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

• URL: http://arxiv.org/abs/2111.12883v3
• Date: Mon, 28 Mar 2022 08:38:21 GMT
• Title: A mathematical formalism of non-Hermitian quantum mechanics and observable-geometric phases
• Authors: Zeqian Chen
• Abstract summary: We present a formalism of non-Hermitian quantum mechanics, following the Dirac-von Neumann formalism of quantum mechanics. Our formalism is nether Hamiltonian-dependent nor basis-dependent, but can recover both PT-symmetric and biorthogonal quantum mechanics.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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