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Physical Review Letters

Question of Unitarity of Foldy-Wouthuysen Transformations and Volkov States in Two-Component Forms

Joseph Kupersztych

Published Feb 19, 1979• 7 citations
Publisher

Abstract

The Foldy-Wouthuysen transformations are not always unitary within the space of solutions of the Dirac equation with boundary conditions. Therefore, imprecise nonrelativistic limits can be obtained for particular interactions if these transformations are always used. This assertion is confirmed by considering the case of plane electromagnetic waves and by comparing with the result given by an exact (closed-form) unitary transformation.

DOI
10.1103/physrevlett.42.483

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The Foldy-Wouthuysen transformations are not always unitary within the space of solutions of the Dirac equation with boundary conditions. Therefore, imprecise nonrelativistic limits can be obtained for particular interactions if these transformations are always used. This assertion is confirmed by considering the case of plane electromagnetic waves and by comparing with the result given by an exact (closed-form) unitary transformation.
The Dirac Hamiltonian for a particle in a nonexplicitly time-dependent field is converted to an even Dirac matrix by means of a single canonical transformation. When the interaction term is an odd Dirac matrix, the transformed Hamiltonian is expressed in a very simple form. An exact transformation is also found for two-particle wave equations of Breit's type. The transformed Hamiltonian is then a uU\mathrm{uU}uU-separating matrix, in Chraplyvy's sense.In the nonrelativistic limit expansions in powers of 1m\frac{1}{m}m1​ or 1c\frac{1}{c}c1​ are made. The approximate wave equations are in agreement with previous transformation results.
We show that care is required in formulating the nonrelativistic limit of generalized Dirac Hamiltonians which describe particles and antiparticles interacting with static electric and/or gravitational fields. The Dirac-Coulomb and the Dirac-Schwarzschild Hamiltonians, and the corrections to the Dirac equation in a non-inertial frame, according to general relativity, are used as example cases in order to investigate the unitarity of the standard and "chiral" approaches to the Foldy-Wouthuysen transformation, and spurious parity-breaking terms. Indeed, we find that parity-violating terms can be generated by unitary pseudo-scalar transformations ("chiral" Foldy-Wouthuysen transformations). Despite their interesting algebraic properties, we find that "chiral" Foldy-Wouthuysen transformations change fundamental symmetry properties of the Hamiltonian and do not conserve the physical interpretation of the operators. Supplementing the discussion, we calculate the leading terms in the Foldy-Wouthuysen transformation of the Dirac Hamiltonian with a scalar potential (of the (1/r)-form and of the confining radially symmetric linear form), and obtain compact expressions for the leading higher-order corrections to the Dirac Hamiltonian in a non-inertial rotating reference frame "Mashhoon term").
The paper describes conditions for transformation from the Dirac representation to the Foldy-Wouthuysen representation. The necessary condition is the block-diagonal transformation of Hamiltonian relative to the upper and lower components of the wave function. The sufficient condition is the wave function transformation law described by relation (6). It has been demonstrated that the unitary transformations offered by the authors of the papers [14], [15], [16], [17] do not satisfy the sufficiency condition (6) and, hence, they are not the Foldy-Wouthuysen transformations. In applications, the matrix elements of any operator in the FW representation can be calculated, according to (6), using the normalized two-component wave functions in the Dirac representation known for the given problem.
A nonrelativistic reduction of the second-order Dirac equation is investigated. An equation K2+[Π0; K−Π̂]=m2+Π̂2 is obtained for a ‘‘kinetic Hamiltonian’’ K such that Φ obeys a nonrelativistic Schrödinger-type equation Π0Φ=KΦ in addition to obeying the second-order Dirac equation. Non-self-adjoint terms in K are removed by a change of representation from Φ to ΦT ≡PΦ, and a closed expression P≡[ 1/2 +(K† −Π̂)(K−Π̂)/2(m)2]1/2 is obtained for P. Similar results are obtained for antiparticle states, and a type of ‘‘factorization theorem’’ for the second-order Dirac equation is obtained in which the second-order Dirac equation is replaced by a pair of uncoupled first-order equations. Since nonperturbative techniques are used throughout, the factorization theorem holds independently of the question of convergence of the Foldy–Wouthuysen series and is valid whenever the equation K2+[Π0; K−Π̂]=m2+Π̂2 admits a solution for K. The connection with the usual linear Dirac equation is investigated, and closed expressions are described for a transformation to a representation of that equation which is ‘‘even’’ in the language of Foldy and Wouthuysen. The transformation described is equivalent to two consecutive ‘‘odd’’ unitary transformations.