Impact of gauge fixing on angular momentum operators of the covariantly quantized electromagnetic field

TL;DR

This paper analyzes how gauge fixing affects angular momentum operators in covariantly quantized electromagnetic fields, clarifying the necessity of gauge-fixing contributions and addressing related conceptual and computational issues.

Contribution

It provides a detailed examination of gauge-fixing contributions to angular momentum operators, emphasizing their essential role and clarifying misconceptions in the covariant quantization framework.

Findings

Gauge-fixing contributions are indispensable for angular momentum operators.

Careful treatment of the indefinite metric space is crucial for correct physical matrix elements.

The study clarifies the role of gauge-fixing terms in the quantum rotation generators.

Abstract

Covariant quantization of the electromagnetic field imposes the so-called gauge-fixing modification on the Lagrangian density. As a result of that, the total angular momentum operator receives at least one gauge-fixing-originated contribution, whose presence causes some confusion in the literature. The goal of this work is to discuss in detail why such a contribution, having no classical interpretation, is actually indispensable. For this purpose, we divide canonical and Belinfante-Rosenfeld total angular momentum operators into different components and study their commutation relations, their role in generation of rotations of quantum fields, and their action on states from the physical sector of the theory. Then, we examine physical matrix elements of operators having gauge-fixing-related contributions, illustrating problems that one may encounter due to careless employment of the resolution of identity during their evaluation. The resolution of identity, in the indefinite-metric space of the covariantly-quantized electromagnetic field, is extensively discussed because it takes a not-so-intuitive form if one insists on explicit projection onto states from the physical sector of the theory. Our studies are carried out in the framework of the Gupta-Bleuler theory of the free electromagnetic field. Relevant remarks about interacting systems, described by covariantly-quantized electrodynamics, are given.