A New Class of non-Hermitian Quantum Hamiltonians with PT Symmetry

TL;DR

This paper extends PT quantum mechanics to include odd time reversal symmetry, broadening the class of non-Hermitian Hamiltonians applicable to fermionic systems and potential fundamental physics models.

Contribution

It generalizes PT quantum mechanics to odd T^2 = -1 case, introduces a Kramer's-like theorem, and discusses applications to fermionic systems and beyond standard model physics.

Findings

Generalization of PT quantum mechanics to odd T^2 = -1

Discovery of an analogue of Kramer's theorem for PT systems

Potential applications in fermionic and condensed matter systems

Abstract

In a remarkable development Bender and coworkers have shown that it is possible to formulate quantum mechanics consistently even if the Hamiltonian and other observables are not Hermitian. Their formulation, dubbed PT quantum mechanics, replaces hermiticity by another set of requirements, notably that the Hamiltonian should be invariant under the discrete symmetry PT, where P denotes parity and T denotes time reversal. All prior work has focused on the case that time reversal is even (T^2 = 1). We generalize the formalism to the case of odd time reversal (T^2 = -1). We discover an analogue of Kramer's theorem for PT quantum mechanics, present a prototypical example of a PT quantum system with odd time reversal, and discuss potential applications of the formalism. Odd time reversal symmetry applies to fermionic systems including quarks and leptons and a plethora of models in nuclear, atomic and condensed matter physics. PT quantum mechanics makes it possible to enlarge the set of possible Hamiltonians that physicists could deploy to describe fundamental physics beyond the standard model or for the effective description of condensed matter phenomena.