The symmetrical rate-equations of particle–wave mechanics.

At various times in his life, Louis de Broglie firmly believed in the coexistence of both particle and an associated wave referring to "the theory of the double solution", and an equation which he called "the guidance formula". In an attempt to account for both particle and wave, the author has proposed a Lorentz invariant alternative to Newton's second law which is developed in Hill (Zeitschrift fur angewandte Mathematik und Physik 69:133–145, 2018; Zeitschrift fur angewandte Mathematik und Physik 70:5–14, 2019; Zeitschrift fur angewandte Mathematik und Physik 70:131–153, 2019; Math Mech Solids 26:263–284, 2020; Math Mech Solids 25: 1763–1777, 2020; Zeitschrift fur angewandte Mathematik und Physik 72:1–14; Mathematics of particle–wave mechanical systems, Springer, Cham, 2022). Here, we summarise some of the major outcomes of this approach, including simple solutions of the proposed model exhibiting both sub-luminal and superluminal behaviour dependent upon the region of space-time, and two symmetrical sets of rate-differential relations for the two Lorentz invariants for a single spatial dimension x. One set follows the particle, while the other follows the wave, revealing a complete symmetry between the one-dimensional spatial physical force f and the force g in the "direction of time". The existence of these symmetrical equations reinforces the parity and interchangeability of particles and waves. [ABSTRACT FROM AUTHOR]