Complex-dimensional Integral and Light-cone Singularities

Abstract

The notion of a complex-dimensional integral is introduced in the complex n-dimensional Minkowski space. Its basic properties, such as Lorentz invariance, are investigated. Complex-dimensional invariant delta functions Δn(_x,m_2), Δ(1)n(_x,m_2), etc. are explicitly calculated in position space. It is proposed to define products of singular functions in the ordinary Minkowski space by analytically continuing the corresponding n-dimensional ones to n=4. The light-cone singularities of [Δ(_x,m_2)]2, Δ(_x,m_2) x Δ(1)(_x,m_2)]2 and Δ(1)(_x,m_2)]2 are shown to be unambiguously determined in this way.