Exact solutions of n-level systems and gauge theories

Abstract

We find a relationship between unitary transformations of the dynamics of quantum systems with time-dependent Hamiltonians and gauge theories. In particular, we show that the nonrelativistic dynamics of spin-$\frac12$ particles in a magnetic field $B^i (t)$ can be formulated in a natural way as an SU(2) gauge theory, with the magnetic field $B^i(t)$ playing the role of the gauge potential A^i. The present approach can also be applied to systems of n levels with time-dependent potentials, U(n) being the gauge group. This geometric interpretation provides a powerful method to find exact solutions of the Schrödinger equation. The root of the present approach rests in the Hermiticity property of the Hamiltonian operators involved. In addition, the relationship with true gauge symmetries of n-level quantum systems is discussed.