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

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.