H. Entanglement or Disentanglement
The 2D structured spin leads to the removal of all entangled states in quantum mechanics. Read on, but first the perspective.
The point about this series is that I have found a Dirac algebra and Dirac Equation which gives a completely different physical interpretation of spin but rests on the same mathematical basis as usual spin. Since measurement cannot distinguish between the two, there is a choice. Fundamentally the choice is between reality and anti-reality. It is also a choice between disentanglement and entanglement.
Since the 2D spin changes into usual spin upon measurement, it seems reasonable that the 2D spin is more fundamental than the usual formulation.
Usual spin is described by the Pauli spin operators and the identity, (I, σ1, σ2, σ3) with a permanent magnetic moment along any direction (say pointing to the surface of a Bloch sphere). The new structured spin is described by (I, σ1, iσ2, σ3) has permanent magnetic moments along the 1 and 3 axes, and these add, giving a total magnitude for the magnetic moment which is √2 larger than can be measured.
What consequences does a larger magnetic moment have on properties that depend upon spin? Think of the clouds of free electrons that are found throughout the universe. Any calculation that involves free spin needs to be examined again.
Bound electrons and entanglement
I have said that the √2 spin only exists when space is isotropic. However when interacting, say with other particles, although the √2 magnetic moment is destroyed, none-the-less the 2D structure remains and this actually removes entanglement from quantum theory!!
So what is entanglement in a nutshell?
“Entanglement” was coined in 1936 by Schrodinger who said of it,
I would not call [entanglement] one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought.
Think of a singlet state, the smallest entangled state. We see this in the hydrogen molecule in high school chemistry classes,
The two electron spins line up, north-south and south-north to give zero angular momentum. Since these electrons are close enough to interact, the singlet state is a local state.
Although two electrons are indistinguishable in Nature, we mortals need labels to keep track of things, so we make them distinguishable,
But if we write a product state, then upon permutation the wave function does not change sign:
However if the state is entangled, then permutation of 1 and 2 is odd, as it must be for fermions,
The reason this state is entangled is that it cannot be written as a product state. The two cannot be separated into a product state. It is “an undivided whole”. But is it?
Now the difficulty with entangled particles is most people believe that when a singlet is separated (e.g.as is done in EPR coincidence experiments), the entangled states remain entangled over any distance. No one knows how. To me it makes no sense. It requires us to accept instantaneous action-at-a-distance.
However if the 2D spin is used, then entanglement disappears from the theory and all entangled states can be written as a sum of product states. First, the figure below is how a singlet might look if formed from the 2D spin,
The repulsion of the charge is just balanced by the attraction of the magnetic moments. The two spins share the same LHV (Local Hidden Variables) which also means they share the same body fixed coordinate frame.
Usual spin is defined by a density operator which for pure states looks like (U. Fano 1957 Rev. Mod. Physics)
Which simply says a single spin has the usual states of |±1><±1|. Of course this is what is observed.
The 2D spin, in contrast, carries not only the diagonal states, but also off-diagonal quantum coherence. These depend upon how we represent the spin. Above, and below, we use the “Z” representation. Above, it is the laboratory frame, Z, and below it is the body frame of the spin, z. This means that the contributions from the “x” axis are off-diagonal, and appear as a coherences. Coherences cannot be directly measured. Here are two states out of the possible eight which you can contrast with the above equations,
This spin not only shows the diagonal polaraizations, but also coherences.
Of course it is easy to switch from the “z” rep. to the “x” rep in which case the diagonal terms become off-diagonal coherences, and the off-diagonal terms become diagonal polarizations. That process just flips the z and x reps showing that there is really no difference in physical interpretation between the two reps. However although experiments are possible in either representation, only one can be measured at a time. This is fundamentally because the two axes do not commute and Heisenberg’s Uncertainty Principle comes into play.
Disentangling the Singlet state
Simply calculate the matrix representation of the singlet by substituting the single spin states. The matrix representation of the singlet in the usual states of equation 2,
Now let’s take the states of two single 2D spins, (s1+++, s2+-+, … etc). and note these are coherent states like above. The two spins that form a product must be in the opposite quadrants (to have opposite angular momentum). But there are four quadrants and therefore four ways the two spins can form a product state. So summing all four contributions gives,
It is easy to plug the Pauli spin matrices into the above and add up all the products to give the expression for the entangled singlet (see end). In other words, the 2D spin shows that there is no entanglement. The singlet state can be written as a sum of product states. This is not possible using usual spin. The 2D spin, however, carries the quantum coherences which just do the trick to allow the states to disentangle.
It might be useful to connect this with the 2D Dirac equation discussed in earlier blogs of this series. Recall that mirror states were formulated, and these are seen in the above equation: each line of that equation has the last subscript as “+” on the first product and “-” on the second. Same for others. Each state operator in the above product is described by a non-hermitian and not-Lorentz invariant equation. If this