number  bosons  vacuum  now  a  least  first  potential  quantum  1  bosons  static  however,  background  Lagrangian  0(g" ),  as  at  i t corresponds  sector,  The  classical the  coupled  In  standard  couples  be  interaction Thus  Our  q<p s o  proportional to  • 0 .  order,  viewed  equation  \p h a s  V(<p)  equations  V - V(*) Y  field  =  V" (</>)  £ X  Dirac that  term  our to  soliton  consider  coordinate  solution  #  .  X  15  It is  known  should  to possess  interpreted is  be m e n t i o n e d fermion  transformations part we  2.1  Collective  static, equation  discuss  we  collective on t h e s e  of the theory.  when  As  based  We  that  ( [ 2 ] , [20])  partner"  will  c a n be  analogously  to this  coordinates  s o l i t o n - f e r m i o n system.  energy  before,  the Lagrangian  solutions  {4> ( x - x (L  0  <p .  0  e R}  to the  theory.  (8) admits  )| x  ti  possibility  p r o p e r t i e s of our  for a  a set of  to the classical  of motion  with  <L (* • * • ) • so  that  the equation  (we  have  The  quantum  standard  with  arbitrarily  , ^ (* • t 00 ) • 0  of motion  fixed  i s integrated to  our sign  theory  described  canonical  variables  the following equal-time  i [ T ( « ) , £ (j)]  f o rV ) .  by t h e L a g r a n g i a n  commutation  • S(»-j)  (8)  and c a n o n i c a l  modes  return  which  of the s o l i t o n  coordinates  fermion  the Lagrangian  the supersymmetry  discussed  finite  point  zero-modes  as the "supersymmetry  possible to define  boson  at this  (8) h a s t h e  relations  ( 9 a )  It  16  • i § c o , n ^ l • [ IT c o , * ( ) ] 3  • [ iW  The  anticommutators  Majorana  fermions,  T h u s we a r e the  , VCjU  The  on  that  our  in Chapter  charge  is,  The  which  gives  t,  a constrained  conjugation  condition  for  .  <0  the  » - r  reads  components  -  by  the  matrix  by  Majorana  to  from the  the  ( 9 b )  fact  that  constraint  procedure,  are  $ = t \p . J  s y s t e m a n d we w i l l  Dirac  V>'s  impose  already  1.  {  given  arise  subject  theory  l f z' is  f  fermions  dealing with  constraints  mentioned  for  • [ irw. * t y 1 • 0  0  • f.  C, d e t e r m i n e d  by  the  condition  17  On  thec l a s s i c a l  instead  level  of theusual  -  we h a v e  Poisson  { * ( « ) , < ( • > }  t o take  brackets  Dirac  brackets  {,} ( [ 2 9 ] , [ 3 0 ] ) :  h ( » ) . f ( i " ) j "  (HO.^j)}  where  so  that  and  o u r new b r a c k e t s  a r e g i v e n by  Ux-j)  analogously  { ^ . n ^ l , . - sc-j) The  theory  Dirac  i sq u a n t i z e d  brackets  compatibility  From set  this  by p o s t u l a t i n g t h e c o r r e s p o n d e n c e  and anticommutators of anticommutators  set  of "collective"  which  between  ensures the  and constraints.  of canonical variables and " i n t e r n a l "  we t r a n s f o r m  variables  t o a new  {$,n,#,#^,X,P} b y  18  def  ining  £  ( x . t )  -  T T ( y . t )  , f ^  ^  =  t  ( x .  x(t))  T T ( x -  •  i  ( x - / ( t ) , t ;  (10a)  (10b)  X ( t ) , t j  (X-X(t)) (t)  (10c) with  and  the  [  equal-time  commutation  T ( o , t(3)j  . .[ i w .  • [ I ( « k n3)]  { HO. * tyj f  relations  are  |(,)]  • [ I w . ^f(j)j  • S('-j)  (1 1a)  (11c)  (1 1d)  19  i [  ? ( 0 ,  X ( t ) j  .  1  f o r *,n,i $* . t  J  The  constraints  as  are  taken  to  §(„,!)  be  • 0  ^(,)TT(*.t)  «  (12)  before.  A Expressions a  formal  power  guarantees and  will  remains is  leaves  the  the  used show  compatible  will  is  obvious;  We obvious  drop  but  the an  will  to  the  our  original  of  standard  integral sign  the  and  [ TT(0, £  The  and  that  our  whenever  (12)  is,  for  the  the  is  notation  integration  c a l c u l a t i o n only  (^)J  (10)  It  verification  simplify  spatial  conserved  zero-modes. by  as  (12)  is  formalism,  operators  denotes  explicit  variables  defined  To  understood  constraint  unchanged.  