The Square-Root Equation

The Square-Root Equation

### Abstract

In this paper, we use the theory of fractional powers of linear operators to construct a general (analytic) representation theory for the massless square-root energy operator of relativistic quantum theory, which is valid for all values of the spin. We find that the operator has a representation as a nonlocal composite of two singularities. At the point of singularity, the terms cancel each other providing a finite result. We then show that the solution of this equation can be used to provide a very interesting general representation for solutions to the wave equation. Unlike the method of spherical means, the representation is independent of the spacial dimension.