Fredrik said:
I think it's obvious enough that it makes sense to think of phase space points as complete sets of properties. I don't think a proof or even a definition is required*. If you want a
reason to think of them that way, then consider the fact that if you know one point on the curve that describes the particle's motion, you can use the force (
) to find all the other. So if you know a point, you know everything.
But you don't know a point, you only know that the particle is in some box. In other words, if I replace classical mechanics the way it is normally described (a theory of impossible precision) with a theory that only talks about intervals, rather than points, do I not have an epistemic version? And here's the real kicker: how is such a theory not completely equivalent to classical mechanics? And what's more, isn't the second theory the one we actually test, not the first one? If the second version is the only version of classical mechanics that ever gets tested, then I claim the second version is the actual theory of classical mechanics, and the first one is just a kind of make-believe version that we only use out of a kind of laziness to talk about the theory that we have actually checked. I like laziness as much as the next guy, but we should at least recognize it. (If we had, we would never have concluded that quantum mechanics was "unclassical", we would have called it what it really is "super-classical." It includes classical physics, and adds more complexity at smaller scales inside the boxes that classical physics never tested.)
As you know it isn't possible to define everything, but more importantly, there are some things that we simply can't avoid treating as more fundamental than other things. For example, the concept of "measuring devices" is more fundamental than any theories of physics, and the concept of natural numbers is more fundamental than even the formal language used to define the set theories that we use to give the term "natural number" a set theoretic definition.
I'm with you up to here.
It seems reasonable to me to take "property" to be one of those things that we consider so fundamental that we don't need to define it.
The problem is not with using "properties" as conceptual devices, we do that all the time-- physics would be impotent without that ability. The issue is what does it mean when we invoke a conceptual device and call it a property. Does it mean that if we knew all the properties, we'd understand the system completely? That's the part I balk at, I see zero evidence of that, and I find it such a complete departure from anything that physics has ever been in the past. I think the more we know about something, the deeper the mysteries about it become-- we never understand it completely, we understand what we didn't understand before and now don't understand something new. So much for properties!
So I ask the same question-- for an approximate theory to work well, why does this require that there be an exact theory underlying it? I think that is a bogus proposition, yet it seems to be the very first assumption of PBR. The crucial assumption is not that the concept of a property might be useful, it is that systems really have properties that determine outcomes. If we strip out that part of the proof, what does it prove now?
Right, but in this context, it's what we know about the ontic states. Like it or not, that's seems to be how these guys are defining it.
Yes, and that is exactly what I think limits the generality of their proof. Let's go back to classical mechanics, and my point that it was never really a theory about points in phase space, it was always a theory about boxes in phase space (since that was all that was ever tested about it). If we had been more careful, and framed classical mechanics that way, then we might have had someone say "of course there really are ontic points inside those boxes, we only use boxes because of our epistemic limits in gathering information about those ontic points."
Indeed, that's what many people did say. Then along comes the hydrogen atom, and oops, those boxes are not boxes of ontic states at all. Why does this always seem to come as a surprise? The whole point of an epistemic treatment is to not pretend we know something we don't know-- like that epistemics is just a lack of information about ontics! If there was ever a lesson of quantum mechanics, it is that epistemics is something potentially much more general than just lack of information about ontics.
This is something that I find confusing. I'm tempted to say "none of it". Suppose that we consider all models that for each measuring device and each member of some set \Lambda assigns a probability P(k|λ,M) to each result k to be "ontic". We have no way of knowing if the ontic states really represent properties, but that also means that nothing will go seriously wrong if we just pretend that we do.
It seems to me the key assumption is that the ontics decide what happens to the system, and the epistemics are just lack of information about the ontics. Could we not prove things about any theory that could be consistent with classical mechanics by making the same assumption, that inside any epistemic "box" in phase space there are ontic points that determine the outcomes of when a hydrogen atom recombines? But quantum mechanics does not respect the ontic points of what people imagined classical mechanics was (but never demonstrated by experiment that it was), yet quantum mechanics does reproduce every experimental prediction that classical mechanics works for. Quantum mechanics is a mathematical structure "at least as good as classical mechanics."
Now, granted, quantum mechanics also makes different predictions at small scales. But that's my point-- I think the real value of the PBR theorem is that it might help us to figure out experiments to test quantum mechanics that quantum mechanics might not get right. If it does that, then it will be a truly valuable theorem. But I don't think it tells us anything about quantum mechanics, any more than proving theorems about ontic points inside boxes in phase space tells us anything about classical mechanics. Classical mechanics never was a theory about ontic points in phase space, it was always, demonstrably, a theory about epistemic boxes in phase space. This is also true of quantum mechanics, with different epistemics. Ultimately, I claim that all theories are built of epistemic primitives, and it is only a kind of laziness that allows us to imagine that any physics theory is ontic.
I think that this is what the HS article does, because their first example of an ontic model (they may have used the term "hidden-variable theory" instead) simply defines \Lambda to be the set of Hilbert subspaces of the Hilbert space of QM.
Expressing quantum mechanics in terms of Hilbert spaces is certainly a useful way to go, just as expressing classical mechanics in terms of points in phase space was. If that is what we mean by quantum mechanics (and that is indeed how it gets defined in the textbooks), then it is definitively ontic, as you point out later. But does this mean that it has to be an ontic theory to work as well as it does? I say no, it should be easy to replace the Hilbert space with a more epistemic version that restricts the theory to what has actually been verified by experiment. Such a theory would be completely equivalent in terms of its experimental justification, but would be much more "honest" (and less lazy but also less parsimonious), because it would not pretend to be an ontic theory when only its epistemic character has actually been tested. It would serve just as well, in every way except parsimony, as the theory we call "quantum mechanics". But we like parsimony, so we use the ontic theory, and that's fine-- as long as we recognize that in choosing parsimony over demonstrability, we have entered into a kind of pretense that we know more than we actually do. Look where that got is in DesCartes' era!
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