# A Bell's theorem mathematical content

1. Mar 30, 2017

### RockyMarciano

Most discussions about Bell's theorem meaning get at some point entangled in semantic and philosophic debates that end up in confusion and disagreement. I wonder if it could be possible to avoid this by reducing the premise, the basic assumption to its bare-bones math content in algebraic/group theoretic/geometric terms trying to abstract it as much as possible from the physical or epistemologic/ontologic implications at a first stage.

I suggest referencing the original 1964 paper by Bell and its numbered equations, the basic premise is summarized in (1), and later in (14) with more pertinent details about the unit vector c.
IMO it establishes the spinorial representation of rotations(double universal cover SU(2)->SO(3) ) as something perfectly grounded on the mathematical structure and topology of the orientable manifold spaces admitting such spin structure, used in all classical physics' theories.
But maybe I'm missing something, or this is not the best way to characterize the main assumption from wich the probabilities are constructed?

2. Mar 30, 2017

### stevendaryl

Staff Emeritus
I'm a little confused by what you're saying. There are three different parts to Bell's proof that QM cannot be explained by a local hidden-variables theory:
1. Derive an inequality about the correlations of distant measurements.
2. Derive the quantum predictions for the correlation in the EPR experiment.
3. Show that 2 violates the inequality in 1.
Are you asking about #2: how to derive the quantum predictions for EPR? I would think so, because #1 has nothing to do with spinors. But if you're talking about #2, what do you mean "the main assumption from which the probabilities are constructed"? The derivation in #2 is just applying the rules of QM, for the special case of 2-component spinors. So I don't know what you would call "the main assumption".

3. Mar 30, 2017

### RockyMarciano

Actually my purpose was to center on the mathematical specification that leads to 1. That's why I mentioned equation (1) in Bell's paper. From that mathematical setting the probabilities and inequality are derived. so I figured it would be good to clarify this starting point avoiding preconceptions.

I get the impression the relation with (2 component) spinors comes from equation (1) in Bell's paper: $A(a, \lambda)=\pm 1 B(b,\lambda)=\pm1$ with outcomes A, B obtained from the iner product of spin component in the particular orientation and a unit vector) and the previous explanatory paragraphs, assuming the spin structure of the Pauli matrices representation for the measurements A and B, basically by implementing the results of spin measurements for arbitrary orientations, that is determining the isomorphism between SO(3) spatial rotations and SU(2) modulo plus or minus the identity. But if there is a more accurate description of the initial mathematical premise what would it be?

Last edited: Mar 30, 2017
4. Mar 30, 2017

### rubi

Bell's theorem (in the CHSH version for simplicity) at full rigor goes as follows:

Theorem (Bell). Let $(\Lambda,\Sigma,\mu)$ be a probability space and $A_\alpha, A_{\alpha'}, B_\beta, B_{\beta'} : \Lambda\rightarrow\{-1,1\}$ be random variables on $(\Lambda,\Sigma,\mu)$. Then $$\left|\left<A_{\alpha} B_{\beta}\right>+\left<A_{\alpha} B_{\beta'}\right>+\left<A_{\alpha'} B_{\beta}\right>-\left<A_{\alpha'} B_{\beta'}\right>\right| \leq 2 \text{.}$$
Proof. For every $\lambda$, either $\left|B_{\beta}(\lambda)+B_{\beta'}(\lambda)\right|=2$ and $\left|B_{\beta}(\lambda)-B_{\beta'}(\lambda)\right|=0$, or exchange $2$ and $0$. Hence, we have:
$\left|\left<A_{\alpha} B_{\beta}\right>+\left<A_{\alpha} B_{\beta'}\right>+\left<A_{\alpha'} B_{\beta}\right>-\left<A_{\alpha'} B_{\beta'}\right>\right|$
$=\left|\int_\Lambda A_{\alpha}(\lambda) B_{\beta}(\lambda)\,\mathrm d\mu(\lambda)+\int_\Lambda A_{\alpha}(\lambda) B_{\beta'}(\lambda)\,\mathrm d\mu(\lambda)+\int_\Lambda A_{\alpha'}(\lambda) B_{\beta}(\lambda)\,\mathrm d\mu(\lambda)-\int_\Lambda A_{\alpha'}(\lambda) B_{\beta'}(\lambda)\,\mathrm d\mu(\lambda)\right|$
$=\left|\int_\Lambda \left(A_{\alpha}(\lambda) B_{\beta}(\lambda)+A_{\alpha}(\lambda) B_{\beta'}(\lambda)+A_{\alpha'}(\lambda) B_{\beta}(\lambda)-A_{\alpha'}(\lambda) B_{\beta'}(\lambda)\right)\,\mathrm d\mu(\lambda)\right|$
$\leq \int_\Lambda \left|A_{\alpha}(\lambda) B_{\beta}(\lambda)+A_{\alpha}(\lambda) B_{\beta'}(\lambda)+A_{\alpha'}(\lambda) B_{\beta}(\lambda)-A_{\alpha'}(\lambda) B_{\beta'}(\lambda)\right|\,\mathrm d\mu(\lambda)$
$\leq \int_\Lambda \left|A_{\alpha}(\lambda)\right| \left|B_{\beta}(\lambda)+B_{\beta'}(\lambda)\right|\,\mathrm d\mu(\lambda)+\int_\Lambda\left|A_{\alpha'}(\lambda)\right| \left|B_{\beta}(\lambda)-B_{\beta'}(\lambda)\right|\,\mathrm d\mu(\lambda)$
$= \int_\Lambda 2\,\mathrm d\mu(\lambda) + \int_\Lambda 0\,\mathrm d\mu(\lambda)= 2$

