# Bell's theorem

1. Jun 17, 2014

### TrickyDicky

When drawing conclusions from this most important and profound theorem, I wonder if somebody has interpreted its proof of the falseness of local realism as implicitly referring to elementary particles as realistic objects.

2. Jun 27, 2014

### jfizzix

I tend to take a conservative approach.

To me, Bell's theorem says that quantum mechanics predicts correlations in certain pairs of highly entangled particles (say, A and B) that cannot be explained by a complete knowledge of everything in the intersection of A's and B's respective light cones.

3. Jun 27, 2014

### atyy

Bell's theorem refers to correlations between "classical" or "macroscopic" experimental outcomes. So as long as one believes that the experimental outcomes in a Bell test are "classical", then the violation of the inequality does rule out local realism.

There are some assumptions that go into this conclusion. For example, it assumes that each measurement produces only one outcome. In many-worlds each measurement has more than one outcome, so the Bell test don't rule out that many-worlds is a local realistic theory.

4. Jun 27, 2014

### TumblingDice

Are you asking if the interpretation casts any bearing on whether elementary particles are real, as opposed to something other than real?

5. Jul 9, 2014

### TrickyDicky

The Bell's inequalities are always mathematically formalized in terms of particle's probabilities, and their effects on detectors wich are identified with the particles themselves. But if one separates these, i.e. doesn't identify the detector outcomes with particle entities, it follows that the inequalities hinge on the assumption of the concept of particles as realistic localized objects for the inequalities to impy local realism.
In this sense the violation of the inequalities by the experiments seems to affect just the forms of local realism depending on that specific particle theoretical conception.
I'm not sure what the objections to this reasoning are, other than it seems to lead to seeing the theorem and the results of the experiments based on it as evidence that no theory with particles as fundamental objects is possible, the local hidden variables would be particles themselves as realistic localized objects and their identification with clicks in detectors.

6. Jul 9, 2014

### stevendaryl

Staff Emeritus
I don't quite agree with that characterization of Bell's theorem. The theorem doesn't actually mention particles at all. It's a theorem about correlations between detector outcomes. It's agnostic about what causes those correlations.

Basically, the general situation is this:
We have three regions of spacetime, $A$, $B$ and $C$. Each region has its own variables (which may be associated with particles, but that's not part of the argument). The assumption is that region $A$ and $B$ have no causal influences on each other, but that region $C$ possibly influences them both. Under those assumptions, we can conclude (for a classical probability model):

The probability of an outcome in $A$ should depend only on variables in $A$ and $C$, and the probability of an outcome in $B$ should only depend on variables in $B$ and $C$. Bell's theorem doesn't require any specific assumptions about the nature of the variables.

7. Jul 9, 2014

### TrickyDicky

Hmmm, that is quite a strong statement, isn't it? I thought local realism was a very important and specific assumption about the nature of the variables in the Bell inequalities. And the theorem may be agnostic about the cause of the correlations but it is not agnostic about identifying individual particles with what detectors detect.
Quoting the Stanford encyclopedia page on the theorem: "Locality is a condition on composite systems with spatially separated constituents, requiring an operator which is the product of operators associated with the individual constituents to be assigned a value which is the product of the values assigned to the factors, and requiring the value assigned to an operator associated with an individual constituent to be independent of what is measured on any other constituent. From his assumptions Bell proved an inequality (the prototype of “Bell's Inequality”) which is violated by the Quantum Mechanical predictions made from an entangled state of the composite system." (my bold)
I agree the word particles is not mentioned in these assumptions but I think it is clearly implicit as localized realistic individual constituents. And I think that is what Bell and most physicists assumed.

Last edited: Jul 9, 2014
8. Jul 9, 2014

### stevendaryl

Staff Emeritus
It makes assumptions about causal influences among variables, but it doesn't say anything about particles.

But I don't see that Bell's theorem has anything specifically to do with the reality of particles.

At the heart of what Bell is doing is assuming that the probability of an outcome can only depend on facts about the causal past of that outcome. Causally separated outcomes can only be correlated due to sharing a common causal past. That's the assumption that Bell used to derive his inequality, and which QM seems to violate.

