  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 $\eta$.
