The Stabilized Poincare-Heisenberg algebra: a Clifford algebra viewpoint

TL;DR

This paper demonstrates how the stabilized Poincare-Heisenberg algebra, crucial for quantum relativistic kinematics, can be derived from the Clifford algebra CL(1,3), offering a more straightforward approach that may inform future theories at the quantum-gravity interface.

Contribution

It introduces a Clifford algebra framework to generate the SPHA, simplifying stability considerations and suggesting a shift to space-time-momentum in quantum gravity research.

Findings

SPHA can be derived from CL(1,3) algebra.

Clifford algebra approach simplifies stability analysis.

Proposes a shift to space-time-momentum framework.

Abstract

The stabilized Poincare-Heisenberg algebra (SPHA) is the Lie algebra of quantum relativistic kinematics generated by fifteen generators. It is obtained from imposing stability conditions after attempting to combine the Lie algebras of quantum mechanics and relativity which by themselves are stable, however not when combined. In this paper we show how the sixteen dimensional Clifford algebra CL(1,3) can be used to generate the SPHA. The Clifford algebra path to the SPHA avoids the traditional stability considerations, relying instead on the fact that CL(1,3) is a semi-simple algebra and therefore stable. It is therefore conceptually easier and more straightforward to work with a Clifford algebra. The Clifford algebra path suggests the next evolutionary step toward a theory of physics at the interface of GR and QM might be to depart from working in space-time and instead to work in space-time-momentum.