Narrow Results

The discrete phase space and continuous time representation of relativistic quantum mechanics are 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 ${\bar{S}}_n^1$, a circle of radius $\sqrt{2n+1}$ where $n\in \{0,1,\dots \}$. We interpret this two-dimensional (2D) Peano circle in our framework as a phase cell inside our 2D discrete phase plane. We postulate that a first quantized Planck oscillator, being very light, and small beyond current experimental detection, occupies this phase cell ${\bar{S}}_n^1$. The time evolution of this Peano circle sweeps out a 2D vertical cylinder analogous to the worldsheet of string theory. Extending this to 3D space, we introduce a $(2+2+2)$-dimensional phase space hyper-tori ${\bar{S}}_{n^1}^1\times {\bar{S}}_{n^2}^1\times {\bar{S}}_{n^3}^1$ 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 fiber 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 quanta of this system can cohabit with the tiny Planck oscillators inside the ${\bar{S}}_{n^1}^1\times {\bar{S}}_{n^2}^1\times {\bar{S}}_{n^3}^1$ 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.

Spanning a planar graph the way D. Hilbert's curve does has various image processing and industrial applications. Spanning a planar graph by two disjoint curves with fractal properties has even more scientific and industrial uses. For example, given two liquids and an active osmosis through membrane between them, we would like to both cool the liquids and to find a cost-effective structure for the osmosis to occur. Another equivalent problem is to expose two liquids to light that passes through a transparent slab as the osmosis between them occurs. Two disjoint curves can be the answer for the required structure. Differences of lengths between the curves can also be useful. A fractal structure is obvious in the lungs, where osmosis of oxygen is vital. Fractal structures are often found in organic osmotic processes in Nature. In this article, a method for spanning a planar graph by two disjoint curves will be presented.