Phase and Group Velocity with a Free Particle Action

The free particle action (relativistic or nonrelativistic) may be written as A = -Et+px. This same expression is also associated with a photon which has no rest mass mo and is linked to the classical wave result E=pv. dA=0 for a photon suggests one may have both a group velocity dE/dp as well as a phase velocity E=pv which are the same and both portions are 0 in dA, so dA=0 overall. Given that dE/dp and E=pv yield the same velocity, one may argue that either a wave or particle approach to a photon holds. This argument is given in (1). We show using dx/dt =x/t for a free particle that this must follow if dA=0.

  For a particle with rest mass mo, a group velocity portion of dA is 0 if dE/dp is taken as dE/dv / dp/dv. The overall dA, however, is not 0 for constant mo. A phase velocity portion is completely at odds with p=Ev. Thus we suggest that the form exp(-iEt+ipx) used in quantum mechanics for a free particle be separated into exp(-Et) and exp(ipx) and not be combined to describe an overall wave.

   Thus for the photon and particle with a mass, dA portions corresponding to velocities are 0. For the photon, both dA(group velocity)=0 and dA(phase velocity)=0 and vgroup=vphase, while for a particle with rest mass only dA (group velocity)=0 and dA (overall) is not zero for fixed mass. Thus the photon and particle with a free mass have a group velocity portion of dA being 0 in common. This suggests a similarity of the two at a “particle” level as group velocity is usually associated with particle motion. In (1) the particle properties of the photon are also stressed.