A quantum model of space-time-matter

TL;DR

This paper proposes a quantum model of space-time-matter based on the Lie algebra of the Lorentz group SO(5,1), replacing traditional commutation relations, and explores its implications and limits.

Contribution

It introduces a novel quantum framework where space-time and matter observables are generators of SO(5,1), unifying them under a new algebraic structure.

Findings

Recover Poincare algebra in specific limits

Describe non-commuting position and momentum observables

Derive equations for spinor representations of SO(5,1)

Abstract

We study a quantum mechanics with the usual postulates but in which the Heisenberg algebra of canonical commutation relations and the Poincare algebra are replaced by the Lie algebra of the homogeneous Lorentz group SO(5,1). It arises from the hypothesis that the above group is the fundamental group of invariance for the laws of physics. The observables of the theory like position, time, momentum, energy, angular momentum and others are the generators of the algebra of the group. Neither position and time observables commute between them, nor momentum and energy observables. The algebra of Poincare quantum mechanics is recovered in the limit in which two parameters, that we physically interpret as the Hubble constant and the Planck mass, are taken to zero and infinite respectively. We consider the equations that are satisfied by the spinor representation of the group.