Why semi-classical electrodynamics is not gauge invariant

TL;DR

This paper demonstrates that semi-classical electrodynamics lacks full gauge invariance because the scalar potential must represent a physical potential, restricting permissible gauge transformations, unlike classical and quantum electrodynamics.

Contribution

It clarifies the limitations of gauge invariance in semi-classical electrodynamics and explains the physical significance of the scalar potential in this context.

Findings

Probability amplitudes are gauge-independent within the restricted class.

Energy differences remain unaffected by gauge choices.

Full gauge invariance does not hold in semi-classical electrodynamics.

Abstract

It is shown that in semi-classical electrodynamics, which describes how electrically charged particles move according to the laws of quantum mechanics under the influence of a prescribed classical electromagnetic field, only a restricted class of gauge transformations is allowed. This lack of full gauge invariance, in contrast to the situation in classical and quantum electrodynamics which are fully gauge invariant theories, is due to the requirement that the scalar potential in the Hamiltonian of wave mechanics represent a physical potential. Probability amplitudes and energy differences are independent of gauge within this restricted class of gauge transformation.