“One day I will take my cobwebs down and have them drycleaned”

There is a section with some of my HS poems, immature of course, but clearly, I am trying out styles and learning from Romantic lyricists, E. E. Cummings, Robert Service, and others.

I wrote three poems while I was working in a logging camp for summer jobs in 1963-4.

I wrote only a few poems from ’65 to 71, when I got my PhD. I will also give two poems by my friend called Nela. I recall her last name as Leya, but it could be Leigh, or something similar.

I did my postdoc in Leiden, Holland, 1972-74. This was an idyllic time for me and my wife. We had two sons born. These poems capture my feelings during that time.

After leaving Holland, I wrote only one poem in the 1980’s.

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Bryan Sanctuary 1967

Sometimes the days are carefree

When springtime fills the air,

And time stands still in blue and green

While dappled shades dance everywhere

To liven up the scene.

Silent boats slip slowly by

Across the hazy blue,

As yellow sand wash softly flat

And clear away the clue

Where two young people sat.

In the timeless flood and ebb,

That leave the two unheeded,

A silent word, a lingered stare

Is all the heart had needed,

To seal the memory there.

Non-locality, going back to Isaac Newton, has always been unacceptable, at least until modern times. Instantaneous action-at-a-distance is physically unpalatable and always belies a deeper theory. So why does almost every physicist believe in it?

To be clear, I do not suggest that Bell’s Theorem is wrong. Bell gave a way of distinguishing classical from quantum correlation, and although there are dissenters, they are few. According to quantum mechanics, to agree with the 2√2 violation of Bell’s Inequalities, the separated spins must remain entangled, and therefore have non-local correlations. No one can explain how particles remain entangled after they have separated.

The point is, I cannot accept that Nature is non-local and not deterministic. So there must be a way, even though most have given up and just accept it. Resolving the EPR paradox is one way to get rid of non-locality. When I found that my spin carries all the correlation that violated Bell’s inequalities without being entangled, it convinced me, but few others. More and deeper arguments are required.

Since the whole concept of non-locality is based upon one experiment, called coincidence photon experiments, if one can explain that data without entanglement, the problem is solved.

Question

1. How do we know that when spin is not measured, it remains in the same state that was observed when measured?

I believe it goes into a different state.

2. How would we know?

I do not think there is any experimental way to know. However the structured spin I found makes a lot of physical sense and each EPR pair carries all the quantum correlation without entanglement. So you have a choice.

3. Since usual spin is firmly established in quantum field theory and the Dirac equation, how can a spin we cannot observe have a hope of being accepted?

As for the fundamental basis of the structured spin, all that is needed is to change the usual Dirac Algebra from,

γ^{0}, γ^{1},γ^{2},γ^{3}

to

γ^{0}, γ^{1}, *i* γ^{2}, γ^{3},

where “*i*” is the imaginary number. Anyone who knows the Dirac equation should easily see that this different Dirac algebra leads to a two dimensional Dirac equation and this has two spin Lorentz invariants which are exactly my structured spin. I will show this in the few entries.

Therefore the 2D structured spin is as firmly based in Physics as usual measured spin.

4. If the 2D spin resolved the EPR paradox and solved the Double Slit experiment, would that be enough?

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Think of NMR (Nuclear Magnetic Resonance) and MRI (Magnetic Resonance Imaging). In these experiments, spins align with magnetic fields and their polarizations are measured.

In quantum mechanics, spin is postulated, but it arises naturally in quantum field theory from the Dirac equation. Everything is clear mathematically, but parts make no sense. You have to accept that Nature is non-local.

**Here are some questions**:

Does spin remain a point particle vector when not observed? Align a bunch of spins in a magnetic field, and then remove the field. Do those individual spins remain as observed: point particles of spin, or do they change state and become something different?

I am saying they become something different. For me, the most appealing aspect of this different spin is it restores local reality to Nature. But it does a lot more and I will come to that later.

If the new structured spin is placed in a magnetic field, it cannot be distinguished from usual spin. Therefore there is no experimental way known that can distinguish the new spin from the usual spin. So who is going to accept something you can’t see it?

