# Intuitive interpretation of negative probabilities

I have heard that in quantum physics negative probabilities show up in certain distributions.

Could you give an example that aids int he intuitional interpretation of a negative probability?

For example, what does $X \sim N(0,1) -\delta$ mean, and how does it relate to $Y \sim N(0,1)$. Can we say anything about the relationship between $Y$ and $X$?

• You might want to have a look at signed measures: en.wikipedia.org/wiki/Signed_measure I've also understood that they are trying to get rid of the negative probabilities by defining observables as Positive Operator Valued Measures insetead of self-adjoint operators. The point being that negative probabilities are considered artifacts produced by the theory instead of something with an actual interpretation. – Matias Heikkilä Apr 20 '16 at 11:09

I don't think anyone has a clear understanding of negative probabilities.

There seems to be a rather good axionmatic background (http://physics.bu.edu/~youssef/quantum/Burgin-1.pdf) and the definition of an extended probability space ("extended" means either positive or negative) is rather well defined mathematically. There are events and anti-events that cancel eachother when elements of a set : $\{w,-w\}\approx\emptyset$.

Honestly I can't validate the content of the acticle personnaly since I read superficially and find it hard to read.

But what does it mean intuitively ?

Assume you have two cases $A$ and $\overline{A}$ (not $A$) each with probability 0.5 to make it simple. If an event $X$ has probabiliy 0.3 when $A$ is true ($P(X|A)=0.3$), and has probability -0.3 when $A$ is false ($P(X|\overline{A})=-0.3$), or equivalently the antievent $-X$ has probability 0.3 when not $A$, then as a whole, X has probability $0$. Thus X never happens.

I don't think anyone on earth understands it further than this.

You can find it explained here by Feynman : https://johncarlosbaez.wordpress.com/2013/07/19/negative-probabilities/ . Maybe the most important motivation for negative probabilities is Quantum Mechanics and most significantly the Wiegner quasi-distribution.

It's like a cancelation happens : $-X$ cancels $X$. This is rather intuitive. But this happens between two cases ($A$ and $\overline{A}$) that cannot happen together. I've never read a text giving an intuivie catch of this. It is directly related to some of the greatest mystery of Quantum Mechanics.