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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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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 F O R T H E H A R M O N I C O S C I L L A T O R A N D A SPIN 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 FLELD B y M A R I O B E R G E R O N 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 S C I E N C E i n THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS W e accept th i s thesis as con fo rm ing to the requ i red s t a n d a r d 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 this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis . for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada DE-6 (2/88) 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 O s c i l l a t o r Coheren t Sta tes 3 2.2 S p i n Cohe ren t S ta tes 6 2.3 P r o p a g a t o r , P a t h In tegra l a n d P a r t i t i o n F u n c t i o n 8 3 Coherent State Path Integral for the Harmonic Oscillator 11 3.1 E x a c t R e s u l t for the D isc re t i sa t i on 11 3.2 C o n t i n u u m L i m i t 12 3.3 Sem ic l ass i ca l A p p r o x i m a t i o n 16 3.4 R e g u l a r i z a t i o n 18 4 Coherent State Path Integral for Spin 26 4.1 D i s c r e t i s a t i o n w i t h S p i n Cohe ren t Sta tes 26 4.2 T h e S c h w i n g e r - B o s o n M o d e l 27 4.3 E q u i v a l e n c e of the two Represen ta t ions 28 i i i 4.4 C o n t i n u u m L i m i t of the S p i n Coheren t States 30 4.5 C o n t i n u u m L i m i t of the S c h w i n g e r - B o s o n M o d e l 32 5 Path Integral for a Charged Particle in a Magnetic Monopole Field 39 5.1 M o n o p o l e V e c t o r P o t e n t i a l 39 5.2 M o n o p o l e A n g u l a r M o m e n t u m 41 5.3 P a t h In tegra l 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 Coheren t S ta te P a t h In tegra l 45 6 Conclusion 47 Bibliography 50 Appendices 51 A Identities for Determinants 51 B Schwinger-Boson Model 54 i v 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 A c k n o w l e d g e m e n t I w o u l d l i ke to t h a n k m y superv iso r , D r . G o r d o n Semenoff , for his e x p l a n a t i o n of var ious sub jec ts i n theore t i ca l phys i cs . O u r m a n y d iscuss ions were ve ry he lp fu l i n u n d e r s t a n d i n g deta i ls I was not f am i l i a r w i t h . In p a r t i c u l a r , he i n t r o d u c e d m e to the sub jec t of coherent s ta te p a t h i n teg ra l , w h i c h , w i t h h is suggest ions a n d c o m m e n t s , led the way to th is thes is . F u r t h e r m o r e , I w i s h to t hank D r . Ian Af f leck a n d his g radua te s tuden t , 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 cons t ruc t i ve d iscuss ions o n th is sub jec t , a n d b y i n t r o d u c i n g to m e the m e t h o d of con tour in teg ra l regu la r i za t i on (chapter 3 . 4 ) . D e p l us , i l me fa i t p la is i r de remerc ier mes parents pou r leur suppo r t c o n t i n u , tou t au l ong de mes etudes a cet te un ivers i te . M a l g r e l a d is tance qu i nous separe , res idant de chaque cote d u p a y s , leur presence et sou t ien a tou jours ete cons tan t . F i n a l l y , I w o u l d l i ke to t h a n k the depa r tmen t of phys ics a n d U B C for accep t i ng m e as one of the i r g radua te s tuden t , a n d the i r suppo r t i n m y work for the last two years . v i i C h a p t e r 1 I n t r o d u c t i o n 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 mechan-ics, 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 par-ticle 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 2.1 Harmonic Oscillator Coherent States From the harmonic oscillator Hamiltonian, in M dimensions, H = Y^=\{^Pk+mT~Q\) — Y^k=\ w{akak + h/2), with ak = -^(\/mu)Qk + i-j^z) and Pk = —ih-^, we can single out a ground state | 0 >: ak | 0 >:= 0, < 0 | PK | 0 >=< 0 | QK | 0 >= 0, H | 0 >= MLO%/2 | 0 > A n d then build up all the eigenstates of the Hamiltonain: \ n i . . . n M >= -7==L—{a\/Vnr • • • ( « M / v ^ ) n w | 0 > M H | nx... nM >— YI uh{nk + 1 / 2 ) | ni... nM > k=i But 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 [<3fc5P/] = iti6ki prevents us from finding eigenstates for both of them. But it is possible 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>= n ( 2 . 1 ) To find such a state, we can start with | p, q >= e~A \ x > and the identity eABe~A = B + jj [A, B] + j{\A, [A, B]] + . . . that stops at the second term if [A, B] = c-number, and 3 Chapter 2. Review of Coherent States and Path Integrals 4 t h e n f ind tha t (2.1) imposes the cond i t i ons : | x > = | 0 > a n d A = r>kQk-9hp*, w h i c h g ives: M \p,q>= e x p { £ TiPkQk ~ qkPk)} | 0 > k=l In te rms o f ak, ak a n d the comp lex va r iab le zk = -^{\/mujQk + ^ § ^ j ) M 1 1 | p,q > = | 2 > = e x p { £ - ( z * a j - * * a f c ) } | 0 > = e x p { - t a + 2 - zU)}\0 > (2.2) k=i b y w o r k i n g w i t h co lumns a > 2 / „ \ W e can eas i ly ve r i f y tha t < z | (Pk -Pk)2 \ z >=< z \ (Qk - qk)2 \ z > = %/2. So | z > is as close as poss ib le to a c lass ica l s ta te . If we use the iden t i t y eAeB = e^A+B+^A'B^2^ w h e n [A, B] = c - n u m b e r , we can rewr i te (2.2) as: ^ _ i , t , *U , n ^ (-^i/v^)" 1 (Wv^ )"" , 2 > = e e T I 0 > = e 2* 2^  7=r=f— 7-—,— | n 1 . . . n M > H - " M = 0 (2.3) F r o m th is we c a n show a \.z >= z \ z >, < z\a) =< z\z* a n d < z \ z' > = e x p { i ( z V - z T z / 2 - z ' V / 2 ) } ft (2.4) a n d also =/n< /• dzdz^ . . [ ™,dzkdz*k. Jj^w{z><zl=JE{^)lz><zl dzkdzt ds f, y r m,nj,...=o K = l £ ((zk/Vh)n"(zt/Vh)< I " i • • • nM >< n\ ... n'M \ = ^2 I ni...nM >< ni...n,M |= I ni...n M=0 (2.5) . 1 Sometimes it will be easier to work with one complex variable z, without any meaningful connection with its real (position) and imaginary (momentum) part. Chapter 2. Review of Coherent States and Path Integrals 5 So the | z > coherent states f o r m an overcomp le te set o f states. T o s u m m a r i z e , the h a r m o n i c osc i l l a to r coherent states have the p roper t ies : • E igens ta tes of the a opera to r : a | z >= z \ z >, so < z \ a \ z >= z. (mean values of pos i t i on a n d m o m e n t u m g iven b y p a n d q.) • A r e no t o r t h o n o r m a l : < z \ z' > ^ 6(z — z') • A r e overcomp le te : / ^'f^M \ z >< z \= I F i n a l l y , let m e prove the i m p o r t a n t f o l l ow ing ident i t y , tha t w i l l be usefu l la te r : F o r a n y c o m p l e x M x M m a t r i x a: e x p { a V a / ^ } | z > = e x p ^ z ^ e ' V - l)z} \ eaz > (2.6) P r o o f : f i rst e ° , < 7 a / f t ( a T z / f t ) = ea^a/h(a*z/h)e-a^a/h x e a ] ( T a / n = ay% (l + o + a2/2l + . . .)zea'aa'h = ( a V z / f t ) e a , < 7 ° / f t S o , us ing (2.3) CO 1 ea<*a/H | z > = e-z<z/2h £ _ L e . W » ( f l t ^ ) » | Q > „=o Vn\ oo i = c-*,*/2h y* - 4 = ( a V * / 7 i ) n | 0 > n =oVn! = e - * , * / 2 » c ( e " * ) t ( e ' * ) / 2 » | > w h i c h gives (2.6). Chapter 2. Review of Coherent States and Path Integrals 6 2.2 S p i n C o h e r e n t S t a t e s In the same sp i r i t as the h a r m o n i c osc i l l a to r coherent s ta tes, a l l three componen ts of the sp in J (Jx, Jy, Jz) c a n no t have def in i te eigenvalues for a g iven s tate of the sys tem. B u t i t is poss ib le t o def ine a state | n > ( ( n ) 2 = 1) such tha t : < ft I J I ft >= stin for a s p i n opera to r J, of s p i n s. T o find such a s ta te | n > , let us def ine a ' g r o u n d s ta te ' | 0 >=| s,s > , a n e igenstate of p ro jec t i on m = 5 of Jz : Jz | 0 >= s% | 0 >. T h e n < 0 | J | 0 >= s%k, where k = (0,0,1), a n d we o b t a i n a coherent s ta te | ft > = | 9, <j> > , us ing spher i ca l coo rd ina tes , b y pe r f o rm ing the app rop r i a te ro ta t ions : 16,4> >= e - W e - * " » / * | o > • (2.7) F o r s p i n 1/2 (where B are the P a u l i ma t r i ces ) : / cos(0/2) \ IM>.=i/2= for B±* \ e«'*sin(0/2) ) ( e " ' * cos(0/2) \ for 6 ± 0 (2.8) V s i n W 2 ) 7 A n d we f i nd <6,<f>\ 0', $ > 1 / 2 = cos(0/2) cos(0'/2) + sin(0/2) sin(0''/2)e-^*-^ for 9 ± TT = cos(0/2) cos(fi72)e*^-*'> + sin(0/2) s in (^ /2 ) for 9^0 w i t h < 0, (f> | d | 9, <j> >i/2= n = i s in 9 cos <f> + j s i n 9 s i n + fc cos 9 Chapter 2. Review of Coherent States and Path Integrals 7 T h e two dif ferent representa t ions are necessary because we need two coord ina te patches to coord ina t i ze sphere. T h e phase fac to r e"^ be tween these two patches is pu re l y t opo log i ca l . It w i l l ho t change the phys ics i n genera l . F o r a rb i t r a r y s p i n s: W e can use the sp inor represen ta t ion for a s p i n : 1 V>(-M) X (2s)l s+cr 8—o \{8 + a)\{8-<T)\ U 2 / So (2*)! 0i(a-c)<j> rr.„s+a c o s * * * ( 0 / 2 ) s i n * " ' 7 (0 /2 ) for 9 ± TT (2s)! ; - > ' ( s + ^ c o s s + < T ( 0 / 2 ) s i n s - < T ( 0 / 2 ) for 0 ^ 0 / : \ |M>.= \(s + a ) ! ( s - <r)\ (2.9) \ '• ) T h e phase fac to r be tween these two patches is n o w e2s*^, where we can recognize 2s has a w i n d i n g n u m b e r . It can be checked tha t < 0, </> | J \ 9,(f> >8= snfi a n d also <9,<f>\9',<f>' >.