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Coherent state path integral for the harmonic oscillator and a spin particle in a constant magnetic field Bergeron, Mario 1989

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COHERENT STATE PATH INTEGRAL FOR T H E HARMONIC  OSCILLATOR  A N D A S P I N P A R T I C L E IN A C O N S T A N T M A G N E T I C F L E L D By MARIO  BERGERON  B . S c . ( P h y s i q u e ) U n i v e r s i t e L a v a l , 1987  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  M A S T E R O F  SCIENCE  in  THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS W e a c c e p t t h i s t h e s i s as c o n f o r m i n g to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA O c t o b e r 1989 ©  M A R I O B E R G E R O N , 1989  In  presenting  degree freely  at  the  available  copying  of  department publication  this  of  in  partial  fulfilment  University  of  British  Columbia,  for  this or  thesis  reference  thesis by  this  his  for  and  scholarly  or  thesis . for  her  of  The University of British Columbia Vancouver, Canada  DE-6 (2/88)  I  I further  purposes  gain  the  shall  requirements  agree  that  agree  may  representatives.  financial  permission.  Department  study.  of  be  It not  that  the  be  Library  an  advanced  shall  permission for  granted  is  for  by  understood allowed  the  make  extensive  head  that  without  it  of  copying my  my or  written  Abstract  The definition and formulas for the harmonic oscillator coherent states and spin coherent states are reviewed in detail.  The path integral formalism is also reviewed with its  relation and the partition function of a sytem is also reviewed. The harmonic oscillator coherent state path integral is evaluated exactly at the discrete level, and its relation with various regularizations is established. The use of harmonic oscillator coherent states and spin coherent states for the computation of the path integral for a particle of spin s put in a magnetic field is caried out in several ways, and a careful analysis of infinitesimal terms (in 1/N where TV is the number of time slices) is done explicitly. The theory of the magnetic monopole and its relation with the spin system are explained, and the equivalence of these two system is established up to infinitesimal order by the introduction of an exterior interaction to the monopole. This gives a new representation of a coherent state path integral in terms of a more familiar Feynman path integral. The coefficient of the topological term in the spin system appears explicitly without ambiguity, as being 2s.  ii  Table of Contents  Abstract  ii  List of Tables  v  List of Figures  vi  Acknowledgement  vii  1  Introduction  1  2  Review of Coherent States and Path Integrals  3  2.1  H a r m o n i c Oscillator Coherent States  3  2.2  S p i n Coherent States  6  2.3  Propagator, P a t h Integral a n d Partition Function  8  3  4  Coherent State Path Integral for the Harmonic Oscillator  11  3.1  E x a c t Result for the Discretisation  11  3.2  Continuum Limit  12  3.3  Semiclassical Approximation  16  3.4  Regularization  18  Coherent State Path Integral for Spin  26  4.1  Discretisation w i t h S p i n Coherent States  26  4.2  T h e Schwinger-Boson M o d e l  27  4.3  E q u i v a l e n c e of the two Representations  28  iii  5  6  4.4  C o n t i n u u m L i m i t of the S p i n Coherent States  30  4.5  C o n t i n u u m L i m i t of t h e S c h w i n g e r - B o s o n M o d e l  32  Path Integral for a Charged Particle in a Magnetic Monopole Field  39  5.1  Monopole Vector Potential  39  5.2  Monopole Angular Momentum  41  5.3  P a t h Integral for a S p i n P a r t i c l e i n a M a g n e t i c F i e l d  43  5.4  C o m p a r i s o n w i t h Coherent State P a t h Integral  45  Conclusion  47  Bibliography  50  Appendices  51  A Identities for Determinants  51  B Schwinger-Boson Model  54  iv  L i s t of T a b l e s  Second order product regularization  L i s t of F i g u r e s  3.1  Discrete and continuous representation of the z variable  .  14  3.2  Contour of integration for the T(x) function of the discrete determinant .  21  vi  Acknowledgement  I w o u l d l i k e t o t h a n k m y s u p e r v i s o r , D r . G o r d o n S e m e n o f f , for his e x p l a n a t i o n of v a r i o u s subjects i n theoretical physics. O u r m a n y discussions were very helpful i n u n d e r s t a n d i n g d e t a i l s I w a s n o t f a m i l i a r w i t h . In p a r t i c u l a r , h e i n t r o d u c e d m e t o t h e s u b j e c t of c o h e r e n t state p a t h integral, w h i c h , w i t h his suggestions a n d c o m m e n t s , led the way to this thesis. F u r t h e r m o r e , I w i s h to t h a n k D r . Ian Affleck a n d his graduate student, L o k C . L e w Y a n V o o n , w i t h w h o m we h a d constructive discussions o n this subject, a n d by i n t r o d u c i n g t o m e t h e m e t h o d of c o n t o u r i n t e g r a l r e g u l a r i z a t i o n ( c h a p t e r  3.4).  D e p l u s , i l m e f a i t p l a i s i r de r e m e r c i e r m e s p a r e n t s p o u r l e u r s u p p o r t c o n t i n u , t o u t a u l o n g d e mes e t u d e s a c e t t e u n i v e r s i t e . M a l g r e l a d i s t a n c e q u i n o u s s e p a r e , r e s i d a n t d e c h a q u e c o t e d u p a y s , l e u r p r e s e n c e et s o u t i e n a t o u j o u r s ete c o n s t a n t . F i n a l l y , I w o u l d l i k e t o t h a n k t h e d e p a r t m e n t of p h y s i c s a n d U B C for a c c e p t i n g m e as o n e of t h e i r g r a d u a t e s t u d e n t , a n d t h e i r s u p p o r t i n m y w o r k for t h e last t w o y e a r s .  vii  Chapter 1  Introduction  The use of path integrals is a very active subject in physics. They have found many applications in quantum field theory, particle physics and condensed matter physics. The foundations have been studied for some time [7,8], and are by now well established. Independently, in nuclear physics and quantum optics [9], some models have been studied using coherent states as a bridge between quantum theory and classical mechanics, obtaining a semi-classical representation of the theories.  The theory of coherent  states is very well known today, and does not present any mysteries by itself. The introduction of coherent states as a tool for the evaluation of path integrals appears as an interesting alternative to the usual \p >, \q > representation. Since these states exhibit classical behevior, we would expect the path integral to be easier to handle and, in particular, easier to approximate. However, their use has some difficulties [1]. A standard Lagrangian possesses a kinetic term which is quadratic in the velocity, like m{f) . 2  With coherent states, instead, we find a Lagrangian having a first order time  derivative only, thus having a very different dynamical behavior and a different set of initial conditions. Actually, the Lagrange equations of motion for the coherent states are equivalent to the Hamilton equations for the coordinate and momentum variables, the latter variables being both included in the coherent state representation. So, a propagator computed with coherent states goes from an initial position and momentum to a final position and momentum, but such coordinates (in phase space) are usually not connected by a classical path! So, the difficulties with the coherent state path integral correspond to  1  Chapter 1. Introduction  2  the inclusion of these non-classical paths into the calculation, for a proper consideration of quantum mechanics. In this thesis, I am going to look at these difficulties closely, and try to find some ways of properly evaluating these path integrals, by first using a discrete path integral , and then carefully examining the continuum limit. In chapter 2 , 1 will start by reviewing the theory of coherent statesand path integrals. This will not cover new results, but is intended to introduce the notation. In chapter 3, I will study the harmonic oscillator coherent state path integral in detail. I will explain different ways of regularizing the path integral, and compare these approximations with the discrete, but exact, path integral. The same work will be done in chapter 4, for the coherent state path integral for a particle of spin s put in a constant magnetic field. For this path integral I use spin coherent states and the harmonic oscillator coherent states alternatively, where their equivalence will be made clear by using various methods. Specifically, for the harmonic oscillator coherent state path integral, the gauge symmetry will be studied and its connection with the topological term, appearing in this path integral, made clear. Furthermore, there has been some question about the coefficient of the topological term. This will be determined unequivocally in my calculation. In chapter 5 , I will review the theory of the magnetic monopole and indicates its use to represent a spin s particle. The path integral for this monopole system will be studied, and its relation with the system of chapter 4 will provide us with a new way of interpreting the regularization of the coherent state path integral.  Chapter 2  Review of Coherent States and Path Integrals  Harmonic Oscillator Coherent States  2.1  F r o m the harmonic oscillator Hamiltonian, in M dimensions, H = Y^=\{^Pk+ T~Q\)  —  m  Y^k=\ w{a a k  k  + h/2), with a = -^(\/mu)Q k  + i-j^z) and P = —ih-^, we can single out  k  k  a ground state | 0 >:  a  k  | 0 >:= 0,  < 0 | P | 0 >=< 0 | Q | 0 >= 0, H | 0 >= MLO%/2 | 0 > K  K  A n d then build up all the eigenstates of the Hamiltonain:  \  n i  ...n  M  >=  - ==L—{a\/Vnr 7  •••  («M/v^)  n  |  w  0>  M  H | n ... x  n  M  >— YI h{n k=i u  k  + 1 / 2 ) | ni...  n  M  >  B u t one of the drawbacks of these states is that they are not eigenstates of either the position or the momentum operator (Q and P). Furthermore, the commutation relation [<3fcP/] = iti6 i prevents us from finding eigenstates for both of them. B u t it is possible 5  k  to define a state, that we will call the coherent state | p,-, qj >, that will have a position and momentum, on average, given by some classical values (p,, qj):  <p,q\Pi\p,°>=Pi,  <p,q\Qj\p,q>=  T o find such a state, we can start with | p, q > = e~  A  n  (  2  .  1  \ x > and the identity e Be~ A  A  ) =  B + jj [A, B] + j \A, [A, B]] + . . . that stops at the second term if [A, B] = c-number, and {  3  4  Chapter 2. Review of Coherent States and Path Integrals  t h e n f i n d t h a t (2.1) i m p o s e s t h e c o n d i t i o n s : | x > = | 0 > a n d A =  r>kQk-9h *, w h i c h p  gives:  M \p,q>= e x p { £ TiPkQk ~ qkPk)} | 0 >  k=l I n t e r m s o f a , a a n d t h e c o m p l e x v a r i a b l e z = -^{\/ Qk + ^ § ^ j ) muj  k  k  k  1 1 | p,q > = | 2 > = e x p { £ - ( z * a j - * * a ) } | 0 > = e x p { - t a M  f c  + 2  -  zU)}\0 >  (2.2)  k=i / „\ by working w i t h columns a  >2  W e c a n e a s i l y v e r i f y t h a t < z | (P -Pk)  \ z >=< z \ (Q - qk) \ z > = %/2. S o | z >  2  2  k  k  is as close as p o s s i b l e t o a c l a s s i c a l s t a t e . I f w e u s e t h e i d e n t i t y e e A  B  = e^  ^ ' ^^  A+B+  A  B  2  w h e n [A, B] = c - n u m b e r , w e c a n r e w r i t e (2.2) a s :  ^  _ i , t , *U , ^ n  2 > =e  e T I 0 > = e 2*  (-^i/v^)"  1  7=r=f—  2^  (Wv^)"" , 7-—,— | n  1  ...n > M  H-"M=0  (2.3) F r o m this w e c a n show  a \.z >= z \ z >, < z\a) =< z\z* a n d < z \ z' > = e x p { i ( z V - z z / 2 ft T  z'V/2)} (2.4)  a n d also /•  dzdz^ .  .  Jj^w  JE ^  {z><zl=  =/n<  dz dzt k  ds  f,  [  ™,dz dz* . k  k  )lz><zl  £ ((zk/Vh) "(zt/Vh)<  y r  m,nj,...=o K = l  =  {  ^2  n  I"i  • ••  I ni...nM >< ni...n,M |= I  n >< n\ ... n' \ M  M  (2.5)  ni...n =0 M  . Sometimes it will be easier to work with one complex variable z, without any meaningful connection with its real (position) and imaginary (momentum) part. 1  Chapter  2.  