lengthy.  in  always  unphysical  canonical  cases  [ H0.1U})]  The  transformation  somewhat  are  independent  commutators  the  i n X.  eliminate  arguments  give  <t> ( x - X ( t ) , t )  expansion  later  with  form  number  that  straightforward we  the  series  that  be to  of  meaning /dx.  two  less  20  [ £M),{ |4f, j PJ  I'i} f  '  L  Using  X  our  = 0  relations  (11 a - f )  Aini)  (  Using  the  formal  power  V  • T  1  we  t «  T  1  7  obtain  ['£(,.*) , ? J series  f o r \}/ t h e c o m m u t a t o r  evaluated  n >o  n  CD  A  n'  >o  (-0 f * 5  can  be  21  HE  z  A  1  • J  0  IT  Rewriting  the  fermion  condition  gives  commutator  with  the  help  fty  of  the  Majorana  | f (3), f (*-*)]  07 / 6t  1  LJf-  .  (  jT  [  ,  TT(V.tr  0  .  P  ,  A  A  ,  22 [ TT ( x - X ) ,  *  J  (j-'x)]  (x-X)  Again  using  t h e power  A 15  P.  [ ?.  K C-x)  1  • ; ^  we  obtain  („.x)  the commutator  [ f ( 0 ,  which  f o r our commutator  I (.-x) ] • i i'(,.x)  Furthermore  I'd)  J  series  Ic,-x)]  simplifies  4  '  1  I z  , $(«j-X')J  -  the terms  (y-X)  4* (* - x)  reduces  i ' ( 0 • c-o ( U y x - 0 - i to  o  to  i ' o o f' ( jf) t  r  23  The  c a l c u l a t i o n s f o r the other  and  need  n o t be  Thus of  we  quantum  We  2.2  of  3  0  We (a)  =  proceed  canonical  transformation  to.a set  to construct  the Hamiltonian  and  the  motion.  (8)  f t '  straightforward  variables.  of  the  energy-momentum  Lagrangian  are  here.  a consistent  collective  Construction  The  where  have  c a n now  equations  reproduced  commutators  3  i s given  1  =  Hamiltonian  tensor by  f x*  obtain  f o r the energy  density  density,  T , , v = 0  0,1  f o r our  24  (b)  f o r t h e momentum  where We  has been  0 I  write  T0I To  T  symmetrized  • $'TT + ; f  ( TT  construct  the Hamiltonian  substitute  The  most  difficult  transformation  the  bulk  ir(«.t)  part  (10b).for  measure  and  this  notation  -if' - i  (14) T oo  i n o u r new  variables  we  i s the evaluation n  i s highly will  of ^ J j x T T  nonlinear.  again  drop  To  (i ^  since reduce  arguments and the  - i ( ^(-/w)  ( ? « . ( r ; . ;*•*•)  ,  n2  A  define  • ir(»-x(t),t)  A  -  cf  f' )  H = Jdx  o f o u r c a l c u l a t i o n s we  integration  IT  _  (10) i n t o ( 1 3 ) .  the  With  in $  (a) a n d (b) as  • :  simply  density  given  by  j + ^ a; T • T a ^  +  c, ^  f  )  25  IT Using  - IE  X  the basic  commutation  >. we  i  The  relation  r e w r i t e II2  -  -  -  - x  i n terms  i  of commutators  ( T <j>' ( £ + 0 t  IT L, £  {  expression  for I  •  (  TT  ft, • CA.  i - : i s more  and anticommutators:  [  complicated:  1  , 1 CI  a a <£  a  +  cx, <f>i 4 i a c  •  so  (15)  [M*'<•«•*>] * i r i  ^hr  =  ^  1  <x a  L  *  x J•  ^cl  ^  A.  CX  r\, a a,  6i  r +  4  CD  n  that  ( <f>* Z t l  +  J  c  ~  l  • ce' ^ *  r~  aa  +  • *  _  A <1 0  +  a  aa"  - <- a  hi  hi  ^  26  2  +  and  we  The  first  can  be  A  have  the  four  A  rk ' <p„ A  4  terms  '  ~a-  ^  Z  1  are  using  I  =  AT  +  following preliminary  evaluated  f L  X T  *  in their  expression  final  form.  for  The  n2:  next  two  terms  (15):  - o• r co  ^ "•  AT  J  '  -  A  ^ '  4 "  ou-  A• I  *. a.  and  ^  a  AT  furthermore,  L  1  "  A * T  A T 1 ' 1 *  O a "  a.  X  (AT)*  '  AT AT  +  A* I  (AT)*  (AT)  4  ^*  W  4. 4. ' CL  27  Together  with  the  next  two  terms  i n our  H -expression 2  we  a,  To  complete  h b r . * J  our a  l i ^ ' * ] '  d  ( ?•  0/  so  n  computation  that  the  -  X 7  of  n2  the  power  series  i  are  reduced  expansion  r  ±^2  ^0 • I the  commutator  becomes  0  and  since  evaluate  - z f ' f ))TT to  I!  