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Stochastic Quantization of Boson and Fermion Fields STOCHASTIC QUANTIZATION OF BOSON AND FERMION FIELDS by GEOFFREY HAYWARD M. A., University of Toronto, Toronto, 1984 B. A., Yale University, New Haven, 1983 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 May 1986 © GEOFFREY HAYWARD, 1986 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 f^k^lCS The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date 1} J<?f{ /an ii ABSTRACT We consider two strategies for stochastic quantization. With the first, one posits an additional time dimension (fictitious time) and describes the evolution of classical fields by means of the Langevin equation. One then evaluates stochastic averages of the field functions. In the limit that the fictitious time goes to infinity, these approach the time ordered correlation functions of canonically quantized field theory. We conclude that, while this strategy successfully describes QED and other quantum field theories, it is contrived and probably lacks deep physical significance. With the second strategy for field quantization, one begins with a classical action in either one or two extra dimensions coupled to an external random source. We review a method of quantizing bosonic fields which uses this strategy. Further• more, we present an analogous method for quantizing fermion fields and a possible new way of quantizing interacting fermion and boson fields. Finally, we discuss ap• plications to quantum mechanics and stochastically quantize the simple harmonic oscillator. iii TABLE OF CONTENTS Abstract ii Table of Contents iii List of Figures . iv Acknowledgments v Introduction 1 Part I: The Langevin Method of Stochastic Quantization 3 11 Review of Canonical Quantization 3 1-2 The Langevin Equation for Bosons: The Method of Parisi and Wu 4 a) The free field two point correlation 7 b) The interacting field two point correlation 8 1-3 Stochastic Quantization of Fermions: The method of Fukai et al 9 1-4 The Method of Breit et al. for Boson and Fermion Fields . 12 1- 5 Summary 14 Part II: Toward a New Stochastic Formalism 15 21 A Field Equation for Bosons: The Method of Aharony et al 16 2 2 A Fermion Field Equation 18 2- 3 A Theory of Interacting Fermions and Bosons? 21 Part III: Applications to Quantum Mechanics 23 Conclusions 27 Footnotes 29 References 30 Appendix: S.R. and the Topolgical Mass in QED-3 ...31 Iv LIST OF FIGURES 1-1.1 Tree diagram expansion to order A3 9 1-1.2 Stochastic Diagrammatic series to order A2 10 ACKNOWLEDGMENTS I am grateful to Dr. Gordon Semenoff for his help and guidance. It was he who introduced me to stochastic quantization and proposed it as a potential area of research. He suggested to me the problem of evaluating the topological mass term in QED-3 and guided me through it. Somehow he patiently endured a series of desperate phone calls to Princeton, and never failed to set me straight when my calculations went awry. I am also indebted to Andre Roberge for his encouragement and ready sug• gestions. He has rare powers of explanation, and I often found myself in need of them. On a more practical note, I am grateful to the mysterious force which, every so often, spontaneously generates a paycheck in my mail box. May it never forsake me. Introduction In recent years, a new method for quantizing field theories has evolved. By coupling a d dimensional classical field to a random external source, we may derive a quantum field in d— 1 dimensions. Why this should be the case is not self evident. To envisage how an external random source might work to quantize a field theory, let us consider an analogy. Imagine shaking out a cloud of dust particles into the air of a room unaffected by gravity. We can approximate the physics of a classical field by considering the behaviour of this cloud of dust. To describe the classical field, we would be interested in the field amplitude as a function of position; to describe a cloud of dust, we concentrate on the velocities of particles as a function of their position. The average velocities of the particles change with time according to what is known as the Langevin equation. This equation has a damping term due to interac• tions between the dust particles and the surrounding air. So we find that, no matter what the initial velocities of the dust particles, they slowly settle down. Eventually, after each dust particle has had a number of random interactions with the atmo• sphere around it, the average velocity of every dust particle is zero. Furthermore, if we then check the average correlations between velocities of different dust particles, we find that they too have settled down-though they will not necessarily be zero. Since the correlations no longer change with time, we can describe them entirely in terms of the distances between the dust particles. In other words, by coupling a four dimensional "field" to an external random source, we can obtain a three dimensional "field" as a long term limit. The above analogy is useful because it highlights how stochastic quantization borrows its methods from non-equilibrium statistical mechanics. The analogy fails to clarify, however, why the classical system should approach a lower dimensional quantum system as a stable limit. The reasons for this mysterious 'quantization' are not simple; they depend on which stochastic method one employs. Here, we will examine some different approaches with an ambition both to establish as general a quantum theory as possible and to uncover some physical significance to the process by which we achieve quantization. In Part I, we discuss a method of quantizing field theories which originated with G. Parisi and Y. WuJ1' In 1981 they developed a way of using a "stochastic", or random distribution to quantize scalar fields. Their work was followed by that of Fukai et al. l2l, who used the technique to quantize fermion fields. It was not until 1983 that Breit, Gupta and Zaks'3] found a way to generalize the method and produce a theory of interacting fermion and boson fields. In Part II, we discuss an alternate method of stochastic quantization. It turns out that this method enjoys some important advantages over the method of Parisi and Wu. We then propose a theory of interacting fermions and bosons which is quite different from that proposed by Breit, Gupta and Zaks. In Part III, we discuss whether stochastic techniques may be used to generate quantum mechanics out of classical mechanics. Specifically, we consider the case of the simple harmonic oscillator. Our calculations suggest a way to generalize the notion of a random averaging. Finally, in an Appendix, we explicitly calculate the topological mass term in QED-3 using stochastic methods. This problem has been something of an enigma because traditional techniques fail to provide an unambiguous result. PART I: The Langevin Method of Stochastic Quantization 3 PART I The Langevin Method of Stochastic Quantization In 1981, G. Parisi and Wu Yongshi developed an alternate way to formulate gauge field theory. Motivated by a desire to avoid the sometimes unwieldy math• ematical baggage of gauge fixing, they adopted techniques from non-equilibrium statistical mechanics. The result is a method of "stochastic quantization" which casts gauge theories in a suggestive physical setting. (1.1) Review of Canonical Quantization As a prelude to our discussion of stochastic quantization (SQ), let us highlight some results of traditional quantum field theory. Consider, for instance, the case of scalar field theory in d Euclidean dimensions. We begin with the action (1.1) and the commutation relation (1.2) From these, we will wish to evaluate correlation functions of the form (1.3) PART I: The Langevin Method of Stochastic Quantization 4 where T(...) signifies ordering with respect to Euclidean time. Calculations of correlation functions of the form (1.3) are usually performed in the limit p —• oo. In this case, only the ground state contributes; (T (tfx), m))3^ -> (o|r |o) oo MzMy)) I (1.4) jd\<t>]e-sM We may now expand the right hand side around the free action. In this way, we generate a perturbative expansion for (T (c£(x)c/>(t/))^. We owe the success of this approach to Wick's Theorem, which allows us to express all terms in the expansion as products of the simple two point correlation. (1.2) The Langevin Equation for Bosons: The Method of Parisi and Wu The insight of Parisi and Wu is to view the correlation (T (4>{X),4>(y))^ as the steady state of a "time" dependent stochastic average {<f>[x,t)<p(y,t')) S2) To make this interpretation, we must introduce a fictitious time t : 0 < t < co. If we also couple the field to a heat reservoir, we can decribe the "time" evolution of <f> by the Langevin equation; where r}(x, t) is a stochastic distribution and |^ acts as a damping termJ3) So we interpret the problem of quantizing the classical field in (d)-dimensions as a problem of non-equilibrium statistical mechanics in (d+l)-dimensions.
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