Higher symmetries of the Schrödinger operator in Newton–Cartan geometry

Abstract

We establish several relationships between the non-relativistic conformal symmetries of Newton–Cartan geometry and the Schrödinger equation. In particular we discuss the algebra 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 Schrödinger equation coupled to a Newtonian potential and generalised Coriolis force. We provide intrinsic Newton–Cartan definitions of Killing tensors and conformal Schrödinger–Killing tensors, and we discuss their respective links to conserved quantities and to the higher symmetries of the Schrödinger 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 cnc(3) on the Newton–Cartan spacetime.