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Stochastic Quantization of the Maxwell Field: Some Comments on the Paper of Guerra and Loffredo.

S. C. Lb

Published Jan 1, 1981• 0 citations
OpenAlex

Abstract

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

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Recently GUERA 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 Fj (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
The problem of a formulation of Nelson's (1967) stochastic mechanics of scalar fields in the context of general relativity is considered. The simple example of the scalar field in the Wightman vacuum state on the Rindler wedge is examined, the stochastic counterpart of the Fulling ambiguity of canonical quantisation is formulated, and the role of the stochastic mechanics of thermal mixtures, as formulated by Guerra and Loffredo (1981), is analysed, in the spirit of Davies (1975) and Unruh (1976), in the solution of the above ambiguity. An overall picture emerges which, both in the explicit example considered here and in its straightforward generalisations to static submanifolds of more general spacetimes, confirms Smolin's point of view that stochastic quantisation is a very natural conceptual frame in which to study the general non-covariance of the distinction between quantum and thermal fluctuations.
We present a succinct way to quantize Maxwell's equations. We begin by discussing the random nature of quantum observables. Then we present the quantum Maxwell's equations and give their physical meanings. Due to the mathematical homomorphism between the classical and quantum cases [1], the derivation of quantum Maxwell's equations can be simplified. First, one needs only to verify that the classical equations of motion are derivable from a Hamiltonian by energy-conservation arguments. Then the quantum equations of motion follow due to homomorphism, which applies to sum-separable Hamiltonians. Hence, we first show the derivation of classical Maxwell's equations using Hamiltonian theory. Then the derivation of the quantum Maxwell's equations follows in an analogous fashion. Finally, we apply this quantization procedure to the dispersive medium case. (This article is written for the classical electromagnetic community assuming little knowledge of quantum theory, an introduction of which can be found in [2, Lecs. 38 and 39].)
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.
We study the stochastic quantization of finite dimensional systems via path-wise calculus of variations with the mean discretized classical action in the general case of electromagnetic interactions. We show that there exists a unique choice of the mean discretized action corresponding to the minimal classical magnetic coupling and derive the general equations of motion by means of a path-wise stochastic calculus of variations. In the case of purely scalar interactions, the total mean energy of the system (which gives the usual quantum mechanical expectation of the Hamiltonian in the canonical limit) works as a Lyapunov functional and the system relaxes on the canonical solutions, represented by Nelson's diffusions, which act as an attracting set. We show that, in presence of a minimal magnetic coupling, the mean energy is no longer a Lyapunov functional. We construct for a simple example, a new Lyapunov functional, and we show that the system can reach the dynamical equilibrium also by absorbing energy from the external magnetic field.
The electromagnetic field is quantized in nonlinear media whose dielectric functions and nonlinear susceptibilities may vary with space. Given the dielectric function and nonlinear susceptibilities, the Hamiltonian of the electromagnetic field is determined completely by this quantization method. From Heisenberg equations we prove that Maxwell's equations are satisfied for the field operators. When the nonlinearity goes to zero, this quantization method returns to the generalized canonical quantization procedure developed in Ref.[1] for linear inhomogeneous media. The general formalism is illustrated by applications to quantization of the second-order and third-order nonlinear quasi-steady-state processes.
The notion that the electromagnetic field is quantized is usually inferred from observations such as the photoelectric effect and the black-body spectrum. However accounts of the quantization of this field are usually mathematically motivated and begin by introducing a vector potential, followed by the imposition of a gauge that allows the manipulation of the solutions of Maxwell's equations into a form that is amenable for the machinery of canonical quantization. By contrast, here we quantize the electromagnetic field in a less mathematically and more physically motivated way. Starting from a direct description of what one sees in experiments, we show that the usual expressions of the electric and magnetic field observables follow from Heisenberg's equation of motion. In our treatment, there is no need to invoke the vector potential in a specific gauge and we avoid the commonly used notion of a fictitious cavity that applies boundary conditions to the field.