Review of Coherent States and Path Integrals  5  So t h e | z > c o h e r e n t states f o r m a n o v e r c o m p l e t e set o f s t a t e s . T o s u m m a r i z e , t h e h a r m o n i c o s c i l l a t o r c o h e r e n t states h a v e t h e p r o p e r t i e s :  •  E i g e n s t a t e s o f t h e a o p e r a t o r : a | z >=  z \ z >, so < z \ a \ z >=  z. ( m e a n v a l u e s  of p o s i t i o n a n d m o m e n t u m g i v e n b y p a n d q.)  •  A r e n o t o r t h o n o r m a l : < z \ z' > ^ 6(z — z')  •  A r e o v e r c o m p l e t e : / ^'f^M  \ z ><  z \= I  F i n a l l y , let m e p r o v e t h e i m p o r t a n t f o l l o w i n g i d e n t i t y , t h a t w i l l b e u s e f u l l a t e r : F o r a n y c o m p l e x M x M m a t r i x a:  | z > = e x p ^ z ^ e ' V - l)z}  exp{aVa/^}  \ ez >  (2.6)  a  P r o o f : first e° = y% a  , < 7 a / f t  ( a z / f t ) = e ^ (a*z/h)e- ^ T  a  a/h  a  (l + o + a /2l + . . .)ze ' ' 2  x e  a/h  a aa h  a ] ( T a / n  = (aVz/ft)e  a , < 7  S o , u s i n g (2.3) CO e  a<*a/H |  z  >  =  e  e  oo  = -* */ ,  1  -z<z/2h £ _ L . W » ( „=o Vn\  l  t ^ ) » |Q >  i  y* - 4 = ( a V * / 7 i ) =oVn!  2h  c  f  n  |0 >  n  =  w h i c h g i v e s (2.6).  -* */2» (e"*) (e'*)/2» | ,  e  t  c  >  °/  f t  Chapter 2. Review of Coherent States and Path Integrals  2.2  6  Spin Coherent States  I n t h e s a m e s p i r i t as t h e h a r m o n i c o s c i l l a t o r c o h e r e n t s t a t e s , a l l t h r e e c o m p o n e n t s o f t h e s p i n J (J , J , J ) c a n n o t h a v e d e f i n i t e eigenvalues f o r a g i v e n s t a t e o f t h e s y s t e m . B u t x  y  z  i t is p o s s i b l e t o define a s t a t e | n >  ( ( n ) = 1) s u c h t h a t : 2  < ft I J I ft >= stin f o r a s p i n o p e r a t o r J, o f s p i n s.  T o find s u c h a s t a t e | n > , l e t u s define a ' g r o u n d s t a t e ' | 0 >=| s,s > , a n e i g e n s t a t e of projection m = 5 of J  z  : J  z  | 0 >= s% | 0 > . T h e n < 0 | J | 0 >= s%k, w h e r e  k = (0,0,1), a n d w e o b t a i n a c o h e r e n t s t a t e | ft > = | 9, <j> > , u s i n g s p h e r i c a l c o o r d i n a t e s , by performing the appropriate rotations: 16,4> >=  e  -W -*"»/* e  |o>  •  (2.7)  F o r s p i n 1/2 ( w h e r e B a r e t h e P a u l i m a t r i c e s ) : /  cos(0/2)  \  IM>.=i/2=  for  B±*  \ e«'*sin(0/2) )  ( e " ' * cos(0/2) \ for  V  s i n  W ) 2  6± 0  (2.8)  7  A n d we find  <6,<f>\ 0', $ >  1 / 2  = cos(0/2) cos(0'/2) + sin(0/2) sin(0''/2)e-^*-^  = cos(0/2) cos(fi72)e*^-*'> + sin(0/2) s i n ( ^ / 2 )  for  f o r 9 ± TT  9^0  with < 0, (f> | d | 9, <j> >i/2= n = i s i n 9 cos <f> + j s i n 9 s i n  +fccos 9  7  Chapter 2. Review of Coherent States and Path Integrals  T h e t w o different r e p r e s e n t a t i o n s are n e c e s s a r y b e c a u s e w e n e e d t w o c o o r d i n a t e p a t c h e s t o c o o r d i n a t i z e s p h e r e . T h e p h a s e f a c t o r e"^ b e t w e e n these t w o p a t c h e s is p u r e l y t o p o l o g i c a l . It w i l l h o t c h a n g e t h e p h y s i c s i n g e n e r a l . F o r a r b i t r a r y s p i n s: W e c a n use t h e s p i n o r r e p r e s e n t a t i o n for a s p i n : 1  V>(-M)  X  s+cr  (2s)l \{8  8—o  + a)\{8-<T)\  U / 2  So (2*)!  0  (2s)! \(s  ;  s +  s + < T  for  7  rr  - '( ^cos >  + a ) ! ( s - <r)\  i(a-c)<j> c o.„s+a s***(0/2) sin*"' (0/2)  (0/2)sin s  /  :  < T  (0/2)  9 ± TT  for  0 ^ 0  \  (2.9)  |M>.= \  '•  )  T h e p h a s e f a c t o r b e t w e e n these t w o p a t c h e s is n o w e *^, w h e r e w e c a n r e c o g n i z e 2s h a s 2s  a w i n d i n g n u m b e r . It c a n b e c h e c k e d t h a t < 0, <> / | J \ 9,(f> > =  snfi a n d a l s o  8  <9,<f>\9',<f>' >.= W e also f i n d < 0, <j> | ( J )  2  2s  [< 9,<f>\9',<{>'>  | 0, <f> >„ - ( <  (2.10)  1/2  9,<f> \ J \ 9,<f> >,)  2  = h [s(s + 1) 2  the m i n i m u m value possible. F i n a l l y , f r o m (2.9), w e c a n e a s i l y f i n d t h e c o m p l e t e n e s s r e l a t i o n : fir r2ir  rr\e,4»<w\.  d<t>sm(9)d9 _ ( 4  .x  This might differfromother authors by a unitary transformation.  2  (2.11)  s ] = 2  s%  8  Chapter 2. Review of Coherent States and Path Integrals  2.3  Propagator, Path Integral and Partition Function  In q u a n t u m mechanics, all the information f o r the evolution of a system can b e stored in the propagator (transition amplitude) between an initial state |  > a n d a final state  | qj > a t a t i m e t — tf — ti l a t e r , g i v e n b y :  K^jit) =<  | e '* | q, > im  Q i  where H is the H a m i l t o n i a n of the system. T h i s allows F e y n m a n , b y using the completeness relation / ^ ? / ^^e"  , p ?  K^ (t) f  /  f t  |p><p|9><<?|=  | p >< q |, t o w r i t e a p a t h i n t e g r a l ( w i t h q = qi a n d q^+\ = 9 / ) : 0  = J dp n  I Po > e - '  < * I  0  P 0 9 l / f t  < ?i | e W * 7  |  p  i  >  e  -*™/»...  iHt  . . . <q \ N  = J T O O / ^  0  e ("+D I p  >< p  h  N  | [ ^ e x p { - i g b ; f e  +  N  |q >  - ft) "  i  f  (^)#(P;,ft)]}  (2.12)  = jdp DpDqex {-y\p -H(p, ))dt} 0  where  ?  H(p,q) = ^ [ ^ f f i ^ + ^ ^ ^ l  q  .  q  is the classical H a m i l t o n i a n , w i t h b o u n d a r y c o n d i -  t i o n s g(f,-) = 9,-, q(t/) = q/. F u r t h e r m o r e i f the H a m i l t o n i a n is o f the f o r m H = ^  + V(q) t h e n we c a n p e r f o r m  the p integrations (being a Gaussian), giving:  =M  j Dqexp{-jJ*'L(q,q)dt}  w h e r e M is a n i n f i n i t e c o n s t a n t , a n d L i s t h e c l a s s i c a l L a g r a n g i a n .  (2.13)  9  Chapter 2. Review of Coherent States and Path Integrals  T h i s n e w r e p r e s e n t a t i o n h a s s o m e d i f f i c u l t i e s . F i r s t o f a l l , t h e r e is t h i s i n f i n i t e c o n s t a n t , <*/N -f 1, t h a t c a n n o t e v e n b e a b s o r b e d i n t h e m e a s u r e o f Dq. F o r e x a m p l e , i f V = 0 we c a n perform the integrations:  w h i c h gives:  T h i s c l e a r l y s h o w s t h e fine t u n e d c a n c e l l a t i o n o f y/N + 1 i n t h i s ( s i m p l e ) case.  For a  m o r e c o m p l i c a t e d s y s t e m , i t c o u l d b e e x p e c t e d t o step t h r o u g h s o m e d i v e r g e n c e s . A l s o , t h e Dq  (= Flit V^'^Tftt ^*.) 1  > c o n t a i n i n g t h e TV f a c t o r , i n d i c a t e s t h e d i f f i c u l t i e s  m e a s u r e  t h a t m i g h t a p p e a r b y p e r f o r m i n g t h e N —> oo l i m i t . I n s t a t i s t i c a l m e c h a n i c s , at a t e m p e r a t u r e T, t h e i n f o r m a t i o n is s t o r e d , i n s t e a d , i n the partition function: Z[/3] = t r ( e - ' » ) ,  8 = ±  S i n c e t h e t r a c e c a n b e r e p r e s e n t e d b y / < q | ( ) | q > dq, o r m o r e g e n e r a l l y / e~ l ipq  q\(  h  <  )\P> ^jt, so t r ( l ) — n u m b e r o f states a v a i l a b l e , w e f i n d :  Z[B] = J  K^iB^dqt  w h i c h s h o w s t h a t t h e p a r t i t i o n f u n c t i o n is t h e i n t e g r a t i o n o v e r t h e i n i t i a l s t a t e o f t h e p r o p a g a t o r t h a t goes a r o u n d a l o o p ( i n i t i a l = f i n a l ) f o r a ' t i m e ' t = i(3fi. So i n terms of the p a t h integral formalism, we can find of u n i t y ( / | p >< p | q >< q | g ) i n e^ .  b y i n s e r t i n g TV r e s o l u t i o n  B y using the F e y n m a n p a t h integral, w e  H  find: N  dp dq k  N  k  fl i 3  / DpDqexp{-  j[  (-pq + H(p, q))dr)  10  Chapter 2. Review of Coherent States and Path Integrals  where now q = limyvr^ooC ^^ ) is a 'temperature derivative', and q(0) = q(/3), a loop in 2  1  the q space for a 'time' /?. Since the variable q, in the propagator (2.13), goes from an initial to a final position, without any condition on the velocity (q) at these boundary points, the same indeterminacy will have to be applied for Z[/3]. The variable q will leave the initial point q(0), goes around a loop and comes back to q((3) = q(0), but it does not mean that the curve will be smooth at the connecting point q(0) : q(0) ^ q{P)- Note that the phase space integration is more complete in this path integral, there is no extra dp integration as in (2.12). 0  Applying this for our example, a free particle (H =  we find from (2.14):  or as we would do in statistical mechanics =  f -p dpdq e  J  as expect.  H  2xh  =  H j E Z r ^ J  Chapter 3  Coherent State Path Integral for the Harmonic Oscillator  3.1  Exact Result for the Discretisation  In this section, we will derive an exact expression for the partition function as a path integral, using resolution of unity and identities of the last chapter for coherent states, and write down the result in a suitable form that will be used to study the continuum limit of this new coherent state path integral. We want to evaluate: Z[0\ = trie-* *) where H = u{o)a + hM/2) 31  -/n(^)<*i«^'"i*>...<*i«^H«.> with (2.6) we find m  =  e  Jg  -»*W  [ TT  _ -pwhM/2 p  - 4 ? ( i - ^ )  e  (k k \ dz  dz  f ^ r  J t}i\( * ) ) 2  in  <  Uk , _  z  Z  l  | - * * / % > ... c  2pwh/N\  (  ti  M  Z  }  kk z  2%  „-0wh/N l+l k+l , „-Pu>h/N k k+ln Z  Z  Z  2%  Also by changing z  k  m  =  —• Z f c e *  e  P«™,2  } w n / W  f  ,  +  h  e  Z  n  we find  1  g (*2*L) - i c  (3.16)  Jfc=i \\2Tnn) J  Note that the condition of periodicity implies the periodicity of the position and momentum as defined by (2.2) and (2.4), in contrast with the usual Feynman path integral. 1  11  12  Chapter 3. Coherent State Path Integral for the Harmonic Oscillator  T h i s gives u s a n e x a c t a c t i o n , a t t h e d i s c r e t e l e v e l , f o r t h i s p a t h i n t e g r a l :  = J2[e^ zlz fc=i  (3.17)  - zlz ]  N  S  k  k+l  These t w o formulas c a n b e checked independently, using the determinants solved i n A p p e n d i x A for a matrix of the form A  o r A'  = 6 - e~ l 8i^j pu% N  itj  itj  =e ^ 6 0u,  itj  N  itj  -  6  i+ltj  ,  because t h e p a t h integral is a G a u s s i a n : -PwhM/2  e  e  [det(A)]M 1  -pw*M/2  (\- -^) V  3.2  Continuum  ^whM/2 ( ^-l)  M  M  e  -0w1i(m +...+m +M/2) _ ( -0H\  p  -[2 inh(^V2)r- „.i„ m  0whM/2  [det(A'))  M  e  _  S  e  l  M  f  £  "  = 1  (o ,n\  p  t  r  (  e  '  (  3  '  1  8  )  Limit  W h e n we are seeking a c o n t i n u u m l i m i t of a p a t h integral, w e want t o keep terms i n t h e s u m m a t i o n o f t h e 3/N o r d e r ( e x c l u d i n g t h e e x t r a 0/N t e r m f o r e a c h ' t i m e ' d e r i v a t i v e ) , so t h a t w e c a n a p p r o x i m a t e t h e s u m m a t i o n b y a n i n t e g r a l JQ dr. F o r t h e p a t h i n t e g r a l s (3.16) o f t h e l a s t s e c t i o n , t h i s m e a n s t h e f o l l o w i n g :  Z[8]  B u t is z\(z — k  z i) k+  0whM/2  «e  JDzdzie-*^i k [z  Zk  %" ><  Zk+l)+  hz  Zk]  r e a l l y b e c o m i n g —z^'zdrl W e c a n e a s i l y c h e c k t h a t  ( f zt'zdTJt = f z^zdr = z*z |J - f z^'zdr = - ( f z^zdr) Jo Jo Jo Jo so t h i s i s p u r e l y i m a g i n a r y . B u t f o r t h e d i s c r e t e c o n t e r p a r t N  N  4( k z  k=l  -  Zk+i)  =  1  k=l  2  1 -  Zk+i)  -  1  ~( k z  -  Zk+iVzk  + •« I  - Zk+i I ] 2  (3.20)  Chapter 3. Coherent State Path Integral for the Harmonic Oscillator  13  c l e a r l y i n d i c a t e s t h e p r e s e n c e o f a r e a l t e r m t h a t is m i s s i n g i n t h e c o n t i n u u m l i m i t ! E v e n i f t h i s r e a l c o n t r i b u t i o n w i l l a p p e a r o n l y t o t h e j3/N o r d e r , i t c o u l d b e r e l e v a n t f o r a convergence of t h e p a t h integral. For example Px/N) P/N  nitLi(l + Px/N)  —• e  P x  as N —• o o , b u t d o i n g t h e a p p r o x i m a t i o n (1 +  « 1 w o u l d give 1 instead. F o r t h e p a t h integral each integration w i l l contain a  term that will bring a finite modification at the end. T o f i n d t h e c o n t i n u u m l i m i t w i t h t h i s e x t r a r e a l t e r m , w e j u s t u s e (3.20) i n t h e p a t h  i n t e g r a l (3.16) w h i c h gives t h e f o l l o w i n g l i m i t :  Z[I3] « where  e  0  u  h  M  '  2  J  DzDz*e-*tf(-  z H + w h z i z +  W  (3.21)  2 ) d t  e = p/N . B e c a u s e o f t h e p r e s e n c e o f t h e N in e we d o n o t r e a l l y e x p e c t a n y  d e p e n d e n c e o n e after t h e e v a l u a t i o n o f t h e p a t h i n t e g r a l , b u t i t s h o u l d h e l p t o ' s m o o t h out' the integrations. Because w e w i l l b e looking for corrections t o t h e order e i n t h e p a t h integral, like i n ( 3 . 