008 The Sub-quantum spin
The sub-quantum spin
Department of Chemistry
The discussion in blog 007 expresses the main concepts of sub-quantum mechanics for spin. In these next few entries I will explain the physical insight sub-quantum mechanics gives.
It is important to be able to use mathematics, which is the logic of the LHS of the brain, to visualize what is going on, using the RHS of the brain. The two must be consistent. Heisenberg said that we observe our natural surroundings and intuitively develop a visualization of what is going on. We do not need a mathematical description macroscopically, unless we need precision, and Classical Mechanics works extremely well.
However down at the microscopic world, we must be guided by the mathematics that allows us to visualize what is going on. If one accepts that sub-quantum mechanics is local (which is of course contrary to Bell’s Theorem) then EPR showed that quantum mechanics is incomplete.
The only real criticism of their findings concerns their locality assumption which is questioned solely on the basis of Bell’s Theorem and one type of experiment. If EPR is correct, than there must be a sub-quantum theory.
Of course John Bell did a lot more than deduce his theorem, in particular clarifying the incorrect conclusion by von Neumann that appeared to rule out hidden variable theories was one. Indeed one of the reasons that Bell developed a sub-quantum theory for spin was to show it was possible, in contrast to what von Neumann had mistakenly deduced. we have seen Bell’s most famous paper before. Let’s see what he did regarding sub-quantum mechanics.
He simply assumed that a pure spin state is described by a single polarization vector p, and a hidden variable lambda. Now all I do is assume that at any instant, spin is not one dimensional, but two dimensional. This makes a lot of difference.
Bell of course was just using the usual quantum model for a spin ½ as being a point particle with intrinsic angular momentum. In contrast, a 2D spin is assumed to have two axes of spin quantization. That is it has structure and is not a point particle. This simple extension of spin, which cannot be described by quantum mechanics, resolves the difficulties with quantum mechanics (that is it removes non-locality, makes new predictions).
In addition the singlet correlations formed from two spins can be written as a product without entanglement, is a local model (showing Bell’s theorem must be wrong); and gives a basis for the anomalies found in the EPR data as I discussed in blog 007.
As many earlier physicists like Lorentz and Einstein wanted, the 2D spin restores determinism and causality to the microscopic treatment of spin.
But is it correct? Since it agrees with experiment, whereas qm does not, then it seems worth pursuing. So to start from the very basic assumption, a spin in the 2D theory frozen in time at some instant is assumed to look like this. It is a real object.
The magnetic moment of an electron, for example, is known to great accuracy and obtaining this value theoretically is a great success of QED. Since the magnetic moment describes a magnetic field, this field is so well defined it is a real property of the spin.
Let us think about an electron for illustrative purposes. Please keep in mind that I am thinking of low energies much less than the electron mass. Since it now has structure, we need a convenient frame of reference, x,y,z, called the spin microframe. It is related to the laboratory frame by a simple rotation in 3D space.
This model says that it has a magnetic moment along the z and x axes and it is assumed that the magnetic moments are of the same magnitude as that of the usual magnetic moment of an electron. At this stage, the small arrow along the y axis is just to define a RH coordinate frame. So the total magnetic moment for a particle is given by the vector sum. The unit vector bisects to spin mircro frame and gives the magnetic moment as square root 2 greater than the usual spin ½ lying along that direction.
That is about all there is to it.
So to summarize, a structure spin, cannot be described by quantum mechanics, and is oriented somewhere in space which defines its unique microframe, one for each spin. Since the usual spin ½ has magnetic moment of magnitude mu, and it is assumed that each of the two magnetic axes has the same magnitude and these sum to give a net magnetic moment of square root two greater than the usual spin. This is directed along a unique axis called n (hat).
However, as soon as a two D spin encounters an EM probe, like a magnetic field, then one axis lines up and the other averages to zero. (insert movie) It is impossible to tell the difference between a 1D spin and the 2D spin when a field is present. Since we must have a probe to measure a spin, clearly the root two magnitude cannot be directly observed. That is why the coincidence experiments are so important for photons: they are sensitive to the 2D spin structure. Without these experiments, the 2D spin would be a mathematical curiosity and nothing more.
But it does agree with all experimental data:
- describes the correlation that leads to the violation of BI without entanglement;
- Agrees with the experimental results of the Stern Gerlach experiment;
- Restores locality to Nature
- Gives a basis for explaining the Anomalies in EPR experiments
- Restores determinism and causality to Nature—He indeed does not roll dice!
In the 2D spin model. The magnetic moment of an electron is given in terms of the Electron g-factor which is accurately known to 11 significant figures, the Bohr magneton and the spin ½ operator is given in terms of the Pauli Spin vector. Therefore up to a constant, the magnetic moment is a quantum mechanical operator being proportional to the Pauli spin vectors.
Therefore, back to our 2D spin. we assume that the magnetic moments are represented by these two Pauli Spin matrices. So let us write the 2D spin in terms of these two Pauli spin componets. Where the magnetic moments are replaced by two components of the Pauli spin matrix. These operators are attributes of the system which describe its state:
So what are the possible observables for a spin ½? We always have the Identity and three components of the Pauli spin vector. There are no more because the any other product of the observables can always be written in terms of only one by the commutation relations. Looking at all such combinations of observable is like saying that the attributes of a system are determined by its Algebra of Observables. What this means for a spin ½ is that only these four operators, and no others, are needed to fully describe a spin, but in what combination.
Let us have a look and see how to proceed.
If you think about it, the only macroscopic property from spin ½ is when they align in a magnetic field giving a paramagnetic polarization in the direction P. Hence the spin ½ density operator was discussed by Fano in 1957 and is given here.
Note that before I used lower case xyz for the spin microframe. Here I am using the laboratory frame upper case XYZ.
A system containing statistically many spins can give rise to a net macroscopic polarization. Its magnitude is proportional to P measured in the laboratory frame. The microframe and the laboratory are related by a rotation.
- When P=1, all the spins are lined up in the direction of P;
- When P<1, not all the spins are lined up and
- When P=0, the spins are randomly oriented and have not net polarization.
This is a statistical treatment and the spin density operator gives the state of an ensemble of spin ½ . But what if the spins are polarized along the Z direction, so PZ is not zero but PX and PY are both zero. Note that there is no error associated with the Z component (it is a pure state) but complete dispersion (no knowledge) for the other two components. This is a consequence of the Heisenberg Uncertainty Principle which states that if the spin is polarized along one direction, say Z, then it cannot be polarized in any other direction.
The system is in a pure state defined by the Z axis. This gives us the usual spin picture from qm. Each spin is a point particle with a permanent magnetic moment pointing in one direction. As mentioned several times, I have no difficulty with this model except when a field is removed and the spins cannot couple to it and align with.
This is a good place to stop. It is now time to discuss the Stern-Gerlach experiment with the goal of showing that qm is a statistical theory about measurement and that spin polarization is a result of the net number of spins pointing up and down.