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  "body": "\n## Definition\n\nLet $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] and let $E$ be a complex vector bundle over $M$.\nLet $D$ be a first-order linear differential operator acting on smooth sections of $E$.\nIn a local chart $x = (x_1, \\ldots, x_n)$, let $(g^{jk}(x))$ be the inverse of the matrix of the metric $g$.\nThe operator $D$ is a *Dirac operator* if, in every local chart and every local trivialisation of $E$,\n$$D^2 = -\\sum_{j,k} g^{jk}(x)\\frac{\\partial^2}{\\partial x_j \\partial x_k} + (\\text{terms of order at most } 1).$$\n\n## Remarks\n\nEquivalently, the principal symbol of $D^2$ is $\\abs{\\xi}^2_g$ times the identity, so $D^2$ is a generalised [[Laplacian|0E0B6RA8K0]].\nIt follows that $D$ is an [[elliptic operator|XY8GEQSWPR]].\n\nExample: 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.\nThen $D^2 = \\sum_{j,k}\\gamma_j\\gamma_k\\partial_j\\partial_k = \\frac{1}{2}\\sum_{j,k}(\\gamma_j\\gamma_k + \\gamma_k\\gamma_j)\\partial_j\\partial_k = -\\sum_j \\partial_j^2$, so $D$ is a Dirac operator.\nFor $n = 1$ and $\\gamma_1 = -i$, this is $D = -i\\, d/dx$.\n\nNon-example: on $\\R$, the operator $D = d/dx$ has $D^2 = +d^2/dx^2$, whose second-order term has the wrong sign.\n\nOn a spin manifold, the Atiyah–Singer Dirac operator acts on the [[spinor bundle|PYYQQKAZ0C]] through Clifford multiplication and the Levi-Civita connection.\n",
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