Symplectic Field Theory of the Galilean Covariant Scalar and Spinor Representations

Abstract: Exploring the concept of the extended Galilei group $\mathcal{G}$, a representation for the symplectic quantum mechanics in the manifold of $\mathcal{G}$, written in the light-cone of a five-dimensional De Sitter space-time, is derived consistently with the method of the Wigner function. A Hilbert space is constructed endowed with a simplectic structure, studying unitary operators describing rotations and translations, whose generators satisfy the Lie algebra of $\mathcal {G}$. This representation gives rise to the Schr\"{o}dinger (Klein-Gordon-like) equation for the wave functions in phase-space, such that the dependent variables have the position and linear momentum contents. Wave functions are associated with the Wigner function through the Moyal product, such that the wave functions represent a quasi-amplitude of probability. We construct the Pauli-Schr\"{o}dinger (Dirac-like) equation in phase-space in its explicitly covariant form. Finally, we show the equivalence between the five dimensional formalism of phase-space with the usual formalism, proposing a solution that recover the non-covariant form of the Pauli-Schr\"{o}dinger equation in phase-space.