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  "body": "\n## Summary\n\nA Dirac operator is a first-order differential operator $D$ whose square $D^2$ is a Laplace-type operator.\nIt is a formal square root of the [[Laplacian|0E0B6RA8K0]], made possible by coefficients in a [[Clifford algebra|A38B1QFK20]].\nDirac introduced such an operator to write a first-order relativistic equation for the electron, the Dirac equation.\nIn geometry, Dirac operators on [[spinor bundles|PYYQQKAZ0C]] are elliptic, and their indices are computed by the Atiyah–Singer index theorem.\n\n## Explanations\n\n### L1 human\n\nThe Laplacian $\\partial_1^2 + \\cdots + \\partial_n^2$ is a sum of squares, and one may ask for a first-order operator whose square is it.\nWith ordinary numbers this is impossible beyond one variable, because the cross terms $2\\partial_j\\partial_k$ do not cancel.\nDirac's idea was to use matrices $\\gamma_j$ whose products anticommute, so the cross terms cancel.\nIn one variable it is easy: $D = -i\\, d/dx$ satisfies $D^2 = -d^2/dx^2$.\nIn the plane, the Cauchy–Riemann operators of complex analysis assemble into a Dirac operator.\n",
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