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arXiv:1712.08759 · Journal of Mathematical Physics

Path integral and spectral representations for supersymmetric Dirac-Hamiltonians

Georg Junker, A. Inomata

Published May 1, 2018• 12 citations
arXivHTMLPDF

Abstract

The resolvent of supersymmetric Dirac Hamiltonian is studied in detail. Due to supersymmetry the squared Dirac Hamiltonian becomes block-diagonal whose elements are in essence non-relativistic Schrödinger-type Hamiltonians. This enables us to find a Feynman-type path-integral representation of the resulting Green’s functions. In addition, we are also able to express the spectral properties of the supersymmetric Dirac Hamiltonian in terms of those of the non-relativistic Schrödinger Hamiltonians. The methods are explicitly applied to the free Dirac Hamiltonian, the so-called Dirac oscillator and a generalization of it. The general approach is applicable to systems with good and broken supersymmetry.

DOI
10.1063/1.5020545

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The resolvent of supersymmetric Dirac Hamiltonian is studied in detail. Due to supersymmetry the squared Dirac Hamiltonian becomes block-diagonal whose elements are in essence non-relativistic Schrödinger-type Hamiltonians. This enables us to find a Feynman-type path-integral representation of the resulting Green’s functions. In addition, we are also able to express the spectral properties of the supersymmetric Dirac Hamiltonian in terms of those of the non-relativistic Schrödinger Hamiltonians. The methods are explicitly applied to the free Dirac Hamiltonian, the so-called Dirac oscillator and a generalization of it. The general approach is applicable to systems with good and broken supersymmetry.
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Some years ago, one of the authors~(MM) revived a concept to which he gave the name of single-particle Dirac oscillator, while another~(CQ) showed that it corresponds to a realization of supersymmetric quantum mechanics. The Dirac oscillator in its one- and many-body versions has had a great number of applications. Recently, it included the analytic expression for the eigenstates and eigenvalues of a two-particle system with a new type of Dirac oscillator interaction of frequency~\\omega. By considering the latter together with its partner corresponding to the replacement of~\\omega by~-\\omega we are able to get a supersymmetric formulation of the problem and find the superalgebra that explains its degeneracy.
Some years ago, one of the authors (MM) revived a concept to which he gave the name single-particle Dirac oscillator, while another of the authors (CQ) showed that it corresponds to a realization of supersymmetric quantum mechanics. The Dirac oscillator in its one- and many-body versions has had a great number of applications. Recently, it included the analytic expression for the eigenstates and eigenvalues of a two-particle system with a new type of Dirac oscillator interaction of frequency omega . By considering the latter together with its partner corresponding to the replacement of omega by - omega , we are able to get a supersymmetric formulation of the problem and find the superalgebra that explains its degeneracy.