Negative probabilities: What they are and what they are for

An observation space mathcal S is a family of probability distributions langle Pi: i∈ I rangle sharing a common sample space Omega in a consistent way. A grounding for mathcal S is a signed probability distribution mathcal P on Omega yielding the correct marginal distribution Pi for every i. A wide variety of quantum scenarios can be formalized as observation spaces. We describe all groundings for a number of quantum observation spaces. Our main technical result is a rigorous proof that Wigner's distribution is the unique signed probability distribution yielding the correct marginal distributions for position and momentum and all their linear combinations.