My goal is to show that this spin makes a lot of physical sense and might resolve old problems and shed light on new ones. For example this spin has states that are mirror images of each other. Could these be the mirror matter that Yang and Lee suggested in 1957? This spin resolves the EPR paradox which means it accounts for all the correlation between EPR pairs without entanglement. As such, it restores locality to Nature. This spin also displays both wave and particle nature. Recall that the wave-particle duality concept says a particle can act either like a wave or like a particle. The states in this spin carry both wave and particle nature at the same time, and it is this property that leads to the resolution of the EPR paradox. I am hoping, will also finally resolve the double slit experiment. (a single particle can interfere with itself.)

All this seems a tall order to fill and of course goes right against the current paradigm. So this is my plan (while trying to get the paper published!!). First I am going to write a few blogs on how this new spin satisfies its own Dirac algebra and has its own Dirac equation. In other words, I am going to put this unobservable spin on the same sure footing as our usual spin ½. Then you have a choice between two formulations based upon the same firm mathematical basis.

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Paul Dirac’s genius was to realize that the Klein-Gordon equation, which conserves energy and mass, would perhaps show more structure if it were a first order differential equation in space-time rather than the second order Klein-Gordon equation. This led to spin and anti-matter for starters. The trick he used required that a set of four matrices (called the gamma matrices) had to anti-commute, (*i* and *j* take values of 0, 1, 2, 3)

Where *I _{4}* is a 4×4 identity matrix and the metric tensor given by,

This means that there is one time variable, the +1, and three spatial variables, the -1. Good old space-time again.

It is not necessary to go into the properties of the gamma-matrices. They are well known. However their representation in terms of the Pauli spin matrices is important for what is to follow, so here they are,

There are two spins here: one for matter and the other for antimatter. From the Dirac equation, spin emerges as a Lorentz invariant defined by the operators {*I _{2}*,σ

**A Different Dirac Algebra**

At the end of the first blog (A) in this series, I said that there is a different way to define the gamma matrices for a spin ½. That is one of the spatial matrices, say γ^{2}, is replaced by (*i* is the imaginary number)

Why not? The matrices still anti-commute, so by definition they form a Dirac algebra. It has a different metric tensor,

which means that there are two time variables and two spatial variables. That is certainly different. It represents a flat structure with only two spatial dimensions, maybe an anyon?

But two times? The first is the usual linear time that differs in different inertial frames. The second is a rotational time which rotates in the plane of the 2D flat space. This is a phase time or a frequency and accounts for the different relative rotations of 2D objects in different inertial frames.

So what does this different Dirac algebra lead to? First spin is now defined by {*I _{2}*,σ

Those Mirror states lead to a 2-dimensional Dirac equation. There is fun with parity too.

At the level of a spin ½ there are only two possibilities which make physical sense {*I*_{2},*σ*_{1},*σ*_{2},*σ _{3}*} and {

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I suppose those who do look at this blog wonder what I am up to because who’s interested in these equations? Well I am not trying to reach the general public right now, but on the other hand the Dirac equation and spin are something all undergraduate physics students study. I have felt for a long time a different look into the foundations of physics is needed. Things have not moved much in areas where entanglement is used as a resource, although papers abound. I believe people are beginning to think there must be some way out of the problems with physics, from dark matter down to our failure to describe the double slit experiment. We must be missing something.

Then of course there is the absurdity of non-locality, my pet peeve.

I believe the way out of many of these problems is to accept that spin has a 2D structure rather than being a point particle. Both structured and point particle spin rest on the same firm mathematical foundations and offer a choice between the status quo, and something that answers a lot of questions, at least for me. So I thought I would write down those ideas in the simplest way, yet detailed enough so that those who have the background can follow.

In the last entry, Part C, “A different Dirac Algebra”, I noted that the small change in the gamma matrix γ^{2} of replacing the Pauli spin operator σ^{2} by *i*σ^{2} makes a big difference. First it changes one of the three spatial variables from our usual 3D real space to 2D space. (I am tempted to say “Welcome to Flatland”). That spatial variable changes into a new time variable, a rotational time, which is frequency or phase.

This new time is analogous to usual linear time but for angular motion. So just as different inertial frames have different linear times, so the flat plane of 2D spin can rotate at different relative frequencies. Spin is a different sort of matter from mass with momentum, having angular momentum rather than linear momentum. There are two types of angular momentum, orbital (like moon moving around the Earth) and spin (like a spinning top). It is not surprising that angular time arises for structured particles.

