Bessel potentials and Green functions on pseudo-Euclidean spaces

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.