The emergent longitudinal wave from space and time derivatives   transformations
  We have shown that a longitudinal wave emerges as a result of general transformations similar to gauge transformations of electrodynamics. The time derivative and the gradient of the gauge function and their alike yield the longitudinal wave. The de Broglie wave associated with the particle motion is a longitudinal one. This wave accompanied the motion of all massive objects. It has a zero magnetic field (or vorticity). The invariance of Dirac equation under these transformations makes the electron to behave as a massless particle while having a mass. 
A. I. Arbab
quant-ph