Usual spin is considered to a point particle, so angular time cannot exist.

As soon as we have a Dirac Algebra, we can immediately write down the Dirac Equation. For the different Dirac Algebra of

the different Dirac equation has two forms

This equation is neither hermitian nor Lorentz invariant* which is usually bad news for any equation, but let’s move on and see where this leads.

The reason it has two forms lies in the fact that space is locally completely isotropic for an isolated particle, say an electron in interstellar space. When a spin interacts with something, like when we try to measure it, then space becomes anisotropic (we turn on a magnetic field (which is a vector) to measure its magnetic moment).

If space is isotropic, then the labels 1 and 3 are indistinguishable and can be permuted without changing the physical meaning of the equation. Therefore if 1 and 3 are interchanged, the 2 term changes sign.

Let *P*_{13} be the operator that permutes these two labels, then

And we get the two forms above for the different Dirac equation.

In fact the operator *P*_{13} is a reflection operator.

Applied to the Dirac equation above it gives

That is, these two states are reflections of each other, see the figure, The operation of reflection via *P*_{13} changes one state into its mirror image. This is exactly the property sort by Yang and Lee to solve the fact that parity is not conserved for the electro-weak force. Using their example, if cobalt atoms undergo beta decay, and you watch it in a mirror, then the magnetic moments are not reflected, and so parity is violated.

This does not make much sense but is an experimental reality.

However when beta decay occurs and we do not observe it, then parity is conserved.

From the above different Dirac equation, the two mirror states exist simultaneously. Therefore we can define states as sums and differences of the two,

These states have definite parity when reflected,

And so parity is conserved in this different Dirac equation, but destroyed when we try to measure those states.

The next step is to change that non-hermitian, non- Lorentz invariant Dirac equation from mirror states to states of definite parity, . When this is done, the resulting equation is both hermitian and Lorentz invariant.

And that makes good physical sense.

Mirror states were postulated to resolve the parity breaking of the electro-weak force and from the different Dirac algebra, they appear naturally. Since there is no external field, the magnetic moments (that bisect the quadrants of the 13 plane), are reflected faithfully. In contrast, in the presence of a magnetic field, the 1 and 3 components cannot be permuted because each axis is generally oriented differently with respect to the field, and are no longer indistinguishable. Hence the mirror states cannot exist when observed, and parity is broken.

I will discuss this different Dirac equation in the next entry.

^{(*) }A non-hermitian equation means that things we observe can be complex (*z=x±iy*), rather than real (*z=x*). An equation that is not Lorentz invariant means that physics at different places in the Universe is different.

In my previous blog, D. A Dirac equation for mirror states, it was shown that the two dimensional Dirac algebra leads to mirror states, ψ^{±}

This can be re-written by combining the two mirror states,

Upon reflection of the 1 and 3 axes the mirror states are interchanged

See the figure,

It follows that the above superpostion gives odd and even parity states,

All this, however, is destroyed by the application of a vector field needed for measurement.

I have a more prosaic view of mirrors states and am not quite able to grasp the notion that there exists a whole mirror world all around us which we cannot see. Rather I just see one state being described in a RH coordinate system and its mirror image in a LH one. From my understanding, Nature does not need a coordinate system, but we do. Therefore the state, and its mirror image are identical to Nature.

The Dirac equation obtained from the 2D Dirac algebra is non-hermitian and not Lorentz invariant, so cannot have solutions that are physical states. They describe a state, and its mirror image. You cannot have one without the other, so they occur together. However simply take the sum and difference of the first equation to obtain the following two equations in terms of states of definite parity,

Now we come to a critical part. The two mirror states are shown in the figure and if you add them, the even parity state is independent of the “2” axis, and if you subtract them, then the odd parity state is independent of the “1” and “3” axes. Looking at the second equation above, clearly the first operator does not depend on the 2 axis and the second operator does not depend upon the 1 and 3 axes. Hence the second equation has two terms that vanish.

This leaves the first equation, and since the even and odd parity states are orthogonal, the two separate into two independent equations. One describes a constant precession,

which gives the rotational or phase time. The second equation is what I am after, which is the two dimensional Dirac Equation,

This equation is both hermitian and Lorentz invariant.

