Fractional Galilean Symmetries

TL;DR

This paper introduces a new class of scale-invariant Galilean algebras using fractional derivatives, including models with dynamical indices relevant to Schrödinger and KPZ equations.

Contribution

It generalizes Galilean algebra representations to fractional derivatives, revealing novel scale-invariant symmetries with potential applications in physics.

Findings

Derived new Galilean algebra representations with fractional derivatives.

Identified algebras with dynamical index z=2 and z=3/2.

Connected the z=3/2 case to the KPZ equation.

Abstract

We generalize the differential representation of the operators of the Galilean algebras to include fractional derivatives. As a result a whole new class of scale invariant Galilean algebras are obtained. The first member of this class has dynamical index $z=2$ similar to the Schr\"odinger algebra. The second member of the class has dynamical index $z=3/2$, which happens to be the dynamical index Kardar-Parisi-Zhang equation.