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arXiv:hep-ph/9503416 · American Journal of Physics

The Foldy–Wouthuysen transformation

John P. Costella, Bruce H. J. McKellar

Published Dec 1, 1995• 101 citations
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Abstract

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.

DOI
10.1119/1.18017

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The Foldy-Wouthuysen transformation

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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.

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

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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.

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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").

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The Foldy-Wouthuysen formalism is presented in a representation-independent manner. It is shown that an algebraic transformation serves to remove the mass term from the Dirac equation for spin-\textonehalf{} particles, thereby providing decoupled two-component equations in a particular (Weyl) representation. Also it is shown that appropriate use of a constant algebraic transformation applied to a Dirac Hamiltonian in the usual (Pauli) representation leads to a nonrelativistic k. p equation used in solid-state theory for which the k-dependent spin-orbit term is absent.

Physical Review•Dec 25, 1968•0 citations
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.
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.
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.
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 Foldy-Wouthuysen formalism is presented in a representation-independent manner. It is shown that an algebraic transformation serves to remove the mass term from the Dirac equation for spin-\textonehalf{} particles, thereby providing decoupled two-component equations in a particular (Weyl) representation. Also it is shown that appropriate use of a constant algebraic transformation applied to a Dirac Hamiltonian in the usual (Pauli) representation leads to a nonrelativistic k. p equation used in solid-state theory for which the k-dependent spin-orbit term is absent.