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  "body": "\n## Definition\n\nLet $M$ be a [[Poisson manifold|A2KQ49Y4NF]] with [[Poisson bracket|2S0MXDT4YH]] $\\set{\\cdot, \\cdot}$, and let $C^\\infty(M)$ be the algebra of real [[smooth functions|0V33BNKZ9S]] on $M$.\nLet $C^\\infty(M)[\\![\\hbar]\\!]$ be the space of [[formal power series|033WGN00BF]] in $\\hbar$ with coefficients in $C^\\infty(M)$.\nA *star product* on $M$ is an $\\R[\\![\\hbar]\\!]$-bilinear associative product $*$ on $C^\\infty(M)[\\![\\hbar]\\!]$ such that:\n\n1. $f * g = \\sum_{n \\ge 0} \\hbar^n B_n(f, g)$ for $f, g \\in C^\\infty(M)$, where each $B_n$ is a bidifferential operator;\n2. $B_0(f, g) = fg$;\n3. $B_1(f, g) - B_1(g, f) = \\set{f, g}$;\n4. $1 * f = f * 1 = f$.\n\nA *deformation quantization* of $M$ is a choice of star product on $M$.\nTwo star products are equivalent when they are equivalent as [[formal deformations|QP92G4ZGJE]] by a series of differential operators.\n\n## Remarks\n\nClause 3 says $f * g - g * f = \\hbar \\set{f, g} + O(\\hbar^2)$.\nMany authors use complex coefficients and ask for $i \\hbar \\set{f, g}$ instead.\nExample: the Moyal product on $\\R^2$ with coordinates $x, p$ and $\\set{f, g} = \\partial_x f \\, \\partial_p g - \\partial_p f \\, \\partial_x g$ is\n$$f * g = \\sum_{n \\ge 0} \\frac{1}{n!} \\left(\\frac{\\hbar}{2}\\right)^n \\sum_{j=0}^{n} (-1)^j \\binom{n}{j} (\\partial_x^{n-j} \\partial_p^{j} f)(\\partial_x^{j} \\partial_p^{n-j} g).$$\nIts term $B_1(f, g) = \\frac{1}{2}\\set{f, g}$ is antisymmetric, so clause 3 holds, and $x * p - p * x = \\hbar$.\nNon-example: the undeformed product $f * g = fg$ satisfies clauses 1, 2 and 4, but fails clause 3 whenever the bracket is nonzero.\n",
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