Comparison of different approaches of finding the positive definite metric in pseudo-Hermitian theories

TL;DR

This paper compares various methods for determining positive definite metrics in pseudo-Hermitian quantum theories, highlighting their differences through explicit examples and discussing phenomena like exceptional points and symmetry breaking.

Contribution

It provides a systematic comparison of approaches to find positive definite metrics in pseudo-Hermitian theories using explicit models.

Findings

Different approaches yield consistent metrics in examples

Exceptional points influence metric properties

Spontaneous symmetry breaking affects metric selection

Abstract

To develop a unitary quantum theory with probabilistic description for pseudo- Hermitian systems one needs to consider the theories in a different Hilbert space endowed with a positive definite metric operator. There are different approaches to find such metric operators. We compare the different approaches of calculating pos- itive definite metric operators in pseudo-Hermitian theories with the help of several explicit examples in non-relativistic as well as in relativistic situations. Exceptional points and spontaneous symmetry breaking are also discussed in these models.