Using  to  w h e r e  M*f  commutators  i t remains  rt . t >  have  28  we  have  Ii  A  IT  * S  r  i  L  AT  Summing  M.  form  only  o n x-X  integration  dx  T  n  terms  gives  the f i n a l  ,  for II : 2  (ATr (*-X(i)j  (A* O  of t h i s that  « CL  J  result  -  *  a  A  (ATJ  .  into  hr:  '  up a l l t h e s e  purpose  n -1  1  t  The  00  rather  permits  a n d commutes variable  dx TT  4  1,  lengthy  calculaton  integration with  X.  over  Thus,  we  x.  was Each  to transform factor  can t r a n s l a t e  x t o x + X and use our c o n s t r a i n t s  dx 7T <jf>  a + a.  n2  depends our  (12):  29  I  A£  /  ,  i  •  The  we h a v e  used  remaining  Thus  Hamiltonian  ,l  *  I  C W ) *  the c l a s s i c a l  soliton  mass M  0  is  evaluated  i s given  Since  we  4  want  consider  We  ( M  by  the orders  want  only  -  *• fy  M ~g~ ,  S/M ~g h e n c e  f  ))  de)  a)  of g  orders  $,\p,n,\p 0  £  p e r t u r b a t i v e expansions  we h a v e 2  # y  ( v c ^ t i )  4  0  M  Z  *  us  that  terms a r e e a s i l y  the t o t a l  H  the fact  I  /  / 4  where  A  i n g f o r our q u a n t i t i e s  involved.  up t o 0 ( g ° )  ,P,X^g ,  and ^ - g "  0  (M +£)- ~ 1  0  l e t  f o r our operators, 1  as before.  g " , n>2.  Thus  we  that i s , have  30  Furthermore  we c a n e x p a n d  Hamiltonian  H decomposes  H  = M  0  + H  +  0  V  and V  2  around  .  Then o u r  as follows H  x  where  (17)  with  H  describes exactly  0  of V  2  this " ( ^ )  This  chapter:  can  x  just  free  momentum  Substituting P that  "position  that  i s already  <p = c o n s t . , 0  standard  at the beginning dependent  background  potential.  at the 0(g°) l e v e l the  on t h e o t h e r  hand, (17)  field.  i n higher  order  i n g.  perturbation theory  Its effects  with the  H . 0  operator  P  =Jdx  T  0  1  i s easily  (14) and p r o c e e d i n g  evaluated.  as before  = P  the total  mass"  nontrivial.  and boson  interactions  (10) i n t o  with  discussed  in a static  soliton  Hamiltonian  we  statement  sector  be i n c l u d e d u s i n g  The  so  a  describes  unperturbed  system  of our theory  t h e vacuum  describes  system  our e a r l i e r  sector  For  H  a boson  and a fermion  confirms  soliton  the situation  we  obtain (18)  momentum  of the system  i s given  by t h e  31  collective  We  momentum  note  commutes take  P  2.3  The  with  P.  t o be a  field  Given function  i n agreement  our Hamiltonian We  the pure  boson  (16) i s i n d e p e n d e n t and P  case.  o f X,  simultaneously  c-number.  equations  f of the f i e l d  particular  with  can d i a g o n a l i z e H  i n q°-order  the Hamiltonian  Hamiltonian  In  that  P,  H  0  the evolution equation  variables  i s given  by  f o r any  the canonical  formalism:  we  (1 )  obtain  A  A.  (19) r  A  IT (2)  A  and +  A  hence and  32 or  (20) (3)  A  L H.. A.  A  |  j  A  .  A  I  Using  the r e p r e s e n t a t i o n  matrices  7  °=a , 2  7  *«/a  5  Jf  7  = - a , , f o r the D i r a c  we can reduce the (pseudo) s c a l a r s t o .  A  ^  A  A t  t  -A t  2  - '  S  the anticommutation. r e l a t i o n s ( 8 ) we o b t a i n  Using  A  A  ,  A  A  i  A  A  V y* v . ^ ] r  "  -f  -  which y i e l d s ,  i  A |  for  A  ,  the fermionic  - - v(fc) i  equation  , 1  i l  of  motion  33 which  is written  in matrix  form as  u  X • V'(* ) u  or  «0  ( ^ i - i f ( o • v'(<U)) V Since of  our  motion  (4) The  Hamiltonian are  0  does  not  i n v o l v e P o r X, t h e i r  (22)  X = P = 0 (20)  equations  shows  that  our  Majorana  We  equations  trivial  a n d (21)  are  familiar  i n 1+1-dimensional  fractionalization  to  H  (21b)  •o  x  t  (21a)  method  i s indeed  i nthe  context  o f charge  ([3],[34]).  