Space versus Time: Unimodular versus Non-Unimodular Projective Ring Geometries?

TL;DR

This paper explores the potential of finite projective geometries over rings, particularly unimodular and non-unimodular types, to model fundamental aspects of space and time in physics, using the projective line over the smallest ring of ternions.

Contribution

It introduces a novel geometric framework linking ring-based projective geometries to physical dimensions and the arrow of time, suggesting a new perspective on space-time structure.

Findings

Demonstrates the fundamental difference between unimodular and non-unimodular geometries.

Proposes a connection between these geometries and physical space-time dimensions.

Outlines how the model relates to the observed 3+1 space-time structure.

Abstract

Finite projective (lattice) geometries defined over rings instead of fields have recently been recognized to be of great importance for quantum information theory. We believe that there is much more potential hidden in these geometries to be unleashed for physics. There exist specific rings over which the projective spaces feature two principally distinct kinds of basic constituents (points and/or higher-rank linear subspaces), intricately interwoven with each other -- unimodular and non-unimodular. We conjecture that these two projective "degrees of freedom" can rudimentary be associated with spatial and temporal dimensions of physics, respectively. Our hypothesis is illustrated on the projective line over the smallest ring of ternions. Both the fundamental difference and intricate connection between time and space are demonstrated, and even the ring geometrical germs of the observed macroscopic dimensionality (3+1) of space-time and the arrow of time are outlined. Some other conceptual implications of this speculative model (like a hierarchical structure of physical systems) are also mentioned.