Where Is Phase Velocity in Minkowski Space?

In the special theory of relativity, massive particles can travel at neither the speed of light c nor faster. Meanwhile, since the photon was quantized, many have thought of it as a point particle. How pointed? The idea could be a mathematical device or physical simplification. By contrast, the preceding notion of wave-group duality has two velocities: a group velocity v_g and a phase velocity v_p. In light v_p = v_g = c; but it follows from special relativity that, in massive particles, v_p > c. The phase velocity is the product of the two best measured variables, and so their product constitutes internal motion that travels, verifiably, faster than light. How does v_p then appear in Minkowski space? For light, the spatio-temporal Lorentz invariant metric is $s^2=c^2{t}^2？x^2？y^2？z^2$ , the same in whatever frame it is viewed. The space is divided into 3 parts: firstly a cone, symmetric about the vertical axis ct > 0 that represents the world line of a stationary particle while the conical surface at s = 0 represents the locus for light rays that travel at the speed of light c. Since no real thing travels faster than the speed of light c, the surface is also a horizon for what can be seen by an observer starting from the origin at time t = 0. Secondly, an inverted cone represents, equivalently, time past. Thirdly, outside the cones, inaccessible space. The phase velocity v_p, group velocity v_g and speed of light are all equal in free space, v_p = v_g = c, constant. By contrast, for particles, where causality is due to particle interactions having rest mass m_o > 0, we have to employ the Klein-Gordon equation with $s^2=c^2{t}^{}$