The gauge-invariant Lagrangian, the Power-Zienau-Woolley picture, and the choices of field momenta in nonrelativistic quantum electrodynamics

TL;DR

This paper demonstrates that the Power-Zienau-Woolley picture in nonrelativistic quantum electrodynamics can be derived from a gauge-invariant Lagrangian, emphasizing the gauge invariance of the transverse vector potential and the freedom in choosing field momenta.

Contribution

It provides a gauge-invariant derivation of the Power-Zienau-Woolley picture and clarifies the role of the transverse vector potential and conjugate momenta choices in nonrelativistic QED.

Findings

The Power-Zienau-Woolley picture is equivalent to a gauge-invariant Lagrangian formulation.

Using the transverse vector potential simplifies handling constraints in nonrelativistic QED.

There is freedom in choosing conjugate momenta, with conventional choices reducing system constraints.

Abstract

We show that the Power-Zienau-Woolley picture of the electrodynamics of nonrelativistic neutral particles (atoms) can be derived from a gauge-invariant Lagrangian without making reference to any gauge whatsoever in the process. This equivalence is independent of choices of canonical field momentum or quantization strategies. In the process, we emphasize that in nonrelativistic (quantum) electrodynamics, the all-time appropriate generalized coordinate for the field is the transverse part of the vector potential, which is itself gauge invariant, and the use of which we recommend regardless of the choice of gauge, since in this way it is possible to sidestep most issues of constraints. Furthermore, we point out a freedom of choice for the conjugate momenta in the respective pictures, the conventional choices being good ones in the sense that they drastically reduce the set of system constraints.