On the geometry of spin 1/2

Abstract

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:S1→S3→S2. 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, Open image in new window , to the Clifford algebra of Minkowski space-time,Cl(M4) withCl(M4) ≅ ℍ(2) for the Lorentz metric (+,−,−,−) andCl(M4) ≅ 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 algebraD16 ≅ ℂ(4).