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Posted by on Feb 17, 2011 in Quantum Mechanics Research | 1 comment

009 Disproof of Bell’s Theorem

Blog 009: Bye-Bye Bell: the end of an era.

Hello anyons: the new era


Han Geurdes Disproves Bell’s Theorem

Joy Christian Disproves Bell’s Theorem Parts 1 and 2

My Sub-quantum 2D spins are Anyons


Bryan Sanctuary

In this entry I will present the evidence that shows that Bell’s inequalities have no consequences for understanding quantum vs. classical correlations. I will also show that spin is a two dimensional anyon.

After a summary, two recent proofs are presented that show BELL’S INEQUALITIES are incorrect due to an over simplistic treatment of spin by Bell.  Finally I relate these approaches to my sub-quantum 2D spins.

One of the proofs is by Han Geurdes in Holland and the other by Joy Christian in England.  These conclusions support my LHV Local Hidden Variable theory that I have presented in blog entries 006 and 007.

I am going to state it unequivocally right at the beginning of this entry what we have found:

Bell’s Theorem is wrong.

His famous inequality, derived in this paper of 1964, upon which Bell’s theorem is based has finally been shown to be incorrect.  Bell’s Inequalities certainly tell us nothing about quantum mechanics or Local Hidden Variables.  They have misled the physics community for the past 50 years. It is the end of an era.

Recall:  Bell’s theorem states:

“No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics”.

This statement is wrong.

Why do people believe Bell’s Inequalities are correct?

As much as scientists have tried to disprove Bell’s theorem, no one has been able to find any errors, at least until now. In my blog 007a I discussed the EPR coincidence experiments upon which the concept of non-local connectivity between EPR pairs is established.

No one understands non-locality.

And no one has been able to find a non-local hidden variable theory to explain the data, and no one ever will. Therefore people are left to accept that quantum mechanics as the most fundamental theory.  Hence physicists accept that Nature is random and non-local.

Non-locality is supposed to be the basis for quantum teleportation. But all this goes against our basic intuition.  Intuition alone, however, is not enough to reject BELL’S INEQUALITIES.  We need a logical proof and alternate ways to explain the data.

Can you understand how “teleportation” works?  Bye the way, sorry to all you pop science fiction buffs, Scotty cannot beam you up, and no one ever will.

We all have trouble understanding Nature. Unless you feel comfortable with non-locality, and it makes sense to you, you should keep the question open. One of my goals here is to dispel those concerns.

However,   today many people state that non-locality as an example of quantum weirdness.  As I mentioned in Blog 006 this word has moved into physics, especially quantum information theory.  It is a word that sums up the confusion of the physics community on this issue.

Here is an example in the prestigious journal Nature written by by Gregor Weihs who in 1998 produced the most definitive EPR coincidence experimental data.  Let us see what he says in the Nature paper:

“Unsurprisingly for connoisseurs of quantum weirdness, entanglement—the mysterious holism in which the state of one quantum object is tied to the state of a second, separate object—is the key to the trick”

Weirdness, mysterious, tricks are words that should not be used to describe a physical phenomenon.

There are no tricks in Nature, only our ignorance as to how Nature works.  Well finally we can banish quantum weirdness and its associated tricks to the annals of Science Fiction forever.  This is why:

Well look at the next part of this entry, 009a in which I describe what Han Geurdes has found.


Blog 009a: Bye-Bye Bell: the end of an era.

Hello anyons: the new era

Blog 009a Han Geurdes disproves Bell’s Theorem

Bryan Sanctuary

In this entry that follows 009, I present the evidence that shows that Bell’s inequalities have no consequences for understanding quantum vs. classical correlations.  Han Geurdes has published a paper (Adv. Studies Theor. Phys., Vol. 4, 2010, no. 20, 945 – 949) which shows usual classical probabilities lead to violation of Bell’s Inequalities. I will say it again, Classical probabilities, not quantum, violate BI.  That is a change.

Han has found that classical probabilities are not restricted to classical physics.  They work for quantum mechanics too. If it is all classical, where are the quantum correlations BI are supposed to quantify?  There are none.

Here’s the abstract:

“Mathematics equivalent to Bell’s derivation of the inequalities, also allows a local hidden variables explanation for the correlation between distant measurements.”

That could not be clearer.  He has proved non-local correlations are not needed. He talking about “ LOCAL hidden variable, not non-local ones.

And let us look at the first paragraph:

“To many the experimental verification of the violation of inequalities is sufficient evidence for completeness of quantum theory.  Here it will be demonstrated that Bell’s form of local hidden correlation can be transformed to violate Bell’s inequalities. “

Han’s results are totally against what Bell obtained.  We can conclude from Geurdes’ paper that:

“Bell’s inequalities do not differentiate between local realism and quantum mechanics. Hence the experiments of Aspects, Weih’s etc. do not differentiate either.”

