The Schrodinger Equation as a Gauge Theory

TL;DR

This paper reformulates the Schrödinger equation as a gauge theory, revealing deep connections with quantum hydrodynamics, topological terms, and boundary phenomena, and explores implications in various physical regimes.

Contribution

It introduces a gauge-theoretic reformulation of the Schrödinger equation, linking quantum mechanics, topology, and gauge fields in a unified framework.

Findings

Establishes local equivalence between Schrödinger equation and gauge theories.

Connects topological terms to physical phenomena like anyons and boundary states.

Relates acoustic memory and soft theorems through a dual two-form gauge description.

Abstract

In this paper, we reformulate 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 gauge formulation, while the global information is carried by the quantization of phase winding around zeros of the wavefunction. We then explore how this correspondence organizes structures on both sides of the duality. On the gauge side, BF couplings to additional one-forms describe electromagnetic coupling, Berry connections, spinor dynamics, projected non-abelian adiabatic connections, and intrinsic holonomy, while Chern--Simons term generate Hopf functionals, charge-flux attachment, and anyonic sectors. 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, large gauge transformations and the corresponding soft theorem, organizing them into an infrared triangle.