It is important to keep in mind that this is one spin oriented relative to some coordinate frame by angles *r*=(Θ,Φ) (the body-fixed frame which are Local Hidden Variables LHV) and refers to this particular spin. In general every spin is oriented differently, and so the spin operator above differs for each spin.

There are two Lorentz invariants, One bisects the 3,1 plane and the other bisects the 3,-1 plane (i.e the even and odd quadrants of the body fixed frame),

We now have another LHV. In addition to the two angles that orient the structured spin in 3D space, *r*=(Θ,Φ), the integer, *n*^{1} =±1 indicates which quadrant the spin is oriented. See the figure,

Since the 1 and 3 axes are assumed to have a magnetic moment of μ the magnitude of the two Lorentz invariants is √2μ. That means that when this spin is not observed, it has a magnitude of √2 larger than observed. This must have some interesting consequences.

It is straight forward to find the spin states from the above Pauli spin operator, These states depend upon the LHV (*r* and and *n*^{1} )

which is the same form as the usual Pauli spin operator, and so it is possible to take over the usual treatment of spin and define the spin operator,

Finally the magnetic moment is root 2 larger than is observed, and it lies along the bisectors of the body fixed frame in the 13 plane.

The purpose of this blog is to show that using a different Dirac Algebra, a 2D Dirac equation is obtained and the spin from this bisects the quadrants of the spin’s body fixed frame. The treatment is as mathematically sound as the usual treatment of spin, and therefore puts the structured spin on the same firm footing as usual spin 1/2.

One therefore has a choice. Accept usual spin that leads to entangled states and a non-local and indeterministic foundation of Nature. Alternately, you can choose the 2D structured spin which gives both a local and realistic view of Nature. Experimentally, the two cannot be distinguished and so the treatment here is not inconsistent with any experimental results.

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Basically Taleb believes that our lack of knowledge is as important as our knowledge. If we know too much, then we make predictions which are usually a continuation of our present history. We really cannot predict: the stock market, world wars, epidemics, down to a sudden illness or a personal crisis: all are black swans. They can indicate either positive of negative events, but in all cases it is their suddenness that takes us by surprise and only then can we start to deal with the impact.

I believe that my 2D structured spin falls into the category of a Black Swan event. Other events in physics are Black Swans: the world is not flat; the Earth is not the center of the universe. Then we have the ultraviolet catastrophe of Black body radiation, explained by Planck, which ushered in quantum theory. There is the impact of relativity, then Black Holes and the Big Bang. These events, and others, shaped our view of Nature and the present paradigm which contends that quantum mechanics (or quantum field theory) is the most basic description of Nature.

I would say that Bell’s Inequities and his theorem (that any Local Hidden Variable Theory be non-local) are also Black Swans, because the existence of quantum correlation leads to the violation of Bell’s Inequalities. This has been interpreted, (incorrectly in my view) that entanglement must persists when entangled states are separated. Hence Nature is to have non-local interactions-how?

Before Bell, ( say in the mid 1960’s) everyone believed Nature to be local, although not deterministic. Today non-locality, or instantaneous-action-at-a-distance, is accepted by most Physicists, and the experimental evidence rests upon one type of experiment (photon-coincidence).

Clearly the non-locality Black Swan is mind-boggling because no-one understands it.

But I believe it to be wrong. This does not mean that I believe Bell’s mathematical proof of his inequalities is wrong, although doubters exist. I also completely believe the experimental results from photon coincidence experiments. I do not believe, however, that the interpretation of the experimental results is correct. My spin gives a local and realistic view of Nature. I think choosing that makes more sense that accepting the absurd.

If my 2D spin were accepted, and people started to study it, I am certain many more problems in physics, unknown to me, could be resolved. What impact is there to Nature if the magnetic moment of an undisturbed electron is √2 larger than can be measured? Certainly I have used this successfully to account for the violation of Bell’s Inequalities without entanglement–that it gives a resolution of the EPR paradox. Similarly I believe that quantum information theory (computing, teleportation and quantum cryptography), are not moving along very well. I believe that emphasis should be placed on controlling the LHV and not entanglement. But there are many more problems that exist: the double slit experiment, dark matter, parity breaking in beta decay and prediction of mirror states;, etc.

