  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.
