THE STABILIZED POINCARE–HEISENBERG ALGEBRA: A CLIFFORD ALGEBRA VIEWPOINT

Abstract

The stabilized Poincare–Heisenberg algebra (SPHA) is a Lie algebra of quantum relativistic kinematics generated by fifteen generators. It is obtained from imposing stability conditions after combining the Lie algebras of quantum mechanics and relativity. In this paper, we show how the sixteen-dimensional real Clifford algebras Cℓ(1,3) and Cℓ(3,1) can both be used to generate the SPHA. The Clifford algebra path to the SPHA avoids the traditional stability considerations. It is conceptually easier and more straightforward to work with a Clifford algebra. The Clifford algebra path suggests that the next evolutionary step toward a theory of physics at the interface of GR and QM might be to depart from working in spacetime and instead to work in spacetime–momentum.