Conformal Extension of Massive Wave Functions

Abstract

We extend from themasslessto themassivecase the canonical conformal mapping of solutions of wave equations in two-dimensional Minkowski spaceM0into the universal coverMof the conformal compactificationMofM0. There is a continuous Poincaré-covariant map from the space of solutions of the Klein–Gordon equation inM0to the dual of the space of analytic vectors for the action of the conformal groupG≃SU(1, 1)×SU(1, 1) on massive fields inM. The extensions satisfy a corresponding wave equation inM. As in the case of massless equations, the extended solutions are periodic under the generatorζof the infinite cyclic center of the conformal groupGinM. The analysis involves the determination of the action of the chronometric temporal evolution, i.e., the one-parameter group of nonlinear transformations generated by the center of the maximal essentially-compact subgroup ofGon the Fourier transforms of wave functions in Minkowski space. The analytic continuation of this evolution to complex times of positive imaginary part is represented by a Hilbert–Schmidt operator whose kernel is determined in closed form. Fields of higher conformal weight associated with other discrete series representations ofGare also treated in this respect.