Path Integral and Spectral Representations for Supersymmetric Dirac-Hamiltonians

TL;DR

This paper develops a path integral and spectral framework for supersymmetric Dirac Hamiltonians, linking their properties to non-relativistic Schrödinger systems, with applications to free particles and oscillators.

Contribution

It introduces a novel method to represent supersymmetric Dirac Hamiltonians using path integrals and spectral analysis, applicable to systems with both broken and unbroken supersymmetry.

Findings

Derived Feynman-type path integral representations for Green's functions.

Expressed spectral properties of Dirac Hamiltonians via non-relativistic counterparts.

Applied methods to free Dirac, Dirac oscillator, and generalized systems.

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 Schr\"odinger-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 Schr\"odinger 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.