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

TL;DR

This paper introduces a Clifford bundle-based approach to derive wave equations for a spin 1/2 fermion in de Sitter space, resulting in two equivalent formulations that extend the Dirac equation to curved spacetime.

Contribution

It develops a novel Clifford bundle framework to formulate and derive wave equations for fermions in de Sitter space, connecting geometric algebra with quantum field theory in curved spacetime.

Findings

Derivation of the DHESS1 wave equation from Casimir invariants.

Derivation of the DHESS2 wave equation using heuristic arguments.

Equivalence of DHESS1 and DHESS2 formulations.

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 $\boldsymbol{g:=-i}^{\ast}\boldsymbol{\mathring{g}%}$ 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 \textbf{DHESS1,}which in previous studies by other authors was simply postulated.$.$Next we derive a wave equation (called \textbf{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 \textbf{DHESS1}and \textbf{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 usualy used by physicists.