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Posted by on Jun 16, 2010 in Quantum Mechanics Research | 0 comments

007 Anomalies in coincidence probabilities in EPR data

Anomalies in the coincidence probabilities in EPR data.

Part 1:  Quantum Limit

Part 2: Sub-quantum results

Part 3a: Non-hermitian states (part 1)

Part 3a: Non-hermitian states (part 2)

Part 4: Sub-quantum ensembles

Part 5: Conclusions

Part 1:  Quantum Limit

The theme of the conference where I presented this paper in Torun Poland, held in June of 22010 is

“Quantum Channels, Quantum Information – Theory & Applications“.

It is often stated that all of quantum information theory rests upon the validity of Bell’s theorem and in this talk I will give an alternate explanation for long distant communication and quantum channels.

In particular I will show that a sub-quantum theory reproduces all the correlations of as well as accounting for the anomalies.  This sub-quantum theory is both local and realistic and it is definitely not classical.

From quantum mechanics it is only possible to calculate the EPR coincidence probabilities using an entangled state.  I will do it from a sub-quantum product state.

So that are the anomalies?  Adenier and Khrennikov re-analyzed the 1998 data of Gregor Weihs.  They find that whereas the overall correlation agrees with experiment, the joint probabilities display anomalies.  Since quantum mechanics is used to calculate these, and they do not agree with quantum mechanics, then the whole of Bell’s theorem is cast into doubt. Adenier and Khrennikov conclude:

“If we follow Bell’s reasoning then both classical and quantum models should be rejected on the basis of the present experimental statistical data.”

In this talk I will present a new sub-quantum theory which resolves the anomalies found in coincidence EPR experiments.  This is the experiment that Clauser Horne Shimony and Holt devised to test Bells’ theorem and to specifically show that quantum mechanics predictions are correct.

Let’s review this experiment again.

A source of two spins in a singlet state remains correlated upon separation.  One spin moves to Alice and the other to Bob who both have filters at specific orientations, angles theta (a) and theta(b).  Sometimes the photons pass the polarizers and sometimes they do not pass.  If both pass, the coincidence is recorded as two clicks as ++.  Likewise is neither pass the filters the coincidence is –, etc.  Over a run of many thousands of coincidences their frequency gives their probabilities.

A particular sum of the coincidences give the correlation and it is this correlation that violates BI.  a and b are the angles of the polarizers at Alice and Bob. Usually Alice’s polarizer is set to zero and Bob’s is rotated through 2pi radians.

A significant result of the sub-quantum theory is it agrees with the experimental results as a product state, with no quantum channels; no entanglement; predicts a new state of spin that quantum mechanics cannot predict and which is supported by the experimental agreement.

Not only does it agree with quantum mechanics, but it also is consistent with the anomalies found.

This experiment is the only one upon which the validity Bell’s theorem rests.

Let’s look at the results from quantum mechanics.  They are obtained from the density operator that describes an entangled singlet.  It is the outer product of the usual singlet state and it is straightforward to calculate the well known results for the correlation and the coincidences to give the usual –cos(theta) and expressions for the coincidences.

But it appears that the experimental data suggests that the coincidences do not agree with quantum mechanics even though in the sum the anomalies tend to cancel.  Contrast that with the results from the sub-quantum theory for the coincidences which are presented here (without proof for now).  It is important to accept that this is not a quantum state but a sub-quantum calculation based upon one pair of spins.  It does not correspond to an ensemble, just two particles with spin.

For comparison, the quantum mechanics result is just ½ which is obtained by ignoring entanglement.  It fails to agree with experiment and must be incorrect.  The quantum result says that the single probabilities are an incoherent superposition of many spins ½ which average to ½.  That is, there is no coherence between them if we take quantum mechanics as a product state.

This expression as a product of two spins is called a bi-particle because the two members have the same orientation and carry the same hidden variables.  That is, when the singlet state was produced at the source, the two entangled spins share common elements.  These hidden variables are:

Nz and nx, both of which are integers with values of +1 and -1.

The angle theta orients a sub-quantum structured spin in 3D space relative to the laboratory frame where the polarizers, Alice’s theta a and Bob’s theta b are oriented.

