Deformation quantization

Remarks

Clause 3 says $f * g - g * f = \hbar \set{f, g} + O(\hbar^2)$. Many authors use complex coefficients and ask for $i \hbar \set{f, g}$ instead. Example: 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 $$f * g = \sum_{n \ge 0} \frac{1}{n!} \left(\frac{\hbar}{2}\right)^n \sum_{j=0}^{n} (-1)^j \binom{n}{j} (\partial_x^{n-j} \partial_p^{j} f)(\partial_x^{j} \partial_p^{n-j} g).$$ Its term $B_1(f, g) = \frac{1}{2}\set{f, g}$ is antisymmetric, so clause 3 holds, and $x * p - p * x = \hbar$. Non-example: the undeformed product $f * g = fg$ satisfies clauses 1, 2 and 4, but fails clause 3 whenever the bracket is nonzero.