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An investigation of coherent state path integrals as applied to a harmonic oscillator and a single spin Voon, Lok Chong Lew Yan 1989

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AN INVESTIGATION OF COHERENT STATE PATH INTEGRALS AS APPLIED TO A HARMONIC OSCILLATOR AND A SINGLE SPIN LOK CHONG LEW YAN VOON B.A. Hons. (Cantab.), England, 1987 A THESIS S U B M I T T E D IN PARTIAL F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in T H E F A C U L T Y O F G R A D U A T E STUDIES D E P A R T M E N T O F PHYSICS We accept this thesis as conforming to the required standard T H E UNIVERSITY O F BRITISH C O L U M B I A June 1989 © LOK CHONG LEW YAN VOON 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 'Pinysi cs>  The University of British Columbia Vancouver, Canada Date Tun<2 \1, I ' lSf DE-6 (2/88) Abstract In this project two steps involved in the handling of path integrals are reexamined in detail for coherent state path integals. They concern the continuum limit approxi-mation and the regularization of the formal path integrals. Restricting oneself to the harmonic oscillator, the technique of time splitting is used to set up the coherent state path integrals and the proper way to pass to the continuum limit is demonstrated. The manipulation of these path integrals calls for regularization procedures and the validity of discrete, Riemann zeta function and 'derivative' regularization methods is observed. A modification to a fermionic theory is briefly mentioned and, finally, the above results are implemented in writing down a path integral for a single spin. Table of Contents Abstract ii List of Figures v Acknowledgement vi 1 INTRODUCTION 1 2 F E Y N M A N PATH INTEGRALS 4 2.1 Configurational Space Representation 4 2.2 Coherent State Path Integrals 8 2.3 The Feynman Ansatz 11 3 T H E Q U A N T U M HARMONIC OSCILLATOR 13 3.1 Feynman Action Principle for Coherent State Representation 13 3.2 The Partition Function 15 3.3 Evaluating Functional Determinants 17 3.4 The Coherent State Path Integral 19 3.5 Reexamining the Continuum Path Integral 24 4 GENERALISATION OF T H E PATH INTEGRAL 28 4.1 A Fermionic System: The Grasmannian Oscillator 28 4.2 Comparison of the Two Oscillators 32 4.3 Field Theory 32 iii 5 APPLICATIONS 34 5.1 A Path Integral for Spin 35 5.2 Berry Phase 40 6 CONCLUSIONS 42 Bibliography 44 A Properties of Coherent States 46 B Properties of the Riemann Zeta Function 48 C Contour Integral Representation of an Infinite Sum 51 iv List of Figures B. l Contours for the integral representation of the Riemann zeta function. . 50 C. 2 Contours for the integral representation of an infinite sum 53 v Acknowledgement First and foremost, I wish to thank my supervisor, Professor Ian Affleck, for bringing to my attention the possibility of relating the spin and harmonic oscillator problems; this has naturally led to the present project. In addition, I benefited immensely from the numerous discussions we had both during the course of the project and during the write-up. I would also like to acknowledge some useful conversations I had with Mario Bergeron and Dr. Gordon Semenoff. In particular, I am grateful to them for introducing to me the technique of zeta function regularization. vi Chapter 1 I N T R O D U C T I O N The path integral formulation of a quantum theory is a very useful technique (especially for constrained systems in quantum field theory). One important property is that, at least for a certain class of Hamiltonians, the propagation of the quantum mechanical state by a contact transformation leads to a Lagrangian formulation of the theory [1,2]. Among other things, this provides an understanding of how initial conditions single out a path from the classical variational principle. The present work concentrates on path integrals which are set up by using coher-ent states of the Heisenberg-Weyl group. Traditionally, coherent state path integrals have been used to treat problems involving collective effects such as quantum optics, nuclear many-body physics and superfluidity. There has been renewed interest in using coherent states as a means of setting up a path integral for spin systems. We are here concerned with some still unsatisfactory aspects of path integration which are relevant to a coherent state (CS) representation. First of all, the traditional continuum path integral in a CS representation has an action which is first order in time derivative. It was first observed by Klauder [3] that the continuum action leads to equations of motion which are, generally, incompatible with the boundary conditions. He proposed to resolve the problem by introducing an infinitesimal second order term into the linear action. I will demonstrate the correctness of Klauder's procedure by first setting up a discrete path integral and showing that the naive continuum time limit (i.e., that leading to a linear action) is ambiguous. This 1 Chapter 1. INTRODUCTION 2 will help us define the proper limiting procedure. I will then try to answer the question of how to make sense out of formal path integrals like the ones mentioned above. Indeed, in addition to being mathematically ill-defined, there is also the problem of the infinities arising from the infinite number of modes of the field theory. The usual practice has been to absorb the infinities into an overall normalization since they are irrelevant to, for example, the perturbative program. A path integral, however, bears some similarity to the thermodynamic partition function (TDPF) in a canonical ensemble Z(T) = Y,e~En/kBT n and in fact this analogy can be made more precise by an analytic continuation of the time variable in the path integral. This, therefore, motivates us to treat the normal-ization factor in the path integral properly. Since the latter is divergent, one needs to introduce regularization schemes in order to isolate the infinities. I will employ three different schemes in evaluating the path integral: a lattice time method, zeta function regularization and a derivative method. The latter will be found to provide a nice interpretation of the correct limiting procedure for a continuum theory. The discrete time method will be seen to lead to an equality between the path integral and the TDPF. For the zeta function regularization, this is only true modulo an infinite energy term but which is physically irrelevant. The above analysis made use of a bosonic simple harmonic oscillator. For complete-ness, I will then show how a fermionic system might be treated in terms of classical variables. Finally, the significance of the above work to spin path integrals will be presented. The latter are usually set up by using SU(2) coherent states. An alternative formulation Chapter 1. INTRODUCTION 3 will be found by using a Schwinger boson representation though the treatment is not fully satisfactory as a representation for all values of the spin quantum number. Chapter 2 F E Y N M A N PATH INTEGRALS 2.1 Conflgurational Space Representation A path integral is a rather formal object which can be used to give an integral formu-lation of a theory. We will be concerned with the context in which this generates a quantum theory from a classical one. Feynman's original conception [1] was to obtain a quantum theory by starting from the 'corresponding' classical Lagrangian. More precisely, the propagator (for quantum mechanics in a conflgurational space represen-tation) from a point x to a point y in a time interval T is given by the functional where H is the Hamiltonian of the system, S the classical action and ~f(t) a path in the space of paths V. Beyond quantum mechanics, the position would be replaced by, for example, a field configuration (in field theory) or a space-like hypersurface (in quantum gravity). In the litterature, the above path integral is commonly referred to as the Feynman path integral (FPI). Before we actually introduce the coherent state representation, it will be necessary to