A Clifford Bundle Approach to the Wave Equation of a Spin 1/2 Fermion in the de Sitter Manifold

Abstract

In this paper we give a Clifford bundle motivated approach to the wave equation of a free spin 1/2 fermion in the de Sitter manifold, a brane with topology ${M=\mathrm {S0}(4,1)/\mathrm {S0}(3,1)}$ living in the bulk spacetime ${{\mathbb{R}^{4,1}}=(\mathring{M}=\mathbb{R}^5,\boldsymbol{\mathring{g}})}$ and equipped with a metric field ${\boldsymbol{g}:\boldsymbol{=}-\boldsymbol{i}^{\ast} \boldsymbol{\mathring{g}}}$ with ${\boldsymbol{i}:M\rightarrow \mathring{M}}$ being the inclusion map. To obtain the analog of Dirac equation in Minkowski spacetime in the structure ${\mathring{M}}$ we appropriately factorize the two Casimir invariants C ₁ and C ₂ of the Lie algebra of the de Sitter group using the constraint given in the linearization of C ₂ as input to linearize C ₁. In this way we obtain an equation that we called DHESS1, which in previous studies by other authors was simply postulated. Next we derive a wave equation (called DHESS2) for a free spin 1/2 fermion in the de Sitter manifold using a heuristic argument which is an obvious generalization of a heuristic argument (described in detail in Appendix D) permitting a derivation of the Dirac equation in Minkowski spacetime and which shows that such famous equation express nothing more than the fact that the momentum of a free particle is a constant vector field over timelike integral curves of a given velocity field. It is a remarkable fact that DHESS1 and DHESS2 coincide. One of the main ingredients in our paper is the use of the concept of Dirac-Hestenes spinor fields. Appendices B and C recall this concept and its relation with covariant Dirac spinor fields usually used by physicists.