Combining Newton’s second law and de Broglie’s particle-wave duality

All matter can exhibit wave-like behaviour, and Louis de Broglie first predicted light to display the dual characteristics as both a collection of particles, called photons, or in some respects as a wave. The particle velocity is the group velocity of the wave, and if the particle velocity u g is subluminal then the associated wave or phase velocity u p through the de Broglie relation u g u p = c 2 is necessarily superluminal. This is believed not to contradict the fact that information cannot be carried faster than the velocity of light c because the wave phase is supposed to carry no energy. However, the superluminal phase velocity may well be physically significant, and here we propose that the sub particle world and the super wave world might be equally important, and that each might exert an influence on the other, such that any mechanical equations must not only be Lorentz invariant but they must also be invariant under the transformation connecting the sub and super worlds. Following this approach, Einstein’s equation E = mc 2 becomes simply E = ( m + m ′ ) c 2 , where m and m ′ are masses given by Einstein expressions arising from the perceived sub and superluminal velocities u g and u p respectively. This modification, although superficially simple, results from non-conventional physics and gives rise to an extension of Newton’s second law, that might well account for the extra energy and mass that is known to exist in the universe, and referred to as dark energy and dark matter. An explicit solution for photons and light predicts a non-zero photon rest-mass m 0 = h ν / 2 c 2 , where h is Planck’s constant and ν is the light frequency. Interestingly, the associated energy of this mass is the zero-point energy, believed to be the lowest energy that a quantum mechanical system may possess.