2 1 ) , i t w i l l b e i m p o r t a n t t o k e e p t r a c k a l s o o f t h e b o u n d a r y c o n d i t i o n s a t 0 a n d /?. T o a p p r o x i m a t e a s u m m a t i o n b y a n i n t e g r a l , i t w i l l b e u s e f u l t o use t h e f o l l o w i n g r e l a t i o n , v a l i d i f / ( r ) i s s m o o t h f r o m 0 t o /?:  /* = /((*  " be) ^ z  k = 1 . . . . N , J2  = f  jfe=i  MdT  + 0(e ) 3  (3.22)  J o  In o u r case, w e w i l l have t o consider t h e variable z , that w e w o u l d like t o represent b y k  a c o n t i n u o u s c u r v e z{r). that t h e value of  z./v+i  T h i s r e p r e s e n t a t i o n a p p e a r s o n t h e figure 3 . 1 . It i s c l e a r , t h e n ,  = z((N  + | ) e ) ^ z\ = z(^t), b e c a u s e Z ( T ) is n o t s m o o t h a t r = 0  a n d T = p, b u t zpj i i s i n s t e a d a n a l y t i c a l l y c o n t i n u e d a w a y . +  If w e w r i t e Zk+i = *k + ez  k  + — z  k  + ...  Chapter 3.  Coherent State Path Integral for the Harmonic Oscillator  F i g u r e 3.1: Discrete a n d continuous representation o f the z variable a n d use t h e i d e n t i t y ( 3 . 2 2 ) , we c a n a p p r o x i m a t e t h e a c t i o n (3.17) b y :  S =  f  [—Z^Z  — \z*Z  Jo  + OjtlZ^Z + ^•U h Z*z]dT 2  + zjv(zjv=l — Ztf)  2  2  2  T h e Z I J ( Z N I — zjy) c o r r e c t i o n c o m e s f r o m t h e f a c t t h a t t h e i n t e g r a l needs a s m o u t h =  f u n c t i o n , t h a t does n o t c o n s i d e r  Z J V + I  ZN+1  = z\,  this has t o be corrected b y ' h a n d ' :  = z(B) + Zz(j)  * i = *(0) + f * ( 0 ) z  T h u s to the e order, we  find  N  =  z{0)  +  0(e) =  +  0(e ) 2  + 0(e ) 2  z(B) +  0(e)  Chapter 3. Coherent State Path Integral for the Harmonic Oscillator  15  T h i s a l l o w s u s t o r e w r i t e t h e a c t i o n as  S = as w e f o u n d i n (3.21).  J  [-2+2 +  t -\z\ + uhz z 2  j  e  + -w a z z]dr 2  2  +  These boundary contributions could be very important,  as i t  w i l l b e seen i n t h e n e x t s e c t i o n . T h e ^u> h z^z t e r m w i l l b e d r o p e d s i n c e i t is o b v i o u s l y 2  2  n e g l i g a b l e c o m p a r e d t o utiz^z. A c t u a l l y , i f w e c o n s i d e r t h e a p p r o x i m a t i o n /(—z^z+uhz^z-rcte \ z \ )dr t o t h e a c t i o n , 2  w h e r e a is a p a r a m e t e r t h a t w o u l d c o n n e c t t h e p a t h i n t e g r a l (3.19) a t a = 0 t o (3.21) a t a = 1 continuously, then the discrete p a t h integral w o u l d contain: N j 2  9  ~  _  k=\  ( k ~ Zk+t)*Zk z  r~a\z -  z  k  z  |] 2  k+1  1  i n s t e a d o f ( 3 . 2 0 ) , w h e r e i t is i m p o r t a n t included i n the calculation.  J  t o check that t h e b o u n d a r y terms h a s been  T h e evaluation of t h e discrete determinant  c a n be done  e x a c t l y b y u s i n g t h e i d e n t i t y ( A . 8 0 ) o f a p p e n d i x A , a n d s i m p l y gives  determinant ^ = 1  (S±I)"( ** C  1) +  ( - ^ * - 1) c  T h i s c l e a r l y i n d i c a t e s t h e effect o f t h e e t e r m . W h e n a = 1, t h e d e t e r m i n a n t is e x a c t l y t h e e x p e c t e d o n e . ( e ^ * - 1) . B u t w h e n 0 < a < 1, t h e s e c o n d t e r m g r o w s , a n d o s c i l l a t e s M  i n s i g n as a f u n c t i o n o f N ! B u t | still negligable.  | is b i g g e r t h a n | ^=1 |, s o t h e s e c o n d t e r m i s  Furthermore we have t o renormalize for the infinite constant (^^•) . A r  B u t a t t h e e x t r e m e case a = 0 , c o r r e s p o n d i n g t o ( 3 . 1 9 ) , w e h a v e so e a c h t e r m h a s t h e s a m e f a c t o r (1/2) , N  2(cosh(/?u>/i) - 1) = [ 2 s i n h ( / 3 w / i / 2 ) ]  2  a n d t h e (—1)  N  (^f^-) = — i ^) 2  = 1/2  of the second t e r m produces  f o r N e v e n a n d 2 sinh(/?a;7j) f o r N o d d . N o t e t h a t  this u n p h y s i c a l oscillation w i l l always b e cancelled b y t a k i n g t h e average.  A l l o f this  Chapter 3.  Coherent State Path Integral for the Harmonic  Oscillator  16  indicates t h e subtlety of t h e p a t h integral (3.19), a n d t h e regularization i n t r o d u c e d b y the e term. I w o u l d also l i k e t o p o i n t o u t t h a t i f w e s t a r t w i t h t h e p a t h i n t e g r a l ( 3 . 1 5 ) , i n s t e a d of ( 3 . 1 6 ) , w e w o u l d o b t a i n t h e p a r t i t i o n f u n c t i o n (3.19) as a c o n t i n u u m l i m i t , w i t h o u t including the e corrections, b u t w i t h  e  ~  P  w  h  M  /  c o m e s b y d o i n g t h e a p p r o x i m a t i o n ujnz\zk+\  2  instead of  P w h M / 2 e  \  T h i s factor actually  « u%z\zk f o r t h e c o n t i n u u m l i m i t . S o  t h i s e x a m p l e i n d i c a t e s h o w s e n s i t i v e t h e d i s c r e t e s u m m a t i o n is t o t h e d i s c r e t e i n d e x (k = l , . . . , i V ) .  A s l i g h t m o d i f i c a t i o n (like k —• k + 1) m i g h t g r e a t l y affect t h e p a t h  integral.  Semiclassical A p p r o x i m a t i o n  3.3  T h e a c t i o n S = /(f ( — z i + whz^z + a | | z \ )dr i n t h e p a t h i n t e g r a l i s e v a l u a t e d a l o n g a T  2  c o n t i n u o u s c u r v e z ( r ) € C f r o m r = 0 t o T = /?, w i t h z ( 0 ) = z(/3), a n d a c l a s s i c a l e q u a t i o n of m o t i o n i s a c u r v e z ( r ) t h a t m i n i m i z e s t h e a c t i o n S t o 5" . T h e o t h e r t r a j e c t o r i e s z ( r ) c  w i l l g i v e a n a c t i o n S > S , t h u s c o n t r i b u t i n g less t o t h e p a t h i n t e g r a l . T h e s e m i c l a s s i c a l c  a p p r o x i m a t i o n i s t o c o n s i d e r o n l y t h e c l a s s i c a l s o l u t i o n s ( / dz e~ ) Sc  c  i n the path integral.  M o r e explicitly, let us w r i t e z = z + z where z is t h e classical s o l u t i o n , w i t h t h e c  c  b o u n d a r y c o n d i t i o n z ( 0 ) = z ( / ? ) = z ( 0 ) = z(/3) = z . S o z w i l l b e a c o m p l e x c u r v e w i t h c  c  0  z ( 0 ) = z(/3) = 0. T h e i n t e g r a t i o n w i l l b e s e p a r a t e d as  So t h e p a r t i t i o n f u n c t i o n is  where  17  Chapter 3. Coherent State Path Integral for the Harmonic Oscillator  = S(z ) + S(z) +./ z^(-z + u}%z - a-z )dr Jo z c  c  c  + / (z\-\-u%z\ -  Jo  c  z  a-z[)zdT  T h e s e c o n d l i n e is o b t a i n e d after i n t e g r a t i o n b y p a r t s a n d t h e u s e o f t h e b o u n d a r y c o n d i t i o n s o n z.  z\,  T h e last t w o integrals are zero b y t h e e q u a t i o n of m o t i o n of z a n d c  so  Z[8] = e^ '  J DzoDzle-fcW  hM 2  DzDz^-^  J  T h e s e m i c l a s s i c a l a p p r o x i m a t i o n is t o d r o p t h e 5 p a t h i n t e g r a l :  ZM  = e  '  Dz Dzle-^So(-^+^^+a^)dr  J  pwhM 2  0  where  —z + u>%z — ct^-z = 0 a n d z\ + Lofiz\ — a^z\ = 0 c  c  c  These two equations of m o t i o n correspond t o the Euler-Lagrange equation (obviously?) for a L a g r a n g i a n :  L = —z^'z  +  uftz^z  -f a | | i | . T h e f a c t t h a t t h e e q u a t i o n s o f m o t i o n 2  of z a n d z\ does n o t s e e m c o m p l e x c o n j u g a t e c o m e s f r o m t h e f a c t t h a t dr = —%dt, so r c  transforms h k e a complex n u m b e r , w h i c h explains the sign change i n z i n t h e equation c  of m o t i o n .  B u t r o r 8 m u s t s t i l l b e c o n s i d e r e d as r e a l , i t is p a n d q o f z t h a t c a n get  complexified i n the search for extremal of the action. B y integrating b y parts | i  | , and 2  c  u s i n g t h e z e q u a t i o n of m o t i o n , we c a n s i m p l i f y S t o c  c  5c = K m a | « J i ,  F o r a = 0 , w e t r i v i a l l y find S and there exists n o solution w i t h  e  |J  = 0. I n f a c t t h e e q u a t i o n o f m o t i o n b e c o m e s l i n e a r ,  c  z (0) = z (8)\ c  S o t h e s e m i c l a s s i c a l a p p r o x i m a t i o n is  c  n o t a p p l i c a b l e , o r a t t h e b e s t i t gives a c o n s t a n t ( / £ ) z o ^ ^ o ° ) - T h i s i n d i c a t e s , a g a i n , t h e e  difficulties of t h e p a t h i n t e g r a l (3.19).  e  F o r a = 1, w e find t w o i n d e p e n d e n t s o l u t i o n s ,  w1ir  and e  - 2 T  /  £  for  z , e~ c  f o r z\. B y c o n s i d e r i n g e v e r y s m a l l a n d t h e b o u n d a r y c o n d i t i o n , w e o b t a i n : z = z [(l c  0  - e-^)e"  2 T  /  £  +  e ^) w%  whT  and e  2 r  /  e  Chapter 3. Coherent State Path Integral for the Harmonic Oscillator  z\ = 4[(1 - - ^ * ) 2 ( r - « A + e  e  c  18  -«*r]  So 5  = 2 | z I e-^ sinh(/3u;ft/2) 2  C  / 2  0  and  Z [/3] = e ™ (3w  ac  /2  J  l f £ l _ - ^ h / 2 i h(/3 ,ft/2) 2e  (2irih)  B  n  M  ll  2e-/ -*/2 inh()9wR/2)  Me  J  s  w h i c h is t h e r i g h t a n s w e r , e x c e p t f o r a f a c t o r e^  .  whM  I n f a c t i t as b e e n s a i d t h a t a  s e m i c l a s s i c a l a p p r o x i m a t i o n f o r a n h a r m o n i c o s c i l l a t o r s h o u l d g i v e t h e r i g h t r e s u l t [1]. A c t u a l l y t h e c o n f u s i o n arises b y t h e a p p r o x i m a t i o n n o t e d i n t h e l a s t s e c t i o n , a c h a n g e of ufizlzk+i t o uhzlzk i n t h e p a t h i n t e g r a l (3.15) w i l l g i v e a n e x t r a f a c t o r e~^  whM  to the  partition function. These two approximations cancel each other t o give a n exact result! It i s o b v i o u s f r o m m y c a l c u l a t i o n t h a t t h e c o r r e c t s t a t e m e n t w o u l d b e :  w h e r e 2(0) = z(B) = 0 .  3.4  Regularization  So f a r , w e w e r e a b l e t o w r i t e d o w n a n e x a c t e x p r e s s i o n f o r t h e d i s c r e t i s a t i o n o f a p a t h i n t e g r a l , a n d e v a l u a t e t h e d e t e r m i n a n t e x a c t l y a f t e r w a r d s . I n m o r e c o m p l i c a t e d cases, i t w i l l be necessary t o evaluate a p a t h integral b y using some approximations, because the d e t e r m i n a n t w i l l n o l o n g e r b e e x a c t l y s o l v a b l e . I n f a c t , these a p p r o x i m a t i o n s w i l l u s u a l l y generate s o m e d i v e r g e n c e s t h a t w e w i l l h a v e t o r e g u l a r i z e b y u s i n g v a r i o u s t e c h n i q u e s . H e r e , I w i l l i n d i c a t e a g e n e r a l w a y t o r e g u l a r i z e these d i v e r g e n c e s a n d t h e u s e o f t h e c o n t o u r i n t e g r a l r e g u l a r i z a t i o n for t h e c o h e r e n t s t a t e p a t h i n t e g r a l o f t h i s c h a p t e r . I w i l l also use t h e e x a c t d e t e r m i n a n t f o u n d e a r l i e r t o f i n d t h e l i m i t a t i o n s o f t h e s e r e g u l a r i z a t i o n s .  Chapter 3. Coherent State Path Integral for the Harmonic Oscillator  19  T h e c o n t i n u o u s a p p r o x i m a t i o n is d e r i v e d i n t h e f o l l o w i n g w a y ; s i n c e t h e p a t h i n t e g r a l uses p e r i o d i c f u n c t i o n s , z ( 0 ) = z(/3), w e c a n use a set o f e i g e n f u n c t i o n s \Pfc(r) =  2 7 r t k T e  ^,  as i n t h e a p p e n d i x A f o r t h e d i s c r e t e case. F o r a g e n e r a l p a t h i n t e g r a l :  / DzDzU-Uo'**»*+°&)' J  *=  L det (a + bf  (3.23)  M  t+  c$)  we obtain for the determinant:  n  det.=  +  k=-N/2  + cRR ] 2  P  (3-24)  P  w h e r e fi/N a p p e a r s b e c a u s e a n i n t e g r a t i o n o f T V slices w i l l g i v e a f a c t o r A T =fi/Ni n f r o n t o f t h e L a g r a n g i a n . I t is a l s o n e c e s s a r y f o r k e e p i n g t h e r i g h t u n i t s .  T h e product  (3.24) is t h e e x p r e s s i o n t h a t needs t o b e r e g u l a r i z e d . B u t before a n a l y s i n g these p r o d u c t s , let m e start f r o m t h e exact f o r m u l a (A.80) f o r t h e p r o d u c t d e r i v e d i n a p p e n d i x A f o r t h e p a t h i n t e g r a l (3.16): w0hM/2  e  ^ det(x) =  ]  TdetF'  =  fl * = fl ( * A  k=i  k=i  e  /N  ^ - ^ e  )  k/N  =*e  (3-25)  1  w h e r e x = flun. F r o m ( 3 . 