- Universita' Cattolica del Sacro Cuore, Brescia
Monday, October 12, 2015
16:15 rinfresco; 16:30 inizio seminario
In this talk we discuss some geometric and hydrodynamical aspects related to the Schroedinger equation in a non-technical manner. First we resume the Madelung - Bohm hydrodynamical approach to quantum mechanics and show that the Schroedinger equation can be written in Hamiltonian form by means of a Poisson bracket of hydrodynamical type (a' la Arnol'd). The ensuing main geometric feature is that the probability current (indeed, its "vorticity") provides an equivariant moment map for the group of measure preserving diffeomorphisms of R^3 (rapidly approaching the identity at infinity) [roughly speaking, a far reaching generalization of the mathematical apparatus describing the dynamics of the rigid body is involved]. This geometrical picture is compared with the one obtained from geometric quantum mechanics, wherein the Schroedinger equation is again portrayed as a Hamiltonian equation on a projective Hilbert space and where hydrodynamical features also naturally crop up. Time permitting, a gauge theoretical reinterpretation of the Schroedinger "classical" Hamiltonian will also be outlined via the introduction of a suitable Maurer-Cartan gauge field, allowing for a simple treatment of the Aharonov-Bohm effect.