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

Temperature-reflection II: Modular Invariance and T-reflection

David A. McGady

In this paper, we present robust evidence that general finite temperature quantum field theory (QFT) path integrals are invariant under reflecting temperatures to negative values (T-reflection), up to a possible anomaly phase. Our main focus is on two-dimensional conformal field theories (2d CFTs) on the two-torus. Modular invariance for 2d CFT path integrals follows from demanding invariance under redundant encodings of the two-torus shape in the path integral. We emphasize that identical logic implies 2d CFTs are invariant under T-reflection, up to phases. We compute T-reflection anomaly phases for certain 2d CFT path integrals via a continuation, and via an extension of modular forms from the upper half-plane to the double half-plane. Crucially, they perfectly agree. Requiring QFT path integrals to be invariant under redundant encodings of the spacetime geometry implies (i) that 2d CFTs are both modular and T-reflection invariant and (ii) that general QFT path integrals are invariant under T-reflection. This quite board argument suggests T-reflection phases may indicate previously unnoticed anomalies and consistency conditions for general QFT.In this paper, we present robust evidence that general finite temperature quantum field theory (QFT) path integrals are invariant under reflecting temperatures to negative values (T-reflection), up to a possible anomaly phase. Our main focus is on two-dimensional conformal field theories (2d CFTs) on the two-torus. Modular invariance for 2d CFT path integrals follows from demanding invariance under redundant encodings of the two-torus shape in the path integral. We emphasize that identical logic implies 2d CFTs are invariant under T-reflection, up to phases. We compute T-reflection anomaly phases for certain 2d CFT path integrals via a continuation, and via an extension of modular forms from the upper half-plane to the double half-plane. Crucially, they perfectly agree. Requiring QFT path integrals to be invariant under redundant encodings of the spacetime geometry implies (i) that 2d CFTs are both modular and T-reflection invariant and (ii) that general QFT path integrals are invariant under T-reflection. This quite board argument suggests T-reflection phases may indicate previously unnoticed anomalies and consistency conditions for general QFT.

arXiv:1806.09873Jun 26, 20184 citations

The Path Integral in Quantum Theory

U. Mosel

Advanced texts in physicsJan 1, 20040 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.