{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $U \\subseteq \\R^d$ be open and let $\\pi^{ij} = -\\pi^{ji}$, $1 \\le i, j \\le d$, be smooth functions on $U$.\nAssume that $\\{f, g\\} = \\sum_{i,j} \\pi^{ij} \\partial_i f \\, \\partial_j g$ satisfies the Jacobi identity.\nSo $U$ is a [[Poisson manifold|A2KQ49Y4NF]] with Poisson bivector $\\pi$.\nFor $n \\ge 0$, let $G_n$ be the set of admissible graphs.\nSuch a graph has aerial vertices $1, \\ldots, n$, ground vertices $L$ and $R$, and $2n$ directed edges.\nEach aerial vertex is the source of exactly two edges, labeled first and second.\nNo edge goes from a vertex to itself, and no two edges have the same source and target.\nTo $\\Gamma \\in G_n$ attach:\n\n1. the bidifferential operator $B_\\Gamma$. Place a copy of $(\\pi^{ij})$ at each aerial vertex, $f$ at $L$ and $g$ at $R$. Let each edge act as a partial derivative $\\partial_i$ on the object at its target, and sum over all indices;\n2. the weight $w_\\Gamma$ defined below.\n\nLet $C_n$ be the space of configurations of $n$ distinct points in the upper half-plane $\\mathbb{H}$, and put $L = 0$ and $R = 1$.\nLet $e_k^1$ and $e_k^2$ be the edges leaving vertex $k$.\nFor an edge $e$, let $\\phi_e$ be the angle at the source of $e$, in the hyperbolic metric of $\\mathbb{H}$.\nIt goes from the geodesic to $\\infty$ to the geodesic to the target of $e$.\nThen\n$$w_\\Gamma = \\frac{1}{n! \\, (2\\pi)^{2n}} \\int_{C_n} \\bigwedge_{k=1}^n \\left( d\\phi_{e_k^1} \\wedge d\\phi_{e_k^2} \\right).$$\n\nLet $C^\\infty(U)[\\![\\hbar]\\!]$ be the space of [[formal power series|033WGN00BF]] in $\\hbar$ with coefficients in $C^\\infty(U)$.\nExtend the following product $\\hbar$-bilinearly to it:\n$$f \\star g = \\sum_{n \\ge 0} \\hbar^n \\sum_{\\Gamma \\in G_n} w_\\Gamma \\, B_\\Gamma(f, g).$$\nThen $\\star$ is associative, and $f \\star g = fg + \\hbar \\{f, g\\} + O(\\hbar^2)$.\nSo $\\star$ is a deformation quantization of $(U, \\pi)$.\n\n## Remarks\n\nKontsevich obtained the formula from his formality theorem: there is an $L_\\infty$ quasi-isomorphism from polyvector fields to polydifferential operators on $\\R^d$.\nThe formula is its explicit form, and associativity follows from Stokes' theorem on compactified configuration spaces, where the boundary terms either cancel or vanish by the Jacobi identity $[\\pi, \\pi] = 0$.\nFor a constant Poisson bivector only graphs without edges between aerial vertices contribute, and the formula gives the Moyal product.\nWith the formality theorem and a globalization argument, Kontsevich showed that every Poisson manifold has a deformation quantization.\nHe also showed that equivalence classes of star products correspond bijectively to equivalence classes of formal Poisson structures.\n\n## Explanations\n\n### L1 human\n\nIn quantum mechanics observables do not commute, while in classical mechanics they do.\nA deformation quantization replaces the ordinary product of functions by a new associative product.\nThe new product depends on a small parameter $\\hbar$, Planck's constant, and its first correction is the Poisson bracket.\nKontsevich's formula writes such a product explicitly for any Poisson structure on a piece of $\\R^d$, as a sum over pictures of graphs with numerical weights given by integrals.\n\n### L2 human\n\nBefore Kontsevich, star products were known for symplectic manifolds, by work of De Wilde and Lecomte and of Fedosov, but not for general Poisson manifolds.\nThe weights $w_\\Gamma$ are hard to compute in general.\nCattaneo and Felder later explained the formula as a perturbative expansion of a topological field theory, the Poisson sigma model, on a disc.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "about": [
      "A2KQ49Y4NF"
    ],
    "kind": "statement",
    "msc": [
      "53D55",
      "53D17"
    ],
    "name": "Kontsevich quantization formula",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q6430406",
      "wikipedia": "Kontsevich quantization formula"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Kontsevich_quantization_formula"
      }
    ],
    "tag": "JZEK70CH3G",
    "uses": [
      "A2KQ49Y4NF",
      "033WGN00BF"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/JZEK70CH3G/",
  "links": [
    {
      "field": "about",
      "target": "A2KQ49Y4NF"
    },
    {
      "field": "uses",
      "target": "A2KQ49Y4NF"
    },
    {
      "field": "uses",
      "target": "033WGN00BF"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/JZEK70CH3G.md",
  "msc": [
    "53D55",
    "53D17"
  ],
  "name": "Kontsevich quantization formula",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/JZE/JZEK70CH3G.md",
  "status": null,
  "tag": "JZEK70CH3G",
  "uri": "m2://JZEK70CH3G",
  "url": "https://math2.tamnd.com/wiki/kontsevich-quantization-formula/",
  "writing": {
    "tag": "JZEK70CH3G",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}