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

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