review the conflgurational FPI. The fundamental property of any propagator is the semigroup law This, in fact, solves the quantum dynamical problem since, for example in quantum mechanics, K is just the transition amplitude in some representation. In principle, 4 integral (2.1) (2.2) Chapter 2. FEYNMAN PATH INTEGRALS 5 therefore, one needs to evaluate K for an infinitesimal time interval (the short time propagator) and then iteration (according to eqn 2.2) gives the expression in eqn 2.1. The FPI in eqn 2.1 has some undesirable features (indeed, it is mathematically ill-defined). In particular, the measure is rather formal and also one would like to, at least in principle, know how to evaluate explicitly the right-hand side of the equation. In usual flat space-time problems, one always normalizes the final answer; inclusion of gravity, however, forces one to evaluate eqn 2.1 exactly since the 'renormalized constant' (see Chapter 3) is related to the cosmological constant. In addition, it is not clear what the space V should be, though the Feynman ansatz is to sum over only continuous paths. It has been conventional to define the FPI as a limiting case of finite-dimensional integrals. It is obvious that different discretized expressions are possible and it is not clear that they will lead to the same FPI. We will illustrate this with respect to the coherent state path integrals (CSPI) in the next chapter. We will here advocate a different procedure: the Feynman time-slicing technique [1]. The idea is not to start with a formal expression like eqn 2.1 but rather to start with a quantum mechanical Hamiltonian H, and use the Trotter product formula and the completeness of the representation to set up a discrete path integral via the kernel of the unitary operator for time translation; this can then be taken over to a continuum expression like eqn 2.1. The expectation is that this will define for us what the measure and space of paths should be. Thus, let us use a complete set of states with the following properties: Jdr1\r1><r}\ = T , <r,\rj'>= 6(V- . (2.3) (For notational convenience, the resolution is written as an integral; for discrete states, one simply reads the integral as a summation.) Now divide the time interval T into Chapter 2. FEYNMAN PATH INTEGRALS 6 n + 1 pieces of length e. Then, one can write K(r7,T;r/,0) = <r7'|e-^T|r7> (2.4) = <i?'l(e-*fi<)B+1|»/> (2-5) = /fn^ )n<»?.-+ik~*fi<i»?.-> (2-6) where rj0 = rj ,r]n+1 = 77'. For most classes of Hamiltonians (see, e.g., Schulman [4] for the mathematical details), and indeed those we will consider, eqn 2.6 is equivalent to JW, T; 17, 0) =. Um / (f[ drj^j f[ <Vi+1\(l - '-He) \m> (2.7) = }™J[T[dVi)I[<Vi+i\rii>e* <",+1"'i> (2-8) i . <r>i4llHl''i> <r)i+1\rii> e ' ' where the limit e —• 0 is also understood (given (n + l)e = T). If, instead, we had started with eqn 2.2 then we would have K(v'i T; 77,0) = Jdi'KW^'^nKii'X-r,^) (2.9) i+iiU+i'i'HiiU) (2.10) J \t=l / i=0 and, therefore, the short-time propagator is K(rn+liti+1;rii,U) =<ili+1\rli> e h <*+il-«> . (2.11) If the basis vectors admit a Fourier representation, then we have <Vi+i\m> = S(Vi+x - Vi) = Jdfj eW™-"-1") (2.12) and eqn 2.8 becomes K(r)', T; rj, 0) = Hm J (f[ dr,^j (j[ dr))j e$ E,n=oW^ +1-".)-eH(J),+1,J7,)] (2.13) Chapter 2. FEYNMAN PATH INTEGRALS 7 with H(r}i+1,rn) = 1 (2.14 <t]i+i\r]i> and eqn 2.13 is known as the discretized phase space path integral. For the specific case of a particle of mass m moving in one-dimension in a potential V(q), H =p2/2m + V(q) ; n = q,fj=p (2.15) and eqn 2.13 becomes K(q',T;q,0) = Jim J (j[d^ {fi e* 2ZoM«+i -^- e f l t e .«> l (2.16) M Jvq^Vp^^Jo^-11^. (2.17) Finally, we can recover eqn 2.1 by doing the momentum integral: K(q',T;q,0) = l i m N(n) f f[ dq, e ^ L ^ £ } " W ' J (2.18) J • i /T f / i M * ) . 4 ^ *i ( 5', i ). (2.19) Our contention is that, in general, eqn 2.17 and 2.19 are defined via eqn 2.16 and 2.18. In this case, however, note that, from a semi classical argument, we find that qi+1-qi = 0(y/i). (2.20) If it is legitimate to interchange the limit and integral operations, then it is obvious that discontinuous paths do not contribute to the kernel because of the large phase variation. Also, paths satisfying eqn 2.20 are non-differentiable. Hence, we deduce the Feynman ansatz that only continuous paths need be summed over. The Feynman action principle has been demonstrated with the Hamiltonian in eqn 2.15 and a quadratic potential. For a more general Hamiltonian, this might not be Chapter 2. FEYNMAN PATH INTEGRALS 8 true (see Ryder [5] for a counter-example) and hence reveals another limitation of this canonical principle. We have already mentioned that eqn 2.1 generates a quantum theory from a classical one but that there exists problems associated with the measure and the non-uniqueness of the discretized expression. The above derivation of eqn 2.16 and 2.18 removes them since we have exact and well-defined expressions. Given the definition in eqn 2.19, we have therefore derived the Feynman rule with a proper measure. This, of course, requires that we keep the limit n —• oo till the end of the calculation, with the contention that this limit exists. Still, it is not completely satisfactory since we necessarily has to start with a quantum Hamiltonian and, therefore, given a classical Hamilton function, operator ordering ambiguities crop up again. 2.2 Coherent State Path Integrals There exists another representation of the path integral which turns out to be very useful. It is based on an overcomplete set of states. Though one still has a resolution of the identity, like in eqn 2.3, the states are no longer orthogonal. The previous derivation of a path integral is valid until eqn 2.8; in particular, eqn 2.12 is no longer true. In general, labelling the states by / and starting from eqn 2.8, we have K(l',T;l,0) = lim f (f[dl)f[<li+1\h>e-^1^ (2.21) T~L ^ OO / \ / J \i=l / «=0 n = J^/(n*je <=° • (2-22) The basic coherent states (CS) we will be interested in are eigenstates of the annihilation operator of a Bose algebra for a one degree of freedom quantum mechanical problem. These states are generated from a fiducial vector (the ground state of a harmonic Chapter 2. FEYNMAN PATH INTEGRALS 9 oscillator) by a unitary transformation: \z>=U(z)\0> , U(z) = eza1-z*a (2.23) where z is a complex number and the a operators satisfy the Heisenberg-Weyl group. Some useful properties of the CS are summarised in Appendix A. Using the resolution of the identity and the overlap (eqn A.172 and A.173A.175), eqn 2.22 becomes K(z>,T;z,0) = KmJ ^ q d / i t e ^ e - * ^ (2.24) = Um J (j[ dfi (z^ e^"=oif [*r+i(*i+i-*0-^ .-+i*0']-efl(«.-+i^ )} (2.25) = Bm / (g^fc)) e * ^ * M ^ M ^ ) l ^ « > } (2.26) where I have restricted myself to a Hamiltonian which is polynomial and in Wick form in the a, aft. This very suggestive, and exact, discrete path integral, if supplemented by the Feynman ansatz and an interchange of the limit with the integrations (collectively known as the continuum approximation), then immediately leads to a Feynman action principle: K(z',T;z,0) = to/(ft«*,(«)) . I E M ' M ^ M ^ ) ' ] - * - « • » } - f ^ ( j [ M « ) y ^ W ^ ) < ^ ) l - ^ } (2.27) = Jm(t)eU>m'Si-'ii-}-''^} (2.28) where Vz(t) = limf[dp(zi) , and (2.29) H(z) = <z\H\z> . (2.30) Chapter 2. FEYNMAN PATH INTEGRALS 10 In § 2.1, I had already alluded to the ambiguity that would arise if one is given the formal Feynman action principle, for example eqn 2.1 or 2.28, and a quantum Hamil-tonian. In essence, the difficulty here is exactly the reverse of the operator ordering ambiguity arising in a canonical quantization scheme. The above CS representation provides a clear illustration of this and its relationship to different discretized expres-sions. To see this, note that if we have a quantum Hamiltonian which admits a diagonal representation H = JdfM(z)h(z)\z><z\ (2.31) (for example, for a second quantized operator, this will be true for any polynomial Hamiltonian already in anti-Wick form) then this leads to a second form for the path integral by a procedure similar to the above: K(z',T;z,0) = <z'\[e-^ny^\z> (2.32) = lim <z'\(l-^r€H) \Z> (2.33) = isa, / (ft * (*<)) ft < *+> i ( i - ith(z')) i* > <2-34) - to /(n4,w) . t i M ^ M - ^ ) " ] - * * } ( 2 . 