Is PT-symmetric quantum mechanics just quantum mechanics in a non-orthogonal basis?

TL;DR

This paper demonstrates that PT-symmetric quantum mechanics is essentially ordinary quantum mechanics expressed in a non-orthogonal basis, with no experimental distinctions for finite systems, challenging claims of its unique advantages.

Contribution

It shows that PT-symmetric quantum mechanics is equivalent to standard quantum mechanics in a non-orthogonal basis, clarifying its physical interpretation and limitations.

Findings

No experimental distinction between PT-symmetric and ordinary quantum mechanics for finite systems

PT-symmetric Hamiltonians are equivalent to Hermitian ones in a non-orthogonal basis

Claims of faster evolution in PT-symmetric quantum mechanics are interpretational issues

Abstract

One of the postulates of quantum mechanics is that the Hamiltonian is Hermitian, as this guarantees that the eigenvalues are real. Recently there has been an interest in asking if $H^\dagger = H$ is a necessary condition, and has lead to the development of PT-symmetric quantum mechanics. This note shows that any finite physically acceptable non-Hermitian Hamiltonian is equivalent to doing ordinary quantum mechanics in a non-orthogonal basis. In particular, this means that there is no experimental distinction between PT-symmetric quantum mechanics and ordinary quantum mechanics for finite systems. In particular, the claim that PT-symmetric quantum mechanics allows for faster evolution than Hermitian quantum mechanics is shown to be a problem of physical interpretation.