That is, Bell’s Theorem is wrong. It is a major result.

Bell is supposed to have showed that the correlation between classical events cannot exceed 2.

But qm violates them, why?  Because Bell made an error, and I will discuss that error in due course.

In the end, it is shown that quantum spin satisfies the corrected version of BI. In fact the correct form of BI has a limit of 2√2 for spin not 2.  Even this is only a special case and BI might take some entirely different value because it fundamentally depends upon the geometry of the data.  I will be clearer on that point in later on in this blog: 009b and c.  But 2√2 is correct of spin ½ not 2.

However when things are simpler than the spin data, the version of BI with a limit of 2, does work.  For example, correlation between different sets of numbers satisfy BI.  They do work in our macroscopic classical world.  They have been tested many times.

For example, think of a set of classically correlated properties:  a population of people have lots of correlation: say, color of skin; color of eyes and color of hair.  If there are correlations between these properties, then they will satisfy BI.  No classical correlations, according to Bell, can exceed the value of 2.

But it is well known that if the filters are set as shown here in the EPR coincidence experiments then these quantum results violate Bell’s Inequalities. (Here I use a spin ½ not photons of spin 1 whence the angles are halved).

No one has been able to explain that “quantum” correlation of 2√2-2=0.828  except to surmise it is due to entanglement:  thatmysterious holism in which the state of one quantum object is tied to the state of a second, separate object—is the key to the trick

Well no, entanglement is NOT key to the trick. Han Geurdes has shown that classical probabilities do the trick.

Quantum mechanics for spin ½  then satisfies the modified BI and so Bell’s theorem is toast.

LHV can exist and can complete quantum mechanics.  Einstein was right:

“He does not play dice!”

However you can be sure that Han Geurdes paper will be scrutinized because there are a lot of people out there who will not like this result.  Unfortunately like and dislike are subjective comments and of little consequence.  People will have to show there are objective errors.

But where did Bell go wrong?   The results of Geurdes are consistent with those of Joy Christian at Oxford, and my LHV model, so we turn to his work now.

In my next entry 009b we will find the reason for Bell’s error, and Christian’s elegant way of correctly account for the elements of physical reality that Bell missed.


Blog 009b: Bye-Bye Bell: the end of an era.

Hello anyons: the new era

Blog 009b: Joy Christian disproves Bell’s Theorem Part 1

Bryan Sanctuary

Now that the problem has been stated in entry 009 and the published results of Geurdes reviewed, along with some consequences in entry 009a, we now turn to the beautiful way in which Joy Christian formulated Bell’s error.

In his disproof of Bell’s theorem, Christian indeed derives the correct value of 2√2 without quantum mechanics and for classical systems. This is what Bell should have got. Christian and Geurdes have found Bell’s errors in two independent studies.

Bell’s error was made in his first equation:  lets look .

This equation of Bell’s does not adequately account for all possible elements of physical reality of a spin.  It simply says a spin can take values of +1 and -1, but that ignores the fact that spin is oriented in some way.

Notice here that Bell says that a product of these two components should agree with quantum mechanics, but that, he says, is impossible. That statement is wrong. Christian showed this by the use of Geometric algebra, also known as Clifford Algebra, to categorize the elements of physical reality of a spin.  Bell missed a lot as we will see.


In a nut shell, all that BELL’S INEQUALITIES show is that the correlation between two points on a line is different between two points on a sphere.  The points on a line could represent the data from hair, eye and skin color all right, but not if the data comes out in a sphere like spin data does. Nonetheless, classical probabilities work on spheres as well as lines, but with a different algebra.

Bell’s first Equation misses important elements of physical reality by ignoring the sphere of data in favour of scalar data.  Spin indeed can take 2 values of +1 and -1 when measured, but these values can point anywhere on the unit sphere, not just along a line.


Let us read the main points of his abstract:

“It is shown that Bell’s theorem fails for Clifford algebra ……. This is made evident by exactly reproducing quantum mechanical expectation value….by means of .. local, deterministic …….variables, …….. Since Clifford product of multivector variables is non-commutative in general, the spin correlations ………violate the CHSH inequality just as strongly as their quantum mechanical counterparts. “

We’ll come back to multi-vectors.

The most important point is that there are situations when classical variables do not commute, whereas Bell’s always do. Christian finds more elements of physical reality, and these are exactly the ones that Bell missed.

And yes, he has had critics too, and he has replied to them all.

Now Clifford algebra is also called geometric algebra and is concerned with the algebra in different geometric spaces.  However the point I want to make is that Bell’s beables are geometrically a unit sphere of zero dimension, denoted by S0.  That is just two points of +1 and -1 on a line.  That is all Bell’s uses.

