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

Z. Haba

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arXiv · 2305.11989

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

Z. Haba

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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.

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Z. Haba. 2023-05-19. Feynman-Kac path integral expansion around the upside-down oscillator. https://arxiv.org/abs/2305.11989

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