Negative probabilities: what are they for?
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Abstract: The title may sound nonsensical, but negative probabilities are profitably used in quantum physics and elsewhere. So what are negative probabilities? Formally, signed probabilities can be defined as special signed measures.
But what is the intrinsic meaning of negative probabilities? We don’t know. The standard frequency-based interpretation of probabilities makes no sense for negative probabilities. There are attempts in the literature to provide meaning for negative probabilities but, in our judgement, the problem is wide open.
Instead, we address a more pragmatic question: What are negative probabilities good for? It is not rare in science to use a concept without understanding its intrinsic meaning.
Consider early uses of complex numbers. The standard quantity-based interpretation of numbers makes no sense for imaginary numbers. And the intrinsic meaning of imaginary numbers wasn’t clear (and is debatable even today). Yet complex numbers were profitably used to solve algebraic equations. “What are numbers and what are they for?” asked Richard Dedekind in 1888.
It turns out that the disparate quantum applications of negative probabilities can be seen as examples of a certain application template. Our first achievement is to make this template explicit. To this end, we introduce observation spaces. An observation space S is a family of (usual) probability distributions P1, P2, … on a common sample space. A question arises whether there is a single probability distribution P (called a grounding for S) which yields all P1, P2, … as marginal distributions. That P may be necessarily signed. Our second achievement is solving the grounding problem for a number of observation spaces of note.