Bessel potentials and Green functions on pseudo-Euclidean spaces

TL;DR

This paper reviews Bessel potentials and Green functions on pseudo-Euclidean spaces, focusing on their properties in Lorentzian signature and applications to quantum field theory, using hypergeometric functions for clarity.

Contribution

It provides a comprehensive analysis of Bessel potentials and two-point functions in pseudo-Euclidean spaces, emphasizing hypergeometric functions and distributional properties, including the tachyonic case.

Findings

Explicit formulas for Bessel potentials using hypergeometric functions

Analysis of distributional properties of two-point functions

Inclusion of tachyonic case in the framework

Abstract

We review properties of Bessel potentials, that is, inverse Fourier transforms of (regularizations of) $\frac{1}{(m^2+p^2)^{\frac{\mu}{2}}}$ on a pseudoEuclidean space with signature $(q,d-q)$. We are mostly interested in the Lorentzian signature $(1,d-1)$, and the case $\mu=2$, related to the Klein-Gordon equation $(-\Box+m^2)f=0$. We analyze properties of various ``two-point functions'', which play an important role in Quantum Field Theory, such as the retarded/advanced propagators or Feynman/antiFeynman propagators. We consistently use hypergeometric functions instead of Bessel functions, which makes most formulas much more transparent. We pay attention to distributional properties of various Bessel potentials. We include in our analysis the ``tachyonic case'', corresponding to the ``wrong'' sign in the Klein-Gordon equation.