Discrete phase space and continuous time relativistic quantum mechanics I: Planck oscillators and closed string-like circular orbits

TL;DR

This paper explores a discrete phase space formulation of relativistic quantum mechanics, revealing closed string-like orbits of constant energy and their geometric properties, with exact solutions for the Planck oscillator and invariance of the Klein-Gordon equation.

Contribution

It introduces a novel discrete phase space framework for relativistic quantum mechanics, solving the Planck oscillator exactly and identifying string-like orbits with geometric and physical significance.

Findings

Exact energy eigenvalues for Planck oscillator in discrete phase space

Existence of closed, string-like circular orbits of constant energy

Invariance of the discrete Klein-Gordon equation under continuous symmetry group

Abstract

The discrete phase space continuous time representation of relativistic quantum mechanics involving a characteristic length $l$ is investigated. Fundamental physical constants such as $\hbar$, $c$, and $l$ are retained for most sections of the paper. The energy eigenvalue problem for the Planck oscillator is solved exactly in this framework. Discrete concircular orbits of constant energy are shown to be circles $S^{1}_{n}$ of radii $2E_n =\sqrt{2n+1}$ within the discrete (1 + 1)-dimensional phase plane. Moreover, the time evolution of these orbits sweep out world-sheet like geometrical entities $S^{1}_{n} \times \mathbb{R} \subset \mathbb{R}^2$ and therefore appear as closed string-like geometrical configurations. The physical interpretation for these discrete orbits in phase space as degenerate, string-like phase cells is shown in a mathematically rigorous way. The existence of these closed concircular orbits in the arena of discrete phase space quantum mechanics, known for the non-singular nature of lower order expansion $S^{\#}$ matrix terms, was known to exist but has not been fully explored until now. Finally, the discrete partial difference-differential Klein-Gordon equation is shown to be invariant under the continuous inhomogeneous orthogonal group $\mathcal{I} [O(3,1)]$ .