That's all. No fancy assumptions about spinors or group theory are needed. QM violates the assumptions of the theorem by not requiring the observables $\hat A_\alpha$, $\hat A_{\alpha'}$, $\hat B_\beta$ and $\hat B_{\beta'}$ to be representable as random variables $A_\alpha, A_{\alpha'}, B_\beta, B_{\beta'} : \Lambda\rightarrow\{-1,1\}$ on a probability space. Non-local theories like BM violate the assumptions by introducing dependence on non-local parameters ($A_{\alpha\beta}, A_{\alpha'\beta}, \ldots, B_{\alpha\beta}, B_{\alpha'\beta},\ldots : \Lambda\rightarrow\{-1,1\}$).

5. Mar 31, 2017

### RockyMarciano

As I said for the purposes of this thread I am interested in describing mathematically, in the simplest manner without loss of generality, exactly what leads to the construction of the space of probabilities in the quote above that is used to construct the Bell inequality just by applying classical probabilities so that the expectation value when measuring A and B can be written $E=\int d\lambda p(\lambda) A(\vec a,\lambda) B(\vec b,\lambda)$, and to QM predictions by applying the Born rule.

The above quoted probability space and random variables assignment can be described for single measurements without loss of generality by the double cover mapping from the unit 3-sphere to the Bloch sphere, that identifies mixed states(interior of the Bloch sphere with pure states(at the sphere's surface) in quantum mechanical terms and is also compatible with the EPR experiment.

If we could agree with this description, or any other that anyone suggested, we'd have a good way to avoid the usual problematic word descriptions of the conditions of the theorem in terms of philosophically charged and semantically ambiguous terms like "local","realism", "definiteness" etc...

6. Mar 31, 2017

### stevendaryl

Staff Emeritus
I don't understand what you're talking about. What Bell is assuming is that
1. There is some variable $\lambda$ (ranging over some set of values $\Lambda$) with a corresponding probability distribution $p(\lambda)$.
2. There are two functions $A(\lambda, \vec{a})$ and $B(\lambda, \vec{b})$ that for any value $\lambda$ in $\Lambda$, and for any unit vectors $\vec{a}$ and $\vec{b}$:
• $A(\lambda, \vec{a}) = \pm 1$
• $B(\lambda, \vec{b}) = \pm 1$
What does that have to do with the Bloch sphere?

There is a technical assumption about measurability involved, which was pointed out by Pitowsky, which is that for all $\vec{a}, \vec{b}, X, Y$

$\{ \lambda | A(\lambda, \vec{a}) = X \wedge B(\lambda, \vec{b}) = Y \}$

is a measurable set.

7. Mar 31, 2017

### rubi

There is no construction of a space of probabilities or anything like that. The assumptions just state that we are in the setting of classical probability theory and the expression for the correlations is just a definition. This suffices to prove the inequality.

Sorry, but it doesn't appear like know what these words mean. At least you are using them in a way that makes no sense.

The theorem I stated above contains absolutely no reference to any philosophical terms. It's all precise mathematics.

This technical condition is contained in my post through the requirement that $A_\alpha,\ldots$ should be random variables. Random variables are required to be measurable functions, which means that the preimages of measurable sets are measurable. The sets you wrote down are exactly those preimages.

8. Apr 1, 2017

### RockyMarciano

I would think that implicit in those assumptions is the possibility that whatever mathematical space of whatever theory those assumptions apply to, it allows for the EPR experiment elements and outcomes to be possible, right? For instance it is implicit in the assumptions the orientability of any theory's manifold that we might want to consider as compatible with the inequality derived from those assumptions.
And since the assumptions were constructed to be able to meet the requirements of the EPR experiment that involved the concept of spin,and measurements in arbitrary orientations in three dimensions of the said spin components, I don't think it is so outrageous to infer that the mathematical space of the theories that are compatible with the assumptions in the theorem must admit a spin structure. Also admitting classical probabilities is implicit as is in any mathematical construct that has the usual logic as foundation.

And that is what goes into the above quoted assumptions and definitions implicitly, since otherwise they wouldn't serve the purpose of minimal requirements to generalize the EPR experiment and test those minimal mathematical assumptions to describe the gedanken experiment and their probabilistic consequences against different theories predictions and real experiments.

This seems very basic but if we can't develop an understanding of the point of departure of the OP it gets really hard to move forward.

9. Apr 1, 2017

### HARSHARAJ

I have recently done a research work on hidden variables in QM, this is a reference, quite simply written and explained.
Lorenzo Maccone, A simple proof of Bell's inequality, American Journal of Physics 81, 854 (2013).