9. Jul 9, 2014

### atyy

Are you asking whether relativistic QFT is also theoretically predicted to violate the Bell inequalities?

10. Jul 9, 2014

### stevendaryl

Staff Emeritus
The way I think of it is in terms of causal connections between regions of spacetime.

In the figure, the triangle whose apex is labeled "Alice" is the collection of all events in the causal past of Alice's experimental result (if we assume relativity, then it's the "backward's lightcone" for the event that is the apex of the triangle). The triangle labeled "Bob" is the collection of all events in the causal past of Bob's result (the backwards lightcone for Bob's result). The various regions of spacetime are numbered:

1. Region 1 is the region immediately prior to Alice's result.
2. Region 2 is the region immediately prior to Bob's result.
3. Region 3 includes events possbily relevant to Alice's experimental setup that are prior to her measurement, but recent enough that they could have no causal influence on Bob's result. This region includes Alice's choice of a detector setting.
4. Region 4 includes events possibly relevant to Bob's experimental setup that are prior to his measurement, but recent enough that they could have no causal influence on Alice's result. This region includes Bob's choice of a detector setting.
5. Region 5 includes events that are possibly relevant to both Bob's result and Alice's result.

Bell's assumption basically is that events in region 1 can be influenced by events in regions 3 and 5, but not on events in regions 2 or 4. Events in region 2 can be influenced by events in regions 4 and 5, but not on events in regions 1 or 3. That's the locality assumption.

Then there is a second assumption, and I'm not sure exactly what the technical name is, but it's something like "completeness of dependencies". Let $F_1, F_2, F_3, F_4, F_5$ be facts about the 5 regions. We are interested in conditional probabilities:

$P(F_1 \wedge F_2\ |\ F_3 \wedge F_4 \wedge F_5)$

the probability of both $F_1$ and $F_2$ being true, given that $F_3, F_4, F_5$ are all true.

If $F_5$ were the complete description of everything there is to know about the common influences of regions 1 and 2, then we assume that probabilities would factor as follows:

$P(F_1 \wedge F_2\ |\ F_3 \wedge F_4 \wedge F_5) = P(F_1\ |\ F_3 \wedge F_5) \cdot P(F_2\ |\ F_4 \wedge F_5)$

I don't think that such a factoring is a law of probability. It's an additional assumption, it seems to me. It certainly holds in any deterministic model, and it holds in the simple sort of local hidden variables models that one is likely to come up with. But whether it holds in every possible local hidden variables model, I'm not sure. I suppose you could just use it as the definition of a local hidden variables model.

11. Jul 9, 2014

### TrickyDicky

Ok, I see what you mean and it is actually not in contradiction with my point.
The theorem by itself is not about particles but about outcomes, it is the situation when those outcomes are identified as being the outcomes of the action of particles, which is almost always assumed in the analysis made by phyisicists that I'm referring to.
The theorem itself is not specific about particles or it would't allow nonlocal interpretations of the outcomes.

12. Jul 9, 2014

### DrChinese

Local, I'm sure. I think the idea was that there could be forces, particles, or whatever present that could take shapes or forms without specific limit as long as it led to a specification of the system which determined an outcome of a counterfactual measurement. Suppose it was a wave and not a particle? What if there were additional unknown particles present? So no particular requirement other than local variables/rules/forces/waves/particles/etc.

13. Jul 9, 2014

### bhobba

First you need be precise on what you mean by realistic object.

Start out understanding what naive reality means in a physical sense. It means properties exist independent of observation. Bells theorem, basically, says you cant have naive realism and locality. You can get rid of one or the other - or even both - but you can't have both.

You are the only one that can determine the relation of your conception of 'realistic' to this. Personally I wimp out and take it as what the math directly tells me - which in QM is both naive realism and locality is kaput - but that's just me.

