# How can I show that the negative region of Wigner quasi-probability distribution is small enough than $h$?

Wigner quasi-probability distribution is the main tools used in the formulation of Quantum mechanics on phase space which is equivalent to usual formulations. This distribution can have negative values at some region, unlike usual probability distributions is non-negative.

Regions of such negative value are provable (by convolving them with a small Gaussian) to be "small": they cannot extend to compact regions larger than a few $$ħ$$, and hence disappear in the classical limit. They are shielded by the uncertainty principle, which does not allow precise location within phase-space regions smaller than $$ħ$$, and thus renders such "negative probabilities" less paradoxical.

This result is very fascinating and so, I would like to find the proofs or the papers mentioned about this, However, it is too tough to find because Wikipedia doesn't cite anythings.

Hence, I would like to know where this result is proven.

• what resources have you consulted thus far? Have you looked at the canonical review papers on the topic (one co-authored by Wigner himself)? A place to start (but which doesn't quite give you what you exactly want) is Case WB. Wigner functions and Weyl transforms for pedestrians. American Journal of Physics. 2008 Oct;76(10):937-46 but maybe you need to start at a more advanced level? Commented Oct 9, 2022 at 18:26
• @ZeroTheHero In fact, I just want to know the proof and prerequisites for understanding that. Maybe what I want to find is just the references. Resource-request is not a quite good tag for my purpose. But, what you recommended looks nice to read. Commented Oct 9, 2022 at 20:15
• for original literature see this question physics.stackexchange.com/q/730869 Commented Oct 9, 2022 at 22:24

I don't know if the following is what you need, but if one smooths Wigner function over a volume $$h$$ of phase space, one obtains Husimi function, which is non-negative everywhere (see, e.g., Ballentine, Quantum mechanics, Chapter 15).