Physical States and Gauge Independence of the Energy-Momentum Tensor in Quantum Electrodynamics

TL;DR

This paper investigates the gauge independence of the energy-momentum tensor in QED using various formalisms, demonstrating gauge invariance of expectation values in asymptotic states and discussing the implications for gauge fixing and quantum field theory.

Contribution

It shows that the expectation values of the energy-momentum tensor in asymptotic states are gauge independent to all orders in QED, and clarifies the role of gauge fixing and invariance in different formalisms.

Findings

Expectation values of the energy-momentum tensor are gauge independent in asymptotic states.

Gauge invariant operators require gauge fixing, linking gauge invariance to gauge choice.

Gauge transformations are unrestricted in the functional and path integral formalism due to the structure of the functional space.

Abstract

Discussions are made on the relationship between physical states and gauge independence in QED. As the first candidate take the LSZ-asymptotic states in a covariant canonical formalism to investigate gauge independence of the (Belinfante's) symmetric energy-momentum tensor. It is shown that expectation values of the energy-momentum tensor in terms of those asymptotic states are gauge independent to all orders. Second, consider gauge invariant operators of electron or photon, such as the Dirac's electron or Steinmann's covariant approach, expecting a gauge invariant result without any restriction. It is, however, demonstrated that to single out gauge invariant quantities is merely synonymous to a gauge fixing, resulting again in use of the asymptotic condition when proving gauge independence. Nevertheless, it is commented that these invariant approaches is helpful to understand the mechanism of the LSZ-mapping and furthermore of quark confinement in QCD. As the final candidate, it is shown that gauge transformations are freely performed under the functional representation or the path integral expression on account of the fact that the functional space is equivalent to a collection of infinitely many inequivalent Fock spaces. The covariant LSZ formalism is shortly reviewed and the basic facts on the energy-momentum tensor are also illustrated.