On the construction of pseudo-hermitian quantum system with a pre-determined metric in the Hilbert space

TL;DR

This paper develops a method to construct pseudo-hermitian quantum systems with a predefined positive-definite metric, enabling real spectra and unitary evolution, with applications to various complex quantum models.

Contribution

It introduces a general approach to realize pseudo-hermitian systems with a specified metric, expanding the class of models with real spectra and unitary dynamics.

Findings

Constructed pseudo-hermitian systems with explicit metrics

Demonstrated real spectra and unitarity in various models

Provided examples including oscillators and spin chains

Abstract

A class of pseudo-hermitian quantum system with an explicit form of the positive-definite metric in the Hilbert space is presented. The general method involves a realization of the basic canonical commutation relations defining the quantum system in terms of operators those are hermitian with respect to a pre-determined positive definite metric in the Hilbert space. Appropriate combinations of these operators result in a large number of pseudo-hermitian quantum systems admitting entirely real spectra and unitary time evolution. The examples considered include simple harmonic oscillators with complex angular frequencies, Stark(Zeeman) effect with complex electric(magnetic) field, non-hermitian general quadratic form of N boson(fermion) operators, symmetric and asymmetric XXZ spin-chain in complex magnetic field, non-hermitian Haldane-Shastry spin-chain and Lipkin-Meshkov-Glick model.