ttn said:
That is absolutely false. To say that all probabilities are descriptions of ignorance, is to say that one insists on determinism -- i.e., that one is unwilling to accept the possibility of an irreducibly stochastic theory. This is just simply wrong as a description of Bell's assumptions. Bell does *not* assume determinism, as he stressed over and over again in his later papers (because people kept insisting that his derivation relied on determinism).
I know that Bell says that. But there are 2 points I wanted to make, one over which we went already many times.
I think that the first point is easy and you'll agree with me:
IF the lambda in Bell's argument IS entirely determining the outcomes, so that the P(A,B|a,b,lambds) are all 1 or 0 (in other words, IF the theory is deterministic) THEN, this function being a complicated way of writing the deterministic dynamics, if that dynamics is local (in the usual sense for a deterministic theory), then Bell locality follows.
The second point is what I called jokingly - a long time ago - Patrick's theorem:
If the theory is NOT deterministic, but follows Bell locality, then one can always EXTEND the theory, by adding "beables" into lambda, to make it into a local, deterministic theory, with the same predictions.
As such, I consider that we can limit ourselves to deterministic theories EVEN IF THAT WAS NOT BELL'S INTENTION.
You make no error by assuming that lambda specifies entirely the outcome (even if in a particular theory you're considering, it doesn't). If a theory is Bell local, and "irreducibly stochastic", you can swap it for a theory which is deterministic, and of which all probabilities are hence probabilities of ignorance of what lambda might be.
To make it more precise, you should say: the probability *that an irreducibly stochastic theory attributes to* something happening at event A is entirely determined by what is in A's past lightcone. The probabilites in "Bell Locality" are *not* epistemic -- they are the fundamental dynamical probabilities that some candidate stochastic theory assigns to things. And what Bell Locality amounts to is the requirement that these probabilities be based on (as, for example, the probabilities in OQM are based on the wave function) stuff/events/beables in the past light cone of the event in question. The probabilities assigned to an event in a Bell Local theory do not depend on stuff outside the past light cone.
Yes, but, as I said, we can now even extend the theory as becoming deterministic, with these probabilities now purely epistemic (about our ignorance of the value of these extra variables, which you called the "stochastic variables" in the previous one, and raised to the rank of beables in the new theory), and we would get exactly the same predictions, and this new, deterministic theory would be entirely local (in the usual sense of the word, which has a meaning for deterministic theories).
It's no more or less an assumption than the definition of "local" for deterministic theories. It's just a more general definition of "local".
Yes, that's all I said: it is an EXTENSION of the concept of local. But - as we discussed already many times before - as much as local for a deterministic theory is entirely clear, with probabilities one has to be careful, because they are not really physical quantities (fields over spacetime, say). An event A can have a certain probability P(A), and after it happened, this reduces to 0 or 1. So P(A) is not really a physical value that can be assigned to a spacetime point, as can, for instance, a temperature or something. P(A) not being a physical field, and always having an epistic component to it, it is not entirely evident how to extend the definition of local to a stochastic theory. However, I'd concur that the Bell definition is by far the most obvious and natural extension of it. I grant that and I do not dispute its reasonableness. But we have to be aware that, outside of the framework of a strictly deterministic theory, the concept of "locality" is not really defined, and hence a choice is to be made.
What I wanted to point out simply is that Bell locality (which is such an extension of the definition of locality, and hence always has some arbitrary component to it over which one can discuss) COINCIDES with the usual word local, when the underlying theory is deterministic and all probabilities are epistemic. And, moreover, in those cases where the underlying theory is not deterministic, we can MAKE it deterministic and still keep locality (in the usual sense) ; that's "Patrick's theorem".
So I READ Bell - even though he didn't have this in mind - as:
The experimental outcomes (or predictions) are COMPATIBLE with an underlying DETERMINISTIC, local theory.
I know that was not the original intention of Bell. If you want to, you can see it as kind of a coincidence, that, using the extension of the concept of local to include stochastic theories, that if such a stochastic theory satisfies it, it can be extended into a deterministic theory that is (normally) local.
If it is seen this way, then this is a rather unfortunate fact, because now the distinction between "fundamentally stochastic but Bell local" and "potentially underlying deterministic and local" is spoiled for ever, them being (by coincidence) equivalent statements.
But coincidence or not, they ARE equivalent statements.
But instead of whining over this (unfortunate) equivalence, which goes against the spirit of Bell, we can take it as an advantage. The nice thing about this is that you even don't have to have the theory. You just have to check whether the Bell conditions hold.
