Carroll-Schr\"odinger Equation

The Poincar\'e symmetry can be contracted in two ways to yield the Galilei symmetry and the Carroll symmetry. The Schr\"odinger equation exhibits the Galilei symmetry and is a fundamental equation in Galilean quantum mechanics. However, the question remains: what is the quantum equation that corresponds to the Carroll symmetry? In this paper, we derive a novel equation, called the Carroll-Schr\"odinger equation, which describes the quantum dynamics in the Carrollian framework. We also introduce the Carroll-Schr\"odinger algebra, which is a conformal extension of the centrally extended Carroll algebra, and show that it is the symmetry algebra of the Carroll-Schr\"odinger equation in two dimensions. We generalize our results to arbitrary dimensions and discuss some possible applications and extensions of our work.