Where Is Phase Velocity in Minkowski Space?

Abstract

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 2 − x 2 − y 2 − z 2 + m o 2 c 2 . Now special relativity requires a complication: v p . v g = c 2 where v g < c and therefore v p > c. In the volume outside the cones, causality due to light interactions cannot extend beyond the cones. However, since v p > c and even v p >> c when wavelength λ is long, extreme phase velocities are then limited in their causal effects by the particle uncertainty σ , i.e. to v g t ± σ / ω , where ω is the particle angular frequency. This is the first time the phase range has been described for a massive particle.