Complex-dimensional Integral and Light-cone Singularities

Abstract

The notion of a complex-dimensional integral is introduced in the complex w-dimensional Minkowski space. Its basic properties, such as Lorentz invariance, are investigated. Complex-dimensional invariant delta functions dn(x',mz), A^n(x w2), 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 ^-dimensional ones to w=4. The light-cone singularities of [^Oc;w2)]2, A(x;mz) r;w2) and [4(1) (x; ra2)]2 are shown to be unambiguously determined in this way. Recently, in quantum field theory, much attention has been paid to complex-dimensional regularization [1]. The momentum-space Feynman integral is regularized by considering it in the complex ^-dimensional space formally. The extension of the dimension 4 to the complex dimension n is easily done in the Feynman-parametric representation of the Feynman integral. The purpose of my talk is to formulate the theory of complex-dimen sional integrals in the general framework and apply it to regularizing singular products in position space. Detailed accounts are presented in my papers [2,3]. The complex ^-dimensional Minkowski space Mn is a product of a one-dimensional Euclidean space R and a complex (n — Y) -dimensional space En~l such that the scalar product in M.n is defined by the difference between the product in R and the scalar product in En~l. Here En~l is an