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Posted by on Jun 18, 2013 in Quantum Mechanics Research | 0 comments

Simple Physics and Optimism. Will Local Realistic Rule?

I am not a local realist by dogma or by virtue of my everyday classical experience.  I am a local realist because I have convinced myself that the EPR (Einstein-Podolsky-Rosen) paradox has a simple resolution. That is, objective reasoning (mathematics and applying the rules of quantum mechanics) led me to a model of an unobserved spin that seems to do the trick. Here I will describe how I stumbled onto this model which reveal new states that exist in free flight. I called their eigenstates “possessed values”.

However this term is used elsewhere  in quantum research, but I could not think of a more descriptive phrase. By “possessed values” I mean that an undisturbed system is completely described by the eigenvalues of pure states none of which have dispersion . The system is ontic, like classical systems, but in a purely quantum way.

Later on I realized that John Bell had used the term “beables”  in exactly the same way, so I now use this term rather than possessed values to describe states that cannot be measured.

Interestingly, this model for a spin ½ actually gives a physical visualization for the Heisenberg Principle, and shows how non-commutation arises when incompatible observables are measured. Another surprise is that a single spin, in my model, displays both “horizontal” and “vertical” components of angular momentum. But these reassuring consequences did not come quickly or easily—it took me sixteen years.

At first, using the usual model for spin, I was trying to violate Bell’s inequalities with a product state, and I never could. As I became more familiar with the literature that surrounds Bell, I was somewhat crestfallen that nothing but persistent entanglement seemed to give the quantum result.

Crestfallen because the absurdity of non-locality had convinced me that there must be an explanation and it eluded me. The logic is this: quantum mechanics is non-local and since non-locality is unacceptable to me, then quantum mechanics must have a deeper level. We could go the John Bell route and reject Special Relativity, but that option does not seem viable either.

I was talking to one celebrated professor, very knowledgeable in the area, who said to me regarding reconciling non-locality, “We are in a realm of physics which is beyond human comprehension.”  I do not agree, and so my obsession grew to find something that would obviate non-locality.

Since Nature is governed by logic, and logic is mathematics, I have complete faith in the ability of the human mind to solve logical problems, although some have too many parameters. This general observation rings true: the more you know about biochemistry, the more complicated it becomes; the more you know about physics, the simpler it gets. People quote Occam’s Razor and KISS (Keep It Simple Stupid)

Permission by Kristin Abkemeier

The reconciliation of EPR turns out to be both physically and mathematically simple,

“Spin has one axis of quantization when you look at it and two when you don’t.”

The Idea that spin has structure

McGill University is a good place to think. In 2006, after being stuck for a long time, I sat back and said to myself something like: “A spin is a mathematically simple system. With two states it is, in fact, the simplest system and purely quantum. What is there to play with?  How much flexibility is there in defining spin? The experiments are clear: pass it through a Stern-Gerlach filter and observe two states. Entangle them in a singlet and the correlation violates Bell’s Inequalities.”

Then I wrote down the observables that characterize spin ½, the three components of the Pauli spin vector  operator: σXYZ. That’s it! There is nothing but this vector operator with Pauli spin matrices for components. Their quantum origin is the Dirac Equation. Everything follows from the laws of quantum mechanics. The Pauli spin matrices obey su(2) algebra and the commutation relations lead to the Heisenberg Uncertainty Principle. The current physical picture of spin is an intrinsic property of matter (a Lorentz invariant) and a point particle with a single axis of quantization. Quantum Electrodynamics gives the g-factor of the electron to twelve significant figures. Surely everything is sewn up?

I wrote the raising and lowering operators down, σx ± i σy, and thought why not choose instead σx, i σy, σz?  Even with the imaginary number, i, added the algebra remained the same. Would everything still work out?  Would the states be non-hermitian? Questions but I realized that apart from these two sets, there is nothing else I can do and still obey the required SU(2) symmetry: it is one way or the other.

My new set, it soon became evident, corresponds to a structured spin rather than the usual point particle: two orthogonal axes carry angular momentum, σx, σz, and the third component is an angular phase σzσx = i σy, acting like a quantum vector cross product.  This set seemed to describe a single spin with a two dimensional flat structure. This means that different spins are oriented differently in space and related to the laboratory frame by rotation angles, which are Local Hidden Variables LHV. This is consistent with the Statistical Ensemble Interpretation whence the state function is considered to be an ensemble average over the LHV.

