Galilean-covariant Clifford algebras in the phase space representation

We apply the Galilean covariant formulation of quantum dynamics to derive the phase-space representation of the Pauli-Schroedinger equation for the density matrix of spin-1/2 particles in the presence of an electromagnetic field. The Liouville operator for the particle with spin follows from using the Wigner-Moyal transformation and a suitable Clifford algebra constructed on the phase space of a (4+1)-dimensional spacetime with Galilean geometry. Connections with the algebraic formalism of thermofield dynamics are also investigated. (author)