Tepper L. Gill and W. W. Zachary
Analytic Representation of Relativistic
Wave Equations II: The Square-Root Operator Case
(320K, pdf)
ABSTRACT. In this paper, using the theory of fractional powers for operators, we construct the most general (analytic) representation for the square-root operator of relativistic quantum theory. We allow for arbitrary, but time-independent, vector potential and mass terms. Our representation is uniquely determined by the Green s function for the corresponding Schr dinger equation. We find that the square-root operator is represented by a nonlocal composite of (at least) three singularities. To our knowledge, this is the first example of a physically relevant operator with these properties. 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 (almost) like the identity outside a Compton wavelength, but has a residual instantaneous connection with all other particles in the universe at each point in time.
In addition to the possibility that the square-root operator may be used to represent the inside of hadrons, it is also possible that the residual attractive (particle) part may be the long sought connection between the internal particle composition and the cause for gravitational interaction in matter. If this view were correct, then we would expect matter and antimatter to be gravitationally attractive among themselves and gravitationally repulsive with each other. This would make physical sense if we take seriously the interpretation of antimatter as matter with its time reversed (as opposed to hole theory).