2 5 ) , w e c a n c o n t e m p l a t e t h e fine t u n i n g o f t h e p r o d u c t o f a l l t h e e i g e n v a l u e s . O n t h e c o m p l e x p l a n e t h e s e eigenvalues f o r m a c i r c l e o f r a d i u s 1, c e n t e r e d a t e / , e q u a l l y x  spaced. T h e i r modulus vary f r o m e l x  N  N  — 1 « x/N < 1 f o r 0 « k <C N, t o e l x  N  +1« 2  for k « N/2. S o a h u g e n u m b e r o f c a n c e l l a t i o n s m u s t b e i n v o l v e d , i n t h i s p r o d u c t , t o give simply e — 1 ! x  W e w i l l n e e d also t h e T(x) f u n c t i o n d e f i n e d b y :  r(x) = -^ln[det(x)] dx  Chapter 3.  Coherent State Path Integral for the Harmonic  JV  E  e  /N  N{eX  x/N  N  _ jriklN)  20  Oscillator  1  = E  N  Q _ 2 *=£ c  B  ( - ) 3  )  26  e* - 1 t h a t can b e evaluated b y contour integral w i t h t h e f u n c t i o n | c o t ( z / 2 ) , w h i c h has poles at z = I'KXI ( n i n t e g e r ) w i t h r e s i d u e 1. T h i s a l l o w s u s t o w r i t e :  r  N  r(x)  = £  \ cot(z/2)  I  feiMfc)  dz  7Y(l_c T<r)27r» 4  w h e r e C(fc) is a n o r i e n t e d l o o p a r o u n d 27rfc. T h e a n a l y s i s o f t h e i n t e g r a n d f(z) s h o w s t h a t 0  i t c o n t a i n s o t h e r s p o l e s a t z = — i x + 2irnN. F u r t h e r m o r e , f(z) b e i n g a p e r i o d i c f u n c t i o n (f(z + 2nN) = f(z)), 2nN  w e c a n t h i n k o f f(z) as a f u n c t i o n o n a c y l i n d e r o f c i r c u m f e r e n c e  oriented along t h e imaginary axis. A n integration along a n imaginary line u p a n d  t h e n d o w n b y a n o t h e r i m a g i n a r y l i n e s h i f t e d b y 2nN w i l l c a n c e l e a c h o t h e r . c h e c k e d t h a t f(z)  —> 0 as Im(z) —> —oo a n d f(z) —• ^  It c a n b e  as Im(z) —• o o . S o b y u s i n g a  c o n t o u r i n t e g r a l , as s h o w n i n f i g u r e 3 . 2 , w e o b t a i n  = - 1^ ~* < ) _ l dR  {  '  Jo  2N  = \ ~5  Z  2iri  c o t  \*< I ) Z  J-* N(l-e^)2TTi  2  o 07)  dz  ( K  '  (-»*/2) =  F o r a c o n t i n u u m l i m i t a p p r o x i m a t i o n , w e have t o find some a p p r o x i m a t i o n s t o t h e p r o d u c t (3.25) t h a t w i l l c o n t a i n a l l t h e p h y s i c a l p r o p e r t i e s o f o u r m o d e l . B u t t h e f i n e t u n i n g o f t h i s p r o d u c t s h o w s t h a t w e c a n e x p e c t a l o t o f d i v e r g e n c e s . T h u s , i n these d i v e r g e n t cases, i t i s n e c e s s a r y t o i n t r o d u c e a r e g u l a r i z a t i o n t h a t w i l l t h r o w a w a y t h e divergent p a r t , b u t keep t h e physics of the m o d e l , o r i n other w o r d s , t h e divergent t e r m should not depend on any physical parameter. F o r o u r p r o b l e m , t h e eigenvalues A * become independent of x f o r large values of k (Afc w 2 + 0(x/N))  so t h e exact expression e  2mk  ^  N  for large k should n o t b e necessary,  Chapter 3.  Coherent State Path Integral for the Harmonic Oscillator  21  Im(z-)  A  Im(z) = oo  — «  C(0)  CO)  e—e-  C(N-1)  -©-  2«(N-1)  •>Re(z) 2«N  z,.2»N  Im(z) = - oo  Figure 3.2: Contour of integration for the T(x) function of the discrete determinant and then we should be able to use the approximation: \2Kik\  e  x  f2nik\  { — r  p  i  1  +  (2mk\  m  ( — j  that I will call the m order approximation, and this should produce some meaningful th  results after regularization. In fact what is going on is the fact that the high frequency modes (large k) are not physically relevent in the path integral, and this enables us to keep only the continuous functions z(r) in the path integral. The higher the approximation is, the better the 'discontinuous' (or fast oscillation) curves will be included properly in the path integral. If the discontinuous curves would contribute as much as the continous ones (or even the classical solutions) then there would be no continuum limit at all! If we use the 1 and 2 st  nd  order approximations to the product (3.25), and use an  odd number of steps (time slices), 27V + 1, so that we have a symmetric product, from  Chapter 3. Coherent State Path Integral for the Harmonic Oscillator  22  k = —N t o N, w e o b t a i n : T  N  det (*) =  = 2:  m  9-n-ik  +  2  (3-29)  I n f a c t , t h e a p p r o x i m a t i o n s (3.28) a n d (3.29) c o r r e s p o n d e x a c t l y t o t h e p a t h i n t e g r a l (3.19) a n d (3.21) r e s p e c t i v e l y , w i t h t h e a p p r o x i m a t i o n (3.23). S o t h i s a p p r o x i m a t i o n f o r t h e d e t e r m i n a n t has s o m e j u s t i f i c a t i o n n o w . L e t us f i r s t s t u d y (3.28). T h e i d e a is t o e v a l u a t e t h e p r o d u c t (3.28) f o r a g i v e n i V , a n d t h e n d i v i d e t h e r e s u l t b y t h e s a m e p r o d u c t , a n d s a m e TV, w i t h x — 0. T h i s d i v i s o r does n o t d e p e n d o n x a t a l l , so i t w i l l n o t c h a n g e a n y t h i n g . A n d s i n c e t h e p r o d u c t b e c o m e s i n d e p e n d e n t o f x f o r l a r g e k, t h i s d i v i s o r c a n c e l s e x a c t l y t h e d i v e r g e n t p a r t t h a t w e w a n t to t h r o w away. M o r e precisely:  k=-N  6  +  2N  1  ~ 2/V+l  J  =  2/Y + l  1}}^2N + V  = wr,U!^W ^ +{  +  (  27V +  r]  r  J  (3  '  30)  S i n c e d e t i ( 0 ) = 0 , w e c a n n o t d i v i d e d e t i ( x ) b y d e t i ( 0 ) , b u t t h e r e l e v a n t f a c t o r i n (3.30) t h a t needs t o b e d i v i d e d i s : ^  T  1  "  i  l t  =  1  2*k  2 j V + l  (2K)  2N  2  J  ~ (2N +  ~ V  }  a n d f u r t h e r m o r e , see [13],  l ™ A h -L(  X  W  sinh  (/) x  2  So  Jim ( i ) » « d * , ( , )  = ^  _  =  2sinh( /2) . I  DET ) i(x  (3.31)  Chapter 3. Coherent State Path Integral for the Harmonic Oscillator  W h e r e DET\(x)  23  is n o w t h e r e g u l a r i z e d d e t e r m i n a n t f o r t h e first o r d e r a p p r o x i m a t i o n , o r  for t h e p a t h i n t e g r a l (3.19). W e see t h a t d e t ( x ) = e l DET {x), x  2  l  right, except f o r t h e s m a l l factor e .  so DET (x) x  is a l m o s t  N o t i c e , a g a i n , t h e u n u s u a l coincidence of this  x/2  f a c t o r a p p e a r i n g h e r e a n d i n f r o n t o f t h e p a t h i n t e g r a l (3.19). If w e w e r e t o forget a b o u t this factor i n (3.19), we w o u l d t h i n k that we have the exact p a r t i t i o n f u n c t i o n . Since t h e divergence is a simple factor independent of x , t h e f u n c t i o n (3.26), T ( x ) , for t h e a p p r o x i m a t i o n (3.28), s h o u l d n o t c o n t a i n a n y divergence at a l l ! 1  N  r (x)=  £  x  *  =  *^ "™" ^^^^  \ cot(z/2) dz  t  t- ~  Uot(z/2)dz  N  f c o t U / 2 j dz  _t Jz=-ix  x — iz  l  y  k=N+i/  [  \ *, = - o o  2ni  x  ^-2^  So hm r (x) = x  N^oo as g i v e n b y DET\(x)  1 V  '  2e - 1 x  i n (3.31). N o t e a l s o t h a t t h e i n t e g r a n d o f t h e c o n t o u r i n t e g r a l i s  n o l o n g e r p e r i o d i c (f(z + 2TTN) ^ f(z)), w h i c h i n d i c a t e s t h a t w e n e e d t h e f u l l R i e m a n n sphere for t h e contour integral. T h e a p p r o x i m a t i o n , t h e n , changes t h e topology of t h e d o m a i n o f i n t e g r a t i o n t h a t is n e e d e d f o r t h e i n t e g r a t i o n ( t h i s w i l l b e t r u e f o r a n y o r d e r ) . N o w w e c a n g o f u r t h e r , a n d a n a l y s e t h e p r o d u c t (3.29). B y u s i n g t h e s a m e m e t h o d t h a t w e u s e d f o r (3.30), w e c a n w r i t e :  /  / x  1  27rJfc 2  l  2  "  N  2  *  RI  N(N + 2x),  ,  x  T h e p r o d u c t (3.33) h a s t h e s a m e d i f f i c u l t y o f (3.30) t h a t d e t ( x ) —* 0 f o r x — 0 . B u t 2  even worse, the last two products do n o t converge t o w e l l k n o w n functions because terms  4  24  Chapter 3. Coherent State Path Integral for the Harmonic Oscillator  0.1  0.5.  1  2  5  10  2  0.1032846  0.5926451  1.434417  4.451911  59.72026  3615.019  det(x)  0.1032875  0.5926791  1.434225  4.451257  59.72432  3615.389  X  DET {x) e-°-  1 8 0 7 l  10000  5000  1000  N  T a b l e 3.1: S e c o n d o r d e r p r o d u c t r e g u l a r i z a t i o n l i k e (jTJi) o r ^ [ * f f i w o u l d n o t decrease t o z e r o as k i n c r e a s e t o N: t  N  N S o t h e o n l y w a y ( b y t h i s m e t h o d ) t o e v a l u a t e (3.29) i s t o a p p l y t h e p r o c e d u r e d i r e c t l y without looking for simple solutions: TV  = lim x  DET (x) 2  =  "  hm x N-*oo  k  2  [2TT(2N + l)k]  + [2TT P] 2  2  A n u m e r i c a l s t u d y o f (3.34) i n d i c a t e s t h a t DET (x)  DET (x) 2  = e  -  7  1  S i n c e DET\(x)  —e  - x  2TT P] 2  2  ^  2  is i n b e t w e e n (3.31), DETi(x),  2  (3.25), t h e exact result.  2 * * * 7 ( 2 t f + 1)]  I-2«* +  [27r(2JV + l)k) + [x(2N + 1) +  1  + 1)]  2  2  I I f-\ =  JJ te-jwo  [x - 2nk + 2Tr k /(2N  -  3  4  ^  and  / det(x), a simple exponential fit for 2  d e t ( x ) s h o w s a v e r y a c c u r a t e r e s u l t f o r 7 = 0.1807 ± 0.001 . M y n u m e r -  i c a l s t u d y is s u m m a r i s e d i n t h e t a b l e 3 . 1 , w h e r e DET (x) 2  is e v a l u a t e d w i t h t h e e q u a t i o n  (3.34) f o r t h e i n d i c a t e d v a l u e o f N.  A s t u d y o f (3.33) w i t h t h e use o f T(x) g i v e s : N  r (x)= 2  £ frW x - 2wik + 2n k /(2N 2  fc  2  + 1)  1 \Jztt-ix  Jzts4iN/  X—  iz +  2(2iV+l)  2iti  fc=—00  du l  i  i  2e -l x  l  +  l  -  2  j  j  ,  ,  ,  \ J N  x-2-rnu-r  TO" + 0(1/N) 2N+1  Chapter 3. Coherent State Path Integral for the Harmonic Oscillator  25  then l i m T (x) =  —  2  S o DET2(x) = e  7  1  7  d e t ( x ) as e x p e c t e d , a n d 7 i s g i v e n b y :  l i m 2Re N-400  =  ( \  1_  du [°°__d  2ir JN u(i — 2N+1-  h m Re f - i l n ( — ^ —  V  /v-00  1  X  X  2N + 1  + - ) \%) u V ;l  7  2  = - a r c t a n ( - ) « 0.18045 7T  TT  as f o u n d e a r l i e r ! If w e w e r e t o c h e c k f o r h i g h e r o r d e r a p p r o x i m a t i o n s , w e w o u l d get b e t t e r  result;  a c t u a l l y i t is m o r e l i k e l y t h a t w e w o u l d find a 7 c o r r e c t i o n a t e a c h o r d e r , b u t w i t h 7 becoming smaller. N o t i c e t h a t t h e p r o d u c t (3.29), w i t h k r u n n i n g f r o m —00 t o 00 i n s t e a d , w o u l d g i v e e x a c t l y t h e r i g h t a n s w e r ( t h e r e w o u l d b e n o 7 c o r r e c t i o n ) . It is v e r y u n l i k e l y , h o w e v e r , t h a t a t a n y o r d e r o f t h e a p p r o x i m a t i o n w e w o u l d o b t a i n a n e x a c t r e s u l t ( l o o k i n g a t (3.25)). T h i s s h o w s t h e i m p o r t a n c e o f k e e p i n g k f r o m — N t o N, f o r 2N +1 s t e p s , i n t h e p r o d u c t . T h e r e has been some study of these p r o d u c t s b y using a R i e m a n n z e t a f u n c t i o n for r e g u l a r i z i n g s o m e i n f i n i t e p r o d u c t s , s u c h as I l ^ - o o ^>  these f o r m u l a s n e e d t h e  m o d i f i c a t i o n j u s t n o t e d , t h a t k m u s t r u n f r o m —00 t o 00 r i g h t a t t h e b e g i n i n g [2,3]. T h i s w o u l d g e n e r a l l y l e a d t o s o m e e r r o r s , as a l r e a d y p o i n t e d o u t here ( t h i s h a s b e e n f o u n d also b y [2]). S o I w i l l n o t e l a b o r a t e o n t h e use o f t h e R i e m a n n Z e t a f u n c t i o n f o r t h e rest of m y w o r k . T o find i f i t is a p p l i c a b l e , w e j u s t l o o k i f t h e p r o d u c t s c o n t a i n t h e v a r i a b l e N, t h a t c o u l d s p o i l i t s c o n v e r g e n c e . I n t h e s e cases, a n u m e r i c a l s t u d y m i g h t gives s o m e a d d i t i o n a l c o r r e c t i o n s t o these p a t h i n t e g r a l s .  Chapter 4  Coherent State Path Integral for Spin  4.1  Discretisation with Spin Coherent States  In t h e last chapter, w e studied t h e properties of t h e h a r m o n i c oscillator coherent states using a n h a r m o n i c oscillator H a m i l t o n i a n . W e where able to obtain a n exact p a t h integral at t h e d i s c r e t e l e v e l . I n t h i s c h a p t e r , w e w o u l d h k e t o s t u d y t h e s p i n c o h e r e n t s t a t e s . I n hope o f finding exact solution, it would b e interesting to consider a simple H a m i l t o n i a n w i t h a spin operator.  