systems  quite general  and  not  This  restricted  fermions.  will  complete  constructing  the  an e x p l i c i t  discussion solution  o f our  t o (20)  quantum  s y s t e m by  a n d (21).  Using  the  "ansatz" A  A  $ (x.-t) (*.t)  v to  separate  the  ("  time  •  1  'tut  * < / > CO t c KOt  . dependence  V*"(0  )  we o b t a i n  fo>)  '  »  h  >  )  <23)  •«r " W  (24)  We  note  that  the  fermionic solutions  can  be e x p r e s s e d  i nterms o f  34 the (  2  1  bosonic one: )  0  v'UJJ ^ • « £  -3* If  »e d e f i n e  the  first  ^  equation  i  ,  ^  turns  .  into  y'  ( < ) J  )  t h e boson  (- V * v"ifc) • v'( so  for $  equation  i  2  i •«£  that  (25)  f 00 - $ provides a solution boson  equation  we c a n e x p a n d annihilation  f o r (21).  (20).  The f i e l d  the f i e l d operators.  Thus  equations  operators We  i tsuffices  t o solve the  a r e now l i n e a r ,  i n terms of c r e a t i o n  o b t a i n f o r t h e boson  and  operator  *(-*) - L L (r,,wr% . r w - 4 ; ; with  £n a s o l u t i o n  runs over spectrum.  both  to equation  the discrete  The ( n o r m a l i z e d )  ft. *  (20) and w  k  «f k  and the.continuous zero-energy  4  •  hence  1  (26)  •  part  T  n  e  s  u  m  of the  solution  (27)  £  35  is  excluded  hence  for  from  we h a v e  properly  operator  with 0tL'  of theorthogonality relation (12),  t h ecompleteness  normalized  £  k  r e q u i r e s somewhat  zero-energy  t  $ because  .  relation  The expansion  more  care.  Equation  (24) admits t h e  solution  W  "  "I* ( "  H  f  V(*J  N a normalization constant. = V(0cl)  f o r t h e fermion  - a n d t h e change  ) (o)  Using  (28a)  equation  of variables  d0 c L  (4)-  = 0 'dx tL  this can  integrated t o  (28b)  To  e l i m i n a t e t h e zero-mode  case.  First  f(*.t) with  we r e p l a c e  analogously  transformation  . X (x-X(M)  a  +  i*  new a n t i c o m m u t a t i o n  Xf  f  relations  equation  ?(«-*W.O  constraint  \ and  we p r o c e e d  • o  t o t h e boson  (10c)  by  (29)  36  [ f.w. v^i • K  - i.^co<t,))  <30b>  (30c)  i M •{ It  •{ ^ . M •  i s straightforward to verify  canonical,  that  \p a n d i//t a  i s a Hilbert  zero-mode,  i// i s now  of  solutions  operators  i s , i t preserves  of  \  that  space  orthogonal  (21) w i t h  •  this  ( 3 0 d )  transformation i s  the Majorana  operator  o  anticommutators f o r  a s s o c i a t e d with the  t o \j/ a n d c a n b e e x p a n d e d 0  fermion  creation  and  i n terms  annihilation  , ^  (31) k  with The  a commutation  For  Using  (for  simplicity  been  evaluated  (  *  )  •  -  t o (24) g i v e n  relations  our standard  analytically.  !  solution  we  f o rf . j  ^"-model  taken  ( 2 5 ) a n d <»>„ •-/ k * * ' .  are fixed  the f i e l d  the c l a s s i c a l  have  by  equations  solution  x =0,g=1) o  by ( 3 0 ) .  c a n be  solved  ( 5 ) , <f> = m - t a n h  the solution  mx  o f (23) has  ([13])  :  —  *x  -  ) i  k  '  ,  Ic *  T R  37  and  equation  (24)  Equation  (26)  in  f o r our  0(g°)  Having a  comment  We  have  and  a  seen  of  viewed  the  be  as  that  (31)  the  full  boson  Lagrangian  (8)  with  potential  that,  the  in  first  zero-mode  order  appears  soliton.  The  interpreted  supersymmetric as  generated  by  and  theory  solutions  fermion  c o n s t r u c t i o n of  i n our  infinitesimally  in a  spinor  the  zero-modes  the  corresponding  give  completed  the  fermionic  existence  show  and  thus  on  yields  the  seems  to  field  operator  be  order.  perturbation theory, as  a  bosonic  direct  operators  in a  consequence  "translation  bosonic of  mode" c a n  translated  soliton.  