= [< 9,<f>\9',<{>'>1/2 2s (2.10) W e also f i n d < 0, <j> | ( J ) 2 | 0, <f> >„ - ( < 9,<f> \ J \ 9,<f> >,)2 = h2[s(s + 1) - s 2 ] = s%2 t he m i n i m u m va lue poss ib le . F i n a l l y , f r o m (2.9), we can eas i ly f i nd t he completeness re la t i on : fir r2ir rr\e,4»<w\. ( 4 . x d<t>sm(9)d9 _ (2.11) 2This might differ from other authors by a unitary transformation. Chapter 2. Review of Coherent States and Path Integrals 8 2.3 Propagator, Path Integral and Partition Function In q u a n t u m mechan i cs , a l l the i n f o rma t i on for the evo lu t i on of a sys tem c a n be s tored i n the p ropaga to r ( t rans i t i on amp l i t ude ) be tween an i n i t i a l s ta te | > a n d a f ina l s ta te | qj > at a t i m e t — tf — ti la te r , g iven b y : K^jit) =< Q i | eim'* | q, > where H is the H a m i l t o n i a n of the sys tem. T h i s a l lows F e y n m a n , b y us ing the completeness re la t ion / ^ ? | p > < p | 9 > < < ? | = / ^ ^ e " , p ? / f t | p >< q |, to wr i t e a p a t h i n teg ra l (w i t h q0 = qi a n d q^+\ = 9 / ) : K^f(t) = J dp0 n < * I I Po > e - ' P 0 9 l / f t < ?i | e W * 7 | p i > e - * ™ / » . . . iHt . . . <qN\ eh("+D I pN >< pN | qf > = J T O O / ^ 0 | [ ^ e x p { - i g b ; f e + i - ft) " ( ^ ) # ( P ; , f t ) ] } . = jdp0DpDqex?{-y\pq-H(p,q))dt} (2.12) where H(p,q) = ^ [ ^ f f i ^ + ^ ^ ^ l is the c lass ica l H a m i l t o n i a n , w i t h b o u n d a r y cond i -t i ons 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 hen we can p e r f o r m the p i n tegra t ions (be ing a G a u s s i a n ) , g i v i n g : = M j Dqexp{-jJ*'L(q,q)dt} (2.13) where M is an in f in i te cons tan t , a n d L is the c lass ica l L a g r a n g i a n . Chapter 2. Review of Coherent States and Path Integrals 9 T h i s new representa t ion has some di f f icu l t ies. F i r s t of a l l , there is th is in f in i te con -s tan t , <*/N -f 1, t ha t can not even be abso rbed i n the measure of Dq. F o r e x a m p l e , i f V = 0 we c a n p e r f o r m t he in tegra t ions : w h i c h g ives: T h i s c lear l y shows the f ine t u n e d cance l l a t i on o f y/N + 1 i n th is (s imple) case. F o r a more c o m p l i c a t e d s y s t e m , i t c o u l d be expec ted to step t h r o u g h some d ivergences. A l s o , t he Dq (= Flit V '^^ Tftt1^*.) m e a s u r e > con ta i n i ng the TV fac tor , i nd ica tes the di f f icu l t ies tha t m igh t a p p e a r b y pe r fo rm ing the N —> oo l i m i t . In s ta t i s t i ca l mechan i cs , at a tempera tu re T, t he i n f o rma t i on is s tored , i ns tead , i n the p a r t i t i o n f unc t i on : Z[/3] = t r ( e - ' » ) , 8 = ± Since the t race can be represented b y / < q | ( ) | q > dq, o r mo re genera l l y / e~ipqlh < q\( )\P> ^jt, so t r ( l ) —number of states ava i lab le , we f i nd : Z[B] = J K^iB^dqt w h i c h shows tha t the p a r t i t i o n f u n c t i o n is the in teg ra t ion over the i n i t i a l s ta te of the p ropaga to r tha t goes a r o u n d a loop ( i n i t i a l = f ina l ) for a ' t ime ' t = i(3fi. So i n te rms of the p a t h in teg ra l f o r m a l i s m , we can f i nd b y inse r t i ng TV reso lu t i on of u n i t y ( / | p >< p | q >< q | g ) i n e^H. B y us ing the F e y n m a n p a t h i n teg ra l , we find: N dpkdqk N /fl3 i DpDqexp{- j[ (-pq + H(p, q))dr) Chapter 2. Review of Coherent States and Path Integrals 10 where now q = limyvr^ooC2^^1) is a 'temperature derivative', and q(0) = q(/3), a loop in 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 dp0 integration as in (2.12). Applying this for our example, a free particle (H = we find from (2.14): or as we would do in statistical mechanics = f e - p H d p d q = H j E Z r ^ J 2xh J as expect. 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-*31*) where H = u{o)a + hM/2) - / n ( ^ ) < * i « ^ ' " i * > . . . < * i « ^ H « . > with (2.6) we find m = e-»*W J g e - 4 ? ( i - ^ ) < Z l | c - * * / % > ... _ p-pwhM/2 [ TT ( dzkdzk \ f ^ r zUk(, _ 2pwh/N\ Zkzk J t}i\(2*in)M) ti } 2% „-0wh/NZl+lZk+l , „-Pu>h/NZkZk+ln 2% + e h n Also by changing zk —• Z f c e * } w n / W , we find 1 m = eP«™,2 f g (*2*L) c- i (3.16) Jfc=i \\2Tnn) J 1Note 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. 11 Chapter 3. Coherent State Path Integral for the Harmonic Oscillator 12 T h i s gives us an exact a c t i o n , at the d iscrete l eve l , for th is p a t h i n teg ra l : S = J2[e^Nzlzk - zlzk+l] (3.17) fc=i These two fo rmu las can b e checked independen t l y , us ing the de te rm inan ts so lved i n A p p e n d i x A for a m a t r i x o f the f o r m Aitj = 6itj - e~pu%lN8i^j or A'itj = e0u,^N6itj - 6i+ltj, because the p a t h i n teg ra l is a G a u s s i a n : e-PwhM/2 e-pw*M/2 e0whM/2 ^whM/2 [de t (A) ]M (\-e-^)M [det(A'))M (e^-l)M 1 _ V p-0w1i(ml+...+mM+M/2) _ f (p-0H\ (o ,n\ - [ 2 S i n h ( ^ V 2 ) r - m „ . i „ = 1 £ " t r ( e ' ( 3 ' 1 8 ) 3 . 2 C o n t i n u u m L i m i t W h e n we are seek ing a c o n t i n u u m l i m i t of a p a t h in teg ra l , we wan t to keep te rms i n the s u m m a t i o n of the 3/N o rder (exc lud ing the e x t r a 0/N t e r m for each ' t i m e ' de r i va t i ve ) , so tha t we c a n a p p r o x i m a t e the s u m m a t i o n b y a n in teg ra l JQ dr. F o r the p a t h in tegra ls (3.16) of the last sec t i on , th is means the fo l l ow ing : Z[8] « e0whM/2 JDzdzie-*^i[zkZk-Zk+l)+%"hz><Zk] B u t is z\(zk — zk+i) rea l l y b e c o m i n g —z^'zdrl W e can eas i ly check tha t ( f zt'zdTJt = f z^zdr = z*z |J - f z^'zdr = - ( f z^zdr) Jo Jo Jo Jo so th is is pu re l y i m a g i n a r y . B u t for the d iscrete conterpar t N N 1 1 1 4(zk - Zk+i) = - Zk+i) - ~(zk - Zk+iVzk + •« I - Zk+i I2] (3.20) k=l k=l 2Chapter 3. Coherent State Path Integral for the Harmonic Oscillator 13 c lear ly ind ica tes the presence of a rea l t e r m tha t is m iss ing i n the c o n t i n u u m l im i t ! E v e n i f th i s rea l c o n t r i b u t i o n w i l l appear on l y to the j3/N o rder , i t cou ld b e re levant for a convergence of the p a t h in teg ra l . F o r examp le nitLi(l + Px/N) —• e P x as N —• oo, bu t do ing the a p p r o x i m a t i o n (1 + Px/N) « 1 w o u l d g ive 1 ins tead . F o r the p a t h i n teg ra l each in teg ra t ion w i l l con ta i n a P/N t e r m tha t w i l l b r i ng a f in i te mod i f i ca t i on at the end . T o f ind the c o n t i n u u m l im i t w i t h th is e x t r a rea l t e r m , we jus t use (3.20) i n the p a t h i n teg ra l (3.16) w h i c h gives the fo l l ow ing l im i t : Z[I3] « e 0 u h M ' 2 J D z D z * e - * t f ( - z H + w h z i z + W 2 ) d t (3.21) . Because of the presence of the N in e we do not rea l l y expec t any dependence o n e after the eva lua t i on of the p a t h i n teg ra l , bu t i t shou ld he lp to ' s m o o t h o u t ' the in tegra t ions . B e c a u s e we w i l l be l ook i ng for cor rec t ions to the order e i n the p a t h i n teg ra l , l i ke i n (3.21) , i t w i l l be i m p o r t a n t to keep t rack also o f the b o u n d a r y cond i t i ons at 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 in teg ra l , i t w i l l be usefu l t o use the fo l l ow ing re l a t i on , v a l i d i f / ( r ) is s m o o t h f r o m 0 to /?: /* = /((* " be) ^ k = 1 . . . . N , J2 = f MdT + 0(e3) (3.22) z jfe=i J o In ou r case, we w i l l have to cons ider the va r iab le zk, tha t we w o u l d l i ke to represent b y a con t i nuous curve z{r). T h i s representa t ion appears o n the figure 3.1. It is c lear , t h e n , t ha t the va lue of z./v+i = z((N + | ) e ) ^ z\ = z(^t), because Z ( T ) is not s m o o t h at r = 0 a n d T = p, bu t zpj+i is i ns tead a n a l y t i c a l l y con t i nued away. If we wr i t e Zk+i = *k + ezk + — zk + . . . where e = p/N Chapter 3. Coherent State Path Integral for the Harmonic Oscillator F i g u r e 3.1: D i sc re te a n d con t inuous representa t ion o f the z va r i ab le a n d use the iden t i t y (3.22) , we c a n a p p r o x i m a t e the ac t i on (3.17) b y : S = f [—Z^Z — \z*Z + OjtlZ^Z + ^•U2h2Z*z]dT + zjv(zjv=l — Ztf) Jo 2 2 T h e Z I J ( Z N = I — zjy) co r rec t ion comes f r o m the fact t ha t the i n teg ra l needs a s m o u t h f u n c t i o n , t ha t does no t cons ider Z J V + I = z\, th is has to b e cor rec ted b y ' h a n d ' : ZN+1 = z(B) + Zz(j) + 0 ( e 2 ) * i = * (0 ) + f * ( 0 ) + 0 ( e 2 ) z N = z{0) + 0(e) = z(B) + 0(e) T h u s to the e o rde r , we find Chapter 3. Coherent State Path Integral for the Harmonic Oscillator 15 T h i s a l lows us to rewr i te the ac t i on as t e S = J [-2+2 + -\z\2 + uhzjz + -w2a2z+z]dr as we found i n (3.21). T h e s e b o u n d a r y con t r i bu t i ons cou ld be ve ry i m p o r t a n t , as i t w i l l be seen i n the nex t sec t ion . T h e ^ u>2h2z^z t e r m w i l l be d r o p e d s ince i t is obv ious l y neg l igab le c o m p a r e d to utiz^z. A c t u a l l y , i f we cons ider the a p p r o x i m a t i o n /(—z^z+uhz^z-rcte \ z \2)dr t o the a c t i o n , where a is a pa rame te r tha t w o u l d connect the p a t h in teg ra l (3.19) at a = 0 to (3.21) a t a = 1 con t inuous ly , t h e n the d iscre te p a t h i n teg ra l w o u l d con ta in : N j 2 9 ~ _ (zk ~ Zk+t)*Zk J r ~ a \ z k - zk+1 |2] k=\ z 1 i ns tead of (3.20), where i t is i m p o r t a n t to check tha t the b o u n d a r y te rms has been i n c l u d e d i n the c a l c u l a t i o n . T h e eva lua t i on of the d iscrete de te rm inan t can be done e x a c t l y b y us ing the iden t i t y (A .80) o f a p p e n d i x A , a n d s i m p l y gives d e t e r m i n a n t 1 ^ = (S±I)"(C** - 1) + ( c - ^ * - 1) T h i s c lear ly ind ica tes the effect of the e t e r m . W h e n a = 1, the de te rm inan t is e x a c t l y the expec ted one . ( e ^ * - 1)M. B u t w h e n 0 < a < 1, t he second t e r m grows , a n d osc i l la tes i n s ign as a f u n c t i o n of N ! B u t | | is b igger t h a n | ^=1 |, so the second t e r m is s t i l l neg l igab le . F u r t h e r m o r e we have to reno rma l i ze for the in f in i te cons tan t ( ^^ • ) A r . B u t at the ex t reme case a = 0, co r respond ing to (3.19), we have (^ f^ -) = — i2^) = 1/2 so each t e r m has t he same fac tor (1/2)N, a n d the (—1)N of the second t e r m p roduces 2(cosh(/?u>/i) - 1) = [2s inh( /3w/ i /2 ) ] 2 for N even a n d 2 sinh(/?a;7j) for N o d d . N o t e tha t th is u n p h y s i c a l osc i l l a t i on w i l l a lways be cance l led b y t a k i n g the average. A l l of th is Chapter 3. Coherent State Path Integral for the Harmonic Oscillator 16 ind ica tes the sub t le ty of the p a t h in tegra l (3.19), a n d the regu la r i za t i on i n t r oduced b y the e t e r m . I w o u l d also l ike to p o i n t out tha t i f we star t w i t h the p a t h i n teg ra l (3.15), i ns tead of (3.16) , we w o u l d o b t a i n the 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 i n c l u d i n g the e cor rec t ions , bu t w i t h e ~ P w h M / 2 i ns tead of e P w h M / 2 \ T h i s fac to r ac tua l l y comes by d o i n g the a p p r o x i m a t i o n ujnz\zk+\ « u%z\zk for the c o n t i n u u m l i m i t . So th is e x a m p l e ind ica tes how sensi t ive the d iscrete s u m m a t i o n is to the d iscrete i ndex (k = l , . . . , i V ) . A s l ight m o d i f i c a t i o n ( l ike k —• k + 1) m i g h t g rea t l y affect the p a t h i n teg ra l . 