Dirac operator

This article defines a property that can be evaluated for a differential operator on a differential manifold (viz a linear map from the space of differentiable functions to itself)

Definition

Given data

Let $M$ be a Riemannian manifold and $E$ a vector bundle over $M$. Let $\Delta$ denote a (the?) Laplacian on $E$.

Definition part

A Dirac operator on $E$ is a differential operator from the sheaf of section of $E$ to itself, whose square is $\Delta$.

In other words, a Dirac operator is a formal squareroot, or a half-iterate, of the Laplacian.