Pseudo-Hermiticity and Generalized PT- and CPT-Symmetries

TL;DR

This paper explores the mathematical structure of pseudo-Hermitian Hamiltonians, extending PT- and CPT-symmetry concepts, and introduces generalized symmetry operators to explain invariance properties in quantum systems.

Contribution

It provides a unified framework for understanding PT- and CPT-symmetries in pseudo-Hermitian Hamiltonians, including new generalized operators and their invariance properties.

Findings

Pseudo-Hermitian operators explain CPT-symmetry in PT-symmetric Hamiltonians.

Introduction of generalized parity, time-reversal, and charge-conjugation operators.

Establishment of PT- and CPT-invariance for diagonalizable pseudo-Hermitian Hamiltonians.

Abstract

We study certain linear and antilinear symmetry generators and involution operators associated with pseudo-Hermitian Hamiltonians and show that the theory of pseudo-Hermitian operators provides a simple explanation for the recent results of Bender, Brody and Jones (quant-ph/0208076) on the CPT-symmetry of a class of PT-symmetric non-Hermitian Hamiltonians. We present a natural extension of these results to the class of diagonalizable pseudo-Hermitian Hamiltonians H with a discrete spectrum. In particular, we introduce generalized parity (${\cal P}$), time-reversal (${\cal T}$), and charge-conjugation (${\cal C}$) operators and establish the ${\cal PT}$- and ${\cal CPT}$-invariance of H.