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arXiv:1306.6922 · Progress of Theoretical and Experimental Physics

Stochastic variational method as quantization scheme: Field quantization of the complex Klein–Gordon equation

T. Koide, T. Kodama

Published Sep 1, 2015• 15 citations
arXivHTMLPDF

Abstract

The stochastic variational method (SVM) is the generalization of the variational approach to systems described by stochastic variables. In this paper, we investigate the applicability of SVM as an alternative field-quantization scheme, by considering the complex Klein–Gordon equation. There, the Euler–Lagrangian equation for the stochastic field variables leads to the functional Schrödinger equation, which can be interpreted as the Euler (ideal fluid) equation in the functional space. The present formulation is a quantization scheme based on commutable variables, so that there appears no ambiguity associated with the ordering of operators, e.g., in the definition of Noether charges.

DOI
10.1093/ptep/ptv127

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The stochastic variational method (SVM) is the generalization of the variational approach to systems described by stochastic variables. In this paper, we investigate the applicability of SVM as an alternative field-quantization scheme, by considering the complex Klein–Gordon equation. There, the Euler–Lagrangian equation for the stochastic field variables leads to the functional Schrödinger equation, which can be interpreted as the Euler (ideal fluid) equation in the functional space. The present formulation is a quantization scheme based on commutable variables, so that there appears no ambiguity associated with the ordering of operators, e.g., in the definition of Noether charges.
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This article develops a variational formulation for the relativistic Klein-Gordon equation. The main results are obtained through an extension of the classical mechanics approach to a more general context, which in some sense, includes the quantum mechanics one. For the second part of the text, the definition of normal field and its relation with the wave function concept play a fundamental role in the main results establishment. Among the applications, we include a model with the presence of electromagnetic fields and also the modeling of a chemical reaction. Finally, in the last section, we present some results about the Spin operator in a relativistic context.
Abstract The discussion of stochasticity inherent in physical phenomena and processes occurring on our world is continued actively at present. For this purpose, various physical models and mathematical descriptions that are based most often on postulated are proposed. As a rule, stochasticity introduced in physical models phenomenologically, as well as its origin and mechanisms of emergence are not considered. However, the problem concerning the origin of stochasticity is far from being solved both in physical and mathematical aspects. In this article, the idea put forth by D.I. Blokhintsev and L.E. Ballentine concerning the emergence of stochasticity in classical and quantum field systems as a consequence of action of the surrounding stochastic background is developed. We are using the Lagrangian formalism and the categorically topological method of description for correct definition of the Lagrangian, action, the Euler–Lagrange equations, Noether currents, and generalized charges in stochastic surroundings. It is shown that Noether currents and conserved quantities become stochastic, and their probabilistic properties are determined by the background characteristics. The change in the action of the background on the observation interval determines the deviation of conserved quantities from their mean values, and the Heisenberg uncertainty relation can be treated as a manifestation of this stochasticity. As an illustration of the operability of a mathematical formulation of the stochasticity origin, we consider a classical scalar field. The proposed approach is naturally generalized to quantum fields. It can be used to obtain stochastic equations describing the behavior of material macroscopic systems in random fields and measurement processes.