3.3 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 T h e ac t i on S = /(f ( — z T i + whz^z + a | | z \2)dr i n the p a t h i n teg ra l is eva lua ted a l ong a con t inuous curve z ( r ) € C f r o m r = 0 to T = /?, w i t h z(0) = z(/3), a n d a c lass ica l e q u a t i o n of m o t i o n is a cu rve z ( r ) tha t m in im i zes the ac t i on S to 5" c. T h e o ther t ra jec tor ies z ( r ) w i l l g ive a n ac t i on S > Sc, thus c o n t r i b u t i n g less to the p a t h i n teg ra l . T h e semic lass ica l a p p r o x i m a t i o n is to cons ider on l y the c lass ica l so lu t ions ( / dzce~Sc) i n the p a t h i n teg ra l . M o r e exp l i c i t l y , let us wr i t e z = zc + z where z c is the c lass ica l so l u t i on , w i t h t he b o u n d a r y cond i t i on z c ( 0 ) = z c ( /?) = z(0) = z(/3) = z 0 . So z w i l l be a c o m p l e x curve w i t h z(0) = z(/3) = 0. T h e in teg ra t i on w i l l be separa ted as So t he p a r t i t i o n f unc t i on is where Chapter 3. Coherent State Path Integral for the Harmonic Oscillator 17 = S(zc) + S(z) +./ z^(-zc + u}%zc - a-zc)dr + / (z\-\-u%z\ - a-z[)zdT Jo z Jo z T h e second l ine is ob ta i ned after i n teg ra t i on b y par ts a n d the use of the b o u n d a r y cond i t i ons on z. T h e last two in tegra ls are zero b y the equa t i on of m o t i o n of zc a n d z\, so Z[8] = e^hM'2 J DzoDzle-fcW J DzDz^-^ T h e semic lass ica l a p p r o x i m a t i o n is to d rop the 5 p a t h in tegra l : ZM = epwhM'2 J Dz0Dzle-^So(-^+^^+a^)dr where —zc + u>%zc — ct^-zc = 0 a n d z\ + Lofiz\ — a^z\ = 0 T h e s e two equat ions of m o t i o n co r respond to the E u l e r - L a g r a n g e equa t i on (obv ious ly? ) for a L a g r a n g i a n : L = —z^'z + uftz^z -f a | | i | 2 . T h e fact t ha t the equat ions of m o t i o n of zc a n d z\ does no t seem comp lex con jugate comes f r o m the fact t ha t dr = —%dt, so r t rans fo rms hke a c o m p l e x n u m b e r , w h i c h exp la ins the s ign change i n zc i n the equa t i on of m o t i o n . B u t r or 8 mus t s t i l l be cons idered as rea l , i t is p a n d q o f z t ha t c a n get comp lex i f i ed i n the search for e x t r e m a l o f the ac t i on . B y in teg ra t i ng b y pa r t s | i c | 2 , a n d us ing the zc equa t i on of m o t i o n , we can s imp l i f y Sc to 5,c = K m a | « J i e |J F o r a = 0, we t r i v i a l l y find Sc = 0. In fact t he equa t i on of m o t i o n becomes l i near , a n d there ex is ts no so lu t i on w i t h zc(0) = zc(8)\ So the semic lass ica l a p p r o x i m a t i o n is not a p p l i c a b l e , or at the best i t gives a c o n s t a n t ( / £ ) z o ^ ^ o e ° ) - T h i s i nd i ca tes , a g a i n , the di f f icu l t ies of the p a t h i n teg ra l (3.19). F o r a = 1, we find two independen t so lu t ions , ew1ir a n d e - 2 T / £ for zc, e~whT a n d e 2 r / e for z\. B y cons ider ing e ve ry s m a l l a n d the b o u n d a r y c o n d i t i o n , we o b t a i n : zc = z0[(l - e - ^ ) e " 2 T / £ + ew%^) Chapter 3. Coherent State Path Integral for the Harmonic Oscillator 18 z\ = 4[(1 - e - ^ * ) c 2 ( r - « A + e - « * r ] So 5 C = 2 | z 0 I2 e -^ / 2 s inh ( / 3u ; f t / 2 ) a n d Zac[/3] = e(3w™/2 J (2irih)Me lf£l_ 2 e-^h /2 Bi nh(/3 l l,ft/2) M 2 e - / J - * / 2 s i n h ( ) 9 w R / 2 ) w h i c h is the r igh t answer , except for a fac tor e^whM. In fact i t as been sa id t ha t a semic lass ica l a p p r o x i m a t i o n for an h a r m o n i c osc i l l a to r shou ld g ive the r igh t resu l t [1]. A c t u a l l y the con fus ion arises b y the a p p r o x i m a t i o n no ted i n the last sec t i on , a change of ufizlzk+i t o uhzlzk i n the p a t h in teg ra l (3.15) w i l l g ive an e x t r a fac to r e~^whM to the p a r t i t i o n f u n c t i o n . T h e s e two a p p r o x i m a t i o n s cance l each o ther to g ive an exac t resu l t ! It is obv ious f r o m m y ca l cu la t i on tha t the correct s ta tement w o u l d be : where 2(0) = z(B) = 0. 3.4 R e g u l a r i z a t i o n So fa r , we were ab le to wr i t e d o w n an exac t express ion for the d isc re t i sa t ion of a p a t h i n t eg ra l , a n d eva lua te the de te rm inan t e x a c t l y a f terwards. In more comp l i ca ted cases, i t w i l l be necessary t o eva lua te a p a t h i n teg ra l b y us ing some a p p r o x i m a t i o n s , because the de te rm inan t w i l l no longer b e exac t l y so lvab le . In fac t , these a p p r o x i m a t i o n s w i l l usua l l y generate some divergences tha t we w i l l have to regu lar ize b y us ing var ious techn iques . He re , I w i l l i nd i ca te a genera l w a y to regu lar ize these d ivergences a n d the use of the con -tou r in tegra l r egu la r i za t i on for the coherent s ta te p a t h i n teg ra l of th is chap te r . I w i l l a lso use the exact de te rm inan t f o u n d ear l ier t o f i nd the l im i t a t i ons of these regu la r i za t ions . Chapter 3. Coherent State Path Integral for the Harmonic Oscillator 19 T h e con t inuous a p p r o x i m a t i o n is der i ved i n the fo l l ow ing way ; s ince the p a t h i n teg ra l uses p e r i o d i c f unc t i ons , z (0 ) = z(/3), we can use a set of e igenfunct ions \Pfc(r) = e 2 7 r t k T ^ , as i n the a p p e n d i x A for the d iscrete case. F o r a genera l p a t h i n teg ra l : / DzDzU-Uo'**»*+°&)' * = L J detM(a + bft+ c$) (3.23) we o b t a i n for the de te rm inan t : d e t . = n + + cRR 2] (3-24) k=-N/2 P P where fi/N appears because a n in teg ra t i on of TV sl ices w i l l g ive a fac tor A T = fi/N i n f ront o f the L a g r a n g i a n . It is a lso necessary for keep ing the r igh t un i t s . T h e p r o d u c t (3.24) is the express ion tha t needs to be regu la r i zed . B u t before ana l ys i ng these p r o d u c t s , let m e star t f r o m the exact f o r m u l a (A .80) for the p r o d u c t de r i ved i n a p p e n d i x A for the p a t h in teg ra l (3.16): ew0hM/2 ^ ] = TdetF' ^ det(x) = fl A* = fl (e*/N - e^k/N) = e* - 1 (3-25) k=i k=i where x = flun. F r o m (3.25), we c a n con temp la te the fine t u n i n g of the p roduc t of a l l the eigenvalues. O n the comp lex p lane these eigenvalues f o r m a c i rc le o f rad ius 1, centered at ex/N, equa l l y spaced . T h e i r m o d u l u s va ry f r o m exlN — 1 « x/N < 1 for 0 « k <C N, to exlN + 1 « 2 for k « N/2. So a huge number o f cance l la t ions mus t be i nvo l ved , i n th is p r o d u c t , t o g ive s i m p l y ex — 1 ! W e w i l l need also the T(x) f unc t i on def ined b y : r(x) = - l^n[det(x)] dx Chapter 3. Coherent State Path Integral for the Harmonic Oscillator 20 JV ex/N N 1 E N{eX/N _ jriklN) = E N Q _ c2 B*=£ ) ( 3- 2 6) e* - 1 tha t can be eva lua ted by con tou r in teg ra l w i t h the func t i on | c o t ( z / 2 ) , w h i c h has po les at z = I'KXI (n in teger) w i t h res idue 1. T h i s a l lows us to wr i te : N r r ( x ) = £ I \ c o t ( z / 2 ) dz feiMfc) 7Y ( l_c 4 T<r)27r» where C(fc) is a n o r ien ted loop a r o u n d 27rfc. T h e ana lys is o f the i n t eg rand f(z) shows t ha t 0 i t con ta ins o thers po les at z = —ix + 2irnN. F u r t h e r m o r e , f(z) be i ng a pe r i od i c f unc t i on (f(z + 2nN) = f(z)), we c a n t h i n k of f(z) as a func t i on on a cy l i nde r o f c i r cumference 2nN o r ien ted a long the i m a g i n a r y ax is . A n in teg ra t i on a long an i m a g i n a r y l ine up a n d t h e n d o w n b y ano the r i m a g i n a r y l ine sh i f ted b y 2nN w i l l cance l each o ther . It can be checked tha t f(z) —> 0 as Im(z) —> —oo a n d f(z) —• ^  as Im(z) —• oo. So b y us ing a con tou r i n teg ra l , as s h o w n i n f igure 3.2, we o b t a i n = - 1^ ~* dR<Z) _ l \*<ZI2) dz (o 07) { ' Jo 2N 2iri J-* N(l-e^)2TTi K ' = \ ~ 5 c o t ( - » * / 2 ) = F o r a c o n t i n u u m l im i t a p p r o x i m a t i o n , we have to f i nd some a p p r o x i m a t i o n s to t he p r o d u c t (3.25) tha t w i l l con ta in a l l t he phys i ca l p roper t ies of ou r m o d e l . B u t the f ine t u n i n g o f th is p r o d u c t shows tha t we can expec t a lot of d ivergences. T h u s , i n these d