Symmetries of the Free Schr\"odinger Equation in the Non-Commutative Plane

TL;DR

This paper explores the symmetries of the free Schr"odinger equation in a non-commutative plane, revealing an infinite-dimensional algebra that could aid in developing non-relativistic higher spin theories.

Contribution

It identifies the full symmetry algebra of the non-commutative Schr"odinger equation and discusses its quantization, extending understanding of non-commutative quantum symmetries.

Findings

Symmetry transformations form an infinite-dimensional Weyl algebra.

The Weyl algebra arises from a two-dimensional Heisenberg algebra.

Finite-dimensional Schr"odinger subalgebra includes dilatation and expansion.

Abstract

We study all the symmetries of the free Schr\"odinger equation in the non-commutative plane. These symmetry transformations form an infinite-dimensional Weyl algebra that appears naturally from a two-dimensional Heisenberg algebra generated by Galilean boosts and momenta. These infinite high symmetries could be useful for constructing non-relativistic interacting higher spin theories. A finite-dimensional subalgebra is given by the Schr\"odinger algebra which, besides the Galilei generators, contains also the dilatation and the expansion. We consider the quantization of the symmetry generators in both the reduced and extended phase spaces, and discuss the relation between both approaches.