On data, or on the stochastic predictions of any theory. IF the Bell conditions hold (on the DATA or PREDICTIONS, not on the supposed machinery of a hypothetical theory), then these DATA OR PREDICTIONS can also be reproduced by a (potentially ugly) local deterministic theory. If not, then you won't find ANY such local deterministic theory, and hence also no Bell-local stochastic theory.
That's Bell's "lambda". (Completeness is of course defined by whatever particular candidate theory one is assessing.) To say that the probabilities in Bell Locality (which are always conditional on this lambda) are ignorance-based is to say that, really, there is some unknown fact which *determines* outcomes, but which is not contained in lambda (hence we only have probabilities, not determinate predictions). But this is all contradictory. That lambda provides a complete state description is *assumed*. You can't just come back later and say "well maybe really it doesn't, and the probabilites are really ignorance-based rather than fundamental". To say that is not some kind of objection to bell's locality criterion -- it's simply to change midstream what theory one is talking about. And if you're too scatterbrained to keep thinking about the same theory through the whole analysis, don't blame it on bell!
No, what I showed, long ago, is that you can EXTEND the lambda of any Bell local theory so as to make it completely deterministic, and that Bell locality is conserved under this operation. Whether you find this an attractive option, and whether the theory resulting has any esthetical or physical appeal is another matter, but it can be done.
So, again, I'm not making any REQUIREMENT of determinism. Maybe I formulated this badly, I didn't want to mean that Bell insisted on determinism. I'm just saying that the way Bell extended the concept of local (which holds for deterministic theories) to stochastic theories, AUTOMATICALLY IMPLIES that a deterministic, local theory is compatible with the predictions.
So, AFTER THE FACT, whether it was intended or not, Bell locality for stochastic theories is EQUIVALENT to requiring the potential of an underlying deterministic local theory.
As such (and probably my wordings were unfortunate), IT IS NOT A RESTRICTION to say that probabilities are ignorance-based in the discussion of Bell locality. Because even if initially they weren't thought to be so (and the theory was irreducibly stochastic), we can swap them for being so (and the NEW, equivalent, theory is now deterministic). So, when talking about Bell locality, there's no NEED (it is not a requirement) to talk about non-epistemic probabilities.
BTW, I have serious conceptual difficulties, as I told you already, with the concept of non-epistemic probabilities, without saying: "things happen". But happily, this concept is NOT NEEDED to discuss Bell's stuff. I'm not saying that it is a RESTRICTION. It is a CONSEQUENCE of the way Bell defined locality for stochastic theories.
And all this was not the point I wanted to make. I wanted to make the point that, PURELY HAVING A SET OF DATA, or HAVING A SET OF PREDICTIONS FROM A BLACK BOX, one can check them against the Bell inequalities. If they don't satisfy these, then THERE IS NO HOPE of finding *either* a Bell-local stochastic theory, OR an underlying local deterministic theory (both being equivalent).
This is because claims were made that the Bell inequalities were somehow based upon underlying physical assumptions of the existence of particles or so. Not at all. A set of DATA, from an experiment, can tell you whether, yes or no, there is ANY HOPE of obtaining them from a Bell-local stochastical theory, or from an underlying deterministic local theory. You do not have to make any assumptions of the physical nature of the cause of these data in order to establish this.
It would constitute a perfect Bell test.
The assumption of particles IS however made in NON-perfect Bell tests as they have been performed up to now, in order to CORRECT for the data (the "efficiency of detection" being intrinsically a particle-related concept).
This is why stochastic electrodynamics can still succeed in making equivalent predictions of the current non-perfect Bell tests. But the day that we will have data from a perfect Bell test, and where these corrections are not needed anymore, we don't need any assumption of particle or whatever in order to check the Bell inequalities. If they are violated, then we know that NO WAY these data are going to come out of a Bell-local stochastic theory, or out of a deterministic local theory.
), but fine. First of all, it are the Copenhagen, Bohm-de Broglie and spontanious collapse model (let me restrict to the realist interpretations) FORMULATIONS which are non-local; this does not imply that the predictions of QM necessarily emerge from a non-local theory. As to my physical intuition, I cannot imagine a theory which is neither realist, neither local and I have met nobody who can make sense of that. It is my deep conviction that ultimately the laws in physics for the CORRECT fundamental degrees of freedom are local, therefore I think the current FORMULATION of QM is incomplete : this is far from saying that QM is wrong.
So it is seems very possible to have a candidate prescription of the real (but not a local one). An interesting paper on free will versus determinism is written by Kochen and Conway : quant-ph/0604079
