Moment map and gauge geometric aspects of the Schroedinger and Pauli equations

In this paper we discuss various geometric aspects related to the Schroedinger and the Pauli equations. First we resume the Madelung - Bohm hydrodynamical approach to quantum mechanics and recall the hamiltonian structure of the Schroedinger equation. The probability current provides an equivariant moment map for the group G = sDiff(R^3) of volume preserving diffeomorphisms of R^3 (rapidly approaching the identity at infinity) and leads to a current algebra of Rasetti-Regge type. The moment map picture is then extended, mutatis mutandis, to the Pauli equation and to generalised Schrodinger equations of the Pauli-Thomas type. A gauge theoretical reinterpretation of all equations is obtained via the introduction of suitable Maurer-Cartan gauge fields and it is then related to Weyl geometric and pilot wave ideas. A general framework accommodating Aharonov-Bohm and Aharonov-Casher effects is presented within the gauge approach. Furthermore, a kind of holomorphic geometric quantization can be performed and yields natural "coherent state" representations of G. The relationship with the covariant phase space and density manifold approaches is then outlined. Comments on possible extensions to nonlinear Schroedinger equations, on Fisher-information theoretic aspects and on stochastic mechanics are finally made.