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Nonrelativistic Wave Equations with Gauge Fields
Authors:M de Montigny  F C Khanna  A E Santana
Institution:(1) Faculté Saint-Jean, University of Alberta, Edmonton, Alberta, Canada;(2) Theoretical Physics Institute, University of Alberta, Edmonton, Alberta, Canada;(3) TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia, Canada;(4) Universidade Federal da Bahia, Campus de Ondina, Salvador, Bahia, Brazil
Abstract:We illustrate a metric formulation of Galilean invariance by constructing wave equations with gauge fields. It consists of expressing nonrelativistic equations in a covariant form, but with a five-dimensional Riemannian manifold. First we use the tensorial expressions of electromagnetism to obtain the two Galilean limits of electromagnetism found previously by Le Bellac and Lévy-Leblond. Then we examine the nonrelativistic version of the linear Dirac wave equation. With an Abelian gauge field we find, in a weak field approximation, the Pauli equation as well as the spin—orbit interaction and a part reminiscent of the Darwin term. We also propose a generalized model involving the interaction of the Dirac field with a non-Abelian gauge field; the SU(2) Hamiltonian is given as an example.
Keywords:Galilean invariance  Riemannian geometry  gauge theory  wave equations
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