ivergent cases, i t is necessary to i n t roduce a regu la r i za t i on tha t w i l l t h r o w away the divergent p a r t , bu t keep t he phys i cs of the m o d e l , o r i n o ther wo rds , the d ivergent t e r m shou ld not d e p e n d on a n y p h y s i c a l pa ramete r . F o r ou r p r o b l e m , the eigenvalues A * become independen t of x for large values of k (Afc w 2 + 0(x/N)) so the exact express ion e2mk^N for large k shou ld no t be 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\ f2nik\ 1 (2mk\m e x p { — r i + ( — j that I will call the m t h order approximation, and this should produce some meaningful 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 1st and 2nd 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 to N, we o b t a i n : N T 9-n-ik m = 2: d e t 2 ( * ) = + (3-29) In fac t , the a p p r o x i m a t i o n s (3.28) a n d (3.29) co r respond exac t l y to the p a t h i n teg ra l (3.19) a n d (3.21) respect ive ly , w i t h the a p p r o x i m a t i o n (3.23). So th is a p p r o x i m a t i o n for the de te rm inan t has some jus t i f i ca t ion now . L e t us f i rst s t u d y (3.28). T h e i dea is to eva lua te the p r o d u c t (3.28) for a g iven i V , a n d t h e n d i v i de the resul t b y the same p r o d u c t , a n d same TV, w i t h x — 0. T h i s d i v i so r does not d e p e n d o n x at a l l , so i t w i l l not change a n y t h i n g . A n d s ince the p r o d u c t becomes independen t o f x for la rge k, th is d i v i so r cancels exac t l y the d ivergent pa r t t ha t we wan t to t h r o w away. M o r e prec ise ly : 6 k=-N2N + 1 ~ 2 / V + l J = 2 /Y + l 1}}^2N + V + ( 2 7 V + r J = wr,U!^W+{^r] (3'30) Since de t i ( 0 ) = 0, we can no t d i v i de d e t i ( x ) b y de t i ( 0 ) , bu t the re levant fac to r i n (3.30) tha t needs to be d i v i d e d is : 1 " 2*k 2 (2K)2N ^ T i t = 1 l 2 j V + l J ~ (2N + } ~ V a n d fu r t he rmore , see [13], l ™ A h -L( X W s i n h( x/ 2) So Jim ( i ) » « d * , ( , ) = ^ _ = 2 s i n h ( I / 2 ) . DETi(x) (3.31) Chapter 3. Coherent State Path Integral for the Harmonic Oscillator 23 W h e r e DET\(x) is now the regu la r ized de te rm inan t for the f i rst order a p p r o x i m a t i o n , or for t he p a t h i n teg ra l (3.19). W e see tha t de t ( x ) = exl2DETl{x), so DETx(x) is a lmos t r i gh t , except fo r the s m a l l fac tor ex/2. N o t i c e , a g a i n , the u n u s u a l co inc idence of th is fac tor appea r i ng here a n d i n f ront of the p a t h i n teg ra l (3.19). If we were to forget abou t th is fac tor i n (3.19), we w o u l d t h i n k tha t we have the exact p a r t i t i o n f u n c t i o n . S ince the d ivergence is a s imp le fac to r i ndependen t of x , the f u n c t i o n (3.26), T ( x ) , for the a p p r o x i m a t i o n (3.28), shou ld not c o n t a i n a n y d ivergence at a l l ! N 1 r x ( x ) = £ *^  "™" ^^ ^^  * t \ cot(z/2) dz Uot(z/2)dz t-N~l = _t f c o t U / 2 j dz y [ -Jz=-ix x — iz 2ni \ , x * = - o o k=N+i/ ^ - 2 ^ So h m r x ( x ) = N^oo 1 V ' 2 ex - 1 as g iven b y DET\(x) i n (3.31). N o t e also tha t the in teg rand of the con tou r i n teg ra l is no longer p e r i o d i c (f(z + 2TTN) ^ f(z)), w h i c h ind ica tes tha t we need the fu l l R i e m a n n sphere for the con tour i n teg ra l . T h e a p p r o x i m a t i o n , t h e n , changes the topo logy of the d o m a i n o f i n teg ra t i on t ha t is needed for the in teg ra t i on ( th is w i l l be t rue for any o rder ) . N o w we c a n go fu r the r , a n d ana lyse the p r o d u c t (3.29). B y us ing the same m e t h o d tha t we used for (3.30), we c a n wr i te : / / x 1 27r2Jfc l 2 " N 2 * RI N(N + 2x), , x T h e p r o d u c t (3.33) has the same d i f f i cu l ty of (3.30) tha t d e t 2 ( x ) —* 0 for x — 0 . B u t even worse, the last two p roduc ts do not converge to we l l k n o w n func t ions because te rms 4 Chapter 3. Coherent State Path Integral for the Harmonic Oscillator 24 N 1000 5000 10000 X 0.1 0.5. 1 2 5 10 DET2{x) 0.1032846 0.5926451 1.434417 4.451911 59.72026 3615.019 e - ° - 1 8 0 7 l d e t ( x ) 0.1032875 0.5926791 1.434225 4.451257 59.72432 3615.389 T a b l e 3.1: Second order p r o d u c t regu la r i za t i on l i ke (jTJi) or ^ [ * t f f i w o u l d no t decrease to zero as k increase to N: N N So the on l y w a y (by th is m e t h o d ) to eva lua te (3.29) is to a p p l y the p rocedu re d i rec t l y w i t h o u t l o o k i n g for s imp le so lu t ions : [x - 2nk + 2Tr2k2/(2N + 1)] TV DET2(x) = l i m x J J te-jwo I - 2 « * + 2 * * * 7 ( 2 t f + 1)] " [27r(2JV + l)k)2 + [x(2N + 1) + 2 T T 2 P ] 2 = h m x I I N-*oo f-\ k = 1 [2TT(2N + l)k]2 + [ 2 T T 2 P ] 2 ^ - 3 4 ^ A n u m e r i c a l s t u d y of (3.34) ind ica tes tha t DET2(x) is i n between (3.31), DETi(x), a n d (3.25) , t he exact resu l t . S ince DET\(x) — e - x / 2 d e t ( x ) , a s imp le e x p o n e n t i a l f i t fo r DET2(x) = e - 7 1 de t (x ) shows a ve ry accura te resul t for 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 the tab le 3 .1 , where DET2(x) is eva lua ted w i t h the equa t i on (3.34) for the i n d i c a t e d va lue of N. A s t u d y of (3.33) w i t h the use of T(x) g ives: N r 2 ( x ) = £ fcfrW x - 2wik + 2n2k2/(2N + 1) \Jztt-ix Jzts4iN/ X — 1 iz + 2(2iV+l) 2iti fc=—00 du l i i l + l - j j , , , 2ex-l 2 \ J N x-2-rnu-r TO" + 0(1/N) 2N+1 Chapter 3. Coherent State Path Integral for the Harmonic Oscillator 25 t h e n l i m T2(x) = — 7 So DET2(x) = e 7 1 de t ( x ) as expec ted , a n d 7 is g i ven b y : du l i m 2Re N-400 ( 1_ [°°__  \ 2ir JN u(i — 2N+1-= h m Re f - i l n ( — ^ — + - ) \%) / v - 0 0 V X X2N + 1 u ; l 7 V 1 2 = - a r c t a n ( - ) « 0.18045 7T TT as f o u n d ear l ier ! If we were to check for h igher order a p p r o x i m a t i o n s , we w o u l d get be t te r resu l t ; ac tua l l y i t is more l i ke ly tha t we w o u l d find a 7 co r rec t ion at each o rder , bu t w i t h 7 b e c o m i n g smal le r . N o t i c e tha t the 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 to 00 i n s tead , w o u l d g ive exac t l y the r igh t answer ( there w o u l d be no 7 co r rec t ion ) . It is very un l i ke l y , however , t ha t at any order of the a p p r o x i m a t i o n we w o u l d o b t a i n a n exact resu l t ( l ook ing at (3.25)). T h i s shows the i m p o r t a n c e of keep ing k f r o m — N to N, for 2N +1 steps, i n t he p r o d u c t . T h e r e has been some s tudy of these p roduc t s b y us ing a R i e m a n n z e t a f u n c t i o n for regu la r i z i ng some in f in i te p r o d u c t s , such as I l ^ - o o >^ these fo rmu las need the m o d i f i c a t i o n jus t n o t e d , tha t k mus t r u n f r o m —00 t o 00 r ight at the beg in i ng [2,3]. T h i s w o u l d genera l l y l ead to some er rors , as a l ready p o i n t e d ou t here ( th is has been f o u n d also b y [2]). So I w i l l no t e labora te on the use of the R i e m a n n Z e t a f u n c t i o n for the rest of m y work . T o find i f i t is app l i cab le , we just look i f t he p r o d u c t s con ta i n the var iab le N, tha t c o u l d s p o i l i ts convergence. In these cases, a n u m e r i c a l s t u d y m igh t gives some a d d i t i o n a l cor rec t ions to these p a t h in tegra ls . Chapter 4 Coherent State Path Integral for Spin 4.1 Discretisation with Spin Coherent States In t he last chap te r , we s tud ied the proper t ies of the h a r m o n i c osc i l l a to r coherent states us ing a n h a r m o n i c osc i l l a to r H a m i l t o n i a n . W e where ab le to o b t a i n an exact p a t h i n teg ra l at t he d iscrete leve l . In th is chap te r , we w o u l d h k e to s t u d y the s p i n coherent s ta tes. In hope o f f i nd ing exact so l u t i on , i t w o u l d be in te res t ing to cons ider a s imp le H a m i l t o n i a n w i t h a s p i n opera to r . M o r e prec ise ly , we w i l l be in terested i n the p a r t i t i o n f u n c t i o n for a s p i n s pa r t i c l e i n a constant magne t i c field B: H = fiB-J , ( J ) 2 = %2s(s + 1) Z[B) = t r ( e " ^ ) = t r ( e - ^ ^ ) W i t h the he lp of the sp in coherent s tates (2.9) a n d (2.11) we can wr i t e the fo l l ow ing p a t h i n teg ra l : m = / ft =|P < Ik, I e-°w I lM, ^ >. J k=l \2a+l) B y the proper t ies of the sp ino r i a l rep resen ta t ion , a n d the use of (2.10) we can t r a n s f o r m i t i n to : J k=l \2s+l) T h i s is abou t as fa r as we can go at the d iscrete leve l . A c o n t i n u u m l im i t s t u d y of (4.35) w i l l fo l low, b u t f i rst let us t r y to ana lyse the same sys tem b y us ing the h a r m o n i c osc i l l a to r coherent s tates. 26 Chapter 4. Coherent State Path Integral for Spin 27 4 . 