Let us maximize the probability at Alice.  It is easy to differentiate the trig terms so we need (theta(a) minus theta) =45 degrees, giving a probability of 1 for Alice’s side.  That is, for any setting of Alice’s polarizer, we pick out, or filter, only one sub-ensemble of spins.

But unless Bob’s filter is the same as Alice’s, which it generally is not, then Bob’s probability will not be maximized because his spin must have the same hidden variables, nz, nx and theta, as Alice.  So we substitute the expression from Alice, (Theta(a) –theta)=45 in Bob’s probability and we agree with the result of quantum mechanics.

Let’s plot them.

Compare again with quantum mechanics.  Alice’s filter is set to zero and bob’s is varied from –pi to +pi.   Both the sub-quantum and quantum mechanics agree but the sub-quantum theory used a specific set of hidden variables.

However it is important to note that to date, no product state, or disentangled state, has ever been able to reproduce all the quantum effects.  Since this is a local hidden variable theory, it shows that Bell’s theorem is incorrect.

There are other settings of the hidden variables other than the one that maximizes the probabilities and gives the quantum result.  Let us look at the anomalies.

These are the joint probabilities and the full correlation as extracted by Khrennikov from Gregor Weihs’ data.  Note that P++ and P- – do not coincide as quantum mechanics predicts.  Also the maxima and minima are shifted, even though these difference cancel when summed appropriately.

Now I should point out that these anomalies were noted by A. Aspect but since the interest was in the full correlation and not the coincidences, and the full correlation agrees with quantum mechanics, it took Khrennikov’s re-analysis to discover these.  Specific experiments to measure the coincidences were not performed. The agreement is therefore qualitative and more experimental data is needed to be quantitative.  Therefore I just want to show the trends.

The most dramatic anomaly is shown in the marginal probabilities which is given by P++ + P+-, and P-+ + P– which from sum each adds up to 0.5 .

We see that there is a marked difference between quantum mechanics  and experiment.

So how does my sub-quantum theory account for the anomalies? Well in a very simple and physically reasonable way which I will discuss next.

Part 2:  Sub-quantum results

Now for the second part of the talk I show how the sub-quantum theory resolves the anomalies. The resolution is very simple and physically reasonable.

As the sub-quantum particles separate, they become misaligned due to spurious local interactions or instrumental effects.  Imagine an entangled source and two sub-quantum particles separate: they are correlated only by their common orientation with no quantum channels.  As the two beams pass through prisms, pass mirrors and are filtered by polarizers, the two bi-particles can get misaligned.

Recall I call the product state of two particles a “bi-particles” if they come from the same singlet state.  In that case they must have the same values of their hidden and therefore are correlated from their source.

Now the hidden variable theta orients a structured spin in 3D space.  The anomalies arise when spurious effects misalign them.

This means that Alice’s and Bob’s orientation can get out of sync a bit which gives a small asymmetry between the two.  Let us take this as symmetric just for the sake of example.

For point zero five pi radians we see that the coincidences become separated and shifted and the marginal shows structure.  The total correlation however remains close to the quantum result.

Now with 0.1 pi radians mismatch, it can be seen that P++ and P- –  are similar to experiment.  The P+- and P-+ coincide quite well which means there is little misalignment in those arms of the experiments.  Clearly the trends are correct.

On the other hand, with a little effort encouraging agreement is found.  Here in an angle range of up to 2 pi the trends are reproduced.  P++,  P–,  P+-  and P-+.

Finally, most dramatically, the marginals qualitatively agree with the sub-quantum results.  This data supports the sub-quantum theory and, as we will see, is experimental evidence for a new state of matter.

So let us look at this sub-quantum theory in more detail.

I have mentioned before (blog 004) that I adhere to the statistical interpretation of quantum mechanics which means that although we might observe single events, we can only calculate the outcome statistically by averaging over many individual particles, i.e. over sub-ensembles.

In fact I will show that the filters in the EPR experiments pick out different sub-ensemble of spins and these sub-ensemble accounts of all the anomalies.

The statistical interpretation is the only one that considers a sub-quantum theory could complete quantum mechanics, but what are these sub-ensembles composed of?

