A Bicycle Built for Two: The Galilean and U(1) Gauge Invariance of the Schr\"odinger Field

TL;DR

This paper demonstrates that the Maxwell equations emerge naturally from local U(1) gauge invariance in non-relativistic quantum mechanics, and shows that the coupled matter and gauge fields are Galilean invariant, emphasizing gauge symmetry's fundamental role.

Contribution

It proves that Maxwell equations are dictated by gauge symmetry regardless of relativistic or non-relativistic context and establishes Galilean invariance of the coupled fields.

Findings

Maxwell equations follow from local U(1) gauge invariance.

The coupled matter and gauge fields are Galilean invariant.

Gauge symmetry determines the structure of fundamental equations.

Abstract

This paper undertakes a study of the nature of the force associated with the local U (1) gauge symmetry of a non-relativistic quantum particle. To ensure invariance under local U (1) symmetry, a matter field must couple to a gauge field. We show that such a gauge field necessarily satisfies the Maxwell equations, whether the matter field coupled to it is relativistic or non-relativistic. This result suggests that the structure of the Maxwell equations is determined by gauge symmetry rather than the symmetry transformation properties of space-time. In order to assess the validity of this notion, we examine the transformation properties of the coupled matter and gauge fields under Galilean transformations. Our main technical result is the Galilean invariance of the full equations of motion of the U (1) gauge field.