Special relativity, de Broglie waves, dark energy and quantum mechanics

In this paper, we connect quantum mechanics with the recent work of the author Hill (Z Angew Math Phys 69:133–145, 2018; Z Angew Math Phys 70:5–14, 2019), suggesting that dark energy arises from the conventional mechanical theory neglecting the work done in the direction of time and consequently neglecting the de Broglie wave energy. Using special relativity and validation through Lorentz invariance, Hill (2018, 2019) develops expressions for the de Broglie wave energy $E$ by making a distinction between particle energy $e=m{c}^2$ and the total work done by the particle W, so that both momentum $p=mu$ and particle energy e contribute to the total work done $W=e+E$. This formulation provides an extension of Newton's second law that is invariant under the Lorentz group and gives work done expressions for $E$ involving the $\log$ function, indicating that large energies might be generated even for slowing mechanical systems. Although inherent in Hill (2018, 2019), here we propose explicitly that the total work done W by a single particle comprises two contributions, namely particle energy e and wave energy $E$; thus, $W=e+E$. Since in any experiment either particles or de Broglie waves are reported, only one of e or $E$ is physically measured, which leads to the expectation that particles appear for $e<E$ and de Broglie waves occur for $E⩽e$, but in either event, both a measurable energy and an unmeasurable energy exist, the latter registering its presence in the form of dark energy. In particular, in this formulation conventional quantum mechanics operates under circumstances such that the spatial physical force $f$ vanishes, and the force g in the direction of time becomes pure imaginary. If both $f$ and g are generated as the gradient of a potential, then the total particle energy is necessarily conserved in a conventional manner. The present paper makes a formal connection between special relativity and quantum mechanics, linking two new invariances of the Lorentz group of special relativity with the corresponding Lorentz invariant differential operators arising in quantum mechanics and the de Broglie particle and wave duality in Hill (2018, 2019) and giving rise to the Klein–Gordon equation of relativistic quantum mechanics.