Later  theory  fermionic  zero-mode  an  the  infinitesimal  we  theory  complete we  have  to  the  d i s c u s s i o n of  find  the  Hilbert  the  soliton  space  sector  f o r our  of  our  operators.  be  will  supersymmetry  transformat ion.  To  the  can  38 2.4  for  The H i l b e r t  space  f o r the soliton  The  space  f o r t h e vacuum  a  free  {|k,...k q.  Hilbert  n  field  ;q,...q >} m  generated  fermion  In by  theory.  from  (k),f  (q).  the s o l i t o n  sector  we  ([28]). space be  state  These  states  states  energy  k; a n d  by t h e b o s o n - a n d  this  the s o l i t o n mechanical  o p e r a t o r s P,X.  a n d momentum  space  w i t h momenta  to enlarge  t h e quantum  f o r the center-of-mass  sumultaneously  t  |P> t o t a k e  form  states  |0 >»|0^>  have  i s t h e s t a n d a r d one  of the Fock  formultiparticle  operators b  a s e t of e x t r a  sector  It consists  a vacuum  sector  They  product  into  space  account  representation are required to  eigenstates  (32)  This of  i s always  X.  We  note  remains  static  state.  Thus  the  product  |0 > » | 0 4  k  The usually As  >»|0  possible  furthermore (X=0),  space 4  {| P  4  our Hamiltonian  that  s o we  the Hilbert  in first  s e t P =0,  space  order 0(g°) the soliton  E(0)=M  4  o  f o r the soliton  ; k , ... k ;q, . . . q > ] w i t h r  w  (16) i s independent  f o r the ground sector vacuum  i s g i v e n by state  >.  problem  of possible  associated  discussed  since  with  d e g e n a r a c i e s of t h e ground  the existence  i n the previous chapter  of t h e f e r m i o n i c the f i r s t - o r d e r  state i s zero-mode.  field  39 equation  (24) admits  a zero-energy  s o l u t i o n so that the fermion  A  operator \p has to be expanded as  V ( * . t ) - t OO £ The  » ?(».t)  .  zero-mode operator a does not have an i n t e r p r e t a t i o n as a  c r e a t i o n operator but i s only r e q u i r e d t o provide a r e p r e s e n t a t i o n of the a l g e b r a  In  a theory with D i r a c fermions these r e l a t i o n s are s u b s t i t u t e d  by the standard anticommutator and the zero-mode operator commute with the Hamiltonian.  The r e p r e s e n t a t i o n of a and a* i s  then given by a doubly degenerate oJ  I  +  | 0. >  •- I 0 >  a  +  T h i s i s not v a l i d  l0 > +  ^ V  e x i s t e n c e of a zero-energy and no d e f i n i t e statement  of  The Hamiltonian (16)  and the r e s u l t i n g crossterms w i l l not The c o n v e n t i o n a l view that the  mode s i g n a l s degeneracy does not hold can be made a t t h i s p o i n t .  s o l u t i o n to t h i s problem i s given by the supersymmetry  the theory.  possess  -- 1 0 . >  A  v a n i s h under commutation with a.  The  |0+> ([3]) with  * O  f o r the present case. A ^  contains b i l i n e a r s  ground s t a t e  a, 1 0. >  - 0  0 >  will  The supersymmetry o p e r a t o r s w i l l  turn out to  j u s t the r i g h t p r o p e r t i e s to p r o v i d e the expected  degeneracies  f o r the s o l i t o n  states.  We w i l l  q u e s t i o n s at the end of the f o l l o w i n g c h a p t e r .  r e t u r n t o these  40  3.  SUPERSYMMETRY  One fact  of  that  the  whose  supersymmetry to  published  I t has mass  Bogomolny  soliton  a  set  of  shown,  at  the  saturated  the  this  exist  field  theory  first  is  breaking  have  order,  Here  of  of  a  been  we  for  that  that  are  no  bound  c o r r e c t i o n s and ([20]).  conserved  c o r r e c t i o n s to  Bogomolny  in  the  theories  several papers  the  is  generators  quantum  quantum  least  quantum  spontaneous  supersymmetry  recall  algebra For  years  s a t u r a t i o n of  remains the  few  concerning  r e c e i v e s no  on  form  p r o p e r t i e s and  last  theory  THEORY  the the  the  the  will  supersymmetry  in  the  sector.  