Negative probabilities are naturally found in the Wigner function (both the original and its discrete variants), the Klein paradox (where it is an artifact of using a one-particle theory) and the Klein-Gordon equation.

Is a general treatment of such quasi-probability distributions, besides naively using 'legit' probabilistic formulas? For example, is there a theory saying which measurements are allowed, so to screen negative probabilities? Additionally, is there an intuition behind negative probabilities?

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    $\begingroup$ Feynman introduced ghosts as "negative probability" in pertubative gauge theories. The main purpose of the ghosts is to cancel the contributions from unphysical polatisations of gauge fields in loops. After Faddeev-Popov we understand them in a different way, but the original idea was just that: "negative probability". $\endgroup$ Commented Oct 11, 2011 at 13:46
  • $\begingroup$ @José: Was not that a negative norm instead? $\endgroup$ Commented Oct 11, 2011 at 14:12
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    $\begingroup$ @Vladimir: Sure, but negative norm implies negative probability. Feynman actually introduced them in the context of gravity and he introduced them by hand to "soak up excess probability" in his own words, I believe. $\endgroup$ Commented Oct 11, 2011 at 14:57
  • $\begingroup$ It is known in QED as indefinite metric and is used to cancel contributions of non physical degrees of freedom (longitudinal and scalar photons). In QED it is the formalism of Gupta-Bleuler. en.wikipedia.org/wiki/Gupta-Bleuler $\endgroup$ Commented Oct 11, 2011 at 15:18
  • $\begingroup$ arxiv.org/abs/1202.3628 This is a very recent paper demystifying the negative probability density in the Wigner Function. I hope this helps. $\endgroup$ Commented Sep 17, 2012 at 15:47

8 Answers 8


One never obtains "negative probability" densities when one discusses single observables. One obtains "negative probability" densities only when one discusses joint distributions of incompatible observables, for which the commutator is non-zero (because they take negative values, they are not probability densities). So, to avoid negative probability densities entirely, only discuss joint probability densities of compatible observables.

There are some states in which some pairs of incompatible observables nonetheless result in positive-valued distributions. The best-known examples are coherent states, for which the Wigner function is positive-definite. This, however, does not extend to all possible observables, so that in a coherent state not all pairs of incompatible observables result in positive-definite joint probability densities.

The failure of joint probabilities to exist for all states means that even though positive-definite densities may exist for particular observables in particular states, it is generally taken to be too much to call any positive-definite joint density that might happen in a special class of states to be a probability density just because it is positive-definite.

There is one quite general way to construct an object that is always positive-definite from a Wigner function, which is by averaging it over a large enough region of phase space. Many attempts to do this in a mathematically general way have been constructed over the years. I personally like Paul Busch's approach (with various co-workers), whose web-site lists two monographs that do this quite nicely:

The Quantum Theory of Measurement
Paul Busch, Pekka Lahti, Peter Mittelstaedt. Springer-Verlag, Berlin
Lecture Notes in Physics, Vol. m2, 1991; 2nd ed. 1996
Operational Quantum Physics
Paul Busch, Marian Grabowski, Pekka Lahti. Springer-Verlag, Berlin
Lecture Notes in Physics, Vol. m31, 1995; corr. printing 1997

I'm certain that other people have other preferences, however. For some, this is a way to reconcile quantum with classical, for others it is not.

There is a quick and dirty way of seeing the relationship between incompatibility and positive-definiteness of putatively positive joint probability densities, which can be found in a paper by Leon Cohen, "Rules of Probability in Quantum Mechanics", Foundations of Physics 18, 983(1988). I trot this out quite regularly, even though it's rarely cited in the literature because it's not very nice mathematics, because it's such elementary mathematics and it influenced my understanding of QM a lot a long time ago (I cited it here, for example, for a not very related Question).

  • $\begingroup$ Great answer. The description of incompatible measurement in terms of negative probabilities renders profound insights into the Bell inequalities. Recently I have dedicated a blog post to this: science20.com/hammock_physicist/… . Main objective was to render the strangeness of quantum mechanics accessible to lay persons. Those interested in an intuitive description of the strange behaviors of entangled systems might like it. $\endgroup$
    – Johannes
    Commented Dec 15, 2012 at 13:55

As Ernesto pointed out in his comment, I've answered your first question here (which was updated on the arXiv and published very recently).

As for the question about the intuition behind negative probabilities, here is my warning if you don't already have tenure: don't go there. As Feynman pointed out (and Dirac much earlier) negative probabilities are a means to an end. What end? Well, regular probability, of course.


