Higher-spin symmetries of the free Schrodinger equation

TL;DR

This paper demonstrates that the symmetry algebra of the free Schrödinger equation can be embedded into a higher-spin algebra, revealing an infinite-dimensional structure and connecting to non-relativistic covariant formulations.

Contribution

It provides an explicit embedding of Schrödinger symmetries into higher-spin algebras and reformulates Vasiliev's equations in a non-relativistic covariant manner.

Findings

Higher-spin algebra encompasses Schrödinger symmetry algebra.

Explicit representation of finite-dimensional subalgebra provided.

Vasiliev's equations adapted for non-relativistic systems.

Abstract

It is shown that the Schrodinger 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 Schrodinger equation. An explicit representation of the maximal finite dimensional subalgebra of the higher spin algebra is given in terms of non-relativistic generators. We show also how to convert Vasiliev's equations into an explicit non-relativistic covariant form, such that they might apply to non-relativistic systems. Our procedure reveals that the space of solutions of the Schrodinger equation can be regarded also as a supersymmetric module.