Quantum mechanics in phase space: I. Unicity of the Wigner distribution function

The joint distribution function in phase space is related to the density matrix by an integral transformation which depends on the rule of correspondence used. All the requirements which can be restrictive for the kernel function defining the transformation are studied. It is shown that the conditions of Galilei invariance, unitarity, reality and normalization lead to the Wigner kernel function in a unique way. Galilei invariance, the requirement that the free particle behaves classically, and the conditions to obtain the correct mixed distributions also lead to the same result.