Derivation of the Power-Zienau-Woolley Hamiltonian in quantum electrodynamics by gauge transformation

The different forms of Hamiltonian for the coupled system consisting of the electromagnetic field and a non-relativistic charged particle are considered in the context of gauge-transformation theory. The conventional Lagrangian of the system in an arbitrary gauge is converted to a new form by transformation to another arbitrary gauge, and a new formulation of the theory is obtained by expressing the new Lagrangian in terms of the initial potentials. Thus different gauge transformations produce different momenta ∏ conjugate to the initial vector potential A , and hence different forms of Hamiltonian. The transformations that produce the Coulomb-gauge and Power-Zienau-Woolley (P. Z. W.) Hamiltonians are considered in detail. It is shown that ∏ is transverse in both cases and only the transverse part of A is accordingly involved in the field quantization; neither the longitudinal part of A nor the scalar potential appears explicitly, the instantaneous Coulomb energies being included via an electronic polarization determined by the gauge generator. The transformations between gauges are illustrated by simple diagrammatic representations of A and ∏ . Compararison with the commonly used unitary transformation derivation of the P. Z. W. Hamiltonian emphasizes the need for a careful reinterpretation of the physical significance of ∏ after the unitary transformation has been made.