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Journal of Mathematical Physics

Normal forms of an abstract Dirac operator and applications to scattering theory

Bernd Thaller

Published Jan 1, 1988• 27 citations
Publisher

Abstract

The unitary transformations which convert an abstract Dirac operator into an ‘‘even’’ (resp. ‘‘odd’’) operator are determined. The problem is formulated and solved completely within the general setup of supersymmetric quantum mechanics. This leads to some apparently new applications in relativistic quantum mechanics, where the transformations are known as the Foldy–Wouthuysen (resp. Cini–Touschek) transformations. The scattering theory for abstract Dirac operators is discussed and the utility of the general theory is illustrated by proving existence of relativistic Mo/ller operators for scattering from long-range magnetic fields.

DOI
10.1063/1.528182

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The unitary transformations which convert an abstract Dirac operator into an ‘‘even’’ (resp. ‘‘odd’’) operator are determined. The problem is formulated and solved completely within the general setup of supersymmetric quantum mechanics. This leads to some apparently new applications in relativistic quantum mechanics, where the transformations are known as the Foldy–Wouthuysen (resp. Cini–Touschek) transformations. The scattering theory for abstract Dirac operators is discussed and the utility of the general theory is illustrated by proving existence of relativistic Mo/ller operators for scattering from long-range magnetic fields.
We give a direct link between covariant description of Dirac particles in the abstract framework of unitary representation of the Poincaré group and description with the help of the Dirac equation. In this context we discuss the spin operator for a relativistic Dirac particle. We show also that the spin operator used in quantum field theory for spin s = 1/2 corresponds to the Foldy-Wouthuysen mean-spin operator. We hope that this formalism will be useful in the theory of relativistic quantum information.
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.
An examination is made of the consequences for the quantum mechanics of spinning particles of equations characteristic of Lorentz-covariant position variables. These equations are commutator analogs of the Poisson bracket equations that express the familiar transformation properties of space-time events in classical mechanics. For a particle of zero spin it is found that the usual canonical coordinate is the unique solution of these equations. For a particle with positive spin there is no position operator which satisfies these equations and has commuting components. For a particle and antiparticle there is a unique solution with commuting components which is valid for all values of the spin and reduces for zero spin to the canonical coordinate. For spin \textonehalf{} this is the Foldy-Wouthuysen transform of the position operator of the Dirac equation. A generalization of the inverse Foldy-Wouthuysen transformation, valid for any value of the spin, appears as a unique unitary transformation which takes this generalized Dirac position to the canonical coordinate. The application of this transformation to the canonical form of the Hamiltonian gives a generalization of the Dirac equation Hamiltonian. This is developed and compared with the literature for spin 1. It gives a nonlocal equation as the spin 1 analog of the Dirac equation.
First it is shown in an abstract mathematical framework of supersymmetric quantum theory that, under some conditions for the perturbation, the wave and the scattering operators on bosonic states have some connections with those on fermionic ones. In particular, a unitary equivalence between the two scattering operators is proved. Then, a class of supersymmetric quantum mechanical models on Rn is discussed and the abstract general results are applied to obtain some more explicit results on the supersymmetric potential scattering.