We apply the Galilean covariant formulation of quantum dynamics to derive the phase-space representation of the Pauli–Schrödinger 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 space–time with Galilean geometry. Connections with the algebraic formalism of thermofield dynamics are also investigated.
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Vianna, J., Fernandes, M. & Santana, A. Galilean-Covariant Clifford Algebras in the Phase-Space Representation. Found Phys 35, 109–129 (2005). https://doi.org/10.1007/s10701-004-1926-5
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DOI: https://doi.org/10.1007/s10701-004-1926-5