3 5 ) and, again, with an interchange of limit and integrations = J V z ( t ) e ^ f o T d t i T . (2.36) Note that the two discrete path integrals 2.26 and 2.35 are both exact and, hence, the two functional forms for the Hamiltonian are different. Indeed, even in the formal path integral they are, in general, different as one can deduce from eqn 2.31: H(z) =<z\H\z>= Jdp(z')h(z')\<z\z'>\2 . (2.37) Chapter 2. FEYNMAN PATH INTEGRALS 11 Whenever they differ, eqn 2.36 would be a more natural choice than eqn 2.26. We will establish their possible validity in the next chapter. However, an attempt at a semiclassical (i.e. h —> 0) solution makes clear one of the fundamantal problems with these two path integrals: there exists two boundary conditions but the two equations of motion are degenerate and first order. Hence, in general, the boundary conditions are incompatible with the equations of motion. This has already been analysed in detail by Klauder [3]. As a final point, the first order (in time derivative) nature of the continuum CS action hints at a possible relationship to the phase space path integral. This can be made formal by noting the operator identities a = ~j^(q + ip) , o) = ^={q-ip). (2.38) If we perform this at the semiclassical level (i.e. a —> z), then the continuum action is S = £dt(pq-h(p,qj) (2.39) where h(p,q) corresponds to H(z) or h(z) and with the appropriate transformation of the argument. 2.3 The Feynman Ansatz We have already indicated that the Feynman ansatz might possibly be valid for the configurational path integral. What is its status for the CS path integral? Going back to an expression like eqn 2.26, the fact that the first two terms in the exponential have no explicit e present means that the latter does not govern the scale of (z,-+i — Zi) even in the large n limit. In particular, we do not have a relationship like eqn 2.20. Hence, the space of paths V might include discontinuous paths. Chapter 2. FEYNMAN PATH INTEGRALS 12 In effect, we see that the very nature of the linear action does not phase out discon-tinuous paths and, therefore, invalidates the continuum approximation which was used to derive that action. The logic seems to be flawed and a mathematical expression for this will now be derived for a specific Hamiltonian. (From now on, we will use the natural set of units: % — &B = 1.) Chapter 3 T H E Q U A N T U M HARMONIC OSCILLATOR In this chapter, we will investigate the various conjectures and assumptions put forth in Chapter 2. To do so, we will find it fruitful, in general, to work with, not the Green's function, but rather the trace of the time evolution operator. In addition, we will go over to imaginary time and thus consider the function space of periodic functions. While the analytic continuation to imaginary time can be viewed as an attempt at a proper mathematical definition of the path integral, our main aim is to make the connection with the partition function in quantum statistical mechanics. We have been able to get a clear elucidation of the above points by using an exact quantum mechanical problem with one degree of freedom: the well-known and trivial simple harmonic oscillator (SHO). Furthermore, this allows a straightforward extension to field theory. 3.1 Feynman Action Principle for Coherent State Representation In § 2.2, we proposed an action principle for the CS representation along the same lines as for the conflgurational representation. We will now justify it for the quantum simple harmonic oscillator (SHO) system in second quantized form. The classical equations of motion for a SHO with frequency u> in terms of a complex variable z are z(t) = -iuz{t) , z*(t) = iuz*(t) (3.40) 13 Chapter 3. THE QUANTUM HARMONIC OSCILLATOR 14 which simply represent the rotation in the z phase space (in real time). The connection to a Lagrangian formulation is by the Euler-Lagrange equation dL d (dL\ n &-sUJ=° <3-41> and it is obvious that the corresponding Lagrangian is 1 ( .dz dz*\ L = - I z — z—-— — LOZ z + constant. (3.42) 2 ^  dt dt J How does this completely classical theory compare with its derivation from the quantum theory? If we allow the interchange of limits and integrations in the path integral and also assume the Feynman ansatz, then the Lagrangian we get has a similar kinetic term to eqn 3.42 but with the Hamiltonians h =u(z*z - 1/2) (3.43) and H = coz*z (3.44) for the two versions of the path integral we have in § 2.2. Hence, we see that all three Lagrangians agree up to a constant, though the constant in eqn 3.43 is u dependent. It should be noted that the classical limit in a path integral is only achieved by a semiclassical approximation (h —-> 0) and hence if we were to put back the /i's in eqn 3.43 and take h —> 0, the three Lagrangians in fact agree up to trivial constants. Thus the semiclassical limit does not allow one to differentiate between the two formal FPI. Chapter 3. THE QUANTUM HARMONIC OSCILLATOR 15 3.2 T h e Part i t ion Function The semiclassical limit failed to provide a clear check on the formal CS path integrals (CSPI). A more convenient method is to evaluate explicitly JdnK^J-^O) (3.45) where (3 = —it. From now on, the term path integral will be mainly restricted to this function instead of the propagators; in any case, the context should be clear. To see why we are interested in this object, let us recall the definition of the thermodynamic partition function (TDPF) in quantum statistical mechanics for a canonical ensemble: Z [[3] = Tre-*3* (3.46) where f3 is the inverse temperature. If the Hamiltonian can be diagonalized in some representation, then the evaluation of Z is trivial. In the language of matrix algebra, the trace is given by the sum of the eigenvalues (for a discrete matrix) and hence, for the SHO Hamiltonian (H = w(afa + 1/2)), we have Z[0] = Y, <n\e~pS\n>= £ e-/M»+i/2) . (3.47) n n However, note that, if we use a continuous representation which is not an eigenstate of the Hamiltonian, then the trace can be written as Z[(3] = Jdrj <71\e-W\V> (3.48) = Jdr,K(r,,P;r,,0) . (3.49) Hence, we see that the TDPF is equivalent to a kernel which has been integrated over all periodic fields (period (3) and in which the time parameter has been analytically continued to imaginary time. Note that one can then interpret the imaginary time Chapter 3. THE QUANTUM HARMONIC OSCILLATOR 16 as corresponding to the inverse temperature. In effect, this gives us a path integral formulation for the partition function. However, since we know the exact TDPF for the SHO, we will use this knowledge to investigate how to handle these path integrals. Thus, from now on, we will only consider path integrals in imaginary time. The actual evaluation of the CSPI is rather more involved than one would have expected. To illustrate some of the techniques involved, we will go back to the conflg-urational path integral. In fact, we would want both to yield the same answer. Hence, a typical example of the kind of object we will be evaluating is the following: which is the conflgurational path integral for our SHO. If this continuum path integral is understood in the sense of eqn 2.18 and 2.19 then, for a finite value of n, we have a finite number of coupled Gaussian integrals and can therefore be done exactly, at least in principle. This is, in fact, the basic example of a path integral found in all text-books (see, for example, Schulman [4]) and the equivalent normalization factor N(n) is for the one-dimensional case. Actually, doing the integrals explicitly does get very messy and an equivalent procedure is to rewrite the solution as a determinant. This arises from the fact that the n integrals can be written as a single one in which the g,-are the components of an n-dimensional vector. The quadratic