Phase space formulation

Summary

The phase space formulation of quantum mechanics works with functions on phase space instead of operators on a Hilbert space. A state is described by its Wigner function, an observable by its Weyl symbol, and the product of operators by the Moyal product. It is equivalent to the Hilbert space formulation. It makes the comparison with classical statistical mechanics direct, since the Moyal bracket tends to the Poisson bracket as $\hbar \to 0$. It is the model case of deformation quantization.