Stochastic Variational Method as a Quantization Scheme II: Quantization of Electromagnetic Fields

TL;DR

This paper explores the stochastic variational method (SVM) for quantizing electromagnetic fields, demonstrating its equivalence to traditional methods in Coulomb gauge and its advantages in Lorentz gauge without indefinite metrics.

Contribution

It introduces a gauge-invariant SVM approach to electromagnetic field quantization, avoiding indefinite metrics and clarifying the relation to canonical quantization.

Findings

SVM reproduces standard results in Coulomb gauge.

SVM quantization in Lorentz gauge avoids indefinite metric issues.

Temporal and longitudinal components behave as c-number functionals.

Abstract

Quantization of electromagnetic fields is investigated in the framework of stochastic variational method (SVM). Differently from the canonical quantization, this method does not require canonical form and quantization can be performed directly from the gauge invariant Lagrangian. The gauge condition is used to choose dynamically independent variables. We verify that, in the Coulomb gauge condition, SVM result is completely equivalent to the traditional result. On the other hand, in the Lorentz gauge condition, SVM quantization can be performed without introducing the indefinite metric. The temporal and longitudinal components of the gauge filed, then, behave as c-number functionals affected by quantum fluctuation through the interaction with charged matter fields. To see further the relation between SVM and the canonical quantization, we quantize the usual gauge Lagrangian with the Fermi term and argue a stochastic process with a negative second order correlation is introduced to reproduce the indefinite metric.