M o r e precisely, we w i l l b e interested i n t h e p a r t i t i o n function for  a s p i n s p a r t i c l e i n a c o n s t a n t m a g n e t i c field B:  H = fiB-J  ,  ( J ) = % s(s + 1) 2  Z[B) = t r ( e " ^ ) =  2  t r ( e - ^ ^ )  W i t h t h e h e l p o f t h e s p i n c o h e r e n t s t a t e s (2.9) a n d (2.11) w e c a n w r i t e t h e f o l l o w i n g p a t h integral:  m  = /ft=|P < J  k=l  Ik, I e-°w  Il , M  ^  >.  \2a+l)  B y t h e p r o p e r t i e s o f t h e s p i n o r i a l r e p r e s e n t a t i o n , a n d t h e use o f (2.10) w e c a n t r a n s f o r m it into: J  k=l  \2s+l)  T h i s i s a b o u t as f a r as w e c a n g o a t t h e d i s c r e t e l e v e l . A c o n t i n u u m l i m i t s t u d y o f (4.35) w i l l follow, b u t first let us t r y to analyse t h e same s y s t e m b y using t h e h a r m o n i c oscillator coherent states. 26  Chapter 4. Coherent State Path Integral for Spin  4.2  27  T h e Schwinger-Boson Model  It is k n o w n i n q u a n t u m m e c h a n i c s t h a t a t w o d i m e n t i o n a l h a r m o n i c o s c i l l a t o r (see s e c t i o n 2.1) s t a t e | n 1 ? n 2 > c a n b e r e p r e s e n t e d i n s t e a d b y t w o n u m b e r s m , s s u c h t h a t n\ s + m,  =  ri2 — s — m, so m r u n f r o m — s t o s b y s t e p o f 1, a n d 2 s i s a p o s i t i v e i n t e g e r .  T h i s looks very m u c h like a spin representation, a n d i n fact t h e representation of a spin s p a r t i c l e b y u s i n g t h e t w o d i m e n t i o n a l h a r m o n i c o s c i l l a t o r is c a l l e d t h e S c h w i n g e r - B o s o n m o d e l , a n d it is explained i n details i n A p p e n d i x B , where we w i l l be using h a r m o n i c oscillator coherent statesin this section. B y using t h e S c h w i n g e r - B o s o n representation w e can evaluate t h e same partition function of last section:  w i t h t h e h e l p o f (2.6) w e o b t a i n  (4.36) T h e A i n t e g r a t i o n i n (4.36) is i n f a c t a c i r c l e i n t h e c o m p l e x p l a n e . A n d it w i l l b e u s e f u l , s o m e t i m e s , t o r e p r e s e n t t h i s c i r c l e w i t h a g i v e n r a d i u s r, i n s t e a d o f r = 1 i n (4.36). T h e p a t h i n t e g r a l (4.36) is t h e n m o d i f i e d t o :  -i^/n(^)r^e {4| ,aM^ A XP  -  )+  [2s  r<A4e-«?S"V,]}  (4.37)  T h e A v a r i a b l e is i n f a c t a g a u g e p o t e n t i a l , t o see t h i s w e c a n v e r i f y t h a t (4.36) a n d (4.37) are i n v a r i a n t u n d e r t h e t r a n s f o r m a t i o n :  z  -> e z tak  k  k  ,  A -> X + a f c  k  h  - ai k+  (4.38)  Chapter 4.  28  Coherent State Path Integral for Spin  It is i m p o r t a n t t o r e a l i z e t h a t w e were a b l e t o o b t a i n , a t t h e d i s c r e t e l e v e l , a n e x a c t p a t h integral w i t h a discrete action: N  S= ^fisihXk k=i  + z\zk - e *zle-^ -*zk+1} iX  B  (4.39)  w h e r e c = 8/N as u s u a l .  4.3  Equivalence of the two Representations  T h e first q u e s t i o n t h a t o n e m a y ask i s : a r e t h e t w o l a s t r e p r e s e n t a t i o n s r e a l l y e q u i v a l e n t ? S i n c e (4.35) does n o t c o n t a i n a n y g a u g e v a r i a b l e , A , t h e first t h i n g w i l l b e t o i n t e g r a t e d i r e c t l y t h i s v a r i a b l e i n (4.36). T o d o s o , let us c a l l w = e * , t h e n , A  k  <"»> N o w we c a n represent z b y : k  z = r e' ,  k  1k  z VK  (4-41)  k  k  W h e r e z is r e s t r i c t e d t o z^z = 1, t h u s z z = r %. T h e 7 i s j u s t a p h a s e f a c t o r , i n f a c t t h e T  2  H o p f phase, c o m i n g f r o m the passage of C  2  t o CP  1  = S . T h e H o p f phase will play an 2  i m p o r t a n t r o l e l a t e r o n . A l s o w e c a n c h e c k t h a t dz^dz = h i r 2  2  s\n(0)drd^d(j)d6. P u t t i n g  3  e v e r y t h i n g i n t o (4.40) g i v e s : r  f* 2  sm(6 )d<f> d6 k  k  k  f°°  d<y (rl) k  d(rl).  2a+1  t  *  x 2 s  v-"> r  (4.42) I n t e g r a t i n g t h e p h a s e 7jt a n d t h e r a d i u s r  k  o f z , t h a t separates c o m p l e t l y i n t h e p a t h k  i n t e g r a l , a n d r e i n t r o d u c i n g a s t a t e \z > l a t e r , gives  w - f f fi ^ j f f " ' w J 0  J  0  k=l  \2s+l)  Chapter 4. Coherent State Path Integral for Spin  •*  fit  -U  fiT  29  " s\n(0 )d<j> d9 , ^ ,  n  k  I J S !  k  _.  _<a . t* _ fi  k  a  a|  i< i>\e-*"""\3l*i  (4.43)  >?•  w h e r e \z > is t h e s a m e as \z > b u t r e s t r i c t e d t o t h e c o n d i t i o n z^z = 1. T h e e q u i v a l e n c e k  of (4.43) a n d (4.35) i s t h e n e x a c t l y d e m o n s t r a t e d . F u r t h e r m o r e , t h i s i n d i c a t e s t h a t t h e |0, <f> > s p i n c o h e r e n t s t a t e c a n b e r e p r e s e n t e d s i m p l y b y a t w o d i m e n s i o n a l h a r m o n i c o s c i l l a t o r c o h e r e n t s t a t e , \z > , w i t h t h e r e s t r i c t i o n z^z = 1. A n o t h e r w a y o f v e r i f y i n g (4.37) d i r e c t l y , a n d t h e n (4.35) b y t h e e q u i v a l e n c e p r o v e d above, can b e done b y using the determinant formulas (A.77) i n appendix A , w i t h t h e p a t h integral (4.37):  Z\ff\ = f TT  d  X  1  k  1  J i\ 2* (re^y<  J  k  where A f ,  = S^jSm^n — re >(e~ £ '' ) 6i j i, lX  J i m n  i  B  ij>mn  ]  w i t h i,j  ff  mn  det[M  i +  = 1 t o N cyclically, a n d  m , n = 1,2. W i t h t h e help of (A.82) a n d ( A . 8 3 ) , we c a n show t h a t d e t [ M ] = (1 -  r V ' S ^ e - ^ X i - r e^L, N  ^  S o b y u s i n g w = re , w e find tXk  k  N m  ~f  Zi wl^ (1 _ - * l * l n f c  )(l - e ¥ l * UU u>j)  {  4  M  )  = 1 Wj  w h e r e t h e c o n t o u r i n t e g r a l i s a c i r c l e o f r a d i u s r a r o u n d t h e o r i g i n . A c t u a l l y w e c a n see that t h e determinant w i l l create a pole i n t h e integration w h e n r = e r = e"^).  -  i  ^ ( a n d also a t  T h i s i n d i c a t e s t h a t t h e G a u s s i a n i n t e g r a l is n o l o n g e r c o n v e r g e n t o v e r t h i s  v a l u e o f r ! T h e r e is n o s u r p r i s e s i n c e t h e A i n t e g r a t i o n has b e e n i n t r o d u c e d t o r e d u c e t h e n u m b e r o f states f r o m a n h a r m o n i c o s c i l l a t o r (oo) t o a s p i n s p a r t i c l e (2s + 1), a n d w e p e r f o r m t h i s A i n t e g r a t i o n after t h e z i n t e g r a t i o n i n (4.44) ( w h i c h c o r r e s p o n d t o a trace). So it just happens that for r < e  t h e i n t e g r a t i o n c o n v e r g e s , we c a n n o t h o p e  for m o r e . T h i s e x p l a i n s w h y t h e p a t h i n t e g r a l (4.36) w o u l d n o t g i v e a c o r r e c t a n s w e r i f we integrate z first.  T h u s i f w e c o m e b a c k t o o u r i n t e g r a l (4.44) a n d i n t e g r a t e wi, w e  Chapter 4. Coherent State Path Integral for Spin  30  encounter two poles a t : w = e  3^.?j; . . . W ) 2  x  a n d t h e i n t e g r a n d v a n i s h e s for  N  —* oo fast e n o u g h t h a t we c a n d e f o r m t h e c o n t o u r o f  i n t e g r a t i o n a n d t h e n u s e t h e r e s i d u e t h e o r y . W h i c h gives dw j e m=/ n 2iriw (1 _ e ^ l ^ l ) + (1 k  1  k  e - ^ l )  s i n h ( ^ ( 2 s + 1)) sinh(^l)  =  )- E  tr(e  B  -/9/i|5|n  as i t s h o u l d .  4.4  Continuum Limit of the Spin Coherent States  T h e p a t h integral has been already calculated a t the discrete level a n d appears at t h e e q u a t i o n (4.35).  N o w w e have t o work o u t a c o n t i n u u m limit a p p r o x i m a t i o n , w h i c h  means that we have t o find a c o n t i n u u m a p p r o x i m a t i o n t o the expression:  < 0 ,<f> | e-*? * 8  k  k  | 6 <f> k+u  k+1  >i  / 2  (4.45)  I n t h e c o n t i n u u m a p p r o x i m a t i o n , w e c a n j o i n t A: t o k +1, c o n t i n u o u s l y , b y u s i n g a T a y l o r series e x p a n s i o n o f | 0 i,<f) i > a r o u n d \0 ,<j) > , u s i n g t h e f i r s t p a t c h i n (2.8), w h e r e , k+  k+  k  k  for s i m p h f i c a t i o n , I w i l l r e m o v e t h e k i n d e x o n a l l t h e v a r i a b l e s :  I 0 i,<f> i >k+  k+  -sin(0/2)0  \9,<f> > +-  ^ (cos(0/2)0^2zsin(0/2)<£)e^  cos(0/2)0 + 2sin(0/2)0 2  \ + •  v  (sin(0/2)0  2  - 4s cos(0/2)0<£ + 4 sin(0/2)<^  2  - 2 c o s ( 0 / 2 ) 0 - 4i sin(0/2)<£)e«*  Chapter 4.  31  Coherent State Path Integral for Spin  A n d a l s o use t h e e x p a n s i o n :  2  8  S o t h a t (4.45) b e c o m e s , a t t h e s e c o n d o r d e r i n e:  l-^B-n+^fi h \B\ +eism\9/2W^ 2 8 2  2  2  8  - l ^ [ B ( c o s ( 0 ) . c o s ( ^ ) 0 " + ism(<f>)6 + i s i n ( 0 ) e * ' ^ ) x  + £ ( c o s ( 0 ) sm(<j>)9 - i cos{<j>)9 + s i n ( 0 ) e ' ^ ) + B (v  z  = exp j - ^ J ?  •n+ ^ - ( \ B \  2  sin(0)0 -  - {B • n) ) + | ( 1 - o o s ( 0 ) ) ^ - ^ ( 0 2  2  2isin (0/2)<^)] 2  + sin  2  - 4 ^ ( s i n ( 0 / 2 ) < £ ) ) - ^ [ B M sin(<£)0 + i s i n ( 0 ) cos(0) cosU)j> + - ^ - ( s i n ( 0 ) c o s ( ^ ) ) ) or 4 dr 2  +B (-icos(<j>)9 + i s i n ( 0 ) cos(0) sm(<j>)<j> + - ^ - ( s i n ( 0 ) sin(<£))) or y  + £ , ( - t s m ( 0 ) ^ + ^(cos(0)))]} 2  = exp  • n + ^L{B  x n)  2  -  + c o s ( 0 ) ) ^ - ^{kf  • l ^ " ^ • *+ T ^ exp  I  O T  *  1  + ^  • (n x S ) }  B  " (*)W}  (- )  C0S  *  4 46  J at the point fc  where the ± 1 has been i n t r o d u c e d for t a k i n g i n t o account the choice of the two patches i n (2.8) (to get t o t h e s e c o n d p a t c h , w e m u l t i p l y b y  e  ^**-**+i)  = e "* -  _ 1  £* -). +  A s it has been e x p l a i n e d i n the last chapter, we have t o correct f o r the b o u n d a r y c o n d i t i o n s , w h e n w e are p a s s i n g f r o m N to N + 1 = 1. W h i c h m e a n s t h a t we h a v e t o add the following term t o the action of the path integral:  i[< M I ^ M > | - < MlJjrlM > M 0  = -j(±l  - cos(0)^ = -J  Q  P  [ ^ ( ± l -  cos(9)))}dT  Chapter 4.  32  Coherent State Path Integral for Spin  M u l t i p l y i n g ( 4 . 4 6 ) , f o r k = 1 t o N, a n d u s i n g t h e i n t e g r a l a p p r o x i m a t i o n gives a c o n t i n u u m limit a p p r o x i m a t i o n , where it could b e noted that the b o u n d a r y t e r m , above, cancels t h e last t e r m i n t h e second bracket i n (4.46), w h i l e the first t e r m vanishes b y t h e b o u n d a r y c o n d i t i o n , n ( 0 ) = n(3).  W e finally obtains the continuum limit approximation  of t h e s p i n c o h e r e n t s t a t e p a t h i n t e g r a l :  Z[B] = J DnS((n)  +  «  (  ^  2  - 1) e x p j- J ^ M ^ l  _  ^  S  x  ^  _  ^  + cos(0))<£ + fihsB • n  .  (  a  x  ^  .  ( 4  .  4 7 )  A s i n ( 3 . 2 1 ) , t h e c t e r m s c a n b e t h o u g h t o f as a r e g u l a t o r f o r t h e p a t h i n t e g r a l . A n d i n f a c t , f o r t h e t w o r e m a i n i n g t e r m s , o n e is B • n w h i c h c a n b e e i t h e r p o s i t i v e o r n e g a t i v e , a n d t h e o t h e r o n e is p u r e l y i m a g i n a r y . T h e n w i t h o u t these e t e r m s t h e r e r e a l l y w o u l d n o t be a n y convergent terms for t h e p a t h integral at a l l , w h i c h w o u l d m a k e t h e c o n t i n u u m approximation meaningless. T h e t e r m JQ (l-\-cos(9))^>dT i n (4.47) a c t u a l l y r e p r e s e n t t h e a r e a o n t h e s p h e r e e n c l o s e d b y t h e v e c t o r n i n i t s c l o s e d l o o p m o t i o n , i n t h e S o u t h p o l e side. W h i l e J j f ( — l - f c o s ( # ) ) ^ > d T r e p r e s e n t m i n u s t h e a r e a seen f r o m t h e N o r t h p o l e . T h e s e t w o t e r m s a l w a y s differ b y a m u l t i p l e o f in, l e a v i n g t h e p a t h i n t e g r a l s i n g l e v a l u e d ( s i n c e 2s is a n i n t e g e r ) . It is v e r y i n t e r e s t i n g t o n o t i c e t h a t t h i s t e r m is p u r e l y t o p o l o g i c a l . I t s r e l a t i o n w i t h t h e H o p f p h a s e w i l l b e m a d e clear at the next section.  