I. What does a Singlet State look like?
When I was a graduate student, and studying quantum mechanics, I came across a statement by Heisenberg which impressed me. We have no trouble visualizing the macroscopic world. It is our common environment and when someone throws a ball to you, you need no Newtonian mechanics to catch it. If you had to catch an electron, well you have no idea without some help. That is because an electron is part of the microscopic world, which is impossible for us to visualize without knowing the equations. This is where Heisenberg came in and said that the only way we can actually visualize the microscopic world is through understanding the equations.
It is through the equations, and only the equations, that we can form a mental picture of microscopic processes. Such mental images are very useful.
As Heisenberg said, we develop “visualizability” of the microscopic world through following the logic of nature which for us is mathematics.
The usual singlet state
In blog H. of this series, I used the usual depiction the singlet state of, say, the hydrogen molecule like:
This certainly gives us a visualization of the singlet state showing the two electron having opposite spin states of up and down thereby cancel out their magnetic moments. On the other hand, as all chemists know, the electron charge on each electron pushes them apart, while the opposite magnetic moments pull them together. A lot of chemistry is based upon this.
Recall from blog H that these two spins in a singlet state are entangled.
If that singlet state is somehow separated, like in the coincident photon experiments, it is assumed that the two spins remain entangled no matter how far apart:
These are the so-called “quantum or EPR channels” which are supposed to exist but no-one can explain. These “quantum channels” are the quantum version of “classical channels” (like a phone line), which are said to be necessary for our understanding of things like “quantum teleportation” as well as playing a major role in most of quantum information theory.
That is, the two spins are believed to retain the singlet state whether they are close together, locally like in a bond, or if the have separated to the other ends of the universe, and are non-local.
Wrong! There is no physical basis that I can see that justifies this view. (help me out here if you can.)
Since EPR believed in locality, I am sure that they would turn in their graves if they knew these non-existent quantum channels were named after them.
The 2D spin.
The structured 2D spin that I have been talking about leads to a very different visualization of the singlet,
In this view, the two electron charges repel and are just balanced by the attraction of the magnetic poles. Recall that there is no magnetic moment of magnitude √2µ here because that state can only exist for free electrons. Singlet states, and higher order states, have interacting electrons. The structured spin gives a physically reasonable representation of the singlet. At least it makes sense to me and fits together nicely.
Consider what happens if this were an electron and a positron. Then the charge on one of them would change from negative to positive and the two charges and the magnetic moments would both be attractive. This leads to a visualization of electron-positron annihilation into energy.
What happens when the singlet above is separated? As soon as the two electrons are far enough away from each other, and in an isotropic environment, the √2 state will appear. Hence the separation will look like this:
There is no entanglement. Note also that the two spins have the same orientation but opposite angular momentum. That is the Local Hidden Variables are the same for the two. This is the 2D spin representation of an EPR pair. These two spins move in opposite directions (conservation of linear momentum) towards filters where they can be detected, and the correlation between them can be measured.
Now the whole point is it is well established (incorrectly in my view) that without entanglement, one can never account for the violation of Bell’s Inequalities: the basis upon which non-locality rests. Therefore if I can show, and I will in later blogs, that this model carries all the quantum correlation without entanglement, then the 2D spin serves as a counter example which disproves Bell’s Theorem.
In the next few blogs, I will show that a product state formed from the 2D spin can, in fact, account for all the correlation between an EPR pair that violates Bell’s Inequalities. No entanglement necessary.
It also contains some unexpected, yet reasonable results. First, indeed it can be shown that the correlation is the same as if the spins were entangled because the larger magnetic moment makes for more correlation. However, note that the two axes of quantization are orthogonal. Since each of those two axes carry magnetic moments which must also be orthogonal, their operators do not commute. Hence only the correlation from one axis (either one) can be detected, but not from the two simultaneously. To do so would violate the Heisenberg Uncertainty principle. Just like position and momentum, two complementary experiments are needed in order to measure along both axes of a 2D spin.
What this means is that only half the correlation can be obtained from a singlet in one experiment. This means that EPR experiments actually satisfy Bell’s inequalities even though it carries all the correlation!
The two complementary experiments on the 2D spin would measure all the correlation, but the results from either are identical to those from usual spin. Naysayers will point to this and say “so you have no way to know if the 2D spin is a better choice than usual spin.” Indeed what is measured is a spin with a single axis even though it has two.
However when you look into it, the 2D spin leads to no disagreements with EPR experiment, but does require a new interpretation of them.
Which spin is correct? “The proof of the pudding is in the eating”, and some of it comes together as I will show.