Possibilistic Physics by Tobias Fritz

Andreas Martin,

thank you for your feedback. You've made a very interesting point.

I agree that most theories of physics are based on the natural numbers. There are a few exceptions, though, for example the causal set approach to quantum gravity. I do not agree, though, that the natural numbers are instrinsically and fundamentally probabilistic. Let me explain why.

First of all, certainly the primes are a very interesting subset of the natural numbers &ndash mathematically. But how are they relevant for fundamental physics?

Then, yes, it seems that the probabilistic Cramér model accurately captures many aspects of the primes. But, as Terry Tao mentions in his structure and randomness in the primes, there are some problems with this model like "most primes are not divisible by 5". If the primes were totally pseudo-randomly probabilistic, exactly 1/5 of them should be divisible by 5. This aspect can indeed better be captured in a possibilistic model. Consider the following possibilities:

[math]\pi_n(n\textrm{ is prime})=\left\{\begin{array}{cl}0&\:\textrm{if }n\textrm{ is divisible by 2, 3 or 5}\\\ast&\:\textrm{otherwise}\end{array}\right.[/math]

So a natural number is considered to be a possible prime if and only if it is not divisible by 2, 3 and 5. This yields the sieve of Eratosthenes.

These are the reasons why I don't believe that the natural numbers are enough justification for the use of probabilities. This, together with the arguments given in the essay, is everything that I can say right now in the favor of possibility. Personally, I don't believe that the concept of possibility is necessarily superior. But I do believe that the concept of probability shouldn't be taken for granted.

Here are some comments by Robert Spekkens on my essay, divided into two posts for length reasons:

Dear Tobias,

Thank you for directing me to your paper. I largely agree withyour perspective. I am also of the opinion that it is best *not* toconsider the toy theory as a probabilistic theory. I have often saidthat all that is required for this theory is modal logic, that is, thelogic of possibility and necessity. Bob Coecke and his students havebeen working on category-theoretic presentations of the toy theory andwithin that framework, one can be a bit more formal about this. I attachan article which I am writing with them and which is available on Bob'swebsite (http://www.comlab.ox.ac.uk/people/bob.coecke/) but not yetpublished. The phrase "possibilistic theory" is actually used anddiscussed therein (see remark 4.5). See also arXiv:0808.1037v1[quant-ph].

Despite the naturalness of thinking of the toy theory as apossibilistic theory, most people's intuitions go the other way and I amconstantly asked why I didn't choose to include arbitrary convexcombinations of the epistemic states. I take it that you are askingyourself the same question. Here is my response.

First, note that allowing such convex sums of epistemic states would, strictly speaking, violate the knowledge-balance principle because for such states one could not say that one has the answer to half the questions in a canonical set. I therefore disagree with your statementon p. 4 that "the knowledge balance principle advocated in [Sp] would still be satisfied".

Second, one cannot close the set of epistemic states under convex combination while simultaneously keeping a restricted notion of coherent combination without doing violence to the quality of the analogy with quantum theory. Here is what I say in section VII.C of my paper.

[quote] We have seen that there are two types of binary operations defined for epistemic states in the toy theory, analogous to convex combinations and coherent superpositions of quantum states. However, these operations are partial; they are not defined for every pair of epistemic states. It might therefore seem desirable to close the set of epistemic states in the toy theory under convex combination with arbitrary probability distributions. In this case, the set of allowed epistemic states for a single elementary system would have the shape of an octahedron in the Bloch sphere picture. Hardy's toy theory, for instance, has this feature [35]. Such a variant of our toy theory has also been consideredby Halvorson [61]. However, there is an important sense in which such a theory is less analogous to quantum theory than the one presented in this paper. The toy theory shares with quantum theory the feature that every mixed state has multiple convex decompositions into pure states, whereas in this modified version, there are many mixed states that have unique decompositions. Similarly, in the toy theory, as in quantum theory, every mixed state has a "purification"-a correlated state between the system of interest and another of equal size such that the marginal over the system of interest is equal to the mixed state in question-whereas in the modifiedversion, there are many mixed states that have only a single purification.

