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Stochastic quantization of boson and fermion fields

Hayward, Geoffrey

Published Jan 1, 2010• 0 citations
Publisher

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. Furthermore, we present an analogous method for quantizing fermion fields and a possible new way of quantizing interacting fermion and boson fields. Finally, we discuss applications to quantum mechanics and stochastically quantize the simple harmonic oscillator.

DOI
10.14288/1.0096700

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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. Furthermore, we present an analogous method for quantizing fermion fields and a possible new way of quantizing interacting fermion and boson fields. Finally, we discuss applications to quantum mechanics and stochastically quantize the simple harmonic oscillator.
The time evolution of quantum fields is shown to be equivalent to a time-symmetric Fokker-Planck equation.Results are obtained using a Q-function representation, including fermion-fermion, boson-boson, and fermion-boson interactions with linear, quadratic, cubic, and quartic Hamiltonians, typical of QED and many other cases.For local boson-boson coupling, the resulting probability distribution is proved to have a positive, time-symmetric path integral and action principle, leading to a forward-backward stochastic process in both time directions.The solution corresponds to a c-number field equilibrating in an additional dimension.Paths are stochastic trajectories of fields in space-time, which are samples of a statistical mechanical steady state in a higher-dimensional space.We derive numerical methods and examples of solutions to the resulting stochastic partial differential equations in a higher time dimension, giving agreement with examples of simple bosonic quantum dynamics.This approach may lead to useful computational techniques for quantum field theory, as well as to new ontological models of physical reality.
The stochastic quantization method is developed in a possible “Heisenberg” operator theory of stochastic processes so designed as to keep a formal analogy to quantum mechanics. The general theory is first formulated for stochastic processes of the Wiener type and then its application to quantization of boson fields is presented. It seems that the operator formalism is convenient to examine implications of the stochastic quantization method. Within the theoretical framework we develop the “Feynman-Dyson” approach to field propagators in perturbation theory. This approach enables us to simplify and systematize practical calculations in comparison with the Langevin equation method. In the Appendix we formulate the Green function approach to field propagators using the characteristic functional.
A procedure of bosonization of Fermions in an arbitrary dimension is suggested. It is shown that a quadratic expression in the fermionic fields after rescaling time t→t/λ2t\to t/λ^2t→t/λ2 and performing the limit λ→0λ\to0λ→0 (stochastic limit), gives rise to a bosonic operator satisfying the boson canonical commutation relations. This stochastic bosonization of Fermions is considered first for free fields and then for a model with three--linear couplings. The limiting dynamics of the bosonic theory turns out to be described by means of a quantum stochastic differential equations.
We consider a quantization of relativistic wave equations which allows to treat quantum fields together with interacting particles at a finite time. We discuss also a dissipative interaction with the environment. We introduce a stochastic wave function whose dynamics is determined by a non-linear Schrödinger-type evolution equation in an additional time parameter. The correct classical limit requires the proper time interpretation of the time parameter. An average over the proper time leads to the conventional quantum field theory of particles which are free at an infinite space separation. We consider models with scalar and vector fields on a pseudoriemannian manifold. A quantization of the Einstein gravity in this approach is briefly discussed.
Recently GUER~A and LOFF~]~D0 (:) have constructed a stochastice elctromagnetic field by applying the stochastic quantization method of Nelson (2) to the classical electromagnetic field. According to their results the (( space part ~) of the Euclidean electromagnetic field F~j (with i, ] = 1, 2, 3) can be regarded as the ground-state stochastic electromagnetic field defined in the real physical time, in complete analogy with the case of the scalar massive boson field (3). In this note we shall comment on a basic assumption contained in the paper of Guerra and Loffredo. The main feature in their construction is that the magnetic field B(~, t) is assumed to be promoted to a random ~iarkov process by the stochastic quantization procedure. The electric field E(~, t) is then split into a couple of functions /~( ~, t) expressible in terms of B(~, t). By considering the classical Maxwell field in terms of normal canonical co-ordinates and employing the stochastic differential equation of Ito type, they obtained the ground-state stochastic magnetic field with zero mean and covarianee given by
In this chapter we review the main ideas of the canonical quantization in nonrelativistic quantum mechanics and optics, which we suppose the reader learned in previous courses. We first consider the least action principle and the action functional for classical many-body systems and also for the Schrödinger wavefunction, interpreted as the complex field of a classical field theory. Then, after having analyzed the Legendre transformation from the Lagrangian to the Hamiltonian formulation, we illustrate the canonical quantization of particles, and finally we extend the approach to the canonical quantization of bosonic and fermionic fields, considering also the role of temperature.