Singularities of adelic Feynman amplitudes

Abstract

We study adelic φ4-theory with propagator, given by homogeneous adelic function. It is shown that almost all ultraviolet and infrared poles of Euclidean Feynman amplitude are cancelled by zeroes of the infinite product of p-adic Feynman amplitudes. Analytic continuation in the degree of homogeneity of general adelic Feynman amplitude is constructed. We prove that all adelic φ4-theory amplitudes can be continued to the half-plane. There are an infinite number of amplitudes whose natural domain of analyticity is given by this half-plane provided the Riemann conjecture about ζ-function zeroes is valid.