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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Gauge invariance, time-dependent Foldy-Wouthuysen transformations, and the Pauli Hamiltonian

Terry Goldman

Published Feb 15, 1977• 38 citations
Publisher

Abstract

The Foldy-Wouthuysen derivation of the generalized Pauli Hamiltonian is reexamined, and described as an operator gauge transformation. The unitary inequivalence of the Pauli and Dirac Hamiltonians is shown to occur, because in general it is only the entire coupled field Hamiltonian which is gauge invariant, not the particle Hamiltonian alone.

DOI
10.1103/physrevd.15.1063

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arXiv:1002.3421•Physical Review A•Jul 22, 2010•13 citations
The Foldy-Wouthuysen derivation of the generalized Pauli Hamiltonian is reexamined, and described as an operator gauge transformation. The unitary inequivalence of the Pauli and Dirac Hamiltonians is shown to occur, because in general it is only the entire coupled field Hamiltonian which is gauge invariant, not the particle Hamiltonian alone.
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 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.
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.
We offer further results and a matured physical interpretation, concerning our invariant algebra P for the Dirac equation discussed in 1983 [ 1 ] and 1996 [ 2 ]. Especially, P may be fully described either with simple spectral theory of the Hamiltonian or with a decoupling modulo order — ∞ of the positive and negative energy parts of the Dirac equation, similar to the Foldy-Wouthuysen transform. There seems to be evidence indicating that only operators in P qualify as observables—i.e., can be measured with arbitrary precision—a feature comparable to the Heisenberg uncertainty relation.
For quantum mechanics of a charged particle in a classical external electromagnetic field, there is an apparent puzzle that the matrix element of the canonical momentum and Hamiltonian operators is gauge dependent. A resolution to this puzzle was recently provided by us [X.-S. Chen et al., Phys. Rev. Lett. 100, 232002 (2008)]. Based on the separation of the electromagnetic potential into pure-gauge and gauge-invariant parts, we have proposed a new set of momentum and Hamiltonian operators which satisfy both the requirement of gauge invariance and the relevant commutation relations. In this paper we report a check for the case of the hydrogen-atom problem: Starting from the Hamiltonian of the coupled electron, proton, and electromagnetic field, under the infinite proton mass approximation, we derive the gauge-invariant hydrogen-atom Hamiltonian and verify explicitly that this Hamiltonian is different from the Dirac Hamiltonian, which is the time translation generator of the system. The gauge-invariant Hamiltonian is the energy operator, whose eigenvalue is the energy of the hydrogen atom. It is generally time dependent. In this case, one can solve the energy eigenvalue equation at any specific instant of time. It is shown that the energy eigenvalues are gauge independent, and by suitably choosing the phase factor of the time-dependent eigenfunction, one can ensure that the time-dependent eigenfunction satisfies the Dirac equation.