PT-Invariance and Indefinite Metric
Mathematical Physics
2009-10-28 v1 math.MP
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.
Cite
@article{arxiv.0910.5017,
title = {PT-Invariance and Indefinite Metric},
author = {Scott Chapman},
journal= {arXiv preprint arXiv:0910.5017},
year = {2009}
}