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  "body": "\n## Summary\n\nDeformation quantization equips the smooth functions on a [[Poisson manifold|A2KQ49Y4NF]] with a noncommutative star product.\nThe star product is a formal power series in a parameter $\\hbar$.\nThe commutator of the star product reproduces the [[Poisson bracket|2S0MXDT4YH]] at first order in $\\hbar$.\nThe Moyal product on $\\R^{2n}$ is the first example.\n[[Kontsevich's formality theorem|XXA9D45E0N]] shows that every Poisson manifold admits a star product.\n\n## Explanations\n\n### L1 human\n\nIn classical mechanics, observables are functions on phase space, and they commute: $fg = gf$.\nIn quantum mechanics, observables are operators, and position and momentum do not commute.\nDeformation quantization keeps the functions but changes their product, adding correction terms in powers of Planck's constant $\\hbar$.\nAt $\\hbar = 0$ the old commutative product comes back.\nFor position $x$ and momentum $p$ the new product satisfies $x * p - p * x = \\hbar$, the formal version of the canonical commutation relation.\n",
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