Quantum states are composed of ontic particles. These particles are governed by the sub-quantum theory:

  • A quantum state is an ensemble of sub-quantum particles usually all prepared in a similar way for experiment-that is the statistical ensemble interpretation of the wave function.
  • The sub-quantum particles have a 2 dimensional structure rather than the point-particle quantum spin
  • Isolated sub-quantum particles display a new state of matter
  • Entanglement is resolved into product states

When considering the sub-quantum concepts, it is necessary to put your interpretation of quantum mechanics aside.  The sub-quantum theory is totally different.

Spin arises from the Dirac equation.  The free particle Dirac equation is obtained by linearizing the relativistic energy from which spin emerges as a Lorentz invariant and therefore an intrinsic property of matter.  However you cannot get the magnetic moment, say for an electron, from the free particle Dirac equation.  You have to add a probe.  In the second form of the Dirac equation, the scalar and vector potentials from the EM force are introduced.

This means that the basic symmetry of the free particle Dirac equation and the particle in the presence of a probe is different.  In the free particle case the environment is isotropic while in the presence of a probe it is anisotropic.

The magnetic moment of an electron is only obtained by using a probe and it successfully gives the correct Landé  g-factor of 2 for the magnetic moment.  This leads to the view that intrinsic angular momentum is as a vector quantity carrying angular momentum along a single axis of quantization.

It is assumed in quantum mechanics that the intrinsic angular momentum found in the presence of a probe also exists in the absence of a probe. The point of divergence between and this sub- is recognizing that it is not valid to assume that the anisotropic solution is valid for the free particle solution.  That is, the vector spin angular momentum determined by a probe does not carry over to the free particle solution because of spontaneous symmetry breaking. Symmetry goes from lower symmetry (a vector) to higher symmetry (a scalar) when the probe is removed.  This is the reverse of most symmetry breaking that is found in physics.  Usually an interaction is turned on due to a potential and the symmetry is lowered and degeneracy is lifted.  How is this done for a free spin?

It is postulated in the sub-quantum theory that the 4 dimensional Dirac spinor space of a spin ½ breaks into two non-hermitian components.  This is the basis of the sub-quantum theory presented here.

That is, the individual particles that form the ensembles that make up the wave function do not obey the usual SU(2) symmetry—they are non-hermitian.

Now this is a radical departure from quantum mechanics, but in all cases, spins states are manifest as hermitian.  However this symmetry reduction introduces additional degrees of freedom to a spin, and as symmetry breaking usually does degeneracy too.

Break here again but I will continue with part three of the talk by showing how sub-quantum particles have structure and how they lead to a new spin state and disentangle the entanglement of quantum mechanics.

Part 3:  Non-hermitian states

For the third part of the talk I discuss the sub-quantum theory in more detail which depends upon non-hermitian states.

In terms of the Pauli spin matrices, these non-hermitian building blocks show two angular momenta and a phase. This is now fundamentally different from quantum mechanics.  The quantity “s” is the state of one ontic particle.  It does not obey quantum mechanics because it is non-hermitian. This is a definition.  Unless it gives the right answers, it should be thrown out.  In fact it gives the right answers and using this, as I will show, resolves all the questions raised by EPR; gives a totally different view of a free particle with spin and accounts not only for quantum mechanics, but also for the anomalies.

Let us compare the quantum hermitian state with this sub-quantum particle state.  In quantum mechanics the statistical or density operator has two pure states of +1 and -1.  The non-hermitian form relates to a single particle, not an ensemble.  That is a major difference.

Now the statistical density operator was first introduced by John von Neumann in 1934 and for spins the form given describes a pure state which can have two components, one lying parallel to the external polarizer oriented in the direction Z and the other one anti-parallel.

Note the presence of this term which is missing in quantum mechanics.  It accounts for coherence but it is very different from the coherence of quantum mechanics.  This coherence is between eigenstates.  However eigenstates are orthogonal and therefore there is no coherence between them from quantum theory.  The situation is different for non-hermitian operators that can have non-orthogonal eigenstates.

Another difference is that quantum mechanics gives the quantum state as polarized in the laboratory Z direction.  In contrast, an isolated sub-quantum particle is oriented in its own body fixed frame of reference, called the spin microframe.  Every sub-quantum particle is in general oriented differently, and each spin microframe can be rotated to the laboratory frame.