2 T h e S c h w i n g e r - B o s o n M o d e l It is k n o w n i n q u a n t u m mechan ics tha t a two d imen t i ona l h a r m o n i c osc i l l a to r (see sec t ion 2.1) s ta te | n 1 ? n 2 > c a n be represented ins tead b y two numbers m , s such tha t n\ = s + m, ri2 — s — m, so m r u n f r o m — s t o s b y step of 1, a n d 2s is a pos i t i ve in teger . T h i s looks ve ry m u c h l i ke a sp in rep resen ta t ion , a n d i n fact the represen ta t ion of a sp in s pa r t i c l e b y us ing the two d i m e n t i o n a l h a r m o n i c osc i l l a to r is ca l led the S c h w i n g e r - B o s o n m o d e l , a n d i t is e x p l a i n e d i n deta i ls i n A p p e n d i x B , where we w i l l be us ing h a r m o n i c osc i l l a to r coherent s ta tes in th is sec t ion . B y us ing the S c h w i n g e r - B o s o n represen ta t ion we can eva lua te the same p a r t i t i o n f u n c t i o n of las t sec t ion : w i t h the he lp of (2.6) we o b t a i n (4.36) T h e A i n teg ra t i on i n (4.36) is i n fact a c i rc le i n the comp lex p lane . A n d it w i l l be use fu l , somet imes , t o represent th is c i rc le w i t h a g i ven rad ius r, i ns tead of r = 1 i n (4.36). T h e p a t h i n teg ra l (4.36) is t hen mod i f i ed to : -i^/n(^)r^eXP{4|[2s,aM^)+A - r < A 4 e - « ? S " V , ] } (4.37) T h e A va r iab le is i n fact a gauge p o t e n t i a l , to see th is we c a n ver i fy t ha t (4.36) a n d (4.37) are invar ian t under the t r a n s f o r m a t i o n : zk -> etakzk , A f c -> Xk + ah - ak+i (4.38) Chapter 4. Coherent State Path Integral for Spin 28 It is i m p o r t a n t to rea l ize t ha t we were ab le to o b t a i n , at the d iscrete leve l , an exact p a t h i n teg ra l w i t h a d iscrete ac t i on : N S= ^fisihXk + z\zk - eiX*zle-^B-*zk+1} (4.39) k=i where c = 8/N as u s u a l . 4.3 Equivalence of the two Representations T h e first ques t ion tha t one m a y ask is : are the two last representa t ions rea l l y equ iva len t? S ince (4.35) does no t con ta in a n y gauge va r iab le ,A , the f irst t h i n g w i l l be to in tegra te d i rec t l y th is va r iab le i n (4.36). T o do so, let us ca l l wk = e , A * , t hen <"»> N o w we c a n represent zk b y : zk = rke,'1k zkVK (4-41) W h e r e z is res t r i c ted to z^z = 1, thus z T z = r2%. T h e 7 is jus t a phase fac to r , i n fact t he H o p f phase , c o m i n g f r o m the passage of C 2 to CP1 = S2. T h e H o p f phase w i l l p l a y an i m p o r t a n t ro le la ter o n . A l s o we c a n check tha t dz^dz = h2i2r3 s\n(0)drd^d(j)d6. P u t t i n g e v e r y t h i n g i n to (4.40) g ives: r f2* sm(6k)d<f>kd6k f°° d<yk(rl)2a+1d(rl). t * x 2 s v-" r> (4.42) I n tegra t ing the phase 7jt a n d the rad ius rk of zk, t ha t separates c o m p l e t l y i n the p a t h i n teg ra l , a n d re i n t r oduc i ng a state \z > l a te r , g ives w - f f fi ^ j f f " ' w -J 0 J 0 k=l \2s+l) Chapter 4. Coherent State Path Integral for Spin 29 •* " s\n(0k)d<j>kd9k , ^ , _<a f i.at*a|_ fit fiT -U n I J S ! _ . i< i>\e-*"""\3l*i >?• (4.43) where \zk > is the same as \z > bu t res t r i c ted to the cond i t i on z^z = 1. T h e equ iva lence of (4.43) a n d (4.35) is then exac t l y demons t ra ted . F u r t h e r m o r e , th is ind ica tes tha t the |0, <f> > sp in coherent s ta te can be represented s i m p l y b y a two d i m e n s i o n a l h a r m o n i c osc i l l a to r coherent s ta te , \z > , w i t h the res t r i c t ion z^z = 1. A n o ther w a y of ve r i f y ing (4.37) d i rec t l y , a n d t h e n (4.35) b y the equ iva lence p roved above , can be done b y us ing the de te rm inan t fo rmu las (A .77 ) i n a p p e n d i x A , w i t h the p a t h i n teg ra l (4.37) : Z\ff\ = f TT d X k 1 1 J J ki\ 2* (re^y< det[Mij>mn] where A f , J i m n = S^jSm^n — relX>(e~i£B''ff)mn6iij+i, w i t h i,j = 1 to N c yc l i ca l l y , a n d m , n = 1,2. W i t h the he lp of (A .82) a n d ( A . 8 3 ) , we c a n show tha t d e t [ M ] = (1 - r V ' S ^ e - ^ X i - rNe^L, ^ So b y us ing wk = retXk, we find N m ~ f Zi wl^ (1 _ c - * l * l n f = 1 Wj)(l - e ¥ l * UU u>j) { 4 M ) where the con tou r i n teg ra l is a c i rc le of rad ius r a r o u n d the o r ig in . A c t u a l l y we can see tha t the de te rm inan t w i l l create a po le i n the in teg ra t i on w h e n r = e - i ^ (and also at r = e " ^ ) . T h i s ind ica tes tha t the G a u s s i a n i n teg ra l is no longer convergent over th is va lue o f r ! T h e r e is no surpr ise s ince the A in teg ra t i on has been i n t r o d u c e d to reduce the n u m b e r o f states f r o m an h a r m o n i c osc i l l a to r (oo) t o a sp in s pa r t i c l e (2s + 1), a n d we p e r f o r m th is A in teg ra t i on af ter the z i n teg ra t i on i n (4.44) (wh i ch co r respond to a t race) . So i t jus t happens tha t for r < e the in teg ra t ion converges, we can not hope for more . T h i s exp la ins w h y the p a t h i n teg ra l (4.36) w o u l d not g ive a correct answer i f we in tegra te z f i rs t . T h u s if we come b a c k to ou r in teg ra l (4.44) a n d in tegra te wi, we Chapter 4. Coherent State Path Integral for Spin 30 encoun te r two poles a t : wx = e 3^.?j; 2 . . . WN) a n d the i n teg rand vanishes for —* oo fast enough tha t we can de fo rm the con tou r o f in teg ra t i on a n d then use the res idue theory. W h i c h gives m=/ n dwk j e + 2iriwk 1 (1 _ e ^ l ^ l ) (1 - e - ^ l ) s i n h ( ^ ( 2 s + 1)) s i n h(^l ) = t r (e )- 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 in teg ra l has been a l ready ca lcu la ted at the d iscrete leve l a n d appears at the equa t i on (4.35). N o w we have to work ou t a c o n t i n u u m l im i t a p p r o x i m a t i o n , w h i c h means tha t we have to f i nd a c o n t i n u u m a p p r o x i m a t i o n to the express ion : < 0k,<f>k | e-*?8* | 6k+u<f>k+1 > i / 2 (4.45) In the c o n t i n u u m a p p r o x i m a t i o n , we can jo in t A: t o k +1, con t inuous ly , b y us ing a T a y l o r series e x p a n s i o n of | 0k+i,<f)k+i > a r o u n d \0k,<j)k > , us ing the f i rst p a t c h i n (2.8), where , for s i m p h f i c a t i o n , I w i l l remove the k i ndex o n a l l t he var iab les : I 0k+i,<f>k+i >- \9,<f> > +-- s i n ( 0 / 2 ) 0 ^ ( c o s ( 0 / 2 ) 0 ^ 2 z s i n ( 0 / 2 ) < £ ) e ^ c o s ( 0 / 2 ) 0 2 + 2 s i n ( 0 / 2 ) 0 v ( s i n ( 0 / 2 ) 0 2 - 4s cos(0/2)0<£ + 4 sin(0/2)<^ 2 - 2 cos (0 /2 )0 - 4i sin(0/2)<£)e«* \ + • Chapter 4. Coherent State Path Integral for Spin 31 A n d also use the expans ion : 2 8 So t ha t (4.45) becomes, at the second order i n e: l-^B-n+^fi2h2\B\2+eism\9/2W^ 2 8 8 - l ^ [ B x ( c o s ( 0 ) . c o s ( ^ ) 0 " + ism(<f>)6 + i s i n ( 0 ) e * ' ^ ) + £ v ( c o s ( 0 ) sm(<j>)9 - i cos{<j>)9 + s i n ( 0 ) e ' ^ ) + Bz(- s in(0)0 - 2isin2(0/2)<^)] = exp j - ^ J ? • n + ^ - ( \ B \ 2 - {B • n)2) + | ( 1 - oos(0 ) )^ - ^ ( 0 2 + s i n 2 - 4 ^ ( s i n 2 ( 0 / 2 ) < £ ) ) - ^ [ B M sin(<£)0 + i s in(0) cos(0) cosU)j> + - ^ - (s in (0) cos(^ ) ) ) o r 4 dr +By(-icos(<j>)9 + i s i n ( 0 ) cos(0) sm(<j>)<j> + -^ - (s in(0) sin(<£))) or + £ , ( - t s m 2 ( 0 ) ^ + ^(cos(0)))]} = exp • n + ^L{B x n ) 2 - + cos (0 ) ) ^ - ^{kf + ^ B • (n x S ) } •exp l ^ " ^ • * + T ^ 1 " C0S(*)W} (4-46) I O T * * J at the point fc where the ± 1 has been i n t r oduced for t a k i n g i n to accoun t the cho ice of the two patches i n (2.8) ( to get t o the second p a t c h , we m u l t i p l y b y e ^ * * - * * + i ) = e - " * _ 1 £ * + - ) . A s i t has been exp la i ned i n the last chap te r , we have to correct for the b o u n d a r y cond i t i ons , w h e n we are pass ing f r o m N to N + 1 = 1. W h i c h means tha t we have to a d d the fo l l ow ing t e r m to the ac t i on of the p a t h i n teg ra l : i[< M I ^ M > | 0- < MlJjr lM > M = -j(±l - c o s ( 0 ) ^ = - J Q P [ ^ ( ± l - cos(9)))}dT Chapter 4. Coherent State Path Integral for Spin 32 M u l t i p l y i n g (4.46) , for k = 1 to N, a n d us ing the in tegra l a p p r o x i m a t i o n gives a con-t i n u u m l im i t a p p r o x i m a t i o n , where i t cou ld be no ted tha t the b o u n d a r y t e r m , above , cancels the last t e r m i n the second bracket i n (4.46), wh i l e the f i rst t e r m vanishes b y the b o u n d a r y c o n d i t i o n , n (0) = n(3). W e f ina l l y ob ta ins the c o n t i n u u m l im i t a p p r o x i m a t i o n of t he sp in coherent s ta te p a t h i n teg ra l : Z[B] = J DnS((n)2 - 1) exp j- J^M^l + cos(0))<£ + fihsB • n + « ( ^ _ ^ S x ^ _ ^ . ( a x ^ . ( 4 . 4 7 ) A s i n (3.21), t he c te rms can be though t of as a regu la tor for the p a t h i n teg ra l . A n d i n fac t , for the two r e m a i n i n g t e rms , one is B • n w h i c h can be e i ther pos i t i ve or nega t i ve , a n d the o ther one is pu re l y imag ina ry . T h e n w i t hou t these e te rms there rea l l y w o u l d not be any convergent te rms for the p a t h in tegra l at a l l , w h i c h w o u l d m a k e the c o n t i n u u m a p p r o x i m a t i o n mean ing less . T h e t e r m JQ (l-\-cos(9))^>dT i n (4.47) ac tua l l y represent the a rea on the sphere enc losed b y the vec to r n i n i ts c losed loop m o t i o n , i n the S o u t h po le s ide. W h i l e J j f ( — l - f c o s ( # ) ) ^ > d T represent m i n u s the area seen f r o m the N o r t h po le . T h e s e two te rms a lways dif fer b y a m u l t i p l e of in, l eav ing the p a t h i n teg ra l s ingle va lued (s ince 2s is a n in teger) . It is ve ry in te res t ing to no t i ce tha t th is t e r m is pu re l y t opo log i ca l . I ts re la t ion w i t h the H o p f phase w i l l b e m a d e c lear at the nex t sec t ion . 