Electromagnetic gauge-freedom and work

TL;DR

This paper addresses the challenge of defining thermodynamic work in electromagnetic systems by focusing on the source's kinetic energy, providing a relativistically consistent formulation of the first law that accounts for gauge invariance and energy separation.

Contribution

It introduces a gauge-invariant definition of work in electromagnetic interactions by considering the source's kinetic energy and derives a relativistically valid first law based on energy-momentum separation.

Findings

Work can be defined via the source’s gauge-invariant kinetic energy.

The formulation is relativistically covariant and causal.

The first law relates thermodynamic work to the Lorentz force on the source.

Abstract

We argue that the definition of the thermodynamic work done on a charged particle by a time-dependent electromagnetic field is an open problem, because the particle's Hamiltonian is not gauge-invariant. The solution of this problem demands accounting for the source of the field. Hence we focus on the work done by a heavy body (source) on a lighter particle when the interaction between them is electromagnetic and relativistic. The work can be defined via the gauge-invariant kinetic energy of the source. We uncover a formulation of the first law (or the generalized work-energy theorem) which is derived from relativistic dynamics, has definite validity conditions, and relates the work to the particle's Hamiltonian in the Lorenz gauge. Thereby the thermodynamic work also relates to the mechanic work done by the Lorentz force acting on the source. The formulation of the first law is based on a specific separation of the overall energy into those of the source, particle and electromagnetic field. This separation is deduced from a consistent energy-momentum tensor. Hence it holds relativistic covariance and causality.