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We have shown that a longitudinal wave emerges as a result of generaltransformations similar to gauge transformations of electrodynamics. The timederivative and the gradient of the gauge function and their alike yield thelongitudinal wave. The de Broglie wave associated with the particle motion is alongitudinal one. This wave accompanied the motion of all massive objects. Ithas a zero magnetic field (or vorticity). The invariance of Dirac equationunder these transformations makes the electron to behave as a massless particlewhile having a mass.

The emergent longitudinal wave from space and time derivatives transformations