I told you the sub-quantum theory is different from quantum mechanics.

There is more.

The sub-quantum spin has internal structure.   Quantum mechanics does not predict any structure.  It treats it as a point particle.  However sub-quantum spins have two orthogonal axes of spin quantization, one determined by the z component of the Pauli spin matrix and the other by the x component.  The structure is therefore two dimensional, with one magnetic moment pointing along the microframe z axis and the other along the x axis.

The last term is not angular momentum but rather a rotation operator.  As such it orients the 2D structure of a spin in 3D space by the commutation relations.  This is another major difference.  Whereas in quantum mechanics the commutation relations lead to the Heisenberg Relations, which show dispersion, in the sub-quantum theory the non-commutation does the opposite:  it orients the 2D particle exactly and specifies it completely with no dispersion, much like the vector cross product orients two vectors in 3D space.

Now I have mentioned that the sub-quantum mechanics is beyond measurement so let us go to the limit and assume that at the briefest instant of time, we can freeze a sub-quantum particle.  In this limit, it looks like a 2D particle.

Finally it is assumed a magnetic moment lies along both axes of quantization.  Without this structure, agreement with experiment is not possible.

Symmetry breaking leads to degeneracy.  Once again this is one spin from an ensemble of many; each oriented in its own microframe. Within a microframe, each spin can be oriented in four ways.  In fact these are resonance structures with the eigenvalues being identical and the wave functions differing by a sign only.  These are degenerate orientations of a two dimensional spin.  The integers simply define the different octants of a microframe.  The integers are hidden variables.

Now these extra hidden variables cannot be seen in the presence of a probe.  That is, if we take our 2D spin and place it in a magnetic field, one axis of quantization lines up and the other precesses so fast that is randomizes leading to the usual spin of quantum mechanics.

Unless sub-quantum theories predict something new, they are simply interesting exercises.  In addition to the ensemble view, there are two major consequences.

The first is that a new spin state is predicted.  If a spin is completely isolated, our usual notion is that it retains its spin vector property.

But not for sub-quantum particles.  They are quite different.  First the two axes of quantization are indistinguishable (not true in the presence of an anisotropic probe) and so superpose to give a hermitian state of root 2 greater in magnitude than usual spin ½.  It lies along a vector that bisects the two dimensional spin.

This root 2 spin is a new state of matter that cannot be predicted from quantum mechanics or the Dirac equation.  It is a unique sub-quantum resonance of a single spin between two internal magnetic moments.  Many examples of resonance are observed in microscopic particles and this is just another.  Recall earlier I mentioned that resonance and indistiguishability are two properties of the microscopic that do not occur in the macroscopic (see Blog entry 5).  The fact this is a pure dispersion free state of one spin.  But symmetry breaking involves degeneracy and here. This leads to a sub-quantum exchange spin in each of the four quadrants.

So the sub-quantum theory predicts that an exchange spin exists in four basically equivalent orientations.  Because the four orientations the spin can take are degenerate, all we can say is that at any instant of time it is in one orientation, and resonates between them.

This picture of an isolated spin is quite different from predicted from quantum mechanics.  It is as if in the absence of any interaction, a spin sort of shuts down into a resonance state of zero angular momentum.

For an isolated spin, Quantum mechanics predicts a spin is a vector magnetic moment and has two states of +1 or -1.  A sub-quantum spin in the absence of all interactions sort of shuts down into a resonance state of net angular momentum of zero.

As I will now show, exactly the same sort of situation occurs for two sub-quantum particles in a singlet state

The second major consequence is that the Bell states are separable. The sum over all the terms leads to cancellation of many terms leaving the usual isotropic singlet state.

The 8 terms organize into four hermitian pairs.  Each pair, being a singlet, has opposite angular momentum, and the phase gives two orientations.

First row:  ++  –, second row +-  -+  third row: -+  +-  and forth row:  — ++.  Each pair has opposite angular momentum.

Each row is hermitian and a product state:  not entangled.  This is the singlet state of one pair of spins.