form in the exponential can then be expressed as a matrix and diagonalization leads to the solution being a product of eigenvalues; in other words, a determinant. The calculation has been done in Schulman [4] and the normalization has been shown to be correct. If we look back at eqn 3.51, we see that the normalization factor actually diverges in the large n limit. This divergence is due to the continuum theory as opposed to one with finite number of modes. (3.50) (3.51) Chapter 3. THE QUANTUM HARMONIC OSCILLATOR 17 The above discussion of the evaluation of eqn 3.50 is well-defined. However, if we understand the equation as being a functional integral, then q(r) lives in an infinite-dimensional function space and the equivalent matrix is an infinite one. It is well known that not all properties of finite dimensional vector spaces carry over to the infinite dimensional ones. Nevertheless, we will follow common practice in physics and introduce a functional determinant such that, for a real function, jVq e~ » So drir)M{r)ir) = d e t - l/2 M_ (353) 3.3 Evaluating Functional Determinants We have seen that one can express the solution of a formal FPI as a functional deter-minant. Two methods are going to be used to evaluate the determinant. They both make use of the fact that, in imaginary time, the paths in the functional integral are all periodic with period /3. This periodicity leads to the discreteness of the eigenvalues of the matrix M in eqn 3.52. Thus, in the case of eqn 3.50, we have (with m = 1) Z[0} = i e t - ^ ( - ^ + A = n (ffl'+»> -1/2 (3.53) / fc=—oo To make sense out of this infinite product, we will employ the Riemann zeta function regularization procedure (see Appendix B and Affleck et al. [6]) and the following identity: / x2 \ sinhz , „K J. For this example, it is trivial and one gets •I CO Z ^ = o • if^ = £ '-'Mn+1/3) • (3-55) 2smh(iyJ n = 0 Chapter 3. THE QUANTUM HARMONIC OSCILLATOR 18 Hence, we see that the formal FPI, together with zeta function regularization, has reproduced the exact result. It is interesting to note that, because of the triviality of the equations, no arbitrary scale has been introduced (generally, a scale is introduced on implementing a regularization scheme). A second method for regulating the determinant in eqn 3.53 follows from the fol-lowing identity: detM = (3.56) and, therefore, ln(detM) =. TrlnM = T . (3.57) This r function is, in general, still formally divergent. However, let us consider again the problem of eqn 3.50. Using periodic functions to represent the matrix, we then observe that the differential of the V function with respect to u2 is dT d f . / d2 du2 T r l n ( - 3 3 + a ; 2 | t (3-58) (3.59) du2 dto2 { \ dr2 2 d 0 0 ' i n £ * + u2 OO I = T 1 • (3-60) This infinite series is clearly convergent. Assuming that one can write down a closed form for the sum, integrating it gives us back the T function. At this stage, however, we will take the slightly unsatisfactory step of throwing away the integration constant (of course, when it cannot be found). Nevertheless, this simply defines for us a new renormalization scheme (which we will call the 'derivative' scheme). This scheme is as good as any provided the integration constant thrown away is independent of any physical parameter (except that it is allowed to be proportional to (3—see eqn 3.109). Here, it is obviously independent of u. Chapter 3. THE QUANTUM HARMONIC OSCILLATOR 19 There exists a standard procedure for summing the infinite series in eqn 3.60. It in-volves finding an equivalent integral representation in a complex plane. This procedure is described in Appendix C. We have shown there that £ , ,1 = f c o t h ^ (3.61) giving r(w) = / du2-^- = 2 In sinh ^  (3.62) J du2, 2 and, hence, Thus we recover the correct partition function again, except for a constant factor (but it is clear that this small discrepancy can be removed by an appropriate choice of the integration constant). Despite the fact that our treatment of the infinities seems a bit arbitrary, the usefulness of the method lies in the complex plane representation of our solution and will be of importance for our interpretation of the corresponding problem involving the CSPI. 3.4 The Coherent State Path Integral We have already set up the general forms for the coherent state path integral in Chap-ter 2. We have also shown how the well-defined discrete path integral can be used to postulate a functional integral form for the CSPI. In addition, we have just demon-strated that this procedure works quite well for the conflgurational path integral. We will now apply the same technique to the CSPI. Note that, as for the conflgurational one, the discrete path integral can be solved exactly. Let us do so to check its correctness, and, for this, we will use the path integral Chapter 3. THE QUANTUM HARMONIC OSCILLATOR 20 in eqn 2.26. For H =w(cta +1/2) (3.64) we have H{zi+uZi) = < ^ H \ z i > = vfa z . + 1/2) . (3.65) <Zi+i\Zi> x / Then n-°° J \i=l / (3.66) and Z[(3} = Jdp(z)K(z,f3;z,0) . (3.67) The propagator can be found explicitly by doing the integrals in eqn 3.66 one by one; for example, by splitting z into its real and imaginary parts and doing each Gaussian integral separately. The result is K(z',f3;z,0) = e - * l * ' l M W a + * " « - ' t o « e - ^ . (3.68) This is indeed the exact propagator since, for example, < z'\z > = K(z', 0; z, 0) = e-H*'l2-§M2+*'** (3.69) and Z[(3] = Jdp(z)K(z,/3;z,0) = Jdp(z) e-N2(i+e-/3<")e-^ (3.70) oo = JT c-/Mn+i/2) . (3.71) n=0 As we remarked in the introduction, the discrete path integral can be used as a definition of the formal FPI. In other words, we are using the lattice cut-off e to regulate the path integral. We again observe that the final result is scale independent and finite (as Chapter 3. THE QUANTUM HARMONIC OSCILLATOR 21 It turns out that we could also have used the second form of the discrete CSPI (eqn 2.35) since the SHO Hamiltonian has a diagonal representation in terms of the coherent states: H = u(afa + l/2) =u(aaft -1/2) (3.72) = UJ j dp(z)(zz* - \/2)\z><z\ . (3.73) Subsequently, the path integral can also be written as (3.74) One can explicitly show that the two discrete CSPI are identical at this stage by using the transformation Z( —> Zi(l + y) in eqn 3.74 and this recovers eqn 3.67. Clearly then, approximating the Hamiltonian in eqn 2.27 into a diagonal form is not justified. This, therefore, casts some doubts on the exactness of previous treatments of the CSPI since they invariably make this unjustified approximation. For our purposes, it will be a lot easier to deal with the path integral in eqn 3.74. If we now do not assume « Z{ but instead write zi+1 = Zi +(zi+i - Zi) (3.75) to go to the continuum path integral , the path integral becomes ZW = Bm /(jj«W«--)) .-^'{H^'-^^M'-'K^)]^-^) = limet!|^(T)e-/o'J<«-*+5lil1+»»-«) . (3.77) Here it has been traditional to discard the e-term, which is equivalent to saying that we take the limit inside the functional integral. We propose, instead, that eqn 3.77 is the correct functional integral and, therefore, neither of eqn 2.28 or eqn 2.36 are correct. Chapter 3. THE QUANTUM HARMONIC OSCILLATOR 22 We can justify the last statement by trying to evaluate the functional integral e 2 jVz{r) e-ffd<''%+"*'>) = Z (3.78) which follows from eqn 2.36. The best way to do this is by using the derivative method introduced in § 3.3. Thus, e-^Z [f3] = det"1 (-±+lJj = c-n«) (3.79) (the difference in the formula is because z(r) is a complex function) and (iT 0 0 1 ^ = T . (3.80) K——OO fi ' Note, however, that in this case the sum is only conditionally convergent (and is a consequence of having an action which is only first order in time derivative). The natural way to treat this kind of series is by what we may call a symmetric regulator: 1 + A:=-oo fi ^w ~ k=l \_ 0 oo = LO 2irik i ' 27rifc ( , . + LO „ h LO (3.81) E 7 • ( 3 - 8 2 ) *=-~(3j*)2+w* But we have already evaluated this series and we, therefore, get Bw ZIP] = T T > s 2 j ; e ^ (3.83) smb. %- n = 0 # Z[P]. (3.84) Hence, we see that our continuum FPI in eqn 2.36 is wrong. As such, the difference is only a shift in the energy spectrum but we note that it is w-dependent (which we will see to have interesting effects when we will consider spin). Also, we have mentioned that, within the context of gravity, it is important to have a consistent treatment of the vacuum energy. Chapter 3. THE QUANTUM HARMONIC OSCILLATOR 23 Let us now see how the new proposal for a CSPI (eqn 3.77) solves the problem. Again the solution is formally Z\p\ = e^det"1 d_ ' dr ed?