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

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.