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

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 $\eta$.

References in corpus (9)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Complex Extension of Quantum Mechanics
• Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
• Exact PT-Symmetry Is Equivalent to Hermiticity
• Extension of PT-Symmetric Quantum Mechanics to Quantum Field Theory with Cubic Interaction
• On Pseudo-Hermitian Hamiltonians and Their Hermitian Counterparts
• PT-Symmetric Versus Hermitian Formulations of Quantum Mechanics
• The disappearing $Q$ operator
• Path-Integral Formulation of Pseudo-Hermitian Quantum Mechanics and the Role of the Metric Operator

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• Action with Acceleration I: Euclidean Hamiltonian and Path Integral
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• Vacuum Stability of the $\mathcal{P}\mathcal{T}$-Symmetric $\left(-{\varphi }^4\right)$ Scalar Field Theory
• Linked $\mathcal{P}\mathcal{T}$-symmetry to Supersymmetry in a class of non-Hermitian Hamiltonians
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