Though trying to describe super-observers who, while not constrained by physical laws, still settle bets about physical systems sounds difficult. For example, Gell-Mann and Hartle make the statement “A set of histories {Cα} is recorded if there are alternatives with projection operators {Rα} at one time that are correlated with the histories.” A super-observer would presumably supernaturally record the one real history while also dismissing the fact that there are is no corresponding projection operator that satisfies equation 3.1 in the paper.

Perhaps the formalism in the paper is limited to demonstrating a consistency between the existence of a real fine-grained history, and the Decoherent Histories procedure used by ordinary, natural, IGUSes trying to learn what they can about the history.

]]>If you are an operationalist, it is sufficient to supply the probabilities for everything that is observable. Clearly, these are going to be ordinary real-valued, positive probabilities, because they are related to relative frequencies, betting odds, or whatever they are in your favourite interpretation of probability, in the usual way. Now, in the course of your calculations you may find it convenient to introduce some unobservable entities. If you can find a calculus that involves assigning exotic probabilities to these unobservable entities, be they negative, complex, or whatever, then that is perfectly fine because the unobservable entities don’t actually exist, and the operationalist can simply view the calculus as a convenient way of summarizing the relations between the probabilities for observable entities. For the operationalist, the whole calculus is just a convenient fiction for getting at the probabilities for observable things.

On the other hand, for the realist, things are not so simple. If you are positing that the unobservable entities actually exist then you have to give some meaning to statements about their probabilities. It is not enough to point out that the probabilities for all observable things are positive. Although that is surely still an important consistency check on the theory, merely demanding this as your definition of what you mean by an exotic probability is an operationalist answer. So, although I can’t say that Gell Mann and Hartle’s proposal is outright inconsistent, I maintain that it is incomplete. They have not done enough to explain what their exotic probabilities mean.

In my view, there is good reason to doubt that exotic probabilities are meaningful in a realist framework. This is because you are positing that the things to which you assign these exotic probabilities actually exist. In Gell-Mann and Hartle, there is a set of fine grained histories, one of which is realized, but the set is assigned an exotic probability measure. What does it mean then if the actual history in one realization of an experiment is assigned a negative or complex probability?

Now, which actual history occurs in any given run of the experiment may be unknowable to me, and it may also be unknowable to anybody who is limited by the same physical theory as me. However, in a realist framework this is not a fundamental limitation on knowledge, but a contingent fact about how the laws of our universe happens to operate. We can certainly imagine a god-like super-observer, not limited by the current laws of physics, who can know all the facts that are true of our universe. No such super-observer may exist in our universe, but the point is that such an entity is logically conceivable within a realist framework. That’s what I meant by “knowable in principle” in my previous comment.

Now, we can imagine how super-observers would reason in the face of uncertainty. Although they can know everything in principle, we can imagine a super-observer who simply hasn’t bothered to look at some part of the universe, and they are interested in making bets about it, perhaps with other super-observers. How should such an entity reason? Well, clearly, they have to use ordinary classical probability theory for exactly the same reasons that we have to use it for observable things, because for them everything is observable in principle. So, in fact, we can see that there must exist an ordinary classical probability measure over all the things that actually exist, observable or not. If you are positing that there are some real unobservable entities that cannot ever be assigned ordinary probabilities then you have a problem with realism. One can still maintain that the exotic probabilities are useful in the operational sense, but it must ultimately be possible to derive them from a theory which does assign ordinary probabilities.

I will admit that there is one big gaping hole in this argument, which is that the foundations of probability are themselves controversial. Many people, myself included, argue that the current foundations are not sufficient to entail that ordinary classical probability is the only rational theory to use for reasoning in the face of uncertainty. However, usually the gaps in these arguments are expressed in operational terms, e.g. we cannot determine whether outcome A of experiment E would have occurred if experiment F was actually performed, because experiments E and F are incompatible, so performing one precludes finding out what the outcome of the other would have been. Therefore, we may say that there is no operational meaning for the joint probability of A and B (where B is an outcome of F), so we don’t have to satisfy an equation like p(A AND B) < = P(A). This is an operational gap in the usual argument for the probability calculus, but if you want to argue from a realist point of view then you have to give me some account of what reality is like that entails nonclassical probabilities and this must be given in terms of the nature of reality itself rather than the nature of my limited access to it. In other words, you have to give an account that would apply to super-observers, rather than just regular ones.

This is why I say that Gell Mann and Hartle’s account is incomplete. It is not necessarily incompletable, but that looks like a very hard task and, at least to my knowledge, nobody has given a realist account of exotic probabilities ever in any context.

Let me just make one more comment. If you want to write quantum theory in terms of exotic probabilities, there are an infinite number of ways to do it. For every basis of the vector space of operators on a Hilbert space, there is a corresponding pseudo-probability function. Why then, should I believe Gell Mann and Hartle’s account rather than any of the other natural ones, such as Dirac’s complex measure, the Wigner function, the Q function, the P function, etc.? This non-uniqueness ought to be troubling to say the least. To repeat my bad Schrödinger joke, this seems like the sort of quantum jumping to conclusions that we ought to be avoiding. The goal is not to come up with an interpretation of quantum theory that works. At this point we have several of those. The goal is to come up with the correct understanding of quantum theory.

]]>“If you are a realist then the fine grained history exists and is knowable in principle”

This looks like a controversial statement to me. A realist might postulate the existence of a fine-grained history for a physical system, but reject the idea that it is knowable in principle, as they might reject the idea that the fine-grained history of the system can be exactly correlated with some arbitrarily sophisticated recording apparatus.

]]>Regarding the Gell-Mann-Hartle paper, I don’t have time to go through it again right now. The short answer is that if your interpretation involves negative probabilities then you cannot claim that it is realist in any straightforward way. The argument that this is OK because the fine-grained histories cannot be observed does not pass muster because that is an argument appropriate for an operationalist rather than a realist. If you are a realist then the fine grained history exists and is knowable in principle, even if it is not known to us. Thus, the standard arguments in the foundations of probability would imply that they have to be assigned non-negative probabilities.

]]>Also (and perhaps you have answered this elsewhere in your blog), do you have sharp criteria an interpretation must satisfy?

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