Discrete phase space and continuous time relativistic quantum mechanics II: Peano circles, hyper-tori phase cells, and fibre bundles

Abstract

The discrete phase space and continuous time representation of relativistic quantum mechanics is further investigated here as a continuation of paper I [1]. The main mathematical construct used here will be that of an area-filling Peano curve. We show that the limit of a sequence of a class of Peano curves is a Peano circle denoted as $\bar{S}^{1}_{n}$, a circle of radius $\sqrt{2n+1}$ where $n \in \{0,1,\cdots\}$. We interpret this two-dimensional Peano circle in our framework as a phase cell inside our two-dimensional discrete phase plane. We postulate that a first quantized Planck oscillator, being very light, and small beyond current experimental detection, occupies this phase cell $\bar{S}^{1}_{n}$. The time evolution of this Peano circle sweeps out a two-dimensional vertical cylinder analogous to the world-sheet of string theory. Extending this to three dimensional space, we introduce a $(2+2+2)$-dimensional phase space hyper-tori $\bar{S}^{1}_{n^1} \times \bar{S}^{1}_{n^2} \times \bar{S}^{1}_{n^3}$ as the appropriate phase cell in the physical dimensional discrete phase space. A geometric interpretation of this structure in state space is given in terms of product fibre bundles. We also study free scalar Bosons in the background $[(2+2+2)+1]$-dimensional discrete phase space and continuous time state space using the relativistic partial difference-differential Klein-Gordon equation. The second quantized field quantas of this system can cohabit with the tiny Planck oscillators inside the $\bar{S}^{1}_{n^1} \times \bar{S}^{1}_{n^2} \times \bar{S}^{1}_{n^3}$ phase cells for eternity. Finally, a generalized free second quantized Klein-Gordon equation in a higher $[(2+2+2)N+1]$-dimensional discrete state space is explored. The resulting discrete phase space dimension is compared to the significant spatial dimensions of some of the popular models of string theory.