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Journal of Physics A Mathematical and General

Gauge invariant wave mechanics and the Power-Zienau-Woolley transformation

R. Guy Woolley

Published Aug 1, 1980• 26 citations
Publisher

Abstract

The wave mechanical formulation of quantum electrodynamics is investigated in an explicitly gauge invariant form and this leads to the connection between manifest charge conservation and the Power-Zienau-Woolley transformation.
DOI
10.1088/0305-4470/13/8/027

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This article deals with the fundamental problem of light-matter interaction in the quantum theory. Although it is described through the vector potential in quantum electrodynamics, it is believed by some that a hamiltonian involving only the electric and the magnetic fields is preferable. In the literature this hamiltonian is known as the Power-Zienau-Woolley hamiltonian. We question its validity and show that it is not equivalent to the minimal-coupling hamiltonian. In this article, we show that these two hamiltonians are not connected through a gauge transformation. We find that the gauge is not fixed in the Power-Zienau-Woolley hamiltonian. The interaction term is written in one gauge whereas the rest of the hamiltonian is written in another gauge. The Power-Zienau-Woolley hamiltonian and the minimal-coupling one are related through a unitary transformation that does not fulfill the gauge fixing constraints. Consequently, they predict different physical results. In this letter, we provide the correct quantum theory in the multipolar gauge with a hamiltonian involving only the physical fields.