The version of locality I adhere to is the cluster decomposition property of QFT. But that only applies to uncorrelated systems - correlated systems are another matter entirely - that's where locality goes bye bye - well can go bye bye depending on your view of entanglement. Even though I believe the state is simply a theoretical device to help us calculate long term averages, and entanglement in that view is just as theoretical, the instantaneous breaking of it in bell type experiments still strikes me as locality is still gone. But Bell says you can keep it if you want.

Thanks
Bill

Last edited: Jul 9, 2014
14. Jul 9, 2014

### .Scott

I don't think you're describing the connection between the many-worlds theory and the Bell inequality very well.

A major point behind the many-worlds theory is that whenever there is a QM "coin flip", you create a "heads" universe and a "tails" universe. But the Bell Inequality brings into question whether there really ever are any such coin flips.

Let's look at how many-worlds sees a common Bell experiment. Our two entangled particles A and B, head towards their detectors. In the case of A, we decide to measure along a 30 degree axis - and so as particle A reaches detector A, the universe splits with one getting spin up and the other spin down. Now consider particle B. B will reach detector B without ever knowing what is happening to A. From B's point of view, the world-splitting created at detector A hasn't happened yet. Just before B reaches its detector, we will set detector B to measure either 30 degrees or 120 degrees. If we pick 120 degrees, things are simple. Both A and B create an independent world split and it doesn't really matter which is first.

But if we pick 30 degrees, we have a bit of a problem: both detector A and B need to split the universe in exactly the same way. One solution would be to allow the split at both A and B to occur and to allow all four worlds to be created, but then we would need to erase two of those worlds once the results between A and B were compared. Alternatively, we can allow the splitting at A and B to be coordinated - using non-local mechanisms.

What the Bell Inequality shows us is that as long as we follow the rules of QM, there will be non-local coordination of apparently random events. This takes an awful lot of wind out of the many-worlds sails. Why do we need to split the universe when we know that there are non-local influences on what we think are random events? We already have an explanation for the apparent randomness - that something or everything, anywhere, anytime in the universe is fair game for deciding QM coin flips.

15. Jul 9, 2014

### TrickyDicky

Ok, all those I consider local realistic objects or individual constituents in the words of the Stanford reference above. If those kind of objects are all what is rejected when it is stated that Bell's theorem proves local realism is incompatible with what we observe in quantum experiments I completely agree. But sometimes a broader concept of local realism is used, what I'm saying is that rejecting that broader concept is a bit of a case of throwing the baby with the bathwater.

For intance, is a classical field in the list of things you consider something local? Because I've seen it sometimes conceptualized as local realistic in the Einsteinian sense, and sometimes as something nonlocal.

16. Jul 9, 2014

### atyy

I've wondered about it too. Let me give an example of a question I'd like to know the answer to: Is non-relativistic Newtonian gravity local or nonlocal in the sense of the Bell theorems?

17. Jul 9, 2014

### Water nosfim

The many worlds are conected in the past , and conected throw colapts , there no randomnes , you just dont see how the futer afect the " randomnes " now by multiply hidden many worlds

18. Jul 9, 2014

### TrickyDicky

No, but QFT in the sense of effective theory or operational tool I don't think that one can reliably answer that question, it doesn't have a well defined ontology, not even enough to have interpretations like quantum mechanics has.

19. Jul 9, 2014

### atyy

There are actually a couple of questions regarding QFT and the Bell inequalities. One is that "local" in QFT is usually defined as operators commuting if they are spacelike, whereas in QM one usually assumes operators that factor according to a tensor product. So the question is whether the Bell inequalities are violated to the same extent in QFT and QM. This is called Tsirelson's problem and is only partially solved.
http://arxiv.org/abs/0812.4305
http://arxiv.org/abs/1008.1142

20. Jul 9, 2014

### stevendaryl

Staff Emeritus
I don't agree with your characterization of many-worlds. The "splitting" is not something that propagates from one point to another. And there is no useful notion of the "number" of possible worlds--that number is always infinite. And there is no need to "erase" possible worlds.

In MW, there is a wave function for the entire universe, and it evolves according to Schrodinger's equation in a continuous fashion--there's no collapse and no splitting. But this single universal wave function can be interpreted as a collection of possible worlds where different things happen.