Matrix Algebras in Non-Hermitian Quantum Mechanics

TL;DR

This paper develops a mathematical framework for non-Hermitian quantum mechanics by mapping commutator algebras onto Hermitian Hamiltonian structures, enabling better simulation of open quantum systems.

Contribution

It introduces a reversible algebraic mapping that allows non-Hermitian quantum dynamics to be formulated using Hermitian Hamiltonians, providing a versatile template for simulations.

Findings

Provides a general algebraic structure for non-Hermitian quantum equations

Enables simulation of open quantum systems with Hermitian-based methods

Establishes a reversible mapping between non-Hermitian and Hermitian formalisms

Abstract

In principle, non-Hermitian quantum equations of motion can be formulated using as a starting point either the Heisenberg's or the Schr\"odinger's picture of quantum dynamics. Here it is shown in both cases how to map the algebra of commutators, defining the time evolution in terms of a non-Hermitian Hamiltonian, onto a non-Hamiltonian algebra with a Hermitian Hamiltonian. The logic behind such a derivation is reversible, so that any Hermitian Hamiltonian can be used in the formulation of non-Hermitian dynamics through a suitable algebra of generalized (non-Hamiltonian) commutators. These results provide a general structure (a template) for non-Hermitian equations of motion to be used in the computer simulation of open quantum systems dynamics.