10. Apr 1, 2017

### stevendaryl

Staff Emeritus
I'm having trouble understanding what you're talking about. Bell's theorem doesn't have anything to do with orientability of manifolds. It doesn't mention manifolds at all. It does assume that there is a notion of events being causally separated. That is, it assumes that there is no causal influence of Alice's measurement on Bob's result, or vice-versa. Yes, it's certainly possible that this assumption is false for actual EPR experiments, because of some weirdness such as FTL communication, or wormholes, or back-in-time causation.

11. Apr 1, 2017

### HARSHARAJ

I would like to mention something. Bell's Whole paper, in a nutshell is like: "Let's calculated expectation value of product of two components of spin(sigma1.unit vec a and sigma2.unit vec b)with a local hidden variable model. Now let's allow the magnets to rotate freely so that by any means measurement done at one won't have any causal effect on the other(experimental locality condition), we also take a pair of entangled particles, a pair of spin half particles in the singlet spin state. Now 1. if EPR were to be correct then these local hidden variables had determined the properties of the particles prior to the experiment, since they were focusing on the causal relationship, and hence the expectation value must be if not equal but an approximation of the original result, but 2. if they are wrong then it will be the opposite, and as Bell showed that 2nd one is the right choice". Of course there is a non local action going on and Bell's paper was chiefly to frame a model as per EPR thought and to test it. That it, a simple 5 page paper.

12. Apr 1, 2017

### Mentz114

Also here https://arxiv.org/abs/1212.5214

13. Apr 1, 2017

### HARSHARAJ

Yes, it's available at the library too, that one was the exact citation by the way, but nevertheless, it was one of the primary paper i read to get into the subject, and it's quite simply written though the vein diagram explaining was a bit superfluous, I came up with a little math for the same, but point is if one goes by I think subsection is "Bell's Theorem" and define two terms and goes through the appendix the whole inequality will be cleared out.

14. Apr 1, 2017

### RockyMarciano

The theorem simply stated: " No 'local realist' physical theory can ever reproduce all QM's predictions". And my goal in this thread is just to pinpoint exactly what is the minimal mathematical structure of the physical theory referred to as "local realist" in the theorem, wich is what the theorem assumes as premise. I really don't know what is hard to understand here.
Sure, nowhere in the theorem are manifolds explicitly mentioned, here is where the concept of implicit enters. You don't think the physical theories alluded in the theorem need the concept of manifold or space? That would be odd, both classical mechanics and relativity to mention some use such mathematical concepts.
How do you suggest to have components of spin measured in different orientations if that space is not orientable? What kind of EPR physical theory would that be?

15. Apr 1, 2017

### rubi

I explained it in my post #4. The minimal mathematical structure required is precisely:
As you can see, it says nothing about manifolds or spin structures or orientations or other fancy mathematics (all of which you have been using in non-sensible way). If you think any additional assumption is needed, then you should be able to point to a line of the proof in my post #4, which you think doesn't follow from the assumptions I quoted.

16. Apr 1, 2017

### stevendaryl

Staff Emeritus
Well, it has nothing to do with orientability of manifolds.

17. Apr 1, 2017

### RockyMarciano

No, I guess you are not paying attention to what I'm writing. The probability space is indeed enough to derive the inequality. But I'm not talking about that or the theorem's proof.
I'm talking about the physical theories referred to in the theorem, the probabillty space and definition of the variables describe concisely the experiment but it is not enough to describe the putative physical theories in which such experiment can take place themselves, rather they limit the possible structures of those physical theories as those in which experiments describable with that probability space and variables can take place, that's what this thread is about.
I'm sorry that you consider objects like manifolds that have been around in physics for over a century as fancy. Not much I can do about it but advise you to read.

Hmm... if you insist that physical theories modelled by non-orientable manifolds are compatible with EPR experiments...good luck to you.

Last edited: Apr 1, 2017
18. Apr 1, 2017

### rubi

The theorem does not refer to any theories. It's a mathematical theorem about certain functions on a probability space.

The inequality holds in any theory, which has four functions $A_{\alpha},A_{\alpha'},B_{\beta}, B_{\beta'}$ with values $\pm 1$. Yes, it is really that simple. The values $\pm 1$ don't even need to refer to spin measurements. They can might as well be answers to yes/no questions or whatever you can think of that can assume values $\pm 1$ or be encoded as values $\pm 1$.

I suppose you should find them fancy, since I can tell from your posts that you have exactly zero clue what these words mean. Otherwise, you would not use them it such a completely meaningless way. None of your sentences makes any mathematical sense, not even with lots of goodwill. So really you should be doing much more reading (browsing Wikipedia and randomly picking up words won't suffice).

Yes, it is perfectly possible to perform an EPR experiment on a non-orientable spacetime.

19. Apr 1, 2017

### stevendaryl

Staff Emeritus
I insist on avoiding using the phrase "non-orientable manifold" when talking about Bell's theorem.

20. Apr 1, 2017

### stevendaryl

Staff Emeritus
I'm wondering if he has been reading Joy Christian.