4.5 C o n t i n u u m L i m i t o f t h e S c h w i n g e r - B o s o n M o d e l In a f irst a p p r o a c h , I w i l l s tar t w i t h the ac t i on (4.39) o f the p a t h in teg ra l (4.36). T h e n , t r y to f ind a covar ian t way to rewr i te th is a c t i o n , i n the same sp i r i t of sec t ion 3.2, t ha t w i l l be su i tab le for a c o n t i n u u m l im i t a p p r o x i m a t i o n . Here , o f course, we m igh t expec t some di f f icul t ies c o m i n g f r o m the gauge va r iab le A, tha t has to be in tegra ted out to rea l ly Chapter 4. Coherent State Path Integral for Spin 33 get a sp in s pa r t i c l e . T h e f irst t h i n g t o do is to find a covar iant der i va t i ve of z , tha t w i l l t rans fo rms i n the same way z t r a n s f o r m u p o n the gauge t r ans fo rma t i on (4.38). If A were zero , th is der i va t i ve shou ld become a n o r m a l der i va t i ve . L i k e i n the last sec t ion , we jus t have to connect zk to zk+i con t inuous ly , bu t b y us ing a covar ian t de r i va t i ve to take i n to accoun t the gauge t e r m e , A * : eiX*zk+1 = etDzk = zk + tDzk + j D 2 z k + . . . (4.48) So now the gauge t r ans fo rma t i on is : z k ^ e i a " z k , \ k ^ \ k + ak-ak+1 , Dzk^eia*Dzk (4.49) . A second i m p o r t a n t po in t is to cons ider a n o n - t r i v i a l H o p f phase i n z . U s u a l l y , we shou ld have z (0) = z(3), bu t let us cons ider ins tead the b o u n d a r y c o n d i t i o n z (0 ) = e ' 7 z ( /3 ) , so tha t the n o r m of z and also i ts representa t ion on the sphere (0,4>) s t i l l agrees at 0 and 3. T h i s phase, 7, does not change a n y t h i n g phys ica l l y , a n d s ince a phase i n z is l o c a l l y u n p h y s i c a l , the 7 phase m i g h t , a n d w i l l , represent on l y a topo log i ca l phase. T o ach ive th is t r a n s f o r m a t i o n , let us cons ider the same gauge t r ans fo rma t i on (4.49) bu t cons ide r ing th is t i m e ct\ a n d ctjq+i has be ing comp le t l y i ndependen t . F u r t h e r m o r e , i n the case of a con t i nuous gauge t r ans fo rma t i on we rea l ly have to cons ider a (0 ) as dif ferent f r o m a(8), i n genera l . T h i s phase is absorbed by the A^ gauge t r ans fo rma t i on (4.49). T h e o n l y p r o b l e m , is tha t usua l l y N N I > - £ A f c k=l k=l Since now ax ^  ajv+i> th is is no longer t rue , a n d we have to correct th i s p r o b l e m b y i m p l e m e n t i n g the l o c a l gauge t r ans fo rma t i on b y a g loba l gauge t r ans fo rma t i on N N N E A* ^  E A* + EK+i - « * ) (4.50) k-l k=l k=l Chapter 4. Coherent State Path Integral for Spin 34 i n the p a t h in teg ra l (4.36). L e t us wr i t e the ac t i on (4.39) w i t h t he -he lp of the covar ian t de r i va t i ve (4.48), to second order i n e: S = £ > M A * + z\zk- z\t-tJZ*-'zk\ + " e ' A l * i ) k=i = £ > i f t A 4 + C-^B • z\ozk - tz\Dzk - t£!L\B\*zlzk - i z t D 2 z k + ^B.ztaDzk} + yDzf0 f^.2si%X uh -> + _ , Jo e 2 - ^-\B\2z^z + iDz*Dz + ^ B • z^Dz]dr (4.51) 8 2 2 B y e x p a n d i n g etD i n power of D, th i s is l i ke e x p a n d i n g i n power of A, because D depends on A b y its de f in i t ion (4.48). S o , at th is s tage, we can re in t roduce A i n (4.51) b y i ts dependence i n D, a n d then in tegra te i t ou t . W e can rewr i te (4.48) as eiX>ei&(zk) = e<D(zk) (4.52) B y e x p a n d i n g o n b o t h s ide we find for D , i n first o rder i n e: D = i ; + i i <4-53> T h i s is an a p p r o x i m a t i o n for A , thus we can no t expect an exac t resul t for the r e m a i n i n g ca l cu la t i on o f the p a t h in teg ra l , bu t th is shou ld g ive a g o o d a p p r o x i m a t i o n . P u t t i n g (4.53) back in to (4.51) g ives: S = / + =-—B • z'Bz — z'z z + z —\B\2z*z Jo e 2 t 8 + | | i | 2 " j & z - zH) + ± A V z + 6-£B . t \ d i + IH^XB • Soz]dT Chapter 4. Coherent State Path Integral for Spin 35 T h e n , at th is po in t , we can use the H o p f phase to set ziz-ziz = 0 (4.54) N o t e tha t (4.54) co r respond on l y to one cons t ra in t because i ts c o m p l e x con jugate gives the same cons t ra in t (un l i ke a cons t ra in t l i ke z^z — 0) . T h e n we have to a d d the new phase (4.50), f$ c W r , to the ac t i on . T h e va lue of a w i l l be eva lua ted la ter . T h i s leave us w i t h the ac t i on ip S= Ldr Jo where L = —(2s% - z]z + ^r-B • z*Bz) + ^-z^z e v 2 ' 2t + 2 s i h a + ^ B • z^z - 2±\3\*zU + | | i | 2 + ^ B • *B = LX + LZ + L, where we w i l l cons ider three par ts for the L a g r a n g i a n , n a m e l y L\ = —[^ + - f ( 2 ^ " z^z + ^ B • z^Bz)f <w6 Z Z Zi (zU-2s%f  L z ~ 2ezU L, = 2siha + s » B . Z - ^ - - ^ ( \ B \ ^ z f - (B • z^Bzf) + | | i | 2 + • z^i T h e Lz c o n t r i b u t i o n to the p a t h i n teg ra l is of the f o r m w h i c h ind ica tes tha t we have z*z = 2sh + 0(hyfe/Ar). T h e n , i f we look at a scale A T >• e, we f i nd tha t z^z is very we l l peaked at 2sh. Howeve r , i f we set A T = e, as we use to do for the d i sc re t i sa t i on , we f ind z^z = h(2s + 0 ( 1 ) ) , so on l y the c lass ica l h'rnit, s —» oo, is we l l peaked . F o r low values of 5, i ns tead , the n o r m of z is less we l l de f ined. Chapter 4. Coherent State Path Integral for Spin 36 In fac t , th is is not such a surpr ise , s ince i f we take the z^z = r2 = R dependence of the p a t h i n teg ra l (4.42), after A has been in tegra ted ou t , R2a a n d f ind an a p p r o x i m a t i o n a r o u n d i ts m a x i m u m : f'(R) = 0 => Ro = 2s , f(Ro) = M^e - 2 ' « , f(Ro) = so flRY^f^)+ 1^-^-1^).) « f(Ro)e-^T- w _ L = e - ^ ^ (4.56) V47T5 w h i c h is jus t equa t i on (4.55) for R = z T z / f t , up to a cons tan t Ln the p a t h i n teg ra l (4.42) we in tegra ted R d i rec t l y s ince it was c o m p l e t l y decoup led to the rest of the p a t h i n teg ra l . Here we no t i ce tha t the t e r m i n c° i n La does no t depend o n z^z at a l l , t hanks to the A in teg ra t i on tha t b rough t cor rect ions i n th is respect . T h e r e m a i n i n g te rms are of the order e, so they do not rea l l y in f luence the p a t h i n teg ra l , they on l y regu lar ize i t . F u r t h e r m o r e , a mo re e labora te expans ion up to order e 2 w o u l d also correct these te rms , as i t d i d for the f irst o rder , s ince we k n o w f r o m (4.42) t ha t the z^z = r2 va r iab le c o m p l e t l y decouples i n the p a t h in tegra l . So i t w o u l d make.sense to s i m p l y set z^z = 2sh fo r the r e m a i n i n g of the c a l c u l a t i o n . T h e n for L\, we f i nd i ts c o n t r i b u t i o n to the p a t h i n teg ra l , af ter cons ider ing the new cons t ra in t above , t o be a s imp le G a u s s i a n : J-*£Ji2T J-°°tA2* i i v ^ F i w h i c h is jus t t he m i s s i n g fac tor i n (4.55) tha t appears i n (4.56)! T h e n we finnaly o b t a i n the p a t h i n teg ra l , af ter resca l l ing z to z^z = ft, Z[3] = J Dz^Dz6(z^z/n-l)expS^-j J*[2si1ia + sfihB • z*az Chapter 4. Coherent State Path Integral for Spin 37 -^^-{\B\\zUf - {B • z^Bz)2) + es\z\2 + esfihB • z+ai^r J (4.57) a n d where z(/?) = 2(0)e,Jo a T , w h i c h is de te rm ined b y the cons t ra in t (4.54). Before c o n t i n u i n g w i t h (4.57), let m e reder ive i t i n a n o ther way , w h i c h is less p h y s i c a l a n d does not represent the s ign i f icance of A , bu t g ives w h a t we want d i rec t ly . In the sec t ion 4.3 we der i ved the equ iva lence of the sp in coherent s ta te a n d h a r m o n i c osc i l l a to r coherent s ta te p a t h i n teg ra l , by in teg ra t i ng the A va r iab le , t he H o p f phase a n d the n o r m r of z i n the la ter p a t h i n teg ra l . T h e resul t appears at the equa t i on (4.43) where the z va r i ab le is res t r i c ted to z^z = h. T h e n we can re in t roduce the fu l l z va r iab le b y inser t ion (by b ru te force!) of a de l t a f u n c t i o n , S(z^z/% — 1), i n the p a t h in teg ra l a n d rewr i te the effect ive L a g r a n g i a n w i t h a n unres t r i c ted z va r iab le : m=I n (ftrw){2s+1)6{zlzk/n - v*/*)8* T h e n we fo l low the same p rocedure , we set zk+i = ec&zk a n d find, u p to second order i n c: z k e zk+i = " • a z k + tZkzk H g—\B\ zkZk 2~~ ' ~2~ where I assumed z^z = h for the first t e r m . B y t a k i n g the l o g a r i t h m of th is exp ress ion , to p u t eve ry th ing i n the e x p o n e n t i a l , a n d i n c l u d i n g the b o u n d a r y cond i t i ons , we o b t a i n the fo l l ow ing p a t h i n teg ra l : 2*2 -^—{\B\2h - (B • z\Bzk)2) - eszlh + esphB • z\azk -2seztzk{-nhB • z\Bzk + z\zk)]d\e-^ ^ * ^ m d r Chapter 4. Coherent State Path Integral for Spin 38 A n d i f we use the H o p f phase to set (z^z — z*z) = 0, w h i c h imp l ies z T i = 0 w i t h z^z = %, we f i nd tha t t he above p a t h in tegra l becomes comp le t l y i den t i ca l w i t h equa t i on (4.57)! N o w , let m e comp le te the der i va t ion of the equa t i on above. F o r the z va r i ab le we can use the same representa t ion as the sp in coherent states (2.8), b y i n c l u d i n g also the H o p f phase: z = Vheia cos(0/2) \ s in(0/2 )e* = V h e i 0 ( -i<t> \ cos(0/2)e I sin(0/2) j for 9 7r for 9^0 T h e n ( z + i - tfz) - h i [ a + - cos(0))$ = 0 w h i c h gives the H o p f phase c o n t r i b u t i o n to the p a t h i n teg ra l a =-(±1 + cos(0))0 F u r t h e r m o r e , i n th is gauge, we c a n checked tha t (4.58) n = z^Bz/% , (n) 2 = 1 , (z^Bz - z1Bz)/% = - i n x n , $z = (£)2/4 (4.59) W e c a n use the fact tha t n(0) = n(/3) and z^'z = 0 to in tegra te b y par ts some te rms i n (4.57). A f t e r t ha t we can use (4.59) to wr i te the p a t h in teg ra l as: Z[3] = J Dn8((nf - 1) exp 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 the same as the sp in coherent s tate p a t h i n teg ra l c o n t i n u u m l i m i t , t ha t appears at the equa t i on (4.47). C h a p t e r 5 P a t h I n t e g r a l f o r a C h a r g e d P a r t i c l e i n a M a g n e t i c M o n o p o l e F i e l d 5 .1 M o n o p o l e V e c t o r P o t e n t i a l In th is chap te r , I w i l l demons t ra te tha t the phys ics o f a charged pa r t i c l e i n the f ie ld of a m a g n e t i c m o n o p o l e is re la ted to the s p i n sys tem tha t we are s t u d y i n g . A magne t i c m o n o p o l e f ie ld is , l i ke a po in t charge e lec t r ic field, g i ven b y the equa t i on : Bm = g (5.61) where g is the magne t i c charge. S ince we w i l l be in teres ted i n the gauge f ie ld ( l i ke A of last sect ion) we want to represent th is f ie ld b y a vector p o t e n t i a l , Am, such tha t r o t t A j = Bm (5.62) M a t h e m a t i c a l l y , we find tha t the 2- fo rm F (Fij = eijkBk) is c losed, dF = 0, a n d i f (5.62) is def ined th roughou t a l l t he space, t h e n F is exac t , F = dA. So the second cohomo logy g roup of the space w i l l i nd i ca te i f there is some B f ie ld t ha t has no so lu t i on for A (c losed 2 - fo rm tha t are not exac t ) . T h e space here is i ? 