Conformal Mapping and its Applications

Abstract

Complex analysis and conformal mapping play a central role in mathematical sciences and theoretical physics. The traditional applications include differential equations, harmonic analysis, potential theory and fluid mechanics. Of particular interest to this study is the complexfied Minkowski space and its corresponding spin space model which is appropriate for the description of quantum field theory. Moreover, for an ambitious scheme to incorporate gravitational field in a quantized form, we introduced the three-dimensional complex projective space as an advanced model whereby points of the complexified Minkowski space are not prime but secondary. In the light of Penrose correspondence these points are complex lines in Twistor space. It has been shown that the conformally invariant zero-rest mass fields, such as weak gravitational field, are represented by contour integrals of holomorphic functions on twistor space.