Negative probabilities: What they are and what they are for

09/22/2020 ∙ by Andreas Blass, et al. ∙ 0

In quantum mechanics, the probability distributions of position and momentum of a particle are normally not the marginals of a joint distribution, that is unless – as shown by Wigner in 1932 – negative probabilities are allowed. Since then, much theoretical work has been done to study negative probabilities; most of this work is about what those probabilities are. We suggest shifting the emphasis to what negative probabilities are for. In this connection, we introduce the framework of observation spaces. An observation space is a family 𝒮 = ⟨𝒫_i: i∈ I⟩ of probability distributions sharing a common sample space in a consistent way; a grounding for 𝒮 is a signed probability distribution 𝒫 such that every 𝒫_i is a restriction of 𝒫; and the grounding problem for 𝒮 is the problem of describing the groundings for 𝒮. We show that a wide variety of quantum scenarios can be formalized as observation spaces, and we solve the grounding problem 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.

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