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Coherent State Path Integral for the Harmonic Oscillator 11 COHERENT STATE PATH INTEGRAL FOR THE HARMONIC OSCILLATOR AND A SPIN PARTICLE IN A CONSTANT MAGNETIC FLELD By MARIO BERGERON B. Sc. (Physique) Universite Laval, 1987 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1989 © MARIO BERGERON, 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 Harmonic Oscillator Coherent States 3 2.2 Spin Coherent States 6 2.3 Propagator, Path Integral and Partition Function 8 3 Coherent State Path Integral for the Harmonic Oscillator 11 3.1 Exact Result for the Discretisation 11 3.2 Continuum Limit 12 3.3 Semiclassical Approximation 16 3.4 Regularization 18 4 Coherent State Path Integral for Spin 26 4.1 Discretisation with Spin Coherent States 26 4.2 The Schwinger-Boson Model 27 4.3 Equivalence of the two Representations 28 iii 4.4 Continuum Limit of the Spin Coherent States 30 4.5 Continuum Limit of the Schwinger-Boson Model 32 5 Path Integral for a Charged Particle in a Magnetic Monopole Field 39 5.1 Monopole Vector Potential 39 5.2 Monopole Angular Momentum 41 5.3 Path Integral for a Spin Particle in a Magnetic Field 43 5.4 Comparison with Coherent State Path Integral 45 6 Conclusion 47 Bibliography 50 Appendices 51 A Identities for Determinants 51 B Schwinger-Boson Model 54 iv List of Tables Second order product regularization List of Figures 3.1 Discrete and continuous representation of the z variable . 14 3.2 Contour of integration for the T(x) function of the discrete determinant . 21 vi Acknowledgement I would like to thank my supervisor, Dr. Gordon Semenoff, for his explanation of various subjects in theoretical physics. Our many discussions were very helpful in understanding details I was not familiar with. In particular, he introduced me to the subject of coherent state path integral, which, with his suggestions and comments, led the way to this thesis. Furthermore, I wish to thank Dr. Ian Affleck and his graduate student, Lok C. Lew Yan Voon, with whom we had constructive discussions on this subject, and by introducing to me the method of contour integral regularization (chapter 3.4). De plus, il me fait plaisir de remercier mes parents pour leur support continu, tout au long de mes etudes a cette universite. Malgre la distance qui nous separe, residant de chaque cote du pays, leur presence et soutien a toujours ete constant. Finally, I would like to thank the department of physics and UBC for accepting me as one of their graduate student, and their support in my work for the last two years. vii Chapter 1 Introduction The use of path integrals is a very active subject in physics. They have found many applications in quantum field theory, particle physics and condensed matter physics. The foundations have been studied for some time [7,8], and are by now well established. Independently, in nuclear physics and quantum optics [9], some models have been studied using coherent states as a bridge between quantum theory and classical 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 > And then build up all the eigenstates of the Hamiltonain: • • • nw \ni...nM >= -7==L—{a\/Vnr («M/v^) | 0 > M u H | nx... nM >— YI h{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 then find that (2.1) imposes the conditions: | x >=| 0 > and A = r>kQk-9hp*, which gives: M \p,q>= exp{£ TiPkQk ~ qkPk)} | 0 > k=l muj In terms of ak, ak and the complex variable zk = -^{\/ Qk + ^§^j) M 1 1 + | p,q >=| 2 >= exp{£ -(z*aj -**afc)}|0 >= exp{-ta 2 - zU)}\0 > (2.2) k=i / „ \ by working with columns a > 2 2 2 We can easily verify that < z | (Pk -Pk) \ z >=< z \ (Qk - qk) \ z >= %/2.
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