If we suppose the standard model is correct, then I have a problem. The standard model is composed of 16 particles )[excluding the Higgg’s boson), and they all have spin of either 1/2 or 1. All these particles are considered to be point particles. Hey, even the Black Hole that exploded to form our universe is treated as a singularity in the theory.

You can imagine the opposition I receive when I say that these 1/2 spin particles (all 12 of them in the standard model) are not point particles, but have a 2D structure. Well, it seems that in the initial Big Bang, what happened in the first 10^{-37} seconds is not understood. Maybe in this early stage, the two axes of my structured spin 1/2 were formed as a condensate to produce those 12 spins. However to suggest that spin has structure, means that the Standard Model needs to be looked at some more.

Another impact of a 2D spin is that it means that Quantum Theory is not the most fundamental description of Nature. Physics is and experimental science, and if the 2D spin exists, then it must lie deeper than quantum mechanics. For me, although measurement is critical for us to obtain knowledge, measurement is simply another interaction in Nature. Nature does not care if we measure, so it seems to me there is nothing fundamentally wrong in assuming there are structures below our ability to measure.

Entanglement, said Schrodinger in 1936, is not a difference between classical and quantum mechanics, but the difference. I agree but go further and state that entanglement is a property of quantum mechanics, but not of Nature.

Finally, of all the other points I can make, the structured 2D spin restores locality and determinism to Nature and this makes me very happy, and I believe would be accepted, especially if many other problems in Physics can be rationalized using the structured spin.

That is why I see my Black Swan as paradigm changing. For me it is a no-brainer to do away with non-locality and indeterminism, but then others, who have not gone through my objective developments, like in parts A to E in this series, will simply accept the status quo and in most cases, usual spin causes no problems.

Still it would be nice to get our understanding of Nature right.

I will add other blogs later in which I will show some aspects of the Coincidence Photon experiments, and how the structured spin leads to two simultaneous coincidences, although only one can be detected. This means that half the EPR correlation cannot be detected and, big surprise, Bell’s Inequalities are no longer violated.

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By changing the Dirac algebra from

a different formulation of spin emerges which rests on the same firm mathematical basis as usual spin. The new algebra has two time variables and two spatial variables. The spatial variables give the 2D Dirac equation and finds the new spin operators as Lorentz Invariants. Besides the usual linear time, the new time is quite different, being a rotational or phase time. Since spin now has structure, it can precess relative to spins in different inertial frames. Hence it plays the same role for angular momentum that linear time plays for linear momentum.

This means that we are presented with a choice: is spin that which we observe, or is it a 2D structured particle which, when measured, is indistinguishable from the former?

In this entry I will describe the 2D spin and how it changes our understanding of Nature.

Quantum mechanics is considered to be the most fundamental theory of Nature. There are many interpretations and the most accepted of these is the Copenhagen primarily due to Nils Bohr, which rests on complementarity: incompatible observables cannot simultaneously exist, like position and momentum. Devise an experiment to measure one, and the other is not detected. One acts like a particle and the other a wave. Einstein, Podolsky and Rosen (EPR) famously disagreed with Bohr and showed that a physical theory must account for all elements of physical reality, thus they asserted quantum theory is incomplete.

Thirty years later, in 1964, Bell proposed his inequalities and theorem that essentially establishes non-local connectivity between separated particles. This repudiates EPR and establishes the persistence of entanglement after particles separate (non-locality) as a resource and property of Nature.

However a search on the Internet using “quantum weirdness” establishes that no-one can give a rational explanation for non-locality. If you accept non-locality, you are an anti-realist. If you do not, then you are a realist. Today the vast majority of physicists are anti-realists.

I assume that readers have a good understanding of usual spin ½ so I do not have to discuss it: a point particle of intrinsic angular momentum displays two states |↑> and |↓>, spin up and spin down in the laboratory frame of reference, see Figure 1.

The reason these states are defined in the laboratory frame is because that is where experiments are performed. This requires turning on a probe which interacts with the spins and filters them into one of their two states. Recall that quantum mechanics cannot predict which state will be found. Rather all that can be known is the probability that a spin will be in one of its two states before filtering. Only after a statistically large number of spins have passed do the results agree with the predictions of quantum mechanics. This is the statistical interpretation. A large number of spins make up a quantum ensemble. Quantum mechanical spin states are ensembles of many spins, not individual spin states.