_ 2dr2 (3.85) (3.86) g i v i n g A,., ~ Viewed as a function of a complex variable z, the following 1 has poles at which, for small e, becomes iz + | z 2 + u> z % z = — ± + 2ew e e 2i . Z , IU) e (3.87) (3.88) (3.89) (3.90) Hence, the analytic structure of eqn 3.88 is similar to that shown in Figure C.2 except that the pole in the lower half-plane has a separation from the real axis which increases as e —> 0. If one now uses the integral representation, the latter pole is seen to contribute and we get and, hence, r(w) = In sinh ^  + ^  . (3.92) The second term is the contribution from the 'movable' pole and is precisely what is needed to correct the previous calculation: Z[/3] = e-W+S? 1 sinh^ " (3.93) Chapter 3. THE QUANTUM HARMONIC OSCILLATOR 24 Again, we have left out the integration constant knowing that it is w-independent. The result of the above calculations is significant. We have found that Z is not well-defined even after the derivative regularization. No such problem arose for Z and we can consider the e-term in the latter to act like a damping term. Furthermore, Z becomes Z if we interchange the functional integration with the limit operation. Therefore, we find that the partition function is non-analytic in the lattice cut-off. This proves that one cannot, strictly speaking, interchange the limit and integration operations; the continuum approximation is not well-defined. The fact that it works for the configurational path integral is then just a peculiarity of that representation; specifically, because the action is intrinsically second order in time derivative. In hindsight, one can actually postulate a formal FPI which makes do with the presence of an e-term. Comparing eqn 3.78 and 3.83, then clearly, gives the correct answer. In fact, this can even be derived from our phase space path integral (eqn 2.17) by the inverse of the transformation we noted in § 2.2. The interest-ing thing is that it looks just like a Feynman action principle where the corresponding classical Hamilton function does not carry the ground state energy of the quantum Hamiltonian. Nevertheless, we have already mentioned that it is really not properly defined and also cannot be naturally transcribed into a discretized form. 3.5 Reexamining the Continuum Path Integral In the previous section, we have derived a continuum path integral by applying the usual rule of differential calculus in going from a discrete variable to a continuous one. It should be clear that keeping a second derivative term is still only an approximation. In fact, we know that a function defined at two neighbouring points is given by the (3.94) Chapter 3. THE QUANTUM HARMONIC OSCILLATOR 25 Taylor series: f(z + h) = f(z) + hf'(z) + ^f'(z) + ^f"'(z) + 0(h4) . (3.95) Basically, the usual continuum approximation has been to stop such an expansion at 0(h) terms; we have demonstrated the need to keep 0{h2) terms too. One might wonder whether further terms might not invalidate our results. Of course, the exact discrete calculation is equivalent to summing the series completely and we have found that it corresponds to an 0(h2) truncation. It is therefore very tempting to see what an 0(h3) truncation might do. To do so, we will go back to the discrete path integral and rewrite the solution in a form which is convenient for such a Taylor expansion. Recall eqn 3.74 which we will rewrite as Z[(3] = 0U = e 2 with One observes g i v i n g B m / ( n < W « ) ) « - ^ W 1 ^ - - ' * ' l Mij =(1 + eu) Sij - Sitj+1 . M ^ = M ^ = [(1 + eu) - e~^} # def^M) = lim 'n-[n/2]-l n ( i + >=-[n/2] €U> — e -1 = lim 'n-[n/2]-l 2irik + -(2irk\ — -e ^ ) 3 + 0(e3) -l -1 (3.96) (3.97) (3.98) (3.99) (3.100) (3.101) Chapter 3. THE QUANTUM HARMONIC OSCILLATOR 26 Thus we see that the two previous continuum path integral treated are just first and second order (in e) truncations. Adding one further term gives _ _ t . (3.102) * = - o o W + — + —) -~6\TJ To sum this we will have to analyse tdz cot 2/2 for various contours. The poles of the integrand are at (3.103) z = kit and (for small e) (-3i ± y/lb) (3.104) and by our now common procedure, we find that = l t h ^ + l ( 3 . 1 0 5 ) du 2 2 2 v J that is, exactly as for the 0(e) truncation. The conclusion is then that the second order term does not contribute. However, if one looks at the details of the algebra, one finds that this is true only for that specific second order term as in eqn 3.102; changing its Taylor coefficient would end us a result different to eqn 3.105. Of course, it is not possible to prove that all further truncated expressions will be exactly the same. Nevertheless, what we have found is that, when going from the discrete to the continuum path integral, the exact discrete expression can be replaced by a continuum one which includes a term linear in e. To conclude, note that the CSPI in eqn 3.77 can also be regularized by the zeta function method. With the help of Appendix B, the following identity sinh(x + y) s'mh(x — y) n k——00 x2 1 + , (kir + iy) sinh2(y) (3.106) Chapter 3. THE QUANTUM HARMONIC OSCILLATOR 27 (see, for example, Gradshteyn & Ryzhik [7]), and two pages of algebra, one obtains id e d2' d e t U+ U ;" 2 ^ n k=—oo 2 sinh ^  2nik e(2itk\' (3.107) (3.108) This result was first derived by Bergeron [6,8]. Contrary to the previous use of the zeta function, this path integral does not quite give the exact result and one has introduced a scale parameter (for finite e). But note that the physically relevant quantity is the free energy and it is given by F = - l n Z (3.109) and the path integral only differs from the partition function by an infinite temperature independent shift in the free energy. Chapter 4 GENERALISATION OF T H E PATH INTEGRAL 4.1 A Fermionic System: The Grasmannian Oscillator The main part of this work has been to analyse the CSPI for a bosonic quantum harmonic oscillator. The latter is based on the canonical commutator. Equally, one can set up fermionic degrees of freedom on a.Fock space by the use of anti-commutators. If we make such a change for a pair of a, a* operators, we end up with what might be called a fermionic oscillator. We will now show how one might set up a path integral formulation for this new problem in terms of classical variables only, that is, our formulation does not require the introduction of Grasmannian quantities. In accordance with our previous discussions, we will derive a path integral formula-tion of the theory by first stating the quantum Hamiltonian and then using the Trotter product formula. For fermions, it does not make much sense talking about a classi-cal theory and this is clearly an impasse in postulating a Feynman action principle. Leaving out any ground state energy term (formally, this can also be achieved from a more general Hamiltonian by a normal ordering process), we will consider the following Hamiltonian: The elementary excitations are represented in the Fock space by the Dirac kets. The H = ujtfb (4.110) where (4.111) 28 Chapter 4. GENERALISATION OF THE PATH INTEGRAL 29 algebra of the operators ensures that the statistics is that of spin-1/2 particles and, hence, n can only be 0 or 1 by the Pauli exclusion principle. By a unit normalization of the state, we effectively have a single fermion in a two-level system. We have chosen this particular Hamiltonian because, as for the bosonic case, we know the TDPF trivially: Z[(3] = £ <n\e-^b\n> (4.112) n=0,l = 1 + e~pw . (4.113) Now we want to choose