Appropriate Inner Product for PT-Symmetric Hamiltonians

TL;DR

This paper clarifies the fundamental inner product structure for PT-symmetric Hamiltonians, showing that the V norm is the most fundamental and relates to the PT norm, with implications for choosing the appropriate inner product in non-Hermitian quantum mechanics.

Contribution

It demonstrates that the V norm is the most fundamental inner product for PT-symmetric Hamiltonians and clarifies its relation to the PT and C norms, providing guidance for their use.

Findings

V norm is always fundamental and chosen by the theory

V norm equals PT norm when PT phase is included

C operator norm generally differs from V norm

Abstract

A Hamiltonian $H$ that is not Hermitian can still have a real and complete energy eigenspectrum if it instead is $PT$ symmetric. For such Hamiltonians three possible inner products have been considered in the literature, the $V$ norm, the $PT$ norm, and the $C$ norm. Here $V$ is the operator that implements $VHV^{-1}=H^{\dagger}$, the $PT$ norm is the overlap of a state with its $PT$ conjugate, and $C$ is a discrete linear operator that always exists for any Hamiltonian that can be diagonalized. Here we show that it is the $V$ norm that is the most fundamental as it is always chosen by the theory itself. In addition we show that the $V$ norm is always equal to the $PT$ norm if one defines the $PT$ conjugate of a state to contain its intrinsic $PT$ phase. We discuss the conditions under which the $V$ norm coincides with the $C$ operator norm, and show that in general one should not use the linear $C$ operator but for the purposes that it is used one can instead use the antilinear $PT$ operator itself.