The  We  the been  bound  concentrate  3.1  and  charges  our  i . e . there  Supersymmetric  the  SOLITON-FERMION  f e a t u r e s of  supersymmetry,  unusual  ([15]-[20])  energy. soliton  remarkable  algebra.  During  mass  1+1-DIMENSIONAL  corresponding  possess  exception.  soliton  most  A  i t possesses  currents  known  FOR  generators  briefly  the  for  definition  the  of  soliton-fermion theory.  an  (N=l)  supersymmetry  ([35]):  given  action  operators  anticommutation  Q  S  0^  the ,  supersymmetry Q  r  *  i.^f)^  algebra  , ^,o,i  i s generated  together  with  the  relations  o  r  (33a)  (33b)  I)  -  o  (33c)  by  a  41  with  P  t h e energy-momentum  e  For  topologically  solitons,  ([16]).  Lagrangian  possesses  generates  to include  "topological  )  2(*,t  corresponding  to  ,  currents  ( jf<fr  t  ; V ( o »  i t scomponents  Q.  ^  r  algebra  generate  . Q,J  [ Q«. T j  ) T  (34)  "supercharge"  a supersymmetry  {  c o n f i g u r a t i o n s , e.g.  d e n s i t y (8)  the corresponding  i.e.  field  S = JdS  The a c t i o n  conserved  j r  and  nontrivial  ( 3 3 b ) h a s t o be m o d i f i e d  charges" our  operator.  Q  with  |j« j ( * , t )  *  topological  a supersymmetry  • -if,  t  \  • [ Q . . ?, ]  *  » (Q° )  6  i ;  charges  algebra  T,  with  y r  T  <35b)  -o  where  T c a n be i n t e g r a t e d t o z e r o trivial, 0  (x=t»)  To using  i s , <pcl(x=t<x>  that = 0  t  verify  f,F:  <p i s a s y m p t o t i c a l l y  0  has t o i n c l u d e the s u r f a c e  ( 3 5 ) we  A  ) = <f> . A s o l i t o n  will  the following identity  B,b a n d  i f the f i e l d  evaluate  field  term  however,  T.  the anticommutator  f o r two b o s o n  with  and fermion  directly operators  42 {Fb,bf} The  = Ff[B,b]  +  {F,f}bB  (37)  generators a  can  be v i e w e d a s c o m p o n e n t s  Q.  i. L d*  J  ' Now a p p l y  The  Kij'f.*  ZL  commutators  (a)  the following matrix  *f A' - i s r  v( )) r  5>pj  L |. r  to the anticommutator  are evaluated  t o be o  * I,. V  product  y  « |i« E (37)  of  S  [f U ) > ( j ) ] - i l  r  $,„  [ v(4>), ity]  43  (b)  SO  •  that the boson product  H  L  can be reduced to  y  Ux-O [ ( *-yV *<yX*)) (y"f - / V V O ) )  IK  T(«)  -i; ^  *  -  v'it)  Ho)  J  J  and for the fermion part  fj, u..ij • - * f p7;.^ ,t  r  r  1  f  *  v(+)  Thus we have as a preliminary result  .  v'(*)  f'  r  f  r  . f , f  -** r;. r  \  44  V ( * Y < r , . / ' ( • ) O V ) ,( / » , .  *  - V'( As  the  last  ordering Formally,  That we and We  choose  term  arising  however,  we our  have  for the  there  terms  a l l these  indeed  familiar  e v a l u a t e the obtain  indicates from  S(o) S.^  <t>)  like  are 7  our  will  of  particular  our  7  0  matrix  terms  \  . m H;' +  (b)  +  \  , 1 ,  the for  operator x-*y.  cancel.  equality  representation  elements  with  \p (x)i//(y)-» ^ 6 ( 0 )  terms  found  problems  •  =  can a ,  (ML,.  be 7  2  =  1  easily =  {Q.  / 03, ,Q.}.  seen 7  5  = "  45  (c)  *  *(o)  f  3L  (d)  ^7) • (ye^)  , o  -  •x'x which  gives  the  final  expression  \ 0  /  f o r the  v'u) (^7 -  +  + 1  , Ho)  i  anticommutators  ) - U7'*^7)  46  whereas  the  right  hand  - X  side  -  I  1  « = o , fl -  (37)  gives  «\  1  1  eo  « . ft  of e q u a t i o n  72  f 0|  r  6  r"  7  and of  the  the other  In Q's  supersymmetry  on  the the  two  next  relation  relations  section  s o l i t o n ground  is  we  (35b)  i s proved.  The  s  -r  verification  straightforward.  will  state.  investigate  the a c t i o n  of  the  47  3.2  Spontaneous  Having our  action  supersymmetry  established we  will  due  to  the  discussion  of  supersymmetry  broken  [35]  and  [36].  