Quantum mechanics in the gaussian wave-packet phase space representation III: From phase space probability functions to wave-functions

Using a mapping formalism, the Wave-Packet Phase Space Representation, we show that for phase space probability distributions that do not violate the Uncertainty Principle one can associate a density matrix, (x, x'), when some conditions are fulfilled. Furthermore if condition Tr ² = 1 is satisfied it becomes possible to find a wave function associated to the probability distribution.