A little bit left-field this but may be of interest. If you want to consider a more abstract setting, then the following paper is of interest from a foundations point-of-view:

R. W. Spekkens, ''Negativity and contextuality are equivalent notions of nonclassicality''

It relates a generalisation of the Wigner function to a generalisation of non-contextual hidden variable theories. It shows that even structure at the more black-box, operational level results in quasi-probability distributions.

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    $\begingroup$ Some recent articles by Chris Ferrie et al. prove the necessity of either negative probabilities or a deformed probability calculus, check out: arxiv.org/abs/0711.2658 and arxiv.org/abs/1010.2701 . If I may point to a paper of mine, demanding positivity from a particular definition of discrete Wigner function (due to Wootters) results in states and operations which are easy to simulate classically: arxiv.org/abs/quant-ph/0506222 $\endgroup$
    – Ernesto
    Commented Oct 11, 2011 at 18:12

There are two works of Feynman about negative probabilities. It is hard to add something to that, if to look for introduction to the subject.

R. P. Feynman, Negative probability in Quantum implications: Essays in honor of David Bohm, edited by B. J. Hiley and F. D. Peat (Routledge and Kegan Paul, London, 1987), Chap. 13, pp 235 – 248.

R. P. Feynman, Simulating physics with computers (Chapter 6), Int. J. Theor. Phys., 21, 467 – 488 (1982).

  • $\begingroup$ As a collector of Feynman works, thanks. I had never even heard of your first reference, which sounds fascinating (Feynman on Bohm?? Intriguing.) $\endgroup$ Commented Mar 30, 2012 at 17:42
  • $\begingroup$ Feynman wrote in this essay: "Trying to think of negative probabilities gave me a cultural shock at first, but when I finally got easy with the concept I wrote myself a note so I wouldn't forget my thoughts." $\endgroup$ Commented Apr 4, 2012 at 20:22
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    $\begingroup$ Alex, thanks. I found a nearly-complete piece of it in an online book sample. Very Feynman in style, with a clearly stated anchor point around which he builds his analysis. And since this just a note, maybe I can get away with a for-dicussion-only observation on @PiotrMigdal's original question?: The simplest self-consistent way to enable negative probabilities is let them represent negative mass-energy states that erase rather than annihilate the positive mass-energy states. Lots of issues, but also lots of fun. Wave packets e.g. become dissolving clouds of +/- pairs with a slight + excess. $\endgroup$ Commented Apr 5, 2012 at 3:07

Negative probabilities are only possible if they are invisible. They can only be associated with joint measurements. However, we must actually forbid joint measurements. This is only possible with the additional property of measurement disturbance aka Heisenberg's uncertainty principle as properly understood. If we measure a marginal value, the very act of measurement must unavoidably change the other marginal distribution. Now, negative probabilities are no longer so clearcut because the measuring apparatus must also be taken into account, and its interaction with the system. In plain old quantum mechanics, the mechanism is entanglement. What is the corresponding analog with negative probabilities?


As Morgan pointed out, extended probabilities, which is the technical name, means joint probability distributions may have negative probabilities, but marginal probabilities never. But this is a stretch. How can it be a joint probability distribution if we can never measure complementary observables simultaneously?

Extended probabilities also means we can have huge cancellations between positive and negative contributions, each on their own adding up to far more than one in absolute value but their difference lies between 0 and 1. An example would be a diffraction grating for the Wigner distribution. Such sensitivity does not arise if all probability contributions are nonnegative.



R. W. Spekkens,

''Negativity and contextuality are equivalent notions of nonclassicality'' from Matty Hoban answer

then https://arxiv.org/abs/0705.2742

...Negative probabilities are found to arise naturally within the model, and can be used to explain the Bell-CHSH inequality violations..

...allowing negative probabilities for the underlying epistemic states...

...That negative probabilitities, in the form of negative values of an appropriate Wigner function, may be used to indicate or explain nonclassical features has been known for a long time**#**...


.-R. Feynman in Quantum Implications, edited by B.J. Hiley and F.D. Peat, Routledge, London (1987).

.-M.O. Scully, H. Walther, and W. Schleich, Phys. Rev. A 49, 1562 (1994)


Additionally, is there an intuition behind negative probabilities?

In my recent article (Entropy 2022, 24(2), 261), I offer a description of quantum particles, such as electrons, as a collection of N+1 point particles with charge +1 (electron charge) and N point particles with charge -1 (electron charge). In particular, I show how one can approximate (in some sense, with an arbitrary precision) a smooth charge density distribution that is not positive everywhere by such a collection. So it is possible that negative probability density for a quantum particle means positive probability density for an antiparticle.