The Moyal product is used to cast the equation for the metric of a non-Hermitian Hamiltonian in the form of a differential equation. For Hamiltonians of the form p2+V(ix) with V polynomial this is an exact equation. Solving this equation in perturbation theory recovers known results. Explicit criteria for the hermiticity and positive definiteness of the metric are formulated on the functional level.

arXiv:quant-ph/0512055Physics Letters BJan 31, 200632 citations

Czechoslovak Journal of Physics

Quasi-hermiticity and the Role of a Metric in Some Boson Hamiltonians

H. B. Geyer, Frederik G. Scholtz, Izak Snyman

Published Oct 1, 2004• 39 citations

Publisher

DOI: 10.1023/b:cjop.0000044007.58266.a0

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H\in M_d(\mathbb C)

with real spectrum, the biorthogonal Gibbs functional

ω_{\rm bi}(A)=Z_{\rm bi}^{-1}\sum_n e^{-βE_n}\langleφ_n|A|ψ_n\rangle

satisfies

ω_{\rm bi}(A^\dagger A)\ge0

for all

A

if and only if

H

is quasi-Hermitian. The proof reconstructs the metric

η

directly from the eigenprojectors of

ω_{\rm bi}

via the Riesz representation theorem, yielding a metric-free criterion for quasi-Hermiticity. Under the quasi-Hermitian hypothesis, we prove that the

η

-Gibbs state

ω_η(A)=Z_η^{-1}{\rm Tr}[ηe^{-βH}A]

satisfies the full analytic KMS condition using the Hadamard three-line theorem and Bari's theorem on Riesz bases. The transported state generally differs from the Gibbs state of the isospectral Hermitian partner whenever

[η,h]\neq0

, so the KMS property cannot be obtained by similarity transformation alone. Finally, within the Haag--Hugenholtz--Winnink programme, we establish the Tomita--Takesaki modular structure of the

η

-Gibbs state in finite dimensions, while the construction of a compatible

C^*

-norm and the proof of

σ

-weak continuity remain open.