On a microscopic representation of spacetime

TL;DR

This paper explores a microscopic geometric model of spacetime using Lie algebra decompositions, linking algebraic structures to physical symmetries like electromagnetism and Lorentz transformations within a hyperbolic space framework.

Contribution

It introduces a novel geometric representation of spacetime based on Lie algebra reductions and projective geometry, connecting algebraic structures to physical symmetries.

Findings

Derives a 5D hyperbolic space with SO(5,1) symmetry.

Decomposes hyperbolic space into subspaces related to electromagnetic and Lorentz symmetries.

Links algebraic structures to a projective geometry over division algebra H.

Abstract

We start from a noncompact Lie algebra isomorphic to the Dirac algebra and relate this Lie algebra in a brief review to low energy hadron physics described by the compact group SU(4). This step permits an overall physical identification of the operator actions. Then we discuss the geometrical origin of this noncompact Lie algebra and 'reduce' the geometry in order to introduce in each of these steps coordinate definitions which can be related to an algebraic representation in terms of the spontaneous symmetry breakdown along the Lie algebra chain su*(4) $\longrightarrow$ usp(4) $\longrightarrow$ su(2)$\times$u(1). Standard techniques of Lie algebra decomposition(s) as well as the (physical) operator identification give rise to interesting physical aspects and lead to a rank-1 Riemannian space which provides an analytic representation and leads to a 5-dimensional hyperbolic space $H_{5}$ with SO(5,1) isometries. The action of the (compact) symplectic group decomposes this (globally) hyperbolic space into $H_{2}\oplus H_{3}$ with SO(2,1) and SO(3,1) isometries, respectively, which we relate to electromagnetic (dynamically broken SU(2) isospin) and Lorentz transformations. Last not least, we attribute this symmetry pattern to the algebraic representation of a projective geometry over the division algebra $H$ and subsequent coordinate restrictions.