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.
Categories: math-ph math.MP