Stochastic Variational Method as Quantization Scheme I: Field Quantization of Complex Klein-Gordan Equation

TL;DR

This paper explores the Stochastic Variational Method as a novel quantization scheme for the complex Klein-Gordon equation, reproducing standard results without operator ordering ambiguities.

Contribution

It introduces a stochastic variational approach to field quantization, providing a new framework that aligns with canonical results and handles anti-particles.

Findings

Reproduces canonical quantization results

Derives the functional Schrödinger equation from stochastic fields

Eliminates operator ordering ambiguities

Abstract

Stochastic Variational Method (SVM) is the generalization of the variation method to the case with stochastic variables. In the series of papers, we investigate the applicability of SVM as an alternative field quantization scheme. Here, we discuss the complex Klein-Gordon equation. In this scheme, the Euler-Lagrangian equation for the stochastic fields leads to the functional Schroedinger equation, which in turn can be interpreted as the ideal fluid equation in the functional space. We show that the Fock state vector is given by the stationary solution of these differential equations and various results in the usual canonical quantization can be reproduced, including the effect of anti-particles. The present formulation is a quantization scheme based on commutable variables, so that there appears no ambiguity associated with the ordering of operators, for example, in the definition of Noether charges.