Moyal product

Remarks

The first terms are $f \star g = fg + \frac{i\hbar}{2}\{f, g\} + O(\hbar^2)$, where $\{\cdot, \cdot\}$ is the Poisson bracket. The product is associative, and $f \star g - g \star f = i\hbar \{f, g\} + O(\hbar^3)$. If $f$ or $g$ is a polynomial, the series stops after finitely many terms, and then $\hbar$ can be given a numerical value. For Schwartz functions and a positive number $\hbar$, an integral formula defines a product $f \star g$ with $\mathrm{Op}^w(f \star g) = \mathrm{Op}^w(f)\, \mathrm{Op}^w(g)$. Here $\mathrm{Op}^w$ is the Weyl quantization, and the series above is the asymptotic expansion of this product as $\hbar \to 0$.

Example: for $n = 1$, $x \star p = xp + \frac{i\hbar}{2}$ and $p \star x = xp - \frac{i\hbar}{2}$, since all derivatives of order $2$ vanish. So $x \star p - p \star x = i\hbar$, which matches $[X, P] = i\hbar$.

Non-example: the ordinary pointwise product $fg$ is the term of order $0$ only; it is commutative, so it does not reproduce the commutator $x \star p - p \star x = i\hbar$.

The Moyal product is a star product in the sense of deformation quantization for the standard Poisson structure on $\R^{2n}$.