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Australian Journal of Physics

Gauge Explicit Quantum Mechanics and Perturbation Theory

Alexander Stewart

Published Oct 6, 1997• 5 citations
Publisher

Abstract

A version of quantum and statistical mechanics, including perturbation theory, is described in which explicit electromagnetic gauge arbitrariness is maintained at every stage. Any gauge may be used for a calculation provided that the wave equation operator is gauge invariant.
DOI
10.1071/p96102

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A novel Hamiltonian scheme for non–relativistic quantum electrodynamics in which the gauge arbitrariness of the field potential is kept explicit is used to study the gauge–dependence properties of various versions of perturbation theory when resonance and line–broadening effects are admitted. Time–dependent perturbation theory is shown to have severe gauge–dependence problems unless ad hoc modifications are made. A time–independent formulation of S–matrix theory is then studied. Far from resonance, the S–matrix is gauge invariant in all orders of perturbation theory due to a very precise cancellation of gauge–dependent terms which requires, among other things, complete sets of intermediate states; energy conservation also has a crucial role. However, an obvious separation of the S–matrix into a resonant (pole) and non–resonant background leads to incomplete cancellation of the gauge–dependent terms. The introduction of the Heitler damping matrix into an integral equation for the T–matrix leads to a gauge–invariant result. This provides the basis for a gauge invariant S–matrix theory of atoms and molecules interacting with electromagnetic radiation that encompasses resonance and damping effects.
The distinction between gauge independence and gauge covariance in electrodynamics is considered for quantities of physical interest. The gauge independence of directly observable quantities, such as the spectra of operators and energy conserving transition rates, is discussed and contrasted with the gauge covariance of other quantities such as scalar products. The quasigauge-invariance of the classical Lagrangian and the invariance of the Euler-Lagrange equations under the addition of a total time derivative to the Lagrangian are discussed. The consequent inherent nonuniqueness of the Lagrangian and Hamiltonian formulations of classical mechanics is pointed out. A physical interpretation of the explicit gauge dependence of classical canonical momenta and of the expectation values of the corresponding quantum mechanical operators is presented. The gauge independence of energy conserving transition rates calculated in the conventional finite-order time dependent perturbation theory is discussed and illustrated. The gauge dependence of the conventional time dependent transition amplitudes in the presence of electromagnetic fields is then discussed, and gauge independent transition amplitudes are constructed. An alternative formulation of the quantum mechanics of charged particles is obtained in terms of a new gauge independent Hamiltonian, which is, as might be expected, unique only within an arbitrary canonical transformation.
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For a quantum mechanical particle in a classical electromagnetic field, the active and passive views of gauge invariance are compared. In the active point of view, both the wavefunction and the potentials are transformed so that the Schrödinger equation is form invariant. In the passive point of view, a change in the representation of the canonical momentum is made which just cancels the change in the gauge of the potentials. The condition that a gauge-invariant operator has the same expectation value in all gauges gives a functional equation, the solution of which shows that gauge-invariant operators depend only on the kinetic momentum or coordinates.