On the Geometry of Spacetime I: baby steps in quantum ring theory

TL;DR

This paper explores a novel algebraic framework for spacetime geometry using asymmetric ring theory, aiming to unify causal and spin structures and introduce an asymmetric quantization approach.

Contribution

It introduces a new geometric model employing two-sided vector spaces and skewed K-algebras, proposing a paradigm shift from traditional fiber bundle methods in spacetime theory.

Findings

Proposes a two-sided vector space model for spacetime

Introduces asymmetric scalar multiplication via skewed K-algebras

Suggests a novel approach to asymmetric quantization of spacetime

Abstract

Vierbeins provide a bridge between the curved space of general relativity and the flat tangent space of special relativity. Both spaces should be causal and spin. We posit intertwining the two symmetries of spacetime bundles asymmetrically; disentangling the non-trivial Id between the base, curved space as a locally ringed space and its Zariski (co-)tangent space. This involves the introduction of a "two-sided vector space" as a section of the smooth, stratified diffeomorphism bundle of spacetime. A change of paradigm from the fiber bundle approach ensues where the bundle space takes an active role and the group actions are implemented through asymmetric "scalar multiplication" by elements of a skewed K-algebra on a free K-bimodule. Accordingly, the left action is augmented from that on the right algebraically by a left-sided algebra automorphism via a left alpha-derivation as a non-central Ore extension of a Weyl algebra. Curiously, summoning the left $\alpha$-derivation in the context of spacetime symmetries may constitute the key to an asymmetric quantization of the theory. Furthermore, it is conjectured that causal and spin structure may be endowed upon the spacetime itself, independently of the tangent space structure.