A mathematical formalism of non-Hermitian quantum mechanics and observable-geometric phases

TL;DR

This paper develops a comprehensive mathematical framework for non-Hermitian quantum mechanics, extending the Dirac-von Neumann formalism to include para-Hermitian operators and non-Hermitian observables, and explores geometric phases in such systems.

Contribution

It introduces a formalism that incorporates para-Hermitian operators, modifies measurement and evolution postulates, and unifies PT-symmetric and biorthogonal quantum mechanics.

Findings

The formalism is basis-independent and Hamiltonian-independent.

It generalizes the Born rule to non-Hermitian observables with metric dependence.

Observable-geometric phases are studied within this new framework.

Abstract

We present a mathematical formalism of non-Hermitian quantum mechanics, following the Dirac-von Neumann formalism of quantum mechanics. In this formalism, the state postulate is the same as in the Dirac-von Neumann formalism, but the observable postulate should be changed to include para-Hermitian operators (spectral operators of scalar type with real spectrum) representing observable, as such both the measurement postulate and the evolution postulate must be modified accordingly. This is based on a Stone type theorem as proved here that the dynamics of non-Hermitian quantum systems is governed by para-unitary time evolution. The Born formula on the expectation of an observable at a certain state is given in the non-Hermitian setting, which is proved to be equal to the usual Born rule for every Hermitian observable, but for a non-Hermitian one it may depend on measurement via the choice of a metric operator associated with the non-Hermitian observable under measurement. Our formalism is nether Hamiltonian-dependent nor basis-dependent, but can recover both PT-symmetric and biorthogonal quantum mechanics, and it reduces to the Dirac-von Neumann formalism of quantum mechanics in the Hermitian setting. As application, we study observable-geometric phases for non-Hermitian quantum systems.