In contrast to Bell, Christian realized that spins can point in two directions over the surface of the usual unit sphere in the real world and this is called a 2-sphere S2 .

Bell’s choice has no orientation, nor left and right handedness, while the 2 sphere of Christian does. I’ll get to that soon.

To see this more clearly, if you look at the standard EPR experiments then each spin has a data set that forms a sphere S2. At each filter, the spin is pointing in some direction on that unit sphere. EPR experiments seek the correlation between the two to give a coincidence data set. Now that data, obtained by correlating the spin data from two 2-spheres giving the coincidence data set that obeys the geometry of a 3-sphere, S3. This is quite different from Bell:


The data set used by Bell in his first equation has too few elements of physical reality whereas Christian has counted them correctly.  Bell misses S2-S0 elements of physical reality.

What does this lead to?

It makes sense that the algebra on different spheres is different.  You know that from when planes fly polar routes. An S2 sphere corresponds to Clifford Algebra Cl(3,0) and this is represented by the Pauli spin matrices.  Bell’s beables are 0-spheres, or scalars.  What are Christian’s?

Christian’s beables are Bell’s Inequalities-vectors and tri-vectors relevant to the 2-sphere, not scalars like Bell used. These two cases can be compared and contrasted.  It is the Bell’s Inequalities-vector choice that makes the difference.

An example of a Bell’s Inequalities-vector is given by the product of two Pauli spin matrices, and a tri-vector is the product of three vectors in terms of the 2×2 Identity matrix.  Note these play important roles in this approach, and the LHV theory I discussed in blogs 006 and 007.

Just a brief primer on Algebraic geometry.  Let’s drop the indices on A and B.

They are still Bell’s Inequalities-vectors. Take the product of two and get this equation that exploits symmetry between Bell’s Inequalities-vectors in terms of the dot and wedge products defined here. The wedge operator is a generalization of the vector cross product.  Whereas the scalar beables of Bell commute, the Bell’s Inequalities-vector beables of Christian’ on a sphere do not commute.   This is an important difference.

Now I mentioned that the Pauli spin operators represent the Clifford Algebra Cl(3,0).  This general expression reduces to the well known relationship between Pauli spin operators, in terms of the Kronecker delta function (i must equal j) and the Levi-Civita totally antisymmetric third rank tensor where if any two components , i,j,k are equal, then epsilon is zero. I,j,k can take values of x,y,z. Epsilon is equal to +1 for an even permutation of i,j,k, and -1 for an odd permutation, i,k,j.

This can also be written as a tri-vector ±I.

This double valued identity operator points in all direction over the unit 2-sphere, so it has direction and orientation. Let’s remove the sphere to see things a Bell’s Inequalitiest more clearly.


Equatorial planes are element of a 1 sphere S1. It follows easily that the identity has two values of +1 and -1 and these two tri-vectors lie along the spin direction.  Look at a different orientation of spin

Once again we get two values, but for a spin now pointing in a different direction.  That is, the elements of physical reality chosen by Christian corrctly specify the possible values a spin can take in different directions.  It makes a lot of sense.

Now we are ready to look at one of Christian’s paper where he discusses this approach. Let us jump to the relevant part,

Ok, well it looks a Bell’s Inequalitiest complicated but it really is not.  First we see the CHSH form of Bell’s Inequalities which can be written in terms of the non-commuting Bell’s Inequalities-vectors  and the filter angles used in EPR experiments, a and b. Now A and B describe spins that are far apart, so since we are dealing with LHV, then A and B must commute. That is a locality assumption.

However A does not commute with A, and B does not commute with B.  If we were using Bell’s scalars, then everything commutes, but not here.  Working out the details, that I skip, an expressions is found in terms of the filter settings, a and b for Bell’s Inequality.  These are the usual vector dot and cross products and they make the difference by giving the correct value of 2√2.  Clearly if those terms did commute, then the answer is what Bell got, 2.  The correct answer is 2√2 for spin ½ .

Let us look for the filter settings that maximize the square root term. It is clear that a must be perpendicular to a’, and b must be perpendicular to b’.  So far so good. We see this in the filter settings that are experimentally found that maximize the CHSH form of BELL’S INEQUALITIES.  But we can go further.

We can write this expression another way using a Bell’s Inequalitiest of vector algebra.  The only way to choose the angles to maximize the correlation is shown here and they are exactly the settings found experimentally and indeed from quantum mechanics.  Note we have used NO quantum mechanics here: only classical algebraic relations on a sphere.  So let us put this together:

Plug in the angles to give the second term of cross products equal to 1.  Christian gets the correct value of 2Ö2  that quantum mechanics predicts and experiments confirm.  In addition the relationship between the filter settings are consistent to the settings that maximize the correct form of BELL’S INEQUALITIES.

