U(1)×SU(2)-gauge invariance of non-relativistic quantum mechanics,and generalized Hall effects

We show that the non-relativistic quantum mechanics of particles with spin coupled to an electromagnetic field has a naturalU(1)×SU(2) gauge invariance. Ward identities reflecting this gauge invariance combined with an assumption of incompressibility of a system of such particles in an appropriate external field and for suitable values of the particle density permit us to determine the form of the effective action of the system as a functional of small fluctuations in the electromagnetic field, in the large-distance-, adiabatic limit. In this limit, the action is found to have a universal form. We present explicit results for two-dimensional, incompressible electron fluids and apply them to derive the equations of linear response theory, describing a variety of generalized Hall effects. Sum rules for the Hall conductivities, magnetic susceptibilities and other quantities of physical interest are found.