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.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.

arXiv:2305.11989May 19, 20230 citations

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.