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Physical Review

Foldy-Wouthuysen Transformation. Exact Solution with Generalization to the Two-Particle Problem

Erik Eriksen

Published Aug 1, 1958• 106 citations
Publisher

Abstract

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 $\mathrm{uU}$-separating matrix, in Chraplyvy's sense.In the nonrelativistic limit expansions in powers of $\frac{1}{m}$ or $\frac{1}{c}$ are made. The approximate wave equations are in agreement with previous transformation results.

DOI
10.1103/physrev.111.1011

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Foldy–Wouthuysen-type reduction of the second-order Dirac equation

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The Foldy–Wouthuysen transformation

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The Foldy-Wouthuysen transformation

Costella, J P, Bruce H. J. McKellar

The Foldy-Wouthuysen transformation of the Dirac Hamiltonian is generally taught as simply a mathematical trick that allows one to obtain a two-component theory in the low-energy limit. It is not often emphasized that the transformed representation is the only one in which one can take a meaningful *classical limit*, in terms of particles and antiparticles. We briefly review the history and physics of this transformation.

•Mar 23, 1995•67 citations
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.
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.
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.