Why Does Phase Velocity Differ in Relativistic and Non-Relativistic Cases?

[qbslug]

So for the phase velocity of a massive particle we have

Vph = Vg/2 for non-relativistic case

Vph = c^2/Vg for the relativistic case

Vg is the group velocity or particle velocity
But there seems to be a contradiction in that for the non-relativistic case the phase velocity is predicted to be less than the group velocity whereas in the relativistic case the phase velocity must be greater than the group velocity (greater than the speed of light even).
Furthermore a taylor series expansion of c^2/Vg will not converge to Vg/2 for small velocities. Whats the deal with this - why are these predictions so vastly different? I am guessing it has something to do with not having absolute speed in relativity but can't connect the dots.

 

[Bill_K]

Different choices for the zero point of energy. The phase velocity in both cases is v_ph = ω/k = E/p. Clearly this will change if a constant is added to E. In relativity, E = √(m²c⁴ + p²c²) ≈ mc² + p²/2m, while in nonrelativistic QM we drop the rest energy mc².

Basically the same as this other thread.