a representation in which the Hamiltonian is not diagonal. Because of the simplicity of the above system, a most general state is |z>=(l-|2|2)1/2|0>+2|l> (4.114) which is just an eigenvector decomposition with normalization constraint; in general, z is a complex number. In fact, we will use the state in eqn 4.114 just as it is, given the following properties: \z> ~ e('-w2)1/26,|0> (4.115) <z\z>> = ( l - | ^ |2)1 / 2( l - |2' |2)1 / 2 + 2V (4.116) r d2z / — \z><z\ = 1. (4.117) J\z\<l 7T Eqn 4.117 is what we need to set up a path integral representation; we note that the measure is the same as that for the bosonic CS, except that this time the integral is only within a unit circle instead of the whole C-plane. The analogy with the bosonic case is further strengthened by eqn 4.116 (which gives us an overcompleteness relation— the overlap is never zero), and eqn 4.115 completes the analogy (where we have not bothered to write down the normalization). Thus, in every sense of the word, we seem to have a fermionic CS. However, recall that the semiclassical limit does not exist (or, at least, does not make sense) and, because there exists an upper bound on the spectrum, Chapter 4. GENERALISATION OF THE PATH INTEGRAL 30 the new states are not right eigenvectors of the annihilation operator. This should not affect the formal development, except that we no longer have a diagonal representation like in eqn 2.31. In any case, eqn 2.8 is still valid: K(z',/3;z,0) = Hm J (j[dfi(zi)) J X i ^ ' + i l ^ and ir, \ _ <Zi+i\H\zi> — zi+lzi (4.118) (4.119) <Zi+i\zi> Unfortunately, in this case, the overlap is not already in exponential form and, hence, we do not have a nice discrete action. Still, if we write jj+i = Zi +(zi+i — z^ — Zi + eii (4.120) and use an expansion in e, we have In <zi+1\zi> In ( l - | zi +i | 2 ) 1 / 2 ( l - \zi\2Y'Z + z*+1Zi 1/2 Uziz*-z*zi] + e-\zi\2(\zi\2-l)+0(e2) and, therefore, The traditional continuum limit gives Z[0] = Jdn(z)K(z,0;z,O) (4.121) (4.122) ff(z',#z,0) = Jim / f f t ^ ) ) e-^iW+t^-ton+Wv+^+'K*)} . (4.123) n _ + 0 ° \t=i / (4.124) (4.125) Surprisingly, we have recovered a continuum path integral which looks just like the bosonic one. There exist two differences though: fermionic nature requires z(r) to be Chapter 4. GENERALISATION OF THE PATH INTEGRAL 31 an anti-periodic function (period /?) and normalization to one particle limits the inte-gration to the interior of the unit circle. One thus have a formal unifying expression for both bosons and fermions and the different partition functions arise from the different function spaces. With the theory of functional determinants of § 3.3, then Z [f3] = e~r (4.126) where T = Trlnf^+wj (4.127) fc=—oo 27rt'(fc + 1/2) h LO (4.128) from a representation in terms of anti-periodic functions. Using symmetric and deriva-tive regulators give Z [[3] <x e^ r (i + e-pw) . (4.129) Again, we observe a /?u>-dependent energy shift from the correct one. Another feature similar to the CSPI is the presence of a damping term in a more exact continuum action and presumably the latter might help in cancelling the energy shift, though one cannot exactly calculate its effect here due to the quartic nature of this term. Our fermion path integral also suffers from the fact that the natural discretized expression, eqn 4.118, cannot be evaluated by explicit integration or by a finite deter-minant method. The functional integral, however, can still be evaluated by the zeta function method. One then find and, hence, d e t ( £ + „ ) = fl ( 2 " ( t ; V 2 ) + «) « 2cosh f (4.130) Z ^ d e t - . j ^ + ^ ^ j L - . (4.131) Chapter 4. GENERALISATION OF THE PATH INTEGRAL 32 This is not the correct result. Since the derivative method did work, the conclusion then seems to be that the zeta function procedure is not right in this case. Though this does not constitute a rigorous proof, it nevertheless strongly suggests that zeta function regularization is not actually a well-behaved procedure. Already, in the pre-vious chapter, we have found that the exactness of the result cannot be guaranteed; presently, it looks like the method might not be applicable to all situations. 4.2 Comparison of the Two Oscillators Notwithstanding the differences mentioned in the previous section, we are tempted to propose a single formal FPI as a formulation for both the bosonic and fermionic where Z(T) is a periodic (anti-periodic) function for the Bose (Fermi) theory. Of course, the previous chapter has taught us that this is so only if we drop all ground state energy terms. Despite the restrictions, it is nice to see that we can reproduce the two spectra from the same formal expression. 4.3 Field Theory We take this opportunity to recall that we have only been treating quantum mechan-ically problem of finite systems. We have not extended our work to field theory. This should not be too difficult and we want to summarize here what might be done. The canonical system in this case would be a free scalar field. In fact, it can be seen as just a collection of harmonic oscillators: oscillators: (4.132) (4.133) k Chapter 4. GENERALISATION OF THE PATH INTEGRAL 33 A bonus when one considers field theory is that one can then use dimensional regular-ization and compare it with the zeta function procedure. Chapter 5 APPLICATIONS In the previous chapters, we have analysed in fair detail formulations of coherent state path integral for two trivial quantum mechanical systems. In addition to carrying out a check on the formalism, however, it would be nice to see possible applicability of the ideas investigated. We will start by considering the problem of writing down a path integral formu-lation for a spin system. Of great interest in condensed matter physics nowadays is to understand the effective low energy behaviour of quantum Heisenberg ferromagnets and antiferromagnets. The latter, in particular, is suspected to provide the proper de-scription of the high-Tc superconductors. The postulated existence of neutral fermions by the RVB group [9] has challenged a proper field theory description. Haldane [10] and Affleck [11] first paved the way by mapping the antiferromagnetic chain onto the non-linear sigma model in the large-s limit by a canonical approach. Recently, how-ever, a number of Russian authors (see, for example, Wiegmann [12]) have succeeded in achieving this mapping in a path integral formulation in the coherent state rep-resentation. We have not reached this far in our work and, instead, we will simply treat the case of a single spin in a magnetic field. By recognizing the Schwinger boson transformation, we will attempt to relate the bosonic oscillator to a spin. We will then briefly mention how one might identify the Berry phase arising from a quantum cyclic evolution from a coherent state path integral. 34 Chapter 5. APPLICATIONS 35 5.1 A Path Integral for Spin We want to investigate an elementary system characterized by only spin degrees of freedom and with a spin quantum number s. There exist path integral formulations based on SU(2) coherent states (for example, by Klauder [3]). We are discussing this problem here, however, because there seems to be a connection with our CSPI for bosonic oscillators. Spin degrees of freedom are related to a pair of harmonic oscillators by the Schwinger coupled boson representation (see, for example, Mattis' book [13]): S=]^aft-a-a (5.134) where S is the spin vector operator, a a 3-vector with Pauli matrices as components, and aft = (c4,c4). In particular, Sz = ^(aj«i - <4a2) = | (ni - n2) (5.135) and the quantum number operator 5 = ^(a{aa + a\a2) = ^(nj + n2) . (5.136) The idea is that, if we start with a pair of these bosonic oscillators and then impose a constraint on the total number of excitations, this should map onto a spin s problem. So if we impose this constraint as an operator Kronecker delta function and then use an integral representation, one can write the partition function as Z[P] = T r j e - ^ ^ a t - a - 2 s)j (5.137) = d6_Tv(-m-ie(a^a-2s)\ ( n