3 , bu t f r o m (5.61) we no t i ce tha t there is a s ingu la r i t y at r = 0. So we have to remove th is p o i n t to get a we l l def ined vec to r po ten t i a l . W h i c h leaves us w i t h R3 — {0} = S2 x [0,oo]. It is k n o w n tha t : H2(S2 x [0,oo]) = H2{S2) = R w h i c h means tha t there is some field B t ha t w i l l have no so lu t i on for (5.62), v a l i d every -where . T h e class to w h i c h Bm be longs ( in i ? ) , i n the cohomo logy g roup , is g i ven b y the 39 Chapter 5. Path Integral for a Charged Particle in a Magnetic Monopole Field 40 in teg ra t i on JsF = JsBm-dS = ATrg T h i s is ac tua l l y the va lue (up to a cons tan t ) of the magne t i c charge! T h e n , for non-zero m a g n e t i c charge, there is no so lu t i on for the vector p o t e n t i a l v a l i d everywhere on the space ( s 2 x [0,oo]). T h i s is k n o w n as the D i r a c s t r i ng (since for any so lu t i on there is a d ivergence a long a s t r i ng s ta r t i ng at the o r ig in a n d go ing to in f in i t y ! ) . T h i s is not the e n d of the s tory , s ince a l l our s tudy has been done c lass ica l ly . In q u a n t u m mecan ics , the vec to r po ten t i a l enters the theory as a gauge f ie ld : Z t y = (V - , # - > e T r x # , A —> A-\-Vx D$ -* e * x / 3 $ (5.63) • —* So the gauge f ie ld A can be changed b y some gauge t r ans fo rma t i on (5.63) b y a quan t i t y V x for a n a rb i t r a r y field x (or A —• A - f dx)- T h i s does no t seem to change the equa t ion (5.62), s ince r o t ( V x ) = 0 (or d(dx) = 0) . In q u a n t u m mecan ics , however , the x appears i n the wave f u n c t i o n i n the exponen t i a l e ^ x . So x does not have to be a f u n c t i o n , bu t s i m p l y a sec t ion of the vec tor b u n d l e ( the wave func t i on o n the space) . In o ther wo rds , X is def ined on l y m o d u l o — . Le t us solve (5.62) i n two pa tches , us ing po la r coord ina tes : Am • dx = g(l — cos(0))d<f> for 0 ^ n A'm-dx = -g(l +cos(6))d<f> for 0^0 T h e s e two vec to r po ten t ia l s differ on l y b y : {Am-A'm)-dx = 2gd<f> = d(2g<l>) or Am=A'm + V(2g<f>) Chapter 5. Path Integral for a Charged Particle in a Magnetic Monopole Field 41 So , as far as q u a n t u m mecan ics is concerned , the Am and A'm gauge f ields are the same, as l ong as 2g<f> is s ingle va lued everywhere m o d u l o 2 - ^ , w h i c h means : ir = ±s for 5 = 0 , 1 / 2 , 1 , 3 / 2 , . . . ' (5.64) n m u s t be fu l f i l l ed , th is is the D i r a c quan t i sa t i on cond i t i on [4]. F o r s i m p l i c i t y I w i l l cons ider ^ = s (by choos ing the app rop r i a te s ign o f e). N o w if we cons ider a charge, e, m o v i n g i n th is m a g n e t i c f ie ld , the H a m i l t o n i a n is s i m p l y g iven b y : W h a t w o u l d be in te res t ing now is to express HQ i n te rms of the angu la r m o m e n t u m , L, of the f ie ld . 5 .2 M o n o p o l e A n g u l a r M o m e n t u m W e shou ld be able to find L i n t e r m of D a n d r. W e have the c o m m u t a t i o n re la t ions : [r,-,rv]=0 , [DurA =(5 t i , [ A - , £ j ] = - * s e 0 f c £ § (5.66) W e expect the angu la r m o m e n t u m to have a c o m m u t a t i o n re la t ion w i t h f a n d D such tha t they t r a n s f o r m as a vec to r [Li,Dj] = ieijkDk , [L,-,rj] = ze, j f c r f c (5.67) In fac t , t he c o m m u t a t i o n re la t ions (5.67), de te rm ined c o m p l e t l y t he c o m m u t a t i o n re-la t ions o f t he componen ts o f L be tween themselves a n d the H a m i l t o n i a n , b y us ing t he J a c o b i i den t i t y a n d the i r r educ t i b i l i t y of the r and D va r iab les : [Li, Lj] = ieijkLk , [Li, H0] = 0 Chapter 5. Path Integral for a Charged Particle in a Magnetic Monopole Field 42 —* T h e f irst choice for L w o u l d be s i m p l y —* 9 —* —* L = —ifif x D = r x (p — eAm) B u t i t tu rns ou t to be w r o n g , m a i n l y because o f the presence of the magne t i c p o t e n t i a l vec to r Am, as i t cou ld be no t i ced a l ready i n (5.66). T h e correct answer is i n fact g i ven b y L = - i h r x D - s h - (5.68) r T h i s express ion for the angu la r m o m e n t u m is ac tua l l y t rue also c lass ica l l y (obv ious ly? ) s ince d • • sfi • d v —(mr x f) = mf x f = r x ( e r x B) = — f x ( f x f ) = — (sh-) dt r J dt r In q u a n t u m mecan ics we jus t have to ver i fy the c o m m u t a t i o n re la t ions (5.67) to conv ince ourse lves. A spec ia l s t udy has to be done to see w h i c h values of / can rea l l y occu r . W e k n o w tha t 21 m u s t be a n in teger s ince L fo l lows the sp in a lgebra . B u t a more care fu l cons t ruc t i on [5] o f the representa t ion o f L def ined b y (5.68) ac tua l l y shows tha t I = s, s + 1, s + 2, • • •. (we mus t have a s ta te \l,s > to cons t ruc t the representa t ion ! ) N o w , we k n o w tha t (L)2 = %2l(l + 1) where / is one o f the va lues men t i oned above . B y us ing the equa t i on (5.68) i ns tead , we find (L)2 = -h2(fxD)2 + h2s2 F u r t h e r m o r e , we have D • D = = D . r ^ - £ ^ e ^ D . = (D.f)(^)(r.D)-(Dxf)(^)(f.D) Chapter 5. Path Integral for a Charged Particle in a Magnetic Monopole Field 43 ,d2 2d, 1 rtx2 ,d2 2d, s2-1(1 + 1) rar r2 or2 r or r2 So, this enable us to write the Hamiltonian (5.65) as: = _ | l ( | l + ^ ) + ffli+i^i!) ( 5. 6 9 ) 2m or2 r or 2mr2 5 . 3 Path Integral for a Spin Particle in a Magnetic Field What would be the Hamiltonian of our magnetic monopole system, if we put it in a constant magnetic field? We already solved the system for a.magnetic potential Am, so a constant magnetic field B, corresponding to a vector potential A = \B x r, will simply shift Am -> Am + \B x r in the Hamiltonian (5.65): = »o- eAm) • (B x f) + £-(B x i f e r* esnr-B e2 , -> Here appears finally the reward of all our work in this chapter, the B • L term that we are studying, but there are sti l l two problems. One is the last two terms in (5.70), they have nothing to do with our model of last chapter. The only simple way of correcting this is to add an interaction potential to the Hamiltonian, that wi l l cancel them. The 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). The contribution of / in the Hamiltonian is given by the term W +1) — 5 2 ) in H0, in (5.69). The smaller / is, the smaller the energy contribution of this term. So if ~2 ^» which means mr2 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 = E3, more precisely we must have 3E„ >^ 1. 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 H0 is well known [6] and 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 0 , such that the particle will stay in the first (ground state) 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 r 8m 2cr2 = 2mrf ' = 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 Es, usually. However, if s —*• oo, the classical limit, we can choose a = r0/\/2s and obtain Eh = E3. In other words, the harmonic potential, that we added here, might 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 = (Eh + E . ) , (L) 2 = h2s(s + 1) (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 + e(Am + A).r--V = jfr2 + \{B x f) • f - + ± 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 jt = in the path integral (5.73), which gives Z\0\ = tx{e~pH) = t r ( e - ^ - ^ - £ ) ) . e - / » ( ^ + ^ ) = J Dr6(r - r0)e~ L E d T 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 ( 5 J 5 ) Chapter 5. Path Integral for a Charged Particle in a Magnetic Monopole Field 46 T h e t opo log i ca l te rms differ b y a m i n u s s i gn , w h i c h is n o r m a l s ince the r differ by a m i n u s s ign (and a cons tan t ) w i t h n , i n d i c a t e d i n (5.75). If eg/% = —s, i ns tead of ou r conven t i on , t hen \i a n d f w i l l change s ign i n (5.75). T h e va lue of fi is not a surpr ise , s ince the H a m i l t o n i a n (5.72) con ta i n th is fac tor i n f ront of the B • L t e rm . T h e very in te res t ing reve la t ion of th is ca l cu l a t i on is the c cor respondence, w h i c h can be rewr i te as: t = 4~ or BE8 = N (5.76) It was a l ready k n o w n tha t /3Ea >• 1 a n d (5.76) ind ica tes th is s ta tement i n t e r m of the N va r iab le . F u r t h e r m o r e , i f we put . (5.76) back in to the 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 the va lue of a = r o / \ / 2 s , va l i d at the c lass ica l l i m i t , we f ind exac t l y the equa t i on (4.55) or (4.56) for the r a d i a l par t of the p a t h in tegra l ! A g a i n the c lass ica l l im i t is ve ry we l l de f ined , bu t the low s p i n l i m i t is more t r i cky , a n d needs some ' a r t i f i c i a l ' cons t ra in ts to b e we l l de f ined. In