Higher spin symmetries of the free Schrödinger equation

Abstract

It is shown that the Schrödinger symmetry algebra of a free particle in d spatial dimensions can be embedded into a representation of the higher spin algebra. The latter spans an infinite dimensional algebra of higher-order symmetry generators of the free Schrödinger equation. An explicit representation of the maximal finite dimensional subalgebra of the higher spin algebra is given in terms of nonrelativistic generators. We show also how to convert Vasiliev's equations into an explicit nonrelativistic covariant form, such that they might apply to nonrelativistic systems. Our procedure reveals that the space of solutions of the Schrödinger equation can be regarded also as a supersymmetric module.