Relativistic wave and particle mechanics formulated without classical mass

The fact that the concept of classical mass plays an important role in formulating relativistic theories of waves and particles is well-known. However, recent studies show that Galilean invariant theories of waves and particles can be formulated with the so-called ‘wave mass’, which replaces the classical mass and allows attaining higher accuracy of performing calculations J.L. Fry and Z.E. Musielak, Ann. Phys. 325 (2010) 1194]. The main purpose of this paper is to generalize these results and formulate fundamental (Poincaré invariant) relativistic theories of waves and particles without the classical mass. In the presented approach, the classical mass is replaced by an invariant frequency that only involves units of time. The invariant frequencies for various elementary particles are deduced from experiments and their relationship to the corresponding classical and wave mass for each particle is described. It is shown that relativistic wave mechanics with the invariant frequency is independent of the Planck constant, and that such theory can attain higher accuracy of performing calculations. The choice of natural units resulting from the developed theories of waves and particles is also discussed.