the p a r t i t i o n f u n c t i o n (5.74), the presence of e ~ ^ E ' = e~N ind ica tes tha t t he p a t h i n teg ra l measure m u s t be renorma l i sed b y e, to exac t l y co r respond to the coherent s ta te p a t h i n teg ra l . O n an o ther h a n d , the e~^Eh come f r o m the h a r m o n i c p o t e n t i a l t ha t we added to the H a m i l t o n i a n , so there is no surpr ise i f we find th is e x t r a t e r m . S ince the k ine t i c energy comes m a i n l y f r o m the s p i n of the pa r t i c l e , thus E„, t he k i ne t i c t e r m ^f(ra) 2 i n the coherent s ta te p a t h i n teg ra l , w i l l con t r i bu te to the order one , w h i c h c a n not rea l l y be neg lec ted i n the ca lcu la t i ons . C h a p t e r 6 C o n c l u s i o n T h e pu rpose of th is work has been the s tudy of p a t h i n teg ra l eva lua ted w i t h coherent s tates. T h i s has been done b y l ook ing at two so lvable p rob lems : the h a r m o n i c osc i l l a to r a n d the pa r t i c l e w i t h sp in i n a cons tan t magne t i c field sys tems. T h e coherent states represent the sys tem so w e l l , tha t the p a t h i n teg ra l van ishes for c lass ica l t ra jec tor ies . T h e rea l p r o b l e m has been , t h e n , to i nc l ude the q u a n t u m t ra jec tor ies i n the p a t h in teg ra l ca l cu la t i ons , to get the q u a n t u m cor rec t ions , i f not a c o m p l e t l y q u a n t u m resu l t . It is c lear tha t the c lass ica l l im i t of these p a t h in tegra ls are ve ry we l l de f ined, bu t it is useless to use p a t h in tegra ls to find on l y c lass ica l so lu t ions . T h e s e q u a n t u m cor rec t ions have been t a k e n in to accoun t , i n m y work , b y keep ing the e te rms in the p a t h in tegra l a n d then t a k i n g the l im i t e —• 0 at the end of the ca l cu la t i ons . T h e s e terms create a b r idge between these c lass ica l a n d pu re l y q u a n t u m p a t h s , b y m a k i n g these t ra jec tor ies s m o o t h enough so tha t we can use a c o n t i n u u m l im i t a p p r o x i m a t i o n . T h e use of a l a t t i ce regu la r i za t i on gives us a way to o b t a i n an exac t d iscrete ac t i on . T h i s m e t h o d has been used before w i t h success [10,11], bu t i ts d iscre te leve l has never been s tud ied ve ry deeply . It seems tha t a carefu l ana lys is of the d isc re t i sa t ion g ives some usefu l cor rec t ions , a n d a l lows an in te res t ing c o m p a r i s o n between var ious con t inuous a p p r o x i m a t i o n s a n d t he exact so lu t i on . T h i s exp la ins w h y the e te rms regu la r i zed the p a t h i n teg ra l app rop r ia te l y , the resu l t i ng p a t h i n teg ra l is closer to i ts d isc re te ve rs ion . F o r the h a r m o n i c osc i l la to r coherent s ta te p a t h i n teg ra l , i t has been no t i ced t ha t the 47 Chapter 6. Conclusion 48 s imp le de f in i t ion of the c lass ica l H a m i l t o n i a n H(z) i n t e r m of the q u a n t u m one H as H(zk) =< Zk\H\zk > , or H(zk) = ^ j J J * * ^ affect the g round s tate i n the p a t h i n teg ra l . It has been demons t ra ted tha t the e\z\2 t e r m helps to regu la r ize the p a t h i n teg ra l . T h i s is pa r t i cu l a r l y apparen t b y go ing back to a d iscrete leve l , at the sect ion 3.2, or a semic lass ica l a p p r o x i m a t i o n , at the sect ion 3.3. T h e a p p l i c a t i o n of the same p rocedure to a sp in s pa r t i c le i n a cons tan t magne t i c field b y the use of s p i n coherent states or h a r m o n i c osc i l l a to r coherent states p r o d u c e d the same c o n t i n u u m limit,• up to order c t e rms , after the i n teg ra t i on of the app rop r i a te var iab les i n the la t te r p a t h in teg ra l . W e were ab le to ex t rac t the m e a n i n g of the Afc var iab les as a gauge field, a n d cons t ruc t a covar iant der iva t ive for the h a r m o n i c osc i l l a to r coherent s tates. T h e topo log ica l a c t i o n , 2 s ( ± l — cos(0))<^>, has been s h o w n to be re la ted to the H o p f phase of th is va r i ab le , z. T h i s gave us a m a p p i n g of th is sp in p a t h i n teg ra l i n to a CP1 m o d e l . T h e coeff ic ient of th is t opo log i ca l ac t i on appea red c lear ly as b e i n g 2s. F u r t h e r m o r e , a s t u d y of magne t i c m o n o p o l e has been rep roduced i n deta i ls . It has been exp la i ned how i t is poss ib le to o b t a i n a sp in s pa r t i c le rep resen ta t ion , us ing a m o n o p o l e field a n d a speci f ic i n te rac t i on : V = - -^(B x r)2 + Ea^jl°£. T h i s i den t i f i ca t i on has been s tud ied before , b u t never up to the e order [14,15]. W e showed a comp le te cor respondence be tween th is monopo le p a t h i n teg ra l , i n pos i t i on space repre-sen ta t i on , a n d the same sp in sys tem p a t h in tegra l us ing coherent s ta tes, i n d i c a t i n g even mo re the re levence of the c terms as a regu la to r . T h e e —• 0 l im i t is i m p o s e d b y a r 0 —• 0 l i m i t , or m —• 0 w i t h ^ fixed, o n the rad ius of the sphere on w h i c h the pa r t i c l e moves a r o u n d the m o n o p o l e . In th is l i m i t , the \m(r)2 w i l l not necessar i ly go to ze ro , s ince the par t i c le m igh t s i m p l y sp in faster , b y conserva t ion of angu la r m o m e n t u m . T h e r a d i a l pa r t o f the m o t i o n of the pa r t i c l e , for low va lue of the s p i n , is no t ve ry we l l peaked at a g i ven rad ius , however , i t comp le t l y decoup led i n the p a t h i n teg ra l . W h i c h 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 . Af f leck , M . B e r g e r o n , L . C. L e w Y a n V o o n &; G . Semenoff , 1 Coherent State Path Integral and the Harmonic Oscillator1, U B C p repr in t 1989 [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 , lThe Magnetic Monopole Fifty Years Later1, I n te rna t iona l S h o o l of Subnuc lea 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 . Y a n g , Nucl. Phys., B 1 0 7 , 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 -H i l l , N e w Y o r k (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 Y o r k (1981) [9] 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 . Serot , Ann. of Phys., 1 8 5 , 138 (1988) [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, 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 , 1 Table of Integrals, Series and Products', A c a d e m i c P r e s s , N e w Y o r k (1980) [14] E . F r a d k i n & M . S tone , I l l ino is p repr in t ( M a r c h 1988) [15] M . S tone , Nucl. Phys., B 3 1 4 , 557 (1989) 50 Appendix A Identities for Determinants /S(: (A.77) Gaussian integrals have the very useful property that: 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 ) = A - B 0 • • 0 -C A - B • • 0 0 -C A • • 0 0 0 0 • • A \ ( , M ( N ) = (N) A - B 0 •• • -C -C A - B •• • 0 0 -C A •• • 0 - B 0 0 •• • A \ I (JV) (A.78) or = A8iti - BSij.i - C6itj+1 i,j = 0 to TV where i,j assume cyclic boundary conditions, N + 1 = 1, iV — 1 = —1, for the M^N) matrix. Let us call DN = det[M(Ar)] , DN = det[M(N)] 51 Appendix A. Identities for Determinants 52 The evaluation of these determinants can be done by using the well known recursive formulas. Expanding along the first line for D^ gives DN = ADN_! - BCDN_2 which can be solved by inserting a solution of the form A^, that produces the following constraint: X2 = AX - BC => A = A± = ]-(A ± VA2 — ABC) Since D\ = A, D2 = A2 — BC, we find the general solution for this determinant identity det[M{N)] = DN= A + * A - A ? + 1 ) (A.79) For DN, we procede in the same way DN = ADN_! - 2BCDN_2 - (BN + CN) with AX± — 2BC = (A+ — A_)(±A-j.) we find for the cyclic determinant D^, the identity: det[M ( j V )] = DN = (X+)N + (X_)N - B N - C N (A.80) We can, furthermore, find the eigenvectors and eigenvalues of the M(JV) matrix, be-cause of its cyclic boundary condition. These are simply This gives another identity for the determinant: DN = f[ Xk = f[(A - Be^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 53 A, B, C, at the same t ime . In o ther wo rds , (SAS-1)^ = £,jA,-, (SBS~l)ij = SijBi a n d (SCS~l)ij = SijCi w i t h the same S m a t r i x . T h e n the de te rm inan t is s i m p l y M det[M(N)] = IJ d e t « [ M ( j v ) ] (A .82 ) t = i where de t^ [M( /v ) ] is the de te rm inan t (A .80) w i t h the use of the Ai, Bi, Ci va r iab les . If C = 0, the p rev ious de te rm inan t (A .79 ) a n d (A .80 ) can be s imp l i f i ed s ign i f i can t l y : det[M{N)] = AN , det[M{N)] = AN - BN A c t u a l l y , i f a l l the A a n d B's are di f ferents, we can s t i l l easi ly p r o v e d , l i ke (A .79 ) or ( A . 8 0 ) , t ha t det In cases of submat r i ces , w a can s t i l l use (A .82) w i t h (A .83 ) , bu t a g a i n , as l ong as we c a n d iagona l i ze a l l the Ak, Bk (k = 1 to A^) at the same t ime . Ax -Bx 0 ' A3 - B N 0 0 0 AN = (U Ak) - (f[ Bk) k=l k=l (A .83 ) A p p e n d i x B S c h w i n g e r - B o s o n M o d e l Let us consider a set of two dimensional creation and destruction operators: [at, a]] =Uij , [o,-, aj] = [at, a)] = 0 , for »,j = l,2 Then we can build up a spin vector, J , by J = a^a (B.84) We can verify the identity U)2 = — ( — + 1) (B-85) 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 0 | W > = 2s|# > In other words, we have to work in a subspace represented by: I* >-> / 2 7 r^e , A( a , a- 2 s)|* > (B.86) JO 27T 54 

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