now  the  We  will  i n the  e x i s t e n c e on  show  presence  breaking  that of  the  N=1  symmetry  central  can  their  be  sector  supersymmetry is  charges.  breaking  modify  an  soliton  spontaneously A  general  found  arguments  for  in references  f o r our  present  case.  First  we  derive three  supersymmetry taking  the  relation  trace  equations  (35a).  g i v e s the  from  our  Multipling  well  known  by  Furthermore  we  deduce  •  CL*  -  Q,4 It  i s obvious  (39)  implies H  which  H  ground  will state  both  now |0>.  and  - ) (38)  (35a)  (39)  T  (40)  equation the  sides  Q4  (38)  quantum  that  H  Bogomolny  is positive. bound  on  Equation  H  > T  i s intended  We  -  from  on  0  H * T  from then  Q  directly  7  formula  H" • i ( {Q..<M t - -M ) : i +  central  (41) to  hold  examine  i n the  these  Obviously  we  weak  sense.  operator have  the  equations  f o r our  alternative  soliton  Q-|0>*0,  48  which  means  Q-|0>=0,  It  that  supersymmetry  i n which  case  i s obvious  i s spontaneously  supersymmetry  from  equations  I t i s n o t so easy  unbroken  o r , e q u i v a l e n t l y , whether  to decide,  the c l a s s i c a l  central  charge  Hamiltonian H|P=0>  T  acts  = M |P=0> o  question  whether  ([l5]-[20]).  level  = j * ^ A  t  l  (cf.  0(g-')  f '  whether  t h e Bogomolny  the solution  reduces  on a p u r e  soliton  state  result  h a s been  I t h a s been  hold  discussed  found  corrections,  so t h a t  the combination  result  already  on t h e d e n s i t y  this  will  Imbimbo  remain Their  fermions  i n lowest  zero-mode breaking  that  Q  bound  remains  0  ( 4 1 ) on H  argument  i s based  order  appears  order  the other  modes  would  to zero,  authors  identical remains  in quantum  zero.  qualitatively  i n perturbation  zero.  The  more  theory  fermion t othe  Goldstone  to renormalize  a process  that  of Goldstone  2.3) c o r r e s p o n d s  have  This  f o r H and T  argued  To g e n e r a t e  The  orders i n  order  on t h e a b s e n c e  (cf. section  exactly  i s saturated.  by s e v e r a l  level  have  with  t o higher  H-T  ,the  1  a  |P=0>  receive  perturbation  corrections  j <*•* <t>  =  0  i s easy: the  to first  to a l l orders  o f t h e Ch-symmetry.  higher  unlikely.  a n d Mukhi  valid  ([19]).  H and T  that  theory  ([19],[20]).  both  will  perturbation  holds  to M  ( 3 2 ) ) and t h e bound  this  perturbation theory  of  however,  i s always  saturated.  On  the  i s preserved.  (38) and (40) t h a t  broken.  is  broken, or  that  modes,  t h e masses  seems  very  49  We  can  breaking by  H'  =  reformulate  i n the H  spectrum Hilbert  now  soliton.  - T. by  a  supersymmetric rewritten  field  the  N=1  due  the  presence  can  section  not  Hamiltonian of  affect  system.  the  This  with  the  more  defined  energy  dynamics  new  equations  H'  or  the  Hamiltonian  general  (39)  and  results (40)  of  with  the  the  of  ( 4 3  a  i s broken  central  the  charge  supersymmetry  state  down  can  ground  be  which  can  state  broken.  to  an  the  In  "N=i"  other  symmetry  to be  possible  state  generators  according  ground  Q  that  we  started  and  Q,  both  0  (35c). taken  as  Q,  does  not  a  degenarate  with  t  +  , Q, |0.> - | 0<> (44)  a, l o 4 > 4  the  in  )  T.  d i s c u s s i o n of  Hamiltonian  ground  f o r the  spontaneously  soliton-sector  The  the  |0 >  T  remains  a, fo > ~ I0_>  Thus  has  (42)  H' W  •  complete  2.4.  