Feynman-Kac path integral expansion around the upside-down oscillator

TL;DR

This paper develops a rigorous path integral approach for quantum systems with potentials perturbing the upside-down oscillator, extending to quantum field theory with complex spatial coordinates, enabling analysis of unbounded potentials.

Contribution

It introduces a method to express real-time path integrals for perturbed upside-down oscillators using Wiener measures and extends this to complex-scaled quantum field theories.

Findings

Path integrals expressed via Wiener measure for specific perturbations.

Extension of the method to quantum field theory with complex coordinates.

Rigorous treatment of unbounded-from-below potentials.

Abstract

We discuss path integrals for quantum mechanics with a potential which is a perturbation of the upside-down oscillator. We express the path integral (in the real time) by the Wiener measure. We obtain the Feynman integral for perturbations which are the Fourier-Laplace transforms of a complex measure and for polynomials of the fotm $x^{4n}$ and $x^{4n+2}$ (where $n$ is a natural number). We extend the method to quantum field theory (QFT) with complex scaled spatial coordinates ${\bf x}\rightarrow i{\bf x}$. We show that such a complex extension of the path integral (in the real time) allows a rigorous path integral treatment of a large class of potentials including the ones unbounded from below.