The phase velocity of waves is never infinite. How do you come to the conclusion that might be so? Also there is no sensible physical interpretation of relativstic wave functions in terms of a single-particle theory. The first who realized this was Dirac who tried to make the Schrödinger way of description of quanta in terms of wave mechanics, working so well in non-relati istic physics, work for relativistic particles. He tried to get a wave function fulfilling an equation of motion which is of first order in the time derivative, and since Lorentz symmetry ties space and time so closely together he had also to assume that it's first order in the spatial derivatives.
However, when analyzing his famous equation (named after him as Dirac equation) for the case of electrons (indeed the wave function turned out to describe particles of spin 1/2, and the electron was known to have spin 1/2 at this time) that interact with an external electric potential he could not make sense of the equation without assuming that also the solutions with negative frequencies are physical. His idea was to let all these state be occupied in the ground state of the system, so that in the free case no electrons can go into such a state of "negative energy". Rather he interpreted probable holes in this Dirac sea as antiparticles (in the case of electrons dubbed positrons) with positive energy moving in the opposite direction. Then he could make sense also of the solutions for interacting particles, and it came out that in fact he dealt with a many-body problem, i.e., if the interaction is strong enough, an electron scattering at the potential might end up creating an electron-positron pair. Thus the electron number and the positron number are not conserved but only the net-charge number. Taking the electric-charge convention the electrons (particles) are negatively and the positrons (antiparticles) are positively charged.
This is a very complicated view on relativistic quantum theory, but it can be made working even for the more complicated case of interacting electrons, positrons and the electromagnetic field. It's in fact a valid way to describe quantum electrodynamics, but it's a quite cumbersome way and not very elegant to work with. That's the more true for the more complicated interactions (strong and weak interactions) of the standard model. That's why nowadays we start right away with the concept of quantum fields which from the very beginning incorporate the possibility that particle number needs not be a conserved quantum number but that it's possible to create and destroy particles in interactions.
At the same time the quantum-field theoretical method automatically takes care of causality and Poincare invariance, and the faster-than-light values of phasevelocities of massive realativistic wave equations is no more an interpretational problem in the modern formulation.
