PT-Invariance and Indefinite Metric

TL;DR

This paper presents a new proof that PT-Invariant non-Hermitian cubic oscillators with imaginary coupling have real eigenvalues, using indefinite metric and Hermitian Hamiltonian correspondence, with implications for vacuum stability in supersymmetric theories.

Contribution

It introduces a novel two-step proof linking PT-Invariance and indefinite metric Hermitian Hamiltonians to establish real eigenvalues.

Findings

PT-Invariant Hamiltonians can be associated with Hermitian Hamiltonians via indefinite metric.

Eigenvalues of the PT-Invariant cubic oscillator are real when analyzed with indefinite metric.

The approach offers insights into vacuum stability in certain supersymmetric gauge theories.

Abstract

A new proof is given for why the non-Hermitian, PT-Invariant cubic oscillator with imaginary coupling has real eigenvalues. The proof consists of two steps. In the first step, it is shown that for many PT-Invariant Hamiltonians, one can define corresponding Hermitian Hamiltonians that have the same eigenvalues when quantized using indefinite metric. The second step is to show that the indefinite-metric eigenvalues of the Hermitian cubic oscillator are real since the norms of its eigenstates do not vanish. The correspondence between PT-Invariant Hamiltonians and indefinite-metric Hermitian Hamiltonians is further discussed as a way to determine vacuum stability for certain supersymmetric gauge theories.