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  "body": "\n## Summary\n\nA spectral triple $(A, H, D)$ consists of an algebra $A$ of operators on a Hilbert space $H$ and an unbounded self-adjoint operator $D$ on $H$.\nThe operator $D$ has compact resolvent, and its commutators with elements of $A$ are bounded.\nIt is the noncommutative version of a compact Riemannian spin manifold together with its [[Dirac operator|3DXGJTEQZH]].\nFrom a spectral triple one recovers a distance, a dimension and an integral, so it encodes metric geometry in operator language.\n\n## Explanations\n\n### L1 human\n\nOn a compact manifold, the Dirac operator is a first-order differential operator that knows the shape of the manifold.\nIts eigenvalues grow at a rate fixed by the dimension, and the size of the commutator $[D, f]$ measures how fast a function $f$ changes.\nA spectral triple keeps only these features: an algebra standing for the functions, a Hilbert space on which they act, and an operator $D$ standing for the Dirac operator.\nThe simplest example is the circle: functions on the circle act on $L^2(S^1)$, and $D = -i \\, d/d\\theta$ has eigenvalues the integers.\nThe algebra need not be commutative, which allows \"spaces\" with no points in the usual sense.\n",
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