The Schrodinger Equation as a Gauge Theory

Abstract

In this paper, we formulate the Schrodinger equation in gauge-theoretic terms. Starting from the Madelung representation, we rewrite the conserved probability current using gauge fields, namely a one-form gauge field in the $(2+1)$-dimensional theory and a two-form gauge field in the $(3+1)$-dimensional theory. This gives a local equivalence between the Schrodinger equation, quantum hydrodynamics and a non-relativistic gauge theory, while the global information is carried by the quantization condition of phase winding around zeros of the wavefunction. We then use this correspondence to study how topological deformations of gauge action and symmetry properties are represented in the wavefunction and fluid descriptions. On the gauge side, BF couplings to additional one-forms account for electromagnetic coupling, Berry connections, spinor dynamics, adiabatic non-abelian Berry connections, and intrinsic holonomy. Chern-Simons term admit, after eliminating the gauge field, a nonlocal realization in terms of wavefunction. This functional retain the topological content of the gauge description, but also contain dynamical contribution. In the presence of boundaries, the topological terms produce edge degrees of freedom and boundary charge algebras. Finally, in the nonlinear regime with a Bogoliubov sound mode, the dual two-form description relates acoustic memory to large gauge transformations and identifies the soft sector expected to complete the corresponding infrared triangle.