Which Green Functions Does the Path Integral for Quasi-Hermitian Hamiltonians Represent?

In the context of quasi-Hermitian theories, which are non-Hermitian in the conventional sense, but can be made Hermitian by the introduction of a dynamically-determined metric $\eta$, we address the problem of how the functional integral and the Feynman diagrams deduced therefrom "know" about the metric. Our investigation is triggered by a result of Bender, Chen and Milton, who calculated perturbatively the one-point function $G_1$ for the quantum Hamiltonian $H=\half(p^2+x^2)+igx^3$. It turns out that this calculation indeed corresponds to an expectation value in the ground state evaluated with the $\eta$ metric. The resolution of the problem turns out be that, although there is no explicit mention of the metric in the path integral or Feynman diagrams, their derivation is based fundamentally on the Heisenberg equations of motion, which only take their standard form when matrix elements are evaluated with the inclusion of $\eta$.