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

TL;DR

This paper explores how the path integral formulation for quasi-Hermitian Hamiltonians inherently encodes the metric ta, revealing that Feynman diagrams implicitly incorporate the ta metric through the Heisenberg equations of motion.

Contribution

It demonstrates that the path integral and Feynman diagrams for quasi-Hermitian theories implicitly include the ta metric via the Heisenberg equations, clarifying their connection.

Findings

Path integrals encode the ta metric without explicit mention.

Feynman diagrams are based on Heisenberg equations valid with ta.

Perturbative calculations match expectation values with the ta metric.

Abstract

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 $\eta$.