So you see what I mean when I say the singlet is like an isolated spin because it has four hermitian terms resonate between the quadrants in a state of zero angular momentum.  It is not hard to imagine that combinations of more spins will lead to a similar sort of Clebsch-Gordan coupling, but I haven’t pursued that.

I write each of the four product states as single terms called bi-particles.  That is each row of the last slide is written as one of the four possible hermitian states that are the biparticles.

Now it is fundamentally important to this theory that when two particles in a bound singlet state separate, they can only decompose into one of the four bi-particles.  It is also assumed that one spin moves towards Alice and the other towards Bob then are only correlated by having microframes which have identical orientations relative to the laboratory frame.

That is why they are called bi-particles.  Notice that this correlation is not due to entanglement.  It is only due to the fact that the two spins have identical orientations and are in the same quadrants of their microframes.

Just to see this explicitly.  Now simply multiply these out and collect terms, and we find that each bi-particles can be written in terms of Pauli spin vectors and reveal, as might be expected from the product of two vectors, the usual isotropic scalar part that is the quantum mechanics result.

In addition there are two vectors, pointing oppositely;  and a symmetric traceless 2nd rank tensor.  These are the extra terms predicted by the sub-quantum theory.  If we sum over the quadrants, the nz and nx, then these terms all cancel leaving only the quantum part.

This is the sub-quantum state for two spins formally entangled.  From this all the correlations for EPR experiments can be calculated.  In particular the coincidences agree with the data and are products with no entanglement.

The quantum result is just a constant of ¼.  But that is not the case here.  Each spin is filtered along a definite direction and defines a sub ensemble with specific orientation.

Summary so far:

  • Sub-quantum spins have 2D structure oriented in its microframe
  • Indistinguishable axes of quantization predict a new exchange state
  • Degeneracy between orientations in a microframe leads to a resonance state of net zero angular momentum
  • The Bell states separate into products of two 2D spins that also display resonance between 4 hermitian states called bi-particles
  • Upon separation, the bi-particle product state remains correlated because both spins have the same orientation

Part 4: Sub-quantum ensembles

In this part I talk about how sub-quantum ensembles are formed and answer questions concerning the filter settings for maximum correlation; resolution of the detection loophole and the limit when the sub- ensemble averages to give the quantum results .

Let’s go back to our product representation before transforming it to the laboratory frame.  This product form agrees with quantum mechanics and accounts for the anomalies.  Notice that it has the form that the filters at Alice and Bob form an inner product with the two axes of quantization for the two spins: one moving to Alice and the other to Bob with no entanglement.  This is one pair of spins, and each pair leaves the singlet source, pair by pair, and each is filtered.

Note that the hidden variables and the spin microframes are the same for each partner of the formally entangled pair.  If they are not from a common source, then the hidden variables and the orientation of the microframes would be different and they spins would not be correlated.

If we set Alice’s angle to any fixed value, then all the spins reaching Alice will define a unique ensemble that maximized the probability which, if you recall, is (theta(a)-theta)=45 degrees.

Without going through the detailed calculation here the joint probabilities from the sub-quantum theory are given by a product. One describes data collected at Bob and the other at Alice.  There is no communication between them.  There are no quantum channels.  It is a product state.  Correlations between them arise because they both come from the same singlet and therefore their microframes and hidden variables coincide.

The form in terms of angles is obtained by introducing the filter angles and transforming the spin from its microframe to the laboratory frame.  Then in addition to the hidden variables nx and nz, there is the angle that orients a 2D spin in the laboratory.  It has also been assumed that there is no component of angular momentum in the direction of linear momentum.  Therefore the azimuthal angle is taken to be zero.

Now let us look at this expression.  There are two cases:  nx = nz and nx=-nz.  The maximum probability is give by + 45 degrees and in the other by -45 degrees.

These are two distinct ensembles and, as we will see are responsible for the detection loophole

Consider the coplanar polarizers at Alice and Bob, with the polarizer angles set at angles “a”and “b”.  Let us put a along the Z axis, then the two ensembles occur at +/- 45 degrees.  Bob’s polarizer cannot be aligned at both positions at the same time.

Hence 50% of the particles cannot be detected.  This is likely the major contribution to the detection loophole.

