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Journal of Physics A Mathematical and General

Gauge transformations, Foldy-Wouthuysen transformations and conservation of energy

Kuo‐Ho Yang

Published Feb 1, 1982• 25 citations
Publisher

Abstract

The conventional interpretation of the quantum mechanical probability amplitudes and probabilities is examined in detail with respect to gauge transformations and the Foldy-Wouthuysen transformations. It is shown that the conventional interpretation is gauge dependent and, when applied to the relativistic (Dirac) and the non-relativistic quantum mechanics, cannot be consistent with itself through the Foldy-Wouthuysen transformations. It is also shown that these two difficulties are not present in a gauge invariant formulation that explicitly incorporates Poynting's theorem and the conservation law of energy into the definition of probability amplitudes.

DOI
10.1088/0305-4470/15/2/016

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The conventional interpretation of the quantum mechanical probability amplitudes and probabilities is examined in detail with respect to gauge transformations and the Foldy-Wouthuysen transformations. It is shown that the conventional interpretation is gauge dependent and, when applied to the relativistic (Dirac) and the non-relativistic quantum mechanics, cannot be consistent with itself through the Foldy-Wouthuysen transformations. It is also shown that these two difficulties are not present in a gauge invariant formulation that explicitly incorporates Poynting's theorem and the conservation law of energy into the definition of probability amplitudes.
The Dirac equation for a charged particle in a time-dependent electromagnetic field is reduced to the Pauli equation in the non-relativistic limit by making a time-dependent Foldy-Wouthuysen transformation in a completely gauge-invariant way. In a time-dependent field, the expectation value of the Hamiltonian is dependent on the gauge, and so it cannot be the energy. The energy is obtained by taking the expectation value of the energy operator, the time rate of change of which is the average of the quantum mechanical power operator. The Hamiltonian formulation of a charged particle in a time-dependent external electromagnetic field is satisfactory when this distinction between the Hamiltonian, which describes the time evolution of the system, and the energy operator, which is gauge invariant, is made.
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.
Hamiltonian operators are gauge dependent. For overcome this difficulty we reexamined the effect of a gauge transformation on Schrödinger and Dirac equations. We show that the gauge invariance of the operator H−iℏ∂∂tH-i\hbar\frac{\partial}{\partial t}H−iℏ∂t∂​ provides a way to find the energy operator from first principles. In particular, when the system has stationary states the energy operator can be identified without ambiguities for non-relativistic and relativistic quantum mechanics. Finally, we examine other approaches finding that in the case in which the electromagnetic field is time independent, the energy operator obtained here is the same as one recently proposed by Chen et al. [1].