annihilate doublet  shift  symmetry  * H  supersymmetry  now  degeneracies  commute  a  i n agreement  0.,-symmetry  the  words,  in  new  to  and  spontaneous  the  i s obviously conserved  0  We  our  of  as  Q -symmetry  to  does  theories,  Q*  whereas  and  energy  Qo'  The  Consider  s t r u c t u r e of  vacuum  problem  corresponds  constant  space  vanishing  This  the  spontaneous  -  T  i os >  breaking  of  supersymmetry  provides  the  50  degeneracy  of  with  fermions  Dirac  We  will  connection fermion  the  conclude  zero-modes  the  boson  classical  have  to  be  as  corresponding the  consequence  on  the  was  set  by  quantum  of  the  consider fields  our  analogous  of are  theory.  not  as  field  theories  mode  zero-mode  infinitesimal to  the  the  the part  0 (x-X) t l  quantum  and  4> (x-x ). t l  invariant  X  x  they  which  0  is  the  be  Since  0  is  soliton  0  ct  infinitesimal  viewed  as  a  symmetry.  supersymmetry conserved  transformation  charges  Q  0  and  ([35],[37])  5  (45a)  (45b)  with  e  an  anticommuting  transformations  -  with  Majorana time  2-spinor.  dependent  then  the  <f>lL v i a a n can  and  far,  on  parameter  translational  corresponding  So  solutions  Furthermore  the  boson  soliton  the  variable  translation  on  and  based  The  translation  boson  note  translationally  operator.  broken  been  classical  the  the  the  has  a  theory.  interpreted  collective  Hence  from  solitons  supersymmetric  of  solutions  a  between  formalism  zero-energy  translation.  Now  a  labelled  considered  induces  in a  operator of  familiar  d i s c u s s i o n with  earlier  symmetry  representative the  this  coordinate  translational  state  ([3],[34]).  mentioned  collective  of  ground  If  we  t-parameter  apply -  to  these the  51  classical  static  we  t h e new  obtain  solutions  solutions (46a)  6i//  o(/  i s - up  (28),  now  to a  transforming  by an  ([38]).  "supertranslated" transformation  the f i e l d s  constraint  This  system  collective for  both  bosonic  c a n now  6  - the zero-energy  B  )  mode  supersymmetry  quantize  these  s o l u t i o n s as b e f o r e  (*)  by  treating  coordinate  £ Ut)  +  V  7  quantized  • 0  the  and  (47)  (  of B a a k l i n i  fermionic  c a n o n i c a l l y as  o r by  ([31]).  4 8 )  .  either  coordinate  systems  4  to  the c o l l e c t i v e and  constant  E.(t) a s a c o l l e c t i v e  ) <U  i n t h e work  (  a  position  generalized  ta  6(*)  • SY («) tCO •  c a n be  constrained  found  •  (*.t)  with  We  parameter  4  infinitesimal  classical  i UO T-  (o)  normalization  generated  transformation  ta  t h e Feynman  A detailed  ([39],[40])  method  parameters  are  path  integral  d i s c u s s i o n can  who,  coordinate  f o r the  be  in addition,  t o the case  present.  where  52 To of  zero-energy  which are  summarize  are  broken  generated  the  fermion  by  by  mode.  to  In  the  quantization  the  theory,  gives  a  be  of  the  0cL,  as  that of  the  existence  the  soliton.  Lagrangian  The  zero-modes  symmetry  in particular,  a  translation  transformation  parameters collective  for  associated  with  coordinates  for  the these the  theory.  i t has  been  information  in particular, fermion  symmetries  supersymmetry  used  shown  infinitesimal  a d d i t i o n , the  can  Conversely,  an  have  to  presence  soliton  and  transformations of  the  we  i s due  applying  boson-mode  zero-modes  results,  solutions  transformation for  the  presence  of  "hidden"  supersymmetry.  that  argued about in a  zero-modes  that the  theory  implies  ([34]).  the  existence  symmetry with the  of  properties  Dirac  fermions  existence  of  a  of  a  the  53  BIBLIOGRAPHY  1]  A.C. S c o t t , F . Y . F . (1973) 1443.  2]  J . Goldstone  3]  R.  4]  J . Goldstone  5]  M.B.  6]  A.J. Niemi  and  7]  T.  Skyrme,  Proc.  RoySoc.,London  8]  E. Witten,  Nucl.  Phys.  9]  R.  Mackenzie and  10  M.  Rice  1 1  W. S u , (1979)  12  R.  Jackiw  and  J . R.  1 3  R.  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