4.5  C o n t i n u u m Limit of the Schwinger-Boson M o d e l  I n a first a p p r o a c h , I w i l l s t a r t w i t h t h e a c t i o n (4.39) o f t h e p a t h i n t e g r a l (4.36). T h e n , t r y t o f i n d a c o v a r i a n t w a y t o r e w r i t e t h i s a c t i o n , i n t h e s a m e s p i r i t o f s e c t i o n 3.2, t h a t will be suitable for a c o n t i n u u m limit a p p r o x i m a t i o n . Here, of course, we might expect some difficulties c o m i n g f r o m the gauge variable A, t h a t has t o b e integrated out t o really  Chapter 4. Coherent State Path Integral for Spin  33  get a s p i n s p a r t i c l e . T h e first t h i n g t o d o i s t o find a c o v a r i a n t d e r i v a t i v e o f z , t h a t w i l l t r a n s f o r m s i n the same w a y z transform u p o n t h e gauge transformation  (4.38).  If A w e r e z e r o , t h i s  d e r i v a t i v e s h o u l d b e c o m e a n o r m a l d e r i v a t i v e . L i k e i n t h e last s e c t i o n , w e j u s t h a v e t o connect z t o z k  continuously, b u t b y using a covariant derivative t o take into account  k+i  the gauge t e r m e * : , A  e *z iX  = e z  = z + tDz  tD  k+1  k  k  + jD z 2  k  k  + ...  (4.48)  So now t h e gauge t r a n s f o r m a t i o n is: z ^e k  .  i a  "z  k  , \  k  ^ \  + a -a  k  k  ,  k+1  (4.49)  Dz ^e *Dz ia  k  k  A s e c o n d i m p o r t a n t p o i n t is t o c o n s i d e r a n o n - t r i v i a l H o p f p h a s e i n z . U s u a l l y , w e  s h o u l d h a v e z ( 0 ) = z(3), b u t let u s c o n s i d e r i n s t e a d t h e b o u n d a r y c o n d i t i o n z ( 0 ) = e ' z ( / 3 ) , so t h a t t h e n o r m o f z a n d also i t s r e p r e s e n t a t i o n o n t h e s p h e r e (0,4>) s t i l l agrees 7  at 0 a n d 3. T h i s p h a s e , 7, does n o t c h a n g e a n y t h i n g p h y s i c a l l y , a n d s i n c e a p h a s e i n z is l o c a l l y u n p h y s i c a l , t h e 7 p h a s e m i g h t , a n d w i l l , represent o n l y a t o p o l o g i c a l p h a s e . T o a c h i v e t h i s t r a n s f o r m a t i o n , l e t u s c o n s i d e r t h e s a m e gauge t r a n s f o r m a t i o n (4.49) b u t c o n s i d e r i n g t h i s t i m e ct\ a n d ctjq+i h a s b e i n g c o m p l e t l y i n d e p e n d e n t .  Furthermore, i n  t h e case o f a c o n t i n u o u s gauge t r a n s f o r m a t i o n we r e a l l y h a v e t o c o n s i d e r a ( 0 ) as different f r o m a(8), i n g e n e r a l . T h i s p h a s e is a b s o r b e d b y t h e A ^ g a u g e t r a n s f o r m a t i o n  (4.49).  T h e o n l y p r o b l e m , is t h a t u s u a l l y N  N  I > - £ A k=l  Since now  a ^ ajv+i> t h i s x  f  c  k=l  is n o l o n g e r t r u e , a n d w e h a v e t o c o r r e c t t h i s p r o b l e m b y  i m p l e m e n t i n g t h e l o c a l gauge t r a n s f o r m a t i o n b y a g l o b a l gauge t r a n s f o r m a t i o n N  N  N  E * ^ E * + EK+i A  k-l  A  k=l  k=l  - «*)  (4.50)  Chapter 4.  Coherent State Path Integral for Spin  34  i n t h e p a t h i n t e g r a l (4.36). L e t us w r i t e t h e a c t i o n (4.39) w i t h t h e - h e l p o f t h e c o v a r i a n t d e r i v a t i v e (4.48), t o s e c o n d o r d e r i n e:  z\t- Z*-'z \ + tJ  S = £ > M A * + z\z k=i  = £ > i f t A  + -^B • z\oz C  4  -iztD z 2  k  Jo B y expanding e  tD  A l  k  - tz\Dz  -  k  + ^B.ztaDzk}  f^.2si%X  -  " e' *i)  k  k  e  ^-\B\ z^z 8  +  uh ->  +  yDzf  _  k  0  ,  2  + iDz*Dz 2  2  t£!L\B\*zlz  + ^ B • z^Dz]dr 2  (4.51)  i n p o w e r o f D, t h i s is l i k e e x p a n d i n g i n p o w e r o f A , b e c a u s e D d e p e n d s  o n A b y i t s d e f i n i t i o n (4.48).  S o , a t t h i s s t a g e , w e c a n r e i n t r o d u c e A i n (4.51) b y i t s  d e p e n d e n c e i n D, a n d t h e n i n t e g r a t e i t o u t . W e c a n r e w r i t e (4.48) as  (4.52)  e >e &(zk) = e< (z ) iX  i  D  k  B y e x p a n d i n g o n b o t h s i d e w e find f o r D , i n first o r d e r i n e:  D  = i ;  +  i  i  <- > 4  53  T h i s is a n a p p r o x i m a t i o n for A , thus w e can n o t expect a n exact result for t h e r e m a i n i n g calculation o f t h e p a t h integral, b u t this should give a g o o d a p p r o x i m a t i o n . (4.53) b a c k i n t o (4.51) g i v e s :  S=  /  Jo e +||i|  2  + =-—B • z'Bz — z'z  z z +  t  2  " j & z - zH) + ± A V z + -£ . \ t  d  2  8  6  B  —\B\ z*z  i  + IH^XB • Soz]dT  Putting  Chapter 4.  Coherent State Path Integral for Spin  35  T h e n , a t t h i s p o i n t , w e c a n use t h e H o p f p h a s e t o set  z z-z z i  (4.54)  = 0  i  N o t e t h a t (4.54) c o r r e s p o n d o n l y t o o n e c o n s t r a i n t b e c a u s e i t s c o m p l e x c o n j u g a t e gives t h e s a m e c o n s t r a i n t ( u n l i k e a c o n s t r a i n t l i k e z^z — 0 ) .  T h e n we h a v e t o a d d t h e n e w  p h a s e ( 4 . 5 0 ) , f$ c W r , t o t h e a c t i o n . T h e v a l u e o f a w i l l b e e v a l u a t e d l a t e r . T h i s l e a v e us w i t h the action  S=  i  p  Jo  Ldr  where L = —(2s% e v  + 2 s i  - z z + ^r-B • z*Bz) + ^-z^z 2 ' 2t ]  h a + ^ B • z^z  - 2±\3\*zU  + ||i|  2  + ^ B  • *B  = L + L + L, X  Z  w h e r e w e w i l l c o n s i d e r t h r e e p a r t s for t h e L a g r a n g i a n , n a m e l y  = —[^ + - f ( ^ <w6 ZZ  L\  2  " z^z + ^ B  • z^Bz)f  Zi  (zU-2s%f L  L, = 2siha + s » B . The L  z  Z  z  ~  2ezU  -^-- ^ ( \ B \ ^ z f  - (B • z^Bzf) + | | i |  2  +  • z^i  c o n t r i b u t i o n t o t h e p a t h i n t e g r a l is o f t h e f o r m  w h i c h i n d i c a t e s t h a t w e h a v e z*z = 2sh + 0(hyfe/Ar).  T h e n , i f we look a t a scale  A T >• e, w e f i n d t h a t z^z i s v e r y w e l l p e a k e d a t 2sh. H o w e v e r , i f w e set A T = e, as w e use t o d o f o r t h e d i s c r e t i s a t i o n , w e f i n d z^z = h(2s + 0 ( 1 ) ) , so o n l y t h e c l a s s i c a l h'rnit, s —» o o , i s w e l l p e a k e d . F o r l o w v a l u e s o f 5, i n s t e a d , t h e n o r m o f z i s less w e l l d e f i n e d .  Chapter 4. Coherent State Path Integral for Spin  36  In f a c t , t h i s i s n o t s u c h a s u r p r i s e , s i n c e i f w e t a k e t h e z^z = r  2  = R dependence of t h e  p a t h i n t e g r a l ( 4 . 4 2 ) , after A has b e e n i n t e g r a t e d o u t ,  R  2a  and find an approximation around its m a x i m u m :  f'(R)  = 0 =>  Ro = 2s , f(Ro) = M ^ e - ' «  , f(Ro) =  2  so  flRY^f^)+  1^-^-1^).) w_ L = e - ^ ^  « f(Ro)e-^T-  (4.56)  V47T5 w h i c h is j u s t e q u a t i o n (4.55) f o r R = z z / f t , u p t o a c o n s t a n t T  Ln t h e p a t h i n t e g r a l  (4.42) w e i n t e g r a t e d R d i r e c t l y since i t w a s c o m p l e t l y d e c o u p l e d t o t h e rest o f t h e p a t h i n t e g r a l . H e r e w e n o t i c e t h a t t h e t e r m i n c° i n L does n o t d e p e n d o n z^z a t a l l , t h a n k s a  to t h e A integration that brought corrections i n this respect. T h e remaining terms are of t h e o r d e r e, so t h e y d o n o t r e a l l y i n f l u e n c e t h e p a t h i n t e g r a l , t h e y o n l y r e g u l a r i z e i t . F u r t h e r m o r e , a m o r e e l a b o r a t e e x p a n s i o n u p t o o r d e r e w o u l d a l s o c o r r e c t these t e r m s , 2  as i t d i d f o r t h e first o r d e r , since w e k n o w f r o m  (4.42)  t h a t t h e z^z = r v a r i a b l e c o m p l e t l y 2  d e c o u p l e s i n t h e p a t h i n t e g r a l . S o i t w o u l d m a k e . s e n s e t o s i m p l y set z^z = 2sh f o r t h e remaining of the calculation. T h e n f o r L\, w e f i n d i t s c o n t r i b u t i o n t o t h e p a t h i n t e g r a l , after c o n s i d e r i n g t h e n e w constraint above, t o be a simple Gaussian:  J-*£Ji  2T  J-°°tA * 2  iiv^Fi  w h i c h is j u s t t h e m i s s i n g f a c t o r i n (4.55) t h a t a p p e a r s i n (4.56)! T h e n w e finnaly o b t a i n t h e p a t h i n t e g r a l , after r e s c a l l i n g z t o z^z = ft,  Z[3]  = J Dz^Dz6(z^z/n-l)expS^-j  J*[2si1ia + sfihB • z*az  37  Chapter 4. Coherent State Path Integral for Spin  -^^-{\B\\zUf and  w h e r e z(/?) = 2(0)e Jo ,  Before continuing w i t h and  a  T  - {B • z^Bz) ) + es\z\ + esfihB • z+ai^r 2  2  , w h i c h is determined b y the constraint  (4.57), l e t m e  J  (4.57)  (4.54).  r e d e r i v e i t i n a n o t h e r w a y , w h i c h is less p h y s i c a l  does n o t r e p r e s e n t t h e s i g n i f i c a n c e o f A , b u t g i v e s w h a t w e w a n t d i r e c t l y . I n t h e s e c t i o n 4.3 w e d e r i v e d t h e e q u i v a l e n c e o f t h e s p i n c o h e r e n t s t a t e a n d h a r m o n i c  oscillator coherent state p a t h integral, b y integrating the A variable, t h e H o p f phase a n d the n o r m r of z i n t h e later p a t h integral.  T h e result appears at t h e equation  (4.43)  w h e r e t h e z v a r i a b l e is r e s t r i c t e d t o z^z = h. T h e n w e c a n r e i n t r o d u c e t h e f u l l z v a r i a b l e b y i n s e r t i o n ( b y b r u t e force!) o f a d e l t a f u n c t i o n , S(z^z/% — 1), i n t h e p a t h i n t e g r a l a n d r e w r i t e t h e effective L a g r a n g i a n w i t h a n u n r e s t r i c t e d z v a r i a b l e :  m  =I n (ftrw)  {2s+1)6{zlzk/n  T h e n w e f o l l o w t h e s a m e p r o c e d u r e , w e set z i  -  = e &z  8  a n d find, u p t o s e c o n d o r d e r i n  c  k+  v*/*) *  k  c:  z  k  e  k+i  z  = "  •  a  z  k  +  tZk k z  H  g—\B\  z Zk k  2~~ '  ~2~  w h e r e I a s s u m e d z^z = h f o r t h e first t e r m . B y t a k i n g t h e l o g a r i t h m o f t h i s e x p r e s s i o n , to p u t everything i n the exponential, and including the b o u n d a r y conditions, we obtain the following path integral:  2*2  -^—{\B\ h 2  -2seztz {-nhB k  - (B • z\Bz ) ) - eszlh + esphB • z\az 2  k  • z\Bz  k  k  + z\z )]d\e-^ ^ * ^ k  m  d  r  Chapter 4.  38  Coherent State Path Integral for Spin  A n d i f we use t h e H o p f p h a s e t o set (z^z — z*z) = 0, w h i c h i m p l i e s z i = 0 w i t h z^z = %, T  w e f i n d t h a t t h e a b o v e p a t h i n t e g r a l b e c o m e s c o m p l e t l y i d e n t i c a l w i t h e q u a t i o n (4.57)! N o w , let m e c o m p l e t e t h e d e r i v a t i o n o f t h e e q u a t i o n a b o v e . F o r t h e z v a r i a b l e w e c a n use t h e s a m e r e p r e s e n t a t i o n as t h e s p i n c o h e r e n t states (2.8), b y i n c l u d i n g also t h e H o p f phase:  z = Vhe  cos(0/2)  ia  \ ( =  V h e  i  for  sin(0/2)e* cos(0/2)e -i<t> \  0  I  7r  9  sin(0/2)  for  9^0  j  Then ( z i - tfz) - h i [ a +  - cos(0))$ = 0  +  w h i c h gives t h e H o p f p h a s e c o n t r i b u t i o n t o t h e p a t h i n t e g r a l  (4.58)  a =-(±1 + cos(0))0 F u r t h e r m o r e , i n this gauge, we can checked that  n = z^Bz/% , ( n ) = 1 , (z^Bz - z Bz)/% = - i n 1  2  x n , $z = (£) /4 2  (4.59)  W e c a n use t h e f a c t t h a t n(0) = n(/3) a n d z^'z = 0 t o i n t e g r a t e b y p a r t s s o m e t e r m s i n (4.57). A f t e r t h a t w e c a n use (4.59) t o w r i t e t h e p a t h i n t e g r a l a s :  Z[3]  = J Dn8((nf  -  1) e x p  j  - J ^ M = F l + cos(0))<£ + fifisB • n  •B • (n x ft)](fr|  (4.60)  w h i c h is e x a c t l y t h e s a m e as t h e s p i n c o h e r e n t s t a t e p a t h i n t e g r a l c o n t i n u u m l i m i t , t h a t a p p e a r s a t t h e e q u a t i o n (4.47).  