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 the 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.
Highlights
We all know that covariantly quantized electrodynamics has some appealing features
If one acknowledges that there are gauge-fixingoriginated contributions to various observables in a covariantly quantized theory, it is of interest to know whether they contribute to physical matrix elements
With the introduction to the Gupta-Bleuler quantization scheme, before focusing on three components of such an operator. Their commutation relations, their role in the generation of rotations of quantum fields, and their action on states from the physical sector of the theory are extensively discussed in Secs
Summary
We all know that covariantly quantized electrodynamics has some appealing features. One is that its free-field photon propagator has the simplest possible form, which facilitates perturbative calculations. If one acknowledges that there are gauge-fixingoriginated contributions to various observables in a covariantly quantized theory, it is of interest to know whether they contribute to physical matrix elements (i.e. the ones computed in states from the physically relevant sector of the theory) This issue was discussed in [18] for the density of the energy-momentum tensor. With the introduction to the Gupta-Bleuler quantization scheme, before focusing on three components of such an operator Their commutation relations, their role in the generation of rotations of quantum fields, and their action on states from the physical sector of the theory are extensively discussed in Secs.
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