Bell’s error was to make the incorrect assumption about spin, thinking it can only take two real values of +1 and -1, where in fact it can take +1 and -1 anywhere on the surface of a sphere. So now we can understand my earlier statement.

Bell’s Inequalities tell us simply that the correlation between two point on a line is different than from between two points on a sphere; respectively 2 and 2√2 .

Finally I will relate my 2D LHV model to Geurdes and Christian’s work, and show that spin is fundamentally an anyon.

Bye-Bye Bell: the end of an era
Hello anyons: the new era

Bryan Sanctuary

Sub-quantum spins are anyons

The repudiation of Bell’s Theorem is a result of a three pronged attack.  First Han Geurdes has shown from a fundamental mathematical approach that quantum correlations do not exist.  BELL’S INEQUALITIES can be violated using classical probaBell’s Inequalitieslities.

Second, Joy Christian has demonstrated that Bell missed a lot of elements of physical reality of a spin.  Including these led to the correct form of BELL’S INEQUALITIES for spin, using no quantum mechanics.  Both of these theories also show that a LHV theory MUST exist to complete quantum mechanics.

My 2D spin discussed in this blog serves as completing quantum mechanics in the sense that EPR envisioned in 1935.  So in this last part of blog 008, I will show how my LHV theory is consistent with Geurdes and Christian, and shows that spin at the most fundamental level is an anyon.

To see the connection to my LHV theory, let us start with the 2 sphere as Christian did, and we see an equatorial planes  which are 1-spheres, S1. Now the unit tri-vectors of Christian define the two spin values as we have seen.  But this can point in any direction in that sphere

There are an infinite number of planes and if you have read my blogs 006 and 007 then each change of coordinates corresponds to a different spin orientation and a different spin microframe x,y,z.

Topologically  the 2-sphere, S2,is made up of equatorial planes which are 1-spheres, S1. So let us look at one plane of the infinite number possible.

I am going to write this slightly differently.  Whereas Christian used a tri-vector in S2 to orient a spin, the 2D sub-quantum spin I have talked about in blogs 006 and 007, uses a Bell’s Inequalities-vector in S1 like this:

So now we see the 2D spin oriented in its microframe.  I have used a right hand rule here so we see that this system has handiness, as well as magnitude, the plane zx, and orientation, by the Bell’s Inequalities-vector is

These are exactly my 2D spins.  In that 2D plane and when no interactions are present, the two components of angular momentum are indistinguishable, and this produces the new state of matter, the exchange √2 spin.


This relates my sub-quantum LHV method to Christian’s and is consistent with Geurdes conclusion that classical probaBell’s Inequalitieslities work.

Let us just rotate them again.

I was talking about this with my colleague in Physics at McGill, Keshav Dasgupta whose field is String Theory, and in our discussion he said that these look a lot like Anyons—anyons exist in a plane (my spin is 2D is in a plane), show fractional or irrational quantum numbers (mine is √2) and display a phase (one spin can be rotated to the other and their orientations differ by a phase, so that when two anyons are interchanged, the phase is neither +1 (bosons) nor -1 (fermions) but any phase, so they are called anyons.

A lot is already known about them, but I will defer that to my next entry. We can summarize the difference between quantum mechanics and sub-quantum local hidden variables of anyon spins.

The quantum view of a spin is a point particle described by the three components of a spin vector that exists both in the presence and absence of a probe. Sub-quantum mechanics replaces this with an anyon spin with a 2D structure that has area, magnitude and handiness.

Bye-Bye Bell: the end of an era Hello anyons the beginning of a new era.

So ok, all this is a Bell’s Inequalitiest of a disaster for Bell, but good news for Einstein. He deserves to look smug, he told us all this must be true in 1935 and finally he has been shown to be correct. Of course Bell’s error has nonetheless helped to focus our understanding, but don’t feel bad for him.  We all make errors, I make a lot, and we should learn from them.

Bell found an error in the work of the famous and brilliant mathematician, John von Neumann, who is also called the father of the modern computer and made enormous contributions to our understanding of quantum mechanics.  But von Neumann did make a Bell’s Inequalitiesg error in his assumptions in an important paper of 1936 which basically incorrectly concluded that no hidden variables of any kind can complete quantum mechanics.  This dashed EPR’s conclusions the year after EPR was published in 1935, and swayed many into believing that quantum mechanics is complete.

Bell found his error in 1966—so there were 30 years of confusion. But let us be reminded of what Bell said about von Neumann’s error:

“The von Neumann proof, if you actually come to grips with it, falls apart in your hands!  There is nothing to it.  It is not just flawed, it is silly! …When you translate (his assumptions) in terms of physical disposition they’re nonsense.  You may quote me on that: The proof of von Neumann is not merely false but foolish!”

Von Neumann was no fool, and neither was Bell.

Welcome to Flatland.

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