Higher Symmetries of the Schr\"odinger Operator in Newton-Cartan Geometry

TL;DR

This paper explores the symmetries of the Schr"odinger equation within Newton-Cartan geometry, establishing links between conformal symmetries, conserved quantities, and twistor theory, and generalizing symmetry algebras to curved spacetimes.

Contribution

It introduces intrinsic Newton-Cartan definitions of Killing tensors and conformal Schr"odinger-Killing tensors, and connects these to higher symmetries and twistor space structures.

Findings

The symmetry algebra of the covariant Schr"odinger equation is generated by curved generalizations of conformal vector fields.

Intrinsic definitions of Killing and conformal Schr"odinger-Killing tensors are provided within Newton-Cartan geometry.

The infinite-dimensional algebra of holomorphic vector fields on twistor space corresponds to a Newton-Cartan symmetry algebra.

Abstract

We establish several relationships between the non-relativistic conformal symmetries of Newton-Cartan geometry and the Schr\"odinger equation. In particular we discuss the algebra $\mathfrak{sch}(d)$ of vector fields conformally-preserving a flat Newton-Cartan spacetime, and we prove that its curved generalisation generates the symmetry group of the covariant Schr\"odinger equation coupled to a Newtonian potential and generalised Coriolis force. We provide intrinsic Newton-Cartan definitions of Killing tensors and conformal Schr\"odinger-Killing tensors, and we discuss their respective links to conserved quantities and to the higher symmetries of the Schr\"odinger equation. Finally we consider the role of conformal symmetries in Newtonian twistor theory, where the infinite-dimensional algebra of holomorphic vector fields on twistor space corresponds to the symmetry algebra $\mathfrak{cnc}(3)$ on the Newton-Cartan spacetime.