Chapter 5  P a t h Integral for a Charged Particle in a Magnetic Monopole Field  5.1  Monopole Vector  Potential  In this chapter, I w i l l demonstrate that the physics o f a charged particle i n the field of a m a g n e t i c m o n o p o l e is r e l a t e d t o t h e s p i n s y s t e m t h a t we are s t u d y i n g . A m a g n e t i c m o n o p o l e f i e l d i s , l i k e a p o i n t c h a r g e e l e c t r i c field, g i v e n b y t h e e q u a t i o n :  Bm = g  (5.61)  w h e r e g is t h e m a g n e t i c c h a r g e . S i n c e w e w i l l b e i n t e r e s t e d i n t h e g a u g e f i e l d ( l i k e A o f l a s t s e c t i o n ) we w a n t t o represent t h i s f i e l d b y a v e c t o r p o t e n t i a l , A , m  such that  rottAj = B  (5.62)  m  M a t h e m a t i c a l l y , w e find t h a t t h e 2 - f o r m F (Fij = eijkBk) is c l o s e d , dF = 0 , a n d i f (5.62) is d e f i n e d t h r o u g h o u t a l l t h e s p a c e , t h e n F is e x a c t , F = dA. S o t h e s e c o n d c o h o m o l o g y g r o u p o f t h e space w i l l i n d i c a t e i f t h e r e is s o m e B f i e l d t h a t h a s n o s o l u t i o n f o r A ( c l o s e d 2 - f o r m t h a t are n o t e x a c t ) . T h e s p a c e here is i ? , b u t f r o m (5.61) w e n o t i c e t h a t t h e r e 3  is a s i n g u l a r i t y a t r = 0. S o w e h a v e t o r e m o v e t h i s p o i n t t o g e t a w e l l d e f i n e d v e c t o r p o t e n t i a l . W h i c h leaves us w i t h R — {0} = S 3  H (S 2  2  2  x [0,oo]. It is k n o w n t h a t :  x [0,oo]) = H {S ) 2  2  = R  w h i c h m e a n s t h a t t h e r e is s o m e field B t h a t w i l l h a v e n o s o l u t i o n f o r (5.62), v a l i d e v e r y w h e r e . T h e class t o w h i c h B  m  belongs ( i n i ? ) , i n t h e cohomology group, is given b y t h e  39  Chapter 5. Path Integral for a Charged Particle in a Magnetic Monopole Field  40  integration  JF  = J B -dS  s  s  m  = ATrg  T h i s is actually the value (up t o a constant) of the magnetic charge! T h e n , for non-zero m a g n e t i c c h a r g e , t h e r e is n o s o l u t i o n f o r t h e v e c t o r p o t e n t i a l v a l i d e v e r y w h e r e o n t h e s p a c e ( s x [0,oo]). T h i s is k n o w n as t h e D i r a c s t r i n g (since f o r a n y s o l u t i o n t h e r e is a 2  divergence along a string starting at the origin and going t o infinity!). T h i s is n o t t h e e n d o f t h e s t o r y , s i n c e a l l o u r s t u d y h a s b e e n d o n e c l a s s i c a l l y . I n q u a n t u m m e c a n i c s , t h e v e c t o r p o t e n t i a l enters t h e t h e o r y as a g a u g e f i e l d :  Z t y = (V -  ,  # - > e T r  x  #  A —> A-\-Vx  ,  D$ -* e * / 3 $  (5.63)  x  • —*  S o t h e g a u g e f i e l d A c a n b e c h a n g e d b y s o m e g a u g e t r a n s f o r m a t i o n (5.63) b y a q u a n t i t y V x f o r a n a r b i t r a r y field x (or A —• A - f dx)- T h i s does n o t s e e m t o c h a n g e t h e e q u a t i o n (5.62), since  r o t ( V x )  = 0 (or  d(dx) = 0 ) .  In q u a n t u m mecanics, however, the  x  appears  i n t h e w a v e f u n c t i o n i n t h e e x p o n e n t i a l e ^ . S o x does n o t h a v e t o b e a f u n c t i o n , b u t x  s i m p l y a section of the vector b u n d l e (the wave function o n the space). I n other w o r d s , X is d e f i n e d o n l y m o d u l o — . L e t u s s o l v e (5.62) i n t w o p a t c h e s , u s i n g p o l a r c o o r d i n a t e s :  A  m  • dx = g(l — cos(0))d<f>  A' -dx  = -g(l +cos(6))d<f>  m  for 0 ^ n f o r 0^0  T h e s e t w o v e c t o r p o t e n t i a l s differ o n l y b y :  {A -A' )-dx m  = 2gd<f> = d(2g<l>)  m  or  A =A' m  m  + V(2g<f>)  Chapter 5. Path Integral for a Charged Particle in a Magnetic Monopole Field  S o , as f a r as q u a n t u m m e c a n i c s is c o n c e r n e d , t h e A  m  as l o n g as 2g<f> i s s i n g l e v a l u e d e v e r y w h e r e m o d u l o  for  ir = ±s n  41  a n d A' g a u g e fields a r e t h e s a m e , m  2  - ^ , which means:  5= 0,1/2,1,3/2,...  '  (5.64)  m u s t b e f u l f i l l e d , t h i s is t h e D i r a c q u a n t i s a t i o n c o n d i t i o n [4]. F o r s i m p l i c i t y I w i l l consider ^ = s ( b y choosing t h e appropriate sign o f e). N o w if w e c o n s i d e r a c h a r g e , e, m o v i n g i n t h i s m a g n e t i c f i e l d , t h e H a m i l t o n i a n i s s i m p l y g i v e n by:  W h a t w o u l d b e i n t e r e s t i n g n o w is t o express HQ i n t e r m s o f t h e a n g u l a r m o m e n t u m , L, of t h e f i e l d .  5.2  Monopole Angular  Momentum  W e s h o u l d b e a b l e t o find L i n t e r m o f D a n d r. W e h a v e t h e c o m m u t a t i o n r e l a t i o n s :  [r,-,rv]=0  ,  [D rA u  =(5  ti  ,  [A-,£j] = -*se  0 f c  £§  (5.66)  W e expect the angular m o m e n t u m t o have a c o m m u t a t i o n relation w i t h f a n d D such t h a t t h e y t r a n s f o r m as a v e c t o r  [L ,D ] i  j  = ie jkD i  k  ,  [L,-,rj] = z e , r jfc  fc  (5.67)  I n f a c t , t h e c o m m u t a t i o n r e l a t i o n s (5.67), d e t e r m i n e d c o m p l e t l y t h e c o m m u t a t i o n r e lations o f t h e components o f L between themselves a n d t h e H a m i l t o n i a n , b y using t h e Jacobi identity a n d the irreductibility of the r and D variables:  [Li, Lj] = ieijkLk ,  [Li, H ] = 0 0  Chapter 5. Path Integral for a Charged Particle in a Magnetic Monopole Field  42  —* T h e first c h o i c e f o r L w o u l d b e s i m p l y  —* 9  —*  —*  L = —ifif x D = r x (p — eA ) m  B u t it turns out t o be w r o n g , m a i n l y because o f the presence of the m a g n e t i c p o t e n t i a l vector A , m  as i t c o u l d b e n o t i c e d a l r e a d y i n (5.66). T h e c o r r e c t a n s w e r i s i n f a c t g i v e n  by (5.68)  L = -ihrxD-sh-  r  T h i s e x p r e s s i o n for t h e a n g u l a r m o m e n t u m i s a c t u a l l y t r u e also c l a s s i c a l l y ( o b v i o u s l y ? ) since d  —(mr  dt  •  sfi  •  d  •  x f) = mf x f = r x ( e r x B) = — f x ( f x f ) = — r dt J  v  (sh-)  r  I n q u a n t u m m e c a n i c s w e j u s t h a v e t o v e r i f y t h e c o m m u t a t i o n r e l a t i o n s (5.67) t o c o n v i n c e ourselves. A s p e c i a l s t u d y h a s t o b e d o n e t o see w h i c h v a l u e s o f / c a n r e a l l y o c c u r . W e k n o w t h a t 21 m u s t b e a n i n t e g e r s i n c e L f o l l o w s t h e s p i n a l g e b r a . B u t a m o r e c a r e f u l c o n s t r u c t i o n [5] o f t h e r e p r e s e n t a t i o n o f L d e f i n e d b y (5.68) a c t u a l l y s h o w s t h a t I = s, s +  1,  s + 2, • • •.  ( w e m u s t h a v e a s t a t e \l,s > t o c o n s t r u c t t h e r e p r e s e n t a t i o n ! ) N o w , w e k n o w t h a t (L)  2  = % l(l + 1) w h e r e / i s o n e o f t h e v a l u e s m e n t i o n e d a b o v e . 2  B y u s i n g t h e e q u a t i o n (5.68) i n s t e a d , w e (L)  2  find  = -h (fxD) 2  + hs  2  2  2  Furthermore, we have  D •D =  =  =  D  .  r  ^ -  £  ^  e  ^  D  (D.f)(^)(r.D)-(Dxf)(^)(f.D)  .  Chapter 5. Path Integral for a Charged Particle  ,d  2  2d,  1  rar  r  ,d  rtx2  2d,  2  or  2  in a Magnetic  2  Monopole  43  Field  s -1(1 + 1) 2  r or  r  2  So, this enable us to write the Hamiltonian (5.65) as:  =  _|l |l ^ (  +  2m or  2  5.3  )+  r or  ffli+i^i!)  (5  .  69)  2mr  2  Path Integral for a Spin Particle in a Magnetic Field  W h a t would be the Hamiltonian of our magnetic monopole system, if we p u t i t i n a constant magnetic field? We already solved the system for a.magnetic potential A , so m  a constant magnetic field B, corresponding to a vector potential A = \B x r, w i l l simply shift A  m  -> A  m  + \B x r i n the Hamiltonian (5.65):  eA ) • (B x f) + £-(B x i f  = »o-  m  e r*  esnr-B  e , -> 2  Here appears finally the reward of all our work in this chapter, the B • L term that we are studying, but there are still two problems. One is the last two terms i n (5.70), they have nothing to do w i t h our model of last chapter. T h e only simple way of correcting this is to add an interaction potential t o the Hamiltonian, that w i l l cancel them. T h e second problem is more important, because we need our particle to have a definite spin, s, and the Hamiltonian (5.70) does not guarantee this constraint (other values of / might appear). T h e contribution of / i n the Hamiltonian is given by the term  W +1)  —  5 2  )  in H , i n (5.69). T h e smaller / is, the smaller the energy contribution of this term. So 0  if ~2 ^»  which means mr  2  is small enough, the values of / > s (where / = s is the  Chapter 5. Path Integral for a Charged Particle in a Magnetic Monopole Field 44  ground state) will not contribute much to the partition function, and we would be able to consider the system in a state I = s with E = E , more precisely we must have 3E„ ^> 1. 3  To accomplish this task we could consider m —• 0, but we do not know if r would not become very large. In fact, a solution of the Hamiltonian H  is well known [6] and  0  indicates an unbounded system! So we really need something that will keep the particle close to the monopole. To do so, we can impose an additional*interaction. Several choices are possible and though many people can study the different potentials, ultimately they should produce the same result.  The idea actually would be to put a particle on a  sphere, r = ro. Physically, this means that we have to put a steep potential (harmonic for example) around r = r , such that the particle will stay in the first (ground state) 0  energy level of that potential. This will just mean a shift of Eh = \h~Wh (PEh > 1) for the remaining part of the Hamiltonian, and a particle constraint to move on a sphere. All together, we will be interested in an interaction potential of the form:  2m  =  2mrf '  =  2**  r  8m  2cr  2  = 2  =  ^  ^  = ^ " V ^  ^  The variable a controls the steepness of the potential, and also indicates the value of the uncertainty of the radius r = ro constraint. We therefore must have a <C ro, which gives Eh  E , usually. However, if s —*• oo, the classical limit, we can choose a = s  r /\/2s 0  and obtain Eh = E . In other words, the harmonic potential, that we added here, might 3  even be present, in (5.70), at the classical limit and responsible for the E„ ground state energy! So we obtain the following Hamiltonian H = H. + V = (E + E h  .  )  ,  ( L ) = h s(s + 1) 2  2  (5.72)  Chapter 5. Path Integral for a Charged Particle in a Magnetic Monopole Field  45  Now, let us write what is the Lagrangian corresponding to (5.72). We already know what to do, since it is simply a particle moving in a magnetic field and a potential, which has a Lagrangian given by L=^(r)  2  = jfr  2  + \{B  + e(A  x f) • f -  +  m  A).r--V  + ± 0  x ff + hs(±l  - cos(0))^  (5.73)  where I droped the harmonic potential term, I will introduce it in the path integral as a S(r — ro) function. Since we are looking for the partition function, we have to introduce the Euclidean Lagrangian, by changing j  t  =  in the path integral (5.73), which  gives Z\0\ = tx{e~ ) = t r ( e - ^ - ^ - ) ) . - / » ( ^ + ^ ) £  pH  e  = J Dr6(r - r )e~  L E d T  0  where  L  °  = W^  2  +  x  "'  l^TT-  ~ ^  B  *^ + ^  "  «»<«>» < ' > 5  74  and f(0) = r(fl) as usual.  5.4  Comparison with Coherent State Path Integral  The comparison of the path integrals (5.74) and (4.60) or (4.47) indicates how similar are these path integrals. In fact, they are identical if we make the correspondence: e  2mrg  _  ' W ' -° £ =  r=  r  n  ,  (5J5)  .  Chapter 5.  Path Integral for a Charged Particle in a Magnetic Monopole Field  46  T h e t o p o l o g i c a l t e r m s differ b y a m i n u s s i g n , w h i c h is n o r m a l s i n c e t h e r differ b y a m i n u s s i g n ( a n d a c o n s t a n t ) w i t h n , i n d i c a t e d i n (5.75).  If eg/% = —s, i n s t e a d o f o u r  c o n v e n t i o n , t h e n \i a n d f w i l l c h a n g e s i g n i n (5.75). T h e v a l u e o f fi i s n o t a s u r p r i s e , s i n c e t h e H a m i l t o n i a n (5.72) c o n t a i n t h i s f a c t o r i n front of t h e B • L t e r m .  T h e v e r y i n t e r e s t i n g r e v e l a t i o n o f t h i s c a l c u l a t i o n is t h e c  correspondence, w h i c h can be rewrite as:  or  t = 4~  BE  8  (5.76)  = N  It w a s a l r e a d y k n o w n t h a t /3E >• 1 a n d (5.76) i n d i c a t e s t h i s s t a t e m e n t i n t e r m o f t h e a  N v a r i a b l e . F u r t h e r m o r e , i f w e p u t . (5.76) b a c k i n t o t h e h a r m o n i c p o t e n t i a l i n (5.71) a n d use t h e v a l u e o f a = r o / \ / 2 s , v a l i d a t t h e c l a s s i c a l l i m i t , w e f i n d e x a c t l y t h e e q u a t i o n (4.55) o r (4.56) f o r t h e r a d i a l p a r t o f t h e p a t h i n t e g r a l ! A g a i n t h e c l a s s i c a l l i m i t is v e r y w e l l d e f i n e d , b u t t h e l o w s p i n l i m i t is m o r e t r i c k y , a n d needs s o m e ' a r t i f i c i a l ' c o n s t r a i n t s to b e well defined. I n t h e p a r t i t i o n f u n c t i o n ( 5 . 