e Z + ) . (5.138) J-mr 2im I J In this section, our canonical system will be a spin interacting with a magnetic field H = B-S = BSZ (5.139) = f (m - n2) . (5.140) Chapter 5. APPLICATIONS 36 (Note that we have not taken the pain to explicitly put in the caret on our operators except for the Hamiltonian; there should be no confusion.) The magnetic field defines a direction in space and we have chosen our coordinate axes so that it is the z-direction. In this case, we have three ways of evaluating the partition function. First, the very trivial method, spin algebra gives us Z[f3] = J£^En= E e~m 0 B (5.141) where m is the eigenvalue of the Sz operator and is, therefore, integer or half-integer. The problem gets slightly less boring when we re-interpret it in terms of Schwinger bosons. So, from expression 5.138, ra = /_ as mr Inn n* d9 n-ir oo 2irn CO p,r=0 = E e p,r=0 n,r de 2im J6[2s-(p+r)] (5.142) (5.143) (5.144) So far, so good. Finally, what we are really after is a path integral representation. Unfortunately, no consistent and rigorous picture has emerged yet from our calculations. Nevertheless, we will present them since there seems to be some intrinsic correctness in the large-s limit. It is straightforward to write down the path integral from the formalism developed in chapter 2 and 3. First note that B 2 H - —a*• az • a = —a • oz • a* B 2 (5.145) and rescaling the Lagrange multiplier 6 = /3\, the path integral is Z [fl = lim r" ilH^m [Vz(T) e- JM*'* + ^ + f "0*+!M2} . (5.146) Chapter 5. APPLICATIONS 37 Here z is a 2-component variable. Our analysis in the previous chapter shows us how to absorb the 0(e) term into a formal FPI: Z  ^ = J"*" jVz(r) e- r<«U+»+f *')' (5.147) with the caveat that a symmetric regulator is used to evaluate the FPI. Note that if we had assumed the traditional continuum approximation, the corresponding path integral would be Z = r " ^e*W hz(r) e~ W ^ + f 4 . (5.148) Hence, what seemed to be an innocent oversight for the SHO is seen here to 'modify' the spin quantum number. If this is correct, it might change the occurence of the Haldane gap [10], as it is presently understood, for a magnetic lattice. Anyway, one can evaluate eqn 5.147 as before, with a new complex (operator) frequency u = i\ + — a2 . (5.149) By continuing A into the complex plane such that Re(u>) > 0, one can integrate out the z variables first and then do the A integral: J C nir/p 2nn W /3 /3d\ k=—oo 2-Kik -n7r s IP 27rn(l - eiA+fi/2)(l - eiX~B/2) (5.150) (5.151) m=—s -mj3B The path integral thus reproduces the correct answer. How does our spin path integral compare with others? In particular, Fradkin & Stone [15] and Affleck [14], among others, have derived a CP1 path integral which bears some similarity with eqn 5.147 and, in so doing, they introduced a unit vector Chapter 5. APPLICATIONS 38 from the beginning. For example, Affleck [14] used the CP1 formulation of the unit vector ft = z*Bz (5.152) with z a unit complex spinor and used, as his coherent state basis, the eigenstates of S • 0 with highest weight: 5-fi|fi>=s|0> . (5.153) The path integral he got is with Z[P] = Jvti(T)e-$odr^) (5.154) dz L = 2sz*— + H(z*oz) . (5.155) dr The problem with eqn 5.147 is that the A integral no longer imposes a delta function constraint since z is no longer restricted to being integer-valued. Actually, before we even treat this problem, we would have to modify eqn 5.147 to make it look more like the CP1 model. In order to do so, we have to recognize that the constraint on Z(T) should really be local in time. Hence, we will need to 'gauge' the Lagrange multiplier. It is then easy to see that we get Z[(3] = lim f [ /d^Zi) <z^\e-^{^)\\Zi> . (5.156) If we now rescale z by l/\/2s + 1 and A by (2s + 1) so that the measure is invariant, Z[(3] = JVz{r) |pA(r)e«.f>*T>C-.J>^ . (5.157) where JVX stands for lim TT / ,n ' . (5.158) n - t c o "=1 J-niA2s+\)lt 2-Krn(2s + 1) Chapter 5. APPLICATIONS 39 The thing is, we still have not recovered a delta function constraint. Remember that (as we found in Chapter 3), strictly speaking, the limit e —» 0 has to be taken at the end of the calculation. However, out of curiosity, let us cheat and put e — > 0 before the game is over. This immediately gives us the much sought for constraint and Z[0] = Jvz(r)s(\z\2 - l) e-f0Pdrz-p.+i)&M.+i/2)B.*], _ ( 5 1 5 9 ) Comparing eqn 5.159 with 5.155 we observe a basic difference: one has a (25 + 1) coefficient for the kinetic (Wess-Zumino) term while the other has a 2.S coefficient. This actually has no effect on the equation of motion for a single spin: w^5 = J2A^5 = fiAJ3 (5.160) dr 0^ for both actions. Yet, it is fairly easy to demonstrate that eqn 5.159 is inadequate. First of all, note that one can project out spin degrees of freedom by considering the problem of a particle on a sphere with a monopole in the limit of vanishing mass (see, for example, Coleman [16]). It turns out that this tells us that the coefficient of the kinetic term has to be 2s (Affleck [8]). Also, if one tries to write down the equation of motion for a Heisenberg ferromagnet, Affleck [14] has shown that his Lagrangian gives the dispersion relation of spin-wave theory: E(k) =\J\sz 1^ - XyX7*j (5.161) where J is the energy scale, z the coordination number at a lattice site and 8 is the translation vector to neighbouring sites. On the other hand, from eqn 5.159, s would get replaced by (s + 1/2). Finally, there have been experimental and numerical confir-mations that the Haldane gap arises as predicted by the Lagrangian of eqn 5.155 for small spin (see, for example, Affleck [17]). Yet, both models are actually weakly consistent. The clue is to recognize that, intrinsic to the drivation of eqn 5.155, is the assumption of large spin. This very Chapter 5. APPLICATIONS 40 assumption can also give us eqn 5.159. Arguably, in this limit, s + 1/2 is the same as s. (A questionable aspect of large parameter theories is exactly how big is big.) To summarize, we have presented another derivation of the CP1 model but, as with previous derivations, it is only valid in the large s limit. 5.2 Berry Phase 'Berry phase' is one of the in-phrases in theoretical physics of the late 1980s. It was originally conceived [18] as resulting from an adiabatic cyclic change but has recently been extended to more general cyclic changes [19]. Experimentally, this phase can be observed in molecular spectra, optical phenomenon, etc ... A comment by Fradkin & Stone [15] in their paper on the spin path integral has prompted us to identify a geometric phase factor in our CSPI. In that respect, let us go back to eqn 2.21 and realise that, for infinitesimal time interval, one can write % >s H mf e l > d ^ > (5.162) and, hence, the kernel </,-+! |/,-> = lim|l-e</t + 1|^:|/i +i>| (5.163) = lime-£</,+l|£|,'+1> . (5.164) Now, instead of eqn 2.22, we write, in the continuum approximation (and noting that, in our Euclidean time formulation, the states are periodic), Z[f3) = | 2 > Z ( r ) c - J > H * l ' > - « W } . ( 5.1 6 5) The first term in the exponential is pure imaginary (as may be more evident in, say, eqn 2.26). It is obvious that, in the semiclassical limit, this term contributes a phase which is due to the cyclic evolution of the state vector. Chapters. APPLICATIONS 41 In order to illustrate the fact that this geometrical phase factor is made up of two distinct contributions, we will briefly continue the review of the path integral derivation of spin chain sigma model correspondence as due to Affleck (§ 5.1, [14]). We are here treating the antiferromagnetic Heisenberg spin chain. From the Lagrangian in eqn 5.155 there are just two more steps. First, one generalises it for a lattice of spin: with (j> • I = 0 and we are using a two-sublattice model. It can then be shown [14] that the imaginary term in the action gives us the Lagrangian density The first term is the topological Berry phase while the second one is kinematical. (5.166) and then it will be useful to decompose fit: —• —* —* Sli = ±<f>(xi) + l(xi) js (5.167) (5.168) Chapter 6 C O N C L U S I O N S The main aim of this project has been to examine the issue of writing down a (con-tinuum) Feynman-type path integral in terms of coherent states for an exactly soluble system. Our analysis shows that a canonical FPI is ambiguous and can be made to reproduce the correct result by an appropriate choice of a regulator, though we do not consider this arbitrariness to be satisfactory. Our approach has been to introduce exact and well-defined discrete path integrals and the above ambiguity is then identified with an incorrect application of the continuum limit. Specifically, the approximations are both too crude. The first one is the one leading to the mathematical difficulty, while the second one can only be justified in the semiclassical limit. A way to resolve these difficulties has been found, involving introducing an infinitesimal quadratic term, and the resulting path integral is shown to give the correct answer (up to trivial constants) for different regularization schemes. The other major part of this project had been an attempt at providing an alterna-tive derivation of the CP1 model for spin. This follows from our work on the quantum and <zi+1\H\zi> <Zi+l\Zi> 42 Chapter 6. CONCLUSIONS 43 harmonic oscillator by analysing the spin system in terms of Bose-like excitations (us-ing a Schwinger boson transformation instead of, for example, the Holstein-Primakov one since the former preserves rotational invariance). The expectation was to have a derivation which would be rigorous for all values of spin. (It turns out that, in previ-ous path integral derivation, use of the naive continuum limit is related to the large s aproximation.) However, this work is still incomplete as yet. Somewhere in the middle of all this, we have used the coherent state path integral to illustrate the Berry phase arising from an antiferromagnetic spin model. Finally, we have also presented a derivation of a fermion path integral which is closely related to the discussion of the bosonic oscillator; this has brought up a possible anomaly in the use of the zeta function regularization. What remains to be done? Clearly, the next logical step is to transcend quantum mechanics and consider field theory proper. Another worthwhile endeavour would be to solve the spin problem. As we mentioned above, the first step is to achieve the CP1 correspondence. Given that, it would be trivial to generalise to a lattice of spins and, for example, provide an alternative mapping of the Heisenberg antiferromagnetic model onto a non-linear sigma model. Bibliography [1] R.P. Feynman Rev. Mod. Phys. 20, 267 (1948). [2] P.A.M. Dirac in 'Selected Papers on Quantum Electrodynamics', J. Schwinger (Ed.), Dover, New York 1958. [3] J.R. Klauder Phys. Rev. D 19(8), 2349 (1979). [4] L.S. Schulman 'Techniques and Applications of Path Integration', John Wiley, New York 1981. [5] L.H. Ryder 'Quantum Field Theory', Cambridge University Press, Cambridge 1985. [6] I.K. Affleck, M. Bergeron, L.C. Lew Yan Voon &: G. Semenoff 'Coherent State Path Integral and the Harmonic Oscillator', UBC Preprint 1989. [7] I.S. Gradshteyn & LM. Ryzhik 'Table of Integrals, Series and Products', Academic Press, New York 1980. [8] Private communication. [9] P.W. Anderson Science 235, 1196 (1987). [10] F.D.M. Haldane Phys. Rev. Lett. 50, 1153 (1983). [11] I.K. Affleck Nucl. Phys. B 257, 397 (1985). [12] P.B. Wiegmann Phys. Rev. Lett. 60(9), 821 (1988). 44 Bibliography 45 [13] D.C. Mattis 'The Theory of Magnetism', Vol. 1, Springer-Verlag, Berlin 1981. [14] I.K. Affleck, unpublished notes. [15] E. Fradkin & M. Stone 'Topological Terms in One and Two-Dimensional Quantum Heisenberg Antiferromagnets', University of Illinois, Urbana, Report No. ILL-TH-88-12, 1988. [16] S. Coleman in 'The Unity of the Fundamental Interactions', A. Zichichi (Ed.), Plenum Press, New York (1981). [17] I.K. Affleck 'Quantum Spin Chains and the Haldane Gap', UBC Preprint 1989 (submitted to J. Phys. C). [18] M.V. Berry Proc. R. Soc. London A 392, 45 (1984). [19] Y. Aharonov & J. Anandan Phys. Rev. Lett. 58, 1593 (1987). [20] R.P. Boas & C. Stutz Am. J. Phys. 39, 745 (1971). Appendix A Properties of Coherent States Given a pair of Bose operators a, a) in a Fock space with vacuum |0> , i.e., a, a = 1 , a|0>=0 (A.169) one can conveniently express an overcomplete set of (coherent) states as z> e 2 a t - 2 * a |o> (zeC) (A.170) (A.171) n=0 Here C represents the entire complex plane. By using eqn A.171 and a polar represen-tation for z, one can easily deduce the resolution of the identity: J dS££Mms.\z><z\ = X Equally easily, one can establish the following: (A.172) <z\z> , * " * - § ( M2 + | * f ) = e 2 .l[z"(z'-z)-4z'-zr} e-,Wo*|*> \e~lwtz> <z'\e-i^at\z> = e-2'*26"iut-HV\2+\*\3) . (A.173) (A.174) (A.175) (A.176) (A.177) The above mathematical properties (especially eqn A.172 and A.173-A.175) are basi-cally all we need to set up a path integral representation. On the physics side, these 46 Appendix A. Properties of Coherent States 47 states are brought in because they are the quantum states closest to a classical descrip-tion of oscillators. For example, eqn A.176 gives the correct equation of motion for electromagnetic fields. Relevant to the states being minimum uncertainty ones is the Poisson distribution of the quanta in these states and with a peak such that <z\n\z>= \z\2 . (A.178) Appendix B Properties of the Riemann Zeta Function The Riemann zeta function can be defined as oo cco = i > ~ * . n ( E 5 + (B- 1 7 9) n=l which is an analytic function for Re(z)> 1. One can obtain an analytic continuation over the complex plane, except for a simple pole at z = 1, by the integral representation C(z) = T ( - \ ~ Z ) I dt (B.180) s v J 2-KI Jc e-'-l v ' and where the contour C is as shown in Figure B.l. One has oo C'(z) = - £ ( h m ) n - 2 (B.181) n=l and, in particular, C(0) = ~ , C'(0) = - i l n 2 7 r . (B.182) The regularization of infinite products stems from oo n(cmm) = c -»ao)+flo)ina (B.183) n=l which can easily be established from the previous equations. What effectively happens is that the infinite product is represented by some function which can then be ana-lytically continued (in some parameter). The method of analytic continuation is also reminiscent of dimensional regularization in field theory. In the zeta function case, the 48 Appendix B. Properties of the Riemann Zeta Function 49 procedure isolates and removes the infinities all in one go. Hence we have, for example, CO -1 n« = -?=• (B-184) n=l V « oo TJ" = V2TT ( B . 1 8 5 ) 71=1 which we will use in the text. Appendix B. Properties of the Riemann Zeta Function 50 -» fleet) Figure B . l : Contours for the integral representation of the Riemann zeta function. Appendix C Contour Integral Representation of an Infinite Sum We will illustrate this technique for the following infinite sum: oo 1 (C.186) oo The trick is to see that this can be represented by a contour integral which can then be evaluated by the method of residues involving a small number of poles (see, for example, Boas Sz Stutz [20]). What we need is a function of a complex variable with poles at z — kit/a, for all integer k. The relevant function is cot az with residues where the contour T is as shown in Figure C.2; simply a large circle whose radius R is then taken to oo. It is obvious that I = 0. Note however, that by the Cauchy integral theorem, one also have R-i{z = kir/a) — 1/a . (C.187) So let us consider the following integral: (C.188) (C.189) This implies (C.190) 51 Appendix C. Contour Integral Representation of an Infinite Sum 52 But the LHS is simply, by the method of residues, and the RHS Hence 1 °° 1 ~ E 77TT (C191) cot tau cot(—zau;) 1 cot iau> ,p , — — 1 — > = : . (C192) 2iu —2iu) iu) £ J — ± = -cothau;. (C.193) * = - ~ ( T ) + <" 2 U> Appendix C. Contour Integral Representation of an infinite Sum 53 * lm(z) Figure C.2: Contours for the integral representation of an infinite sum. 


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