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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $\\mathbb{R}_{0,n}$ be the real [[Clifford algebra|A38B1QFK20]] generated by $e_1, \\ldots, e_n$ with the relations $e_j^2 = -1$ and $e_j e_k = -e_k e_j$ for $j \\ne k$.\nThe *Dirac operator* on $\\R^n$ is $D = \\sum_{j=1}^{n} e_j \\dfrac{\\partial}{\\partial x_j}$, acting on differentiable functions from open subsets of $\\R^n$ to $\\mathbb{R}_{0,n}$.\n*Clifford analysis* is the study of functions from open subsets of $\\R^n$ to $\\mathbb{R}_{0,n}$ by means of $D$ and related [[Dirac operators|3DXGJTEQZH]].\nIts main objects are the [[monogenic functions|646PNPT1R6]], the solutions of $Df = 0$.\n\n## Remarks\n\nThe anticommutation relations give $D^2 = -\\Delta$, where $\\Delta$ is the [[Laplacian|0E0B6RA8K0]] of $\\R^n$: the mixed terms $e_j e_k \\partial_j \\partial_k + e_k e_j \\partial_k \\partial_j$ cancel.\nSo every monogenic function of class $C^2$ has harmonic components.\n\nExample: for $n = 1$, $\\mathbb{R}_{0,1} = \\R + \\R e_1$ with $e_1^2 = -1$ is a copy of $\\C$, and $D = e_1 \\partial_x$ only sees one real variable.\nTo recover complex analysis one uses instead the operator $\\partial_{x_0} + e_1 \\partial_{x_1}$ on $\\R^2$, which is twice the [[Wirtinger derivative|WMXDVF36ZT]] $\\partial_{\\bar z}$; many authors work with this operator $\\partial_{x_0} + \\sum_j e_j \\partial_{x_j}$ on $\\R^{n+1}$.\n\nNon-example: harmonic analysis of real-valued functions alone is not Clifford analysis, since it uses only $\\Delta$ and not its first-order factor $D$.\n",
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