Dirac operator

Remarks

Equivalently, the principal symbol of $D^2$ is $\abs{\xi}^2_g$ times the identity, so $D^2$ is a generalised Laplacian. It follows that $D$ is an elliptic operator.

Example: on $\R^n$ with the flat metric, let $\gamma_1, \ldots, \gamma_n$ be complex $N \times N$ matrices with $\gamma_j\gamma_k + \gamma_k\gamma_j = -2\delta_{jk} I$, and let $D = \sum_j \gamma_j \partial_j$ act on $\C^N$-valued functions. Then $D^2 = \sum_{j,k}\gamma_j\gamma_k\partial_j\partial_k = \frac{1}{2}\sum_{j,k}(\gamma_j\gamma_k + \gamma_k\gamma_j)\partial_j\partial_k = -\sum_j \partial_j^2$, so $D$ is a Dirac operator. For $n = 1$ and $\gamma_1 = -i$, this is $D = -i\, d/dx$.

Non-example: on $\R$, the operator $D = d/dx$ has $D^2 = +d^2/dx^2$, whose second-order term has the wrong sign.

On a spin manifold, the Atiyah–Singer Dirac operator acts on the spinor bundle through Clifford multiplication and the Levi-Civita connection.