[quote] The problem with the modified theory is that although convex combination has been extended to a full binary operation rather than a partial one, the coherent binary operations have not been so extended. Moreover, although one has allowed arbitrary weights in the convex combinations, one has not allowed the analogue of arbitrary amplitudes and phases for the coherent binary operations. It is likely that a better analogy with quantum theory can be obtained only if both operations are generalized. Unfortunately, it is unclear how to do so in a conceptually well-motivated way. [end quote]

Third, I don't interpret the probabilities in the toy theory as subjective degrees of belief nor as objective propensities but rather as relative frequencies. My epistemic states describe the relative frequencies of ontic states in an ensemble of similarly preparedsystems, or, equivalently, what an agent knows about the ontic state of a single system drawn from this ensemble. Unlike the subjective Bayesians, I believe that the epistemic state that an agent uses is justified and so in this sense the probabilities are not subjective. See below for some additional comments on the interpretation of probability. The upshot is that to prepare an arbitrary probability distribution over epistemic states, I would argue that one requires something like a physical die that samples from that distribution. The question then arises about whether such a die could exist in a world governed by the toy theory. Suppose we begin by presuming only those epistemic statesthat satisfy the knowledge-balance principle. What dice can be built out of these? A single elementary system in the toy theory can be used as an unbiased coin flip. Similarly, using a pair of elementary systems one can achieve a distribution of the form (1/4,1/4,1/4,1/4). Below is an example of how to achieve this. However, to get a biased distribution, such as (1/4,3/4), by a measurement on an epistemic state satisfying the knowledge-balance principle would require an invalid measurement. Again, see below.

Another strategy would be to prepare a single elementary system, call it A, in the epistemic state of maximal ignorance and then implement acontrolled operation from A to another system, call it B, that shifts the ontic state of B if the ontic state of A is 2,3 or 4 and leaves the ontic state of B invariant if the ontic state of A is 1. One can show, however, that this sort of controlled operation also violates the knowledge-balance principle.

Note that if one had access to classical dice and classical conditional operations, then clearly any convex sum could be implemented. However, if we imagine that the dice are themselves made out of the elementarysystems of the toy theory, and are subject to the same sort ofrestricted dynamics, then this sort of conditional operation ispossible. The toy theory does not "play well" with classical mechanics.

The principle to which I'm appealing here is that any dice or operationsconditioned on dice that are presumed to be available as externalresources must also be internal resources. Only then can the cutbetween internal and external degrees of freedom be moved arbitrarily. Such motility strikes me as a consistency condition that one would want to impose on any theory claiming universal applicability.

[...]

Let me end with a comment concerning how you describe my work. Your statement that my toy theory "is inconsistent in the usual probabilistic interpretation, but is a perfectly fine example of a possibilistictheory" suggests to the reader of your article that the interpretation that I proposed for it (presumably the "usual" one) is probabilistic. I hope that the discussion above and in particular the content of sectionVII.C of my paper make it clear that my favored interpretation is in fact much closer to the possibilistic one that you expound.

[...]

All the best,

Rob

Dear Eckard,

thank you for your comment. Although I'm aware of John Baez' work, I don't know what "his negative probability" is. Could you point me to a reference?

Obviously, probabilities in the sense of asymptotic frequencies are always non-negative. And asymptotic frequencies are what the Born predicts. No negative probabilities here.

best, Tobias

Dear Andreas Martin,

unfortunately I'm having trouble retrieving the paper you mentioned... no online access and not in our library.

But I am currently reading your essay, which seems to contain some intriguing observations, and I will need a little more time to fully grasp it.

So do you mean that probability emerges from a Kripke model of modal logic, just as it emerges in many-worlds interpretations of quantum mechanics? If yes, then that implies that -- in modal logic -- probability is not a fundamental concept, but merely a derived quantity. If this is what you mean when you write "this function naturally leads to probabilities", then we agree: probabilities then are an extremely useful, but non-fundamental, concept.

Also remember that I'm not claiming possibilistic physics to be realistic in any sense; I'm just trying to question established concepts and see what one can do without them.

(By the way, both the long delay in posting Spekkens' email and the missing spaces therein are completely my bad.)

Although it's been a few years since this essay has been written, it seems that there is still some interest in it and it is being read. So I would like to mention a few additional things about the possibilistic Bell inequality of the essay.

It turns out that this possibilistic Bell inequality had been well-known before and is actually due to Lucien Hardy. Nowadays it is known as "Hardy's paradox", and many papers have been written about it.

On the general idea of possibilistic Bell inequalities, there are also notes of a talk "Horn Formulas as Linear Inequalities and Nonlocality Paradoxes" on my webpage,

http://perimeterinstitute.ca/personal/tfritz/2011/horn_formulas.pdf

and a paper that I coauthored,

http://arxiv.org/abs/1105.1819

On a related note, Abramsky and Hardy have subsequently explored "logical" Bell inequalities,

http://arxiv.org/abs/1203.1352