When I wrote down that set, it felt like a bit of a Eureka moment. I took my scribbled notes that had this idea, dated them, signed them, plasticized them and gave them to my daughter, who was studying physics.

I started to study this new set and it was a rollercoaster: whereas before I got bogged down and frustrated; with this new set everything fell into place easily. Calculations worked out the first time, and doubts vanished. Heisenberg, I once read, said that the only way to visualize the microscopic is through the equations. And so a very simple picture emerged of a spin in free flight.

I will not elaborate on that picture at this time except to say that basic properties of Nature should be simple: a simple way to resolve the oldest question in quantum mechanics is to assume spin has two orthogonal axes of spin quantization. When measured, one axis aligns and the other randomizes, leaving the usual observed spin states.  See the figure:

Depiction of how half the correlation is lost in the presence of a probe: (a) The two body-fixed axes of spin quantization, z and x (heavy lines).  The two arrows depict two possible orientations of a vector probe field in the laboratory frame, one closer to the z axis and the other closer to the x axis (only one can exist at any instant): (b) The case when the probe is closer to the z axis. Then the 2D spin deterministically evolves until the z axis aligns with the probe while the x component randomize in the XY plane. (c) The same as case (b) but now the probe is closer to the x axis which now aligns in the X direction and the z component randomize in ZY plane.

Depiction of how half the correlation is lost in the presence of a probe: (a) The two body-fixed axes of spin quantization, z and x (heavy lines). The two arrows depict two possible orientations of a vector probe field in the laboratory frame, one closer to the z axis and the other closer to the x axis (only one can exist at any instant): (b) The case when the probe is closer to the z axis. Then the 2D spin deterministically evolves until the z axis aligns with the probe while the x component randomize in the XY plane. (c) The same as case (b) but now the probe is closer to the x axis which now aligns in the X direction and the z component randomize in ZY plane.

So What?

A point I want to make is that I have convinced myself but, so far, no one else. I understand why but I could not be so vociferous had I a single doubt that this spin model nicely ties together all those things that used to bug me. There have been times when some new challenge would threaten my model. I could see the house of cards collapsing. During those times before the problem resolved, I was almost despondent, apprehensive and incommunicable. One by one all the difficulties I came across subsided. Confidence grows as obstacles are overcome, but I know that there still might be something I have missed, and the model will fail.

But if the model does stand up, what have I accomplished? There is presently no experimental way to distinguish between one or two axes of quantization. Including counterfactual coincidences can be rejected out of hand and quantum mechanics still viewed as complete. The mechanics of applying quantum will not change. Having the option to accept local reality might have a salving effect on those generations of physicists who have been brought up believing the Copenhagen Interpretation and Bell. One can sigh a sigh of relief that non-locality is history and Nature is both real and deterministic, at least for spin. That would make a lot of people happy.

My optimism

Either way, there appears to be a logical way our of non-local indetermism, and this view of the foundations of quantum mechanics might (should) lead to new insight. There is a lot of things that I do not know, but my sense is that when some scientists out there start to think of spin with two axes, it might lead to new interpretations.  I think non-locality has held back a lot of new ideas.  What are the consequences for, say, the Standard Model, (all 16 particles have spin of 1 or 1/2 (except Higg’s Boson))?  What about other incompatible observables?  Will we get a local realistic view for everything?

It would open a new challenge to those in quantum information: rather than looking upon entanglement as a resource, it should be viewed as a property of quantum mechanics, not Nature.  Attention will then turn to trying to use and control the LHVs. It will no longer be necessary to consider EPR pairs are entangled and remain connected by intangible “quantum channels” that jump between different light cones. EPR pairs will be seen as correlated biparticles. Whereas EPR pairs involve two particles, quantum “teleportation” is a coincidence experiment involving three particles, and can be explained by similar local realistic arguments.

One also has to accept that Nature is not the way we see it. The model, if right, shows that experiments on a spin ½ can only measure half the polarization. Philosophically we have been brought up to believe that we can measure everything, but for some things not all at once. By definition, we cannot ever measure beables. Nonetheless, it is generally accepted that by experimenting we gain a complete characterization of all the properties of a system. Now the situation has changed. We actually can miss stuff by the act of measurement.

Spin emerges from the solution of the Dirac equation which must include an interaction to some external probe. Spin does not emerge in the absence of the probe field and hence it is unlikely this model can be formulated in Quantum Field Theory.

 

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