Carroll-Schr\"odinger Equation

The Poincar\'e symmetry can be contracted in two ways to yield the Galilei symmetry and the Carroll symmetry. The well-known 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 in two dimensions, called the ``Carroll-Schr\"odinger equation'', which describes the quantum dynamics in the Carrollian framework. We also construct the so-called ``Carroll-Schr\"odinger algebra'' in two dimensions, which is a conformal extension of the centrally extended Carroll algebra with a dynamical exponent of $z=1/2$. We demonstrate that this algebra is the symmetry algebra of the Carroll-Schr\"odinger field theory. Moreover, we apply the method of canonical quantization to the theory and utilize it to compute the transition amplitude. Finally, we discuss higher dimensions and identify the so-called ``generalized Carroll-Schr\"odinger equation''.