# Have Dirac's negative probabilities been furthered in modern probability theory? [closed]

Recently I read that P.A.M. Dirac has not only devised negative energy states and the delta function but even negative probabilities:

Since I am not a probability theorist, my question is this: Has this approach been widened and has it been given an axiomatic base, perhaps by adapting Kolmogorov's axioms? And does anyone know of an intuitive picture like the Dirac-sea of negative energy-states?

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## closed as off-topic by Marc Palm, Andrey Rekalo, Steven Landsburg, Ramiro de la Vega, David WhiteJul 17 '13 at 14:47

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question does not appear to be about research level mathematics within the scope defined in the help center." – Marc Palm, Steven Landsburg, Ramiro de la Vega, David White
If this question can be reworded to fit the rules in the help center, please edit the question.

If you interpret "negative probability" literally as "signed measure", then yes, this is too elementary for this site. On the other hand, physicists use "negative probability" to refer to something much weirder than measures taking negative values. For example, here is an excerpt from the Wikipedia article on the Glauber-Sudarshan P-representation: en.wikipedia.org/wiki/… – Tom LaGatta Jul 17 '13 at 13:59
> Even if P does behave like an ordinary probability distribution, however, the matter is not quite so simple. According to Mandel and Wolf: "The different coherent states are not [mutually] orthogonal, so that even if $\mathcal P(\alpha)$ behaved like a true probability density [function], it would not describe probabilities of mutually exclusive states." – Tom LaGatta Jul 17 '13 at 13:59
Dirac wrote: "Negative energies and probabilities should not be considered as nonsense. They are well-defined concepts mathematically, like a negative of money." (Dirac, P. (1942): "The Physical Interpretation of Quantum Mechanics," Proc. Roy. Soc. London, (A 180), 1–39.) – John Baez Jul 19 '13 at 15:34

## 1 Answer

Kolmogorov's axioms for probabilities are used to justify formulating probability theory in terms of measure theory. Mathematically, the theory of measures that take negative or even complex values is well-developed. So, to the extent that probability theory is just measure theory, you can say a lot is known about negative probabilities.

But probability theory is not just measure theory; it adds its own distinctive ideas. And I'm not aware of significant extensions of these distinctive extra ideas to negative probabilities, even though not just Dirac but Feynman broached the idea of doing so.

As the above link notes, the Wigner distribution associated to a function $\psi \in L^2(\mathbb{R}^n)$ is an interesting example of something that conceptually resembles a probability distribution, but takes negative values.

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is there a Lorentz-invariant version of the Wigner distribution? Do you have a good reference for quasiprobability distributions from the measure-theoretic point of view? – Tom LaGatta Jul 17 '13 at 13:48