7 4 ) , t h e p r e s e n c e o f e ~ ^ ' = e~ E  N  indicates that t h e p a t h  i n t e g r a l m e a s u r e m u s t b e r e n o r m a l i s e d b y e, t o e x a c t l y c o r r e s p o n d t o t h e c o h e r e n t s t a t e p a t h i n t e g r a l . O n a n o t h e r h a n d , t h e e~^  Eh  come from the harmonic potential that we  added t o t h e H a m i l t o n i a n , so t h e r e is n o s u r p r i s e i f w e find t h i s e x t r a t e r m . S i n c e t h e k i n e t i c e n e r g y comes m a i n l y f r o m t h e s p i n o f t h e p a r t i c l e , t h u s E„, t h e k i n e t i c t e r m ^f(ra) i n t h e c o h e r e n t s t a t e p a t h i n t e g r a l , w i l l c o n t r i b u t e t o t h e o r d e r o n e , 2  w h i c h can not really be neglected i n the calculations.  Chapter 6  Conclusion  T h e p u r p o s e of t h i s w o r k h a s b e e n t h e s t u d y of p a t h i n t e g r a l e v a l u a t e d w i t h c o h e r e n t s t a t e s . T h i s h a s b e e n d o n e b y l o o k i n g at t w o s o l v a b l e p r o b l e m s : t h e h a r m o n i c o s c i l l a t o r a n d t h e p a r t i c l e w i t h s p i n i n a c o n s t a n t m a g n e t i c field s y s t e m s . T h e c o h e r e n t states r e p r e s e n t t h e s y s t e m so w e l l , t h a t t h e p a t h i n t e g r a l v a n i s h e s for c l a s s i c a l t r a j e c t o r i e s .  T h e real p r o b l e m has been, t h e n , to include the  quantum  t r a j e c t o r i e s i n t h e p a t h i n t e g r a l c a l c u l a t i o n s , t o get t h e q u a n t u m c o r r e c t i o n s , if n o t a c o m p l e t l y q u a n t u m r e s u l t . It is c l e a r t h a t t h e c l a s s i c a l l i m i t of t h e s e p a t h i n t e g r a l s are v e r y w e l l d e f i n e d , b u t it is useless t o use p a t h i n t e g r a l s t o find o n l y c l a s s i c a l s o l u t i o n s . These q u a n t u m corrections have been taken into account, in m y work, by keeping t h e e t e r m s i n t h e p a t h i n t e g r a l a n d t h e n t a k i n g t h e l i m i t e —• 0 at t h e e n d of t h e c a l c u l a t i o n s . T h e s e t e r m s c r e a t e a b r i d g e b e t w e e n these c l a s s i c a l a n d p u r e l y  quantum  p a t h s , b y m a k i n g these t r a j e c t o r i e s s m o o t h e n o u g h so t h a t w e c a n use a c o n t i n u u m l i m i t approximation. T h e use of a l a t t i c e r e g u l a r i z a t i o n gives us a w a y t o o b t a i n a n e x a c t d i s c r e t e a c t i o n . T h i s m e t h o d h a s b e e n u s e d b e f o r e w i t h success been studied very deeply.  [10,11], b u t  its discrete level has never  It seems t h a t a c a r e f u l a n a l y s i s of t h e d i s c r e t i s a t i o n g i v e s  some useful corrections, and allows an interesting comparison between various continuous approximations and the exact solution. T h i s explains w h y the e terms regularized the p a t h i n t e g r a l a p p r o p r i a t e l y , t h e r e s u l t i n g p a t h i n t e g r a l is closer t o its d i s c r e t e v e r s i o n . F o r t h e h a r m o n i c o s c i l l a t o r c o h e r e n t s t a t e p a t h i n t e g r a l , it h a s b e e n n o t i c e d t h a t t h e  47  Chapter 6.  48  Conclusion  s i m p l e d e f i n i t i o n o f t h e c l a s s i c a l H a m i l t o n i a n H(z) i n t e r m of t h e q u a n t u m o n e H as  H(zk) =< Zk\H\zk  >, or  H(zk) = ^ j J J * * ^  affect t h e g r o u n d s t a t e i n t h e p a t h i n t e g r a l .  It has b e e n d e m o n s t r a t e d t h a t t h e e\z\ t e r m h e l p s t o r e g u l a r i z e t h e p a t h i n t e g r a l . T h i s is 2  p a r t i c u l a r l y a p p a r e n t b y g o i n g b a c k t o a d i s c r e t e l e v e l , at t h e s e c t i o n 3 . 2 , or a s e m i c l a s s i c a l a p p r o x i m a t i o n , at t h e s e c t i o n 3.3. T h e a p p l i c a t i o n of t h e s a m e p r o c e d u r e t o a s p i n s p a r t i c l e i n a c o n s t a n t m a g n e t i c field  b y t h e use of s p i n c o h e r e n t states o r h a r m o n i c o s c i l l a t o r c o h e r e n t states p r o d u c e d  t h e s a m e c o n t i n u u m limit,• u p t o o r d e r c t e r m s , after t h e i n t e g r a t i o n of t h e a p p r o p r i a t e variables in the latter p a t h integral.  W e w e r e a b l e t o e x t r a c t t h e m e a n i n g of t h e Afc  v a r i a b l e s as a g a u g e field, a n d c o n s t r u c t a c o v a r i a n t d e r i v a t i v e for t h e h a r m o n i c o s c i l l a t o r c o h e r e n t s t a t e s . T h e t o p o l o g i c a l a c t i o n , 2 s ( ± l — cos(0))<^>, h a s b e e n s h o w n t o b e r e l a t e d t o t h e H o p f p h a s e of t h i s v a r i a b l e , z. T h i s g a v e us a m a p p i n g of t h i s s p i n p a t h i n t e g r a l i n t o a CP  1  m o d e l . T h e coefficient of t h i s t o p o l o g i c a l a c t i o n a p p e a r e d c l e a r l y as b e i n g  2s. F u r t h e r m o r e , a s t u d y o f m a g n e t i c m o n o p o l e h a s b e e n r e p r o d u c e d i n d e t a i l s . It h a s b e e n e x p l a i n e d h o w i t is p o s s i b l e t o o b t a i n a s p i n s p a r t i c l e r e p r e s e n t a t i o n , u s i n g a m o n o p o l e field a n d a s p e c i f i c i n t e r a c t i o n : V =  -  -^(B  x r)  2  This  + E ^jl°£. a  i d e n t i f i c a t i o n h a s b e e n s t u d i e d b e f o r e , b u t n e v e r u p t o t h e e o r d e r [14,15]. W e s h o w e d a complete correspondence between this monopole p a t h integral, i n position space representation, a n d the same spin system p a t h integral using coherent states, i n d i c a t i n g even m o r e t h e r e l e v e n c e o f t h e c t e r m s as a r e g u l a t o r . T h e e —• 0 l i m i t is i m p o s e d b y a r l i m i t , or m —• 0 w i t h ^  fixed,  0  —• 0  o n t h e r a d i u s of t h e s p h e r e o n w h i c h t h e p a r t i c l e m o v e s  a r o u n d t h e m o n o p o l e . In t h i s l i m i t , t h e \m(r)  2  w i l l not necessarily go to zero, since the  p a r t i c l e m i g h t s i m p l y s p i n f a s t e r , b y c o n s e r v a t i o n of a n g u l a r m o m e n t u m . T h e r a d i a l p a r t o f t h e m o t i o n of t h e p a r t i c l e , for l o w v a l u e of t h e s p i n , is n o t v e r y w e l l p e a k e d at a g i v e n r a d i u s , h o w e v e r , it c o m p l e t l y d e c o u p l e d i n t h e p a t h i n t e g r a l .  Which  Chapter 6. Conclusion  49  gives us a unambiguous path integral for the tangential motion. In the future, it would be interesting to apply this path integral method to some statistical models, like the Heisenberg model or the spin chain, that are currently under study, since they could be relevant to high temperature superconductors. The comparison of some of these studies, like [12], and the ones using coherent state path integrals might gives some new insights into these models.  Bibliography  [1]  J . R . K l a u d e r , Phys. Rev., D 1 9 , 2349  (1979)  [2] L o k C . L e w Y a n V o o n , U B C thesis 1989 [3] I. K . A f f l e c k , M . B e r g e r o n , L . C. L e w Y a n V o o n &; G . S e m e n o f f ,  1  Coherent State  Path Integral and the Harmonic Oscillator , U B C p r e p r i n t 1989 1  [4]  P . A . M . D i r a c , Proc:Roy. Soc, A 1 3 3 ,  60 (1931)  [5] S . C o l e m a n , The Magnetic Monopole Fifty Years Later , I n t e r n a t i o n a l S h o o l o f l  1  S u b n u c l e a r P h y s i c s L e c t u r e , ' E t t o r e M a j o r a n a ' , 1981  [6] T. T. W u h  C . N. Yang,  Nucl. Phys.,  B107,  365 (1976)  [7] R . P . F e y n m a n & A . R . H i b b s , ' Quantum mechanics and Path Integrals', M c G r a w Hill, New York  (1965)  [8] L . S . S c h u l m a n , ''Techniques and Applications of Path Integral?, J o h n W i l e y , N e w York  [9]  (1981)  R . J . G l a u b e r , Phys. Rev., 1 3 1 ,  2766 (1963)  [10]  R . E . P u g h , Phys. Rev., D 3 3 ,  1027 (1986)  [11]  R . J . F u r n s t a h l & B . D . S e r o t , Ann. of Phys., 1 8 5 , 138  [12]  D . V . K h v e s h c h e n k o k A . V . C h u b u k o v , Sov. Phys. JETP,  (1988) 66, 1088  (1987)  [13] I. S . G r a d s h t e y n Sz I. M . R y z h i k , Table of Integrals, Series and Products', A c a d e m i c 1  Press, N e w Y o r k  (1980)  [14] E . F r a d k i n & M . S t o n e , I l l i n o i s p r e p r i n t ( M a r c h  [15]  M . S t o n e , Nucl. Phys., B 3 1 4 ,  557 (1989)  50  1988)  Appendix A  Identities for Determinants  Gaussian integrals have the very useful property that:  /S(:  (A.77)  Thus, the evaluation of various determinants might give an alternative verification of some Gaussian path integrals. In most cases, the MM matrix is non-zero for \k —1\ = 0 or 1 only, so let us study the determinant of  M  ( N )  =  • • 0 \  A  - B  0  -C  A  - B  •  • 0  0  -C  A  •  • 0  0  0  0  •  •  A  (  ,  M  ( N )  =  (N)  0  •• •  -C  A  - B  -C  A  - B  ••  •  0  0  -C  A  ••  •  0  - B  0  0  •• •  A  \  I  (JV)  (A.78) or = A8 iti  BSij.i  -  C6  i,j  itj+1  = 0 to TV  where i,j assume cyclic boundary conditions, N + 1 = 1, iV — 1 = —1, for the M^N) matrix. Let us call D  N  =  det[M r)] (A  ,  51  D  N  =  det[M( )] N  52  Appendix A. Identities for Determinants  The evaluation of these determinants can be done by using the well known recursive formulas. Expanding along the first line for D^ gives D  = AD _!  N  -  N  BCD _2 N  which can be solved by inserting a solution of the form A^, that produces the following constraint: X = AX - BC  A = A± = ]-(A ± VA — ABC)  =>  2  2  Since D\ = A, D = A — BC, we find the general solution for this determinant identity 2  2  det[M ] {N)  = D= N  A  *  +  - A? )  (A.79)  +1  A  For DN, we procede in the same way D  N  = AD _! N  - 2BCD _ N  2  - (B  N  +  C) N  with AX± — 2BC = (A+ — A_)(±A-j.) we find for the cyclic determinant D^, the identity: det[M ] = D (jV)  N  = (X+)  N  + (X_)  N  - B  N  - C  N  (A.80)  We can, furthermore, find the eigenvectors and eigenvalues of the M(JV) matrix, because of its cyclic boundary condition. These are simply  This gives another identity for the determinant: D  N  = f[ X = f[(A - Be^ k  k  - Ce-^ ) k  (A.81)  If the A, B and C are not simple complex numbers, but submatrices, we can still solve the determinant under one condition: that we can diagonalize all three submatrices,  Appendix A. Identities for Determinants  A,  53  B, C, a t t h e s a m e t i m e . I n o t h e r w o r d s , (SAS )^  = £,jA,-, (SBS~ )ij  -1  (SCS~ )ij l  l  = SijBi a n d  = SijCi w i t h t h e s a m e S m a t r i x . T h e n t h e d e t e r m i n a n t i s s i m p l y  M det[M ] (N)  = IJ d e t « [ M j v ) ]  (A.82)  (  t=i  w h e r e d e t ^ [ M ( / v ) ] is t h e d e t e r m i n a n t ( A . 8 0 ) w i t h t h e use o f t h e Ai, Bi, Ci v a r i a b l e s . If C = 0, t h e p r e v i o u s d e t e r m i n a n t ( A . 7 9 ) a n d ( A . 8 0 ) c a n b e s i m p l i f i e d s i g n i f i c a n t l y :  det[M ] {N)  = A  N  ,  det[M ] {N)  = A  -  N  B  N  A c t u a l l y , i f a l l t h e A a n d B's are differents, w e c a n s t i l l e a s i l y p r o v e d , l i k e ( A . 7 9 ) o r (A.80), that A  -B  x  det  x  0  'A  3  0 0  = (U A ) - (f[ k  k=l  -B  N  0  B) k  (A.83)  k=l  AN  I n cases o f s u b m a t r i c e s , w a c a n s t i l l use ( A . 8 2 ) w i t h ( A . 8 3 ) , b u t a g a i n , as l o n g as w e c a n d i a g o n a l i z e a l l t h e Ak, B  k  (k = 1 t o A^) a t t h e s a m e t i m e .  Appendix B  Schwinger-Boson  Model  Let us consider a set of two dimensional creation and destruction operators: [a , a]] =Uij t  , [o,-, aj] = [at, a)] = 0 , for »,j = l,2  Then we can build up a spin vector, J , by J = a^a  (B.84)  U) = — ( — + 1)  (B-85)  We can verify the identity 2  This indicates that we can use (B.84) to represent a spin s angular momentum operator on a set of states, |\P >, build up by a], if we have the constraint  0  t | W > = 2s|# > 0  In other words, we have to work in a subspace represented by: I* >-> / JO  27r  ^e ( - )|* > ,A  27T  54  a,a  2s  (B.86)  

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