A review of de Broglie particle–wave mechanical systems

Abstract

The existence of the so-called ‘dark’ issues of mechanics implies that our present accounting for mass and energy is incorrect in terms of applicability on a cosmological scale, and the question arises as to where the difficulty might lie. The phenomenon of quantum entanglement indicates that systems of particles exist that individually display certain characteristics, while collectively the same characteristic is absent simply because it has cancelled out between individual particles. It may therefore be necessary to develop theoretical frameworks in which long-held conservation beliefs do not necessarily always apply. The present paper summarises the formulation described in earlier papers (Hill, JM. On the formal origin of dark energy. Z Angew Math Phys 2018; 69:133-145; Hill, JM. Some further comments on special relativity and dark energy. Z Angew Math Phys 2019; 70: 5–14; Hill, JM. Special relativity, de Broglie waves, dark energy and quantum mechanics. Z Angew Math Phys 2019; 70: 131–153.), which provides a framework that allows exceptions to the law that matter cannot be created or destroyed. In these papers, it is proposed that dark energy arises from conventional mechanical theory, neglecting the work done in the direction of time and consequently neglecting the de Broglie wave energy $\mathcal{E}$ . These papers develop expressions for the de Broglie wave energy $\mathcal{E}$ by making a distinction between particle energy $e=m{c}^2$ and the total work done by the particle $W=e+\mathcal{E}$ , that which accumulates from both a spatial physical force $\mathbf{f}$ and a force $g$ in the direction of time. In any experiment, either particles or de Broglie waves are reported, so that only one of $e$ or $\mathcal{E}$ is physically measured, and particles appear for $e<\mathcal{E}$ and de Broglie waves occur for $\mathcal{E}<e$ , but in either event both a measurable and an immeasurable energy exists. Conventional quantum mechanics operates under circumstances such that $\mathbf{f}$ vanishes and $g$ becomes purely imaginary. If both $\mathbf{f}$ and $g$ are generated as the gradient of a potential, the total particle energy is necessarily conserved in the conventional manner.