For operators which are functions of coordinates or momenta only, such as or , the Weyl transform agrees with the classical function corresponding to . This can not be the case in general, as is clear from the non-commutativity of the operator product. For example the Weyl transform of is .

The crucial question is: what does the operator product look like in phase space? This question was answered by Moyal in [Moy49] by introduction of the star product.

Let be the Weyl transforms of the operators . Then the Weyl transform of the operator is given by the Moyal star product :

(6.49) |

The -multiplication of c-number phase space functions is in complete isomorphism to Hilbert-space operator multiplication.

The von Neumann equation

(6.50) |

becomes the quantum Liouville (or Wigner) equation:

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