Choosing theta as 45 degrees gives exactly the results of quantum mechanics. That is of the many possible sub-ensembles which can be filtered, only the 45 degree sub-ensembles reproduce quantum mechanics.  When theta is not 45 degrees, then the probabilities will not be maximized and anomalies are found.

Quantum mechanics appears to correspond to the sub-ensembles of a system which maximize the probability.

Recall the filter settings that maximize the violation of the CHSH form of BI have the four filter setting differ by 45 degrees, as predicted by the sub-quantum theory.

In support of this is a paper by Karl Gustafson (Noncommutative Trigonometry and , in Advances in Deterministic and Stochastic Analysis (N.Chuong, P.Ciarlet, P.Lax, D.Mumford, D.Phong, eds), World Scientific (2007), 341-360.) from the U of Boulder in which he analyzed the CHSH violations.  Using non-commutative trigonometry he found that a vector of root two is responsible.  It has identical properties new exchange spin discussed here.   In the next entry I will summarize briefly.

Part 5:  Conclusions

I will now draw some conclusions.

  • Due to the presence of the root two spin, the locality assumption is intact.
  • The sub-quantum theory presented here is local and realistic, providing an interpretation of all known coincidence experiments.
  • Entanglement becomes a property of quantum theory but not of Nature.
  • A single sub-quantum spin has 2D structure.

As I mentioned before, a fundamental difference from quantum mechanics is the sub-quantum theory must allow for coherence (interference) between eigenstates.  This is not possible in the formulation of quantum mechanics because hermitian operators can only have orthogonal eigenstates.

In contrast, non-hermitian states have non-orthogonal eigenstates and hence can interfere, for example this sort of coherence between two non-orthogonal spin states gives a non-hermitian state of the type I am discussing.

Neither quantum field theory nor the Dirac equation can predict the exchange spin state.  To include the sub-quantum theory, QFT need to be generalized to state operators rather than vector states.  Only then can the non-hermiticity be introduced.

  • A local realistic sub-quantum theory accounts for the main features of the experimental data and resolves the anomalies.
  • Predicts a new exchange spin state and the Bell states disentangle.
  • More experimental data is needed to quantify this new state of spin.
  • Need not be concerned about the locality or detection loopholes.


What are some of the other consequences of this approach which completes quantum mechanics? One can only speculate and so my final comments are simply ideas that arise as interesting notions pop up along the way, and might fit.  I have not delved into these, and my list is incomplete and needs you to expand it.

  • First, quite prosaically, the 1935 EPR paper can be exonerated  and we can conclude that the objective reality of position and is resolved  since now non-locality can be put into its rightful place in history like cold fusion.
  • Completing quantum mechanics wrt to spin has led to a number of new properties and concepts that could lead to the resolution of difficulties with modern physics.  Spin is ubiquitous and occurs in many particles.  It is a fundamental intrinsic property of matter and of many individual nuclear particles.  Of course when spin is in the presence of a probe, or other interaction, it obeys the statistical results of quantum mechanics, but in free flight between collisions, a spin goes into its resonant state of zero angular momentum.
  • We have shown that the coincidence probabilities are indeed sensitive to the 2D spin structure as seen from the filter settings of 45 degrees.  Could this mean we have to re-consider collisions and Feynman Diagrams?  What about the Standard Model? What about other experiments, like the 0.7 anomaly noticed in Ballistic electron experiments?  0.7 is the inverse of the square root of 2. How does symmetry breaking and non-hermitian state affect its formulation?  Could the existence of distinct sub-ensembles have an impact on the Dark Matter problem?  What about the cosmological constant?  How do we formulate QFT in terms of Hilbert-Schmidt space and what will the consequences be?  Will this affect string theory?

There is another point.  As enthusiastic as I am about high energy physics that require huge investments to build instruments like the Large Hadron Collider, it seems that some answers also lie at the low energy limit that can be done in a small lab with some good optical equipment.  Bi-particles need much more work.

In the sub-quantum domain resonance, indistiguishability and most of all objective reality play dominant roles.

These and many more issues need to be re-considered but what is most striking is a totally different conceptual approach is needed in physics which comes under the truism:

“He does not roll dice”.


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