All the necessary elements for the description of the spin degree of freedom of a non relativistic spin 1/2 particle (e.g. the electron) are contained in the geometry of the complex Hopf bundleh:S ¹→S ³→S ². This bundle has also the necessary information for the geometrical construction of the symmetry group of the standard model,U(1)×SU(2)×SU(3). The passage from the Schroedinger-Pauli equation to the relativistic Dirac equation is equivalent, geometrically, to the passage from the Clifford algebra of ordinary Euclidean space, , to the Clifford algebra of Minkowski space-time,Cl(M ⁴) withCl(M ⁴) ≅ ℍ(2) for the Lorentz metric (+,−,−,−) andCl(M ⁴) ≅ ℝ(4) for the Lorentz metric (−,+,+,+), where ℝ, ℂ and ℍ are the real, complex and quaternionic numbers, respectively. The physical equivalence of the two metrics leads to the complexification of the corresponding Clifford algebras, giving the physical Dirac algebraD ¹⁶ ≅ ℂ(4).