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Journal of Physics A Mathematical and General

The Foldy-Wouthuysen representation revisited

K. Ohta

Published Feb 1, 1987• 2 citations
Publisher

Abstract

On the basis of the approach of Eriksen and Kolsrud (1960), the author derives a canonical transformation which diagonalises the Dirac Hamiltonian for a particle interacting with an external electromagnetic field. It is shown that an explicitly gauge invariant electromagnetic interaction can be obtained without recourse to the 1/m expansion. When expanded in 1/m, the result coincides with the Foldy-Wouthuysen interaction.

DOI
10.1088/0305-4470/20/2/024

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On the basis of the approach of Eriksen and Kolsrud (1960), the author derives a canonical transformation which diagonalises the Dirac Hamiltonian for a particle interacting with an external electromagnetic field. It is shown that an explicitly gauge invariant electromagnetic interaction can be obtained without recourse to the 1/m expansion. When expanded in 1/m, the result coincides with the Foldy-Wouthuysen interaction.
We apply Kutzelnigg's method for the Foldy-Wouthuysen (FW) transformation of the Dirac-Pauli Hamiltonian. The exact FW transformations exist and agree with those obtained by Eriksen's method for two special cases. In the weak-field limit of static and homogeneous electromagnetic fields, the long-held speculation is rigorously proven, by mathematical induction on the orders of 1/c1/c1/c in the power series, that the FW transformed Dirac-Pauli Hamiltonian is in full agreement with the classical counterpart, which is the sum of the orbital Hamiltonian for the Lorentz force equation and the spin Hamiltonian for the Thomas-Bargmann-Michel-Telegdi equation.
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.