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  "body": "\n## Definition\n\nLet $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$ with a [[spin structure|RSPZ9TCFQ8]].\nLet $S \\to M$ be the complex [[spinor bundle|PYYQQKAZ0C]], with its Clifford multiplication $T M \\otimes S \\to S$, $v \\otimes \\psi \\mapsto v \\cdot \\psi$, which satisfies $v \\cdot v \\cdot \\psi = -g(v, v) \\, \\psi$, as in the [[Clifford algebra|A38B1QFK20]].\nLet $\\nabla^S$ be the spin connection on $S$, the connection induced by the [[Levi-Civita connection|CAZYGE1T4E]].\nThe *Dirac operator* of $M$ is the operator $D \\colon \\Gamma(S) \\to \\Gamma(S)$ given by\n$$D \\psi = \\sum_{i=1}^{n} e_i \\cdot \\nabla^S_{e_i} \\psi,$$\nwhere $e_1, \\ldots, e_n$ is any local orthonormal frame of $TM$.\n\n## Remarks\n\nThe right side does not depend on the frame, because it is the contraction of $\\nabla^S \\psi \\in \\Gamma(T^*M \\otimes S)$ with Clifford multiplication, after identifying $T^*M$ with $TM$ by the metric.\nThe principal symbol of $D$ at $\\xi$ is Clifford multiplication by the vector dual to $\\xi$, up to a factor $i$ in some conventions, and its square is $-\\abs{\\xi}^2$.\nSo $D$ is an [[elliptic operator|MT9P2X85V8]] and a [[Dirac operator|3DXGJTEQZH]] in the general sense.\nIt is formally self-adjoint for the Hermitian metric on $S$.\nIn even dimension $S = S^+ \\oplus S^-$, and $D$ maps $\\Gamma(S^\\pm)$ to $\\Gamma(S^\\mp)$.\n\nExample: on $\\R^n$ with the flat metric and the trivial spin structure, $S = \\R^n \\times \\C^N$, the spin connection is the ordinary derivative, and $D = \\sum_j \\gamma_j \\partial_j$, where $\\gamma_j$ are complex $N \\times N$ matrices with $\\gamma_j \\gamma_k + \\gamma_k \\gamma_j = -2 \\delta_{jk} I$.\nFor $n = 1$ one can take $N = 1$ and $\\gamma_1 = i$, which gives $D = i \\, d/dx$ and $D^2 = -d^2/dx^2$.\n\nNon-example: the complex projective plane $\\C P^2$ with the Fubini–Study metric has no spin structure, since its second Stiefel–Whitney class is nonzero.\nSo it has no spinor bundle, and the definition does not apply; one uses a spin$^c$ structure instead.\n\n## Explanations\n\n### L1 human\n\nThe Dirac operator of a spin manifold differentiates a spinor field in each direction of an orthonormal frame and multiplies by the matching Clifford element, then sums.\n",
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