PT symmetry in relativistic quantum mechanics

TL;DR

This paper extends PT symmetry concepts from nonrelativistic quantum mechanics to relativistic quantum mechanics by redefining time reversal as an operation on the time coordinate operator, demonstrating real energy spectra in non-Hermitian models.

Contribution

It introduces a method to implement PT symmetry in relativistic quantum mechanics by treating time as an operator and shows non-Hermitian Hamiltonians can have real spectra.

Findings

Relativistic models with PT symmetry have real eigenvalues.

Time reversal is redefined as an operation on the time coordinate operator.

Non-Hermitian Hamiltonians in relativistic models can be physically meaningful.

Abstract

In nonrelativistic quantum mechanics and in relativistic quantum field theory, time t is a parameter and thus the time-reversal operator T does not actually reverse the sign of t. However, in relativistic quantum mechanics the time coordinate t and the space coordinates x are treated on an equal footing and all are operators. In this paper it is shown how to extend PT symmetry from nonrelativistic to relativistic quantum mechanics by implementing time reversal as an operation that changes the sign of the time coordinate operator t. Some illustrative relativistic quantum-mechanical models are constructed whose associated Hamiltonians are non-Hermitian but PT symmetric, and it is shown that for each such Hamiltonian the energy eigenvalues are all real.