  In this paper, we use the theory of fractional powers of linear operators to
construct a general (analytic) representation theory for the square-root energy
operator of relativistic quantum theory, which is valid for all values of the
spin. We focus on the spin 1/2 case, considering a few simple yet solvable and
physically interesting cases, in order to understand how to interpret the
operator. Our general representation is uniquely determined by the Green's
function for the corresponding Schrodinger equation. We find that, in general,
the operator has a representation as a nonlocal composite of (at least) three
singularities. In the standard interpretation, the particle component has two
negative parts and one (hard core) positive part, while the antiparticle
component has two positive parts and one (hard core) negative part. This effect
is confined within a Compton wavelength such that, at the point of singularity,
they cancel each other providing a finite result. Furthermore, the operator
looks like the identity outside a few Compton wavelengths (cutoff). To our
knowledge, this is the first example of a physically relevant operator with
these properties.
