Deconstructing non-Dirac-hermitian supersymmetric quantum systems

TL;DR

This paper presents a method to construct non-Dirac-hermitian supersymmetric quantum systems that are isospectral with hermitian systems, expanding the class of exactly solvable models with real spectra using pseudo-hermitian algebraic techniques.

Contribution

It introduces a general framework for building non-Dirac-hermitian supersymmetric systems with real spectra, including explicit examples like Pauli, super-conformal, and Calogero models.

Findings

Constructed non-Dirac-hermitian supersymmetric models with real spectra.

Demonstrated equivalence to Dirac-hermitian systems via isospectrality.

Provided explicit solvable examples of such systems.

Abstract

A method to construct non-Dirac-hermitian supersymmetric quantum system that is isospectral with a Dirac-hermitian Hamiltonian is presented. The general technique involves a realization of the basic canonical (anti-)commutation relations involving both bosonic and fermionic degrees of freedom in terms of non-Dirac-hermitian operators which are hermitian in a Hilbert space that is endowed with a pre-determined positive-definite metric. A pseudo-hermitian realization of the Clifford algebra for a pre-determined positive-definite metric is used to construct supersymmetric systems with one or many degrees of freedom. It is shown that exactly solvable non-Dirac-hermitian supersymmetric quantum systems can be constructed corresponding to each exactly solvable Dirac-hermitian system. Examples of non-Dirac-hermitian (i) non-relativistic Pauli Hamiltonian, (ii) super-conformal quantum system and (iii) supersymmetric Calogero-type models admitting entirely real spectra are presented.