Note that the 2D spin has two orthogonal axes of spin quantization, and these do not commute. Hence axes 1 and 3 (*x* and *z*) are incompatible which means the two cannot be measured simultaneously. This is why the 2D spin looks the same as usual spin when measured. The 2D spin gives a clear example of the Heisenberg Uncertainty relations.

The main difference suggested here is that the ensembles are made up of individual spins which have a 2D structure rather than being point particles. Once again, this is not postulated, but follows from the new Dirac algebra. Since they have structure, the spins can be conveniently viewed in their individual body fixed frames of reference, see Figure 2.

In the body fixed frame (*x,y,z*), a spin has two possible orientations, which bisect the (*x,z*)-plane in the even and odd quadrants. The unit vectors are given by,

Since the body fixed frame is related to the laboratory frame by a rotation by θ,φ and its orientation within each body fixed frame is given by *n*_{1}=±1, the parameters (θ,φ,*n*_{1}) are Local Hidden Variables (LHV). Averaging over those LHV is the same as ensemble averages, and must retrieve the quantum mechanical results. In a later blog, I will show this to be true by computer simulation.

If each axis, (*x,z*), carries a magnetic moment of magnitude µ, then the magnitude of the magnetic moment that bisects the quadrants is √2 µ. One question to ask is what are the consequences of this larger magnetic moment? The reader might have ideas.

As seen in Figure 3, the application of an external field destroys the 2D spin. Recall that the mirror states and the states of definite parity (Part E of this series), depend upon the indistinguishably of the 1 and 3 labels (the *x* and *z* axes) within the 2D Dirac equation, and this requires that space be isotropic.

If the field is oriented between 0 and 45 degrees in the even quadrant, then the *z* axis lines up with the field and the *x *axis precesses in the plane perpendicular to the field and averages away, see Figure 3. Between 45 and 90 degrees, the x axis lines up and the z axis averages away.

Hence one important result is that the 2D spin is deterministic. We know from its orientation before it is filtered whether it is in the up or down state.

However the axis that precesses are quantum coherences and these are phase randomized away and make no contribution to measurement. This means the act of measurement destroys the polarization associated with the axis that precesses. Upon measurement, one must accept that only one axis can be measured so that any experiment can only detect half the spin polarization present in the system. This is simply a manifestation of the Heisenberg Uncertainty relations: the two spin axes of the 2D spin carry angular momentum which do not commute.

When not measured, however, 2D spin generally has differently oriented spin operator for every spin in the ensemble. Whereas the usual spin states observed are either up or down states, each 2D spin has a spin operator oriented along either one of the two bisecting directions. Rather than the usual two pure states from usual spin, the 2D spin displays four pure states: two for each orientation in its body frame, (Figure 2 pure states along each of the directions **n**^{n1}).

The Pauli spin operator associated with the 2D spin is mathematically the same as usual spin. There are two Lorentz invariants of the 2D Dirac equation,

It is easy to find the eigenstates for this operator which depend on the LHV,

The states are given by

which are super-positions of the x and the z axes in the body fixed frame (not the usual laboratory frame). None-the-less they have the same usual representations as usual spin,

but once again in the body frame.

The usual approach to structured particles is to transform the states from the body fixed frame into the laboratory frame, where experiments are done. This is obtained by a simple rotation by angles θ,φ . Averaging over these angles for a specific quadrant must give the ensemble averaged result from quantum mechanics.

Although the mathematical basis for both usual and 2D spin is equivalent, two very different views of Nature emerge. The choice between the two spins will be made on the ability of one to resolve problems, and which is more physically appealing. In the following blogs, I will show that entanglement is not needed to account for the violation of Bell’s inequalities. I believe that this spin will shed light on the Double Slit experiment and perhaps other problems unknown to me.

The 2D spin does not only exist in an isotropic environment. Higher states that are entangled in quantum mechanics are not entangled if the 2D spin states are used. The singlet can be written as a sum of products. Therefore entanglement is not a property of Nature, but it is a valid approximation and a useful property of quantum mechanics.

Adopting 2D spin and a local realistic view of Nature is unlikely to interfere with the current success of quantum mechanics for most problems. It does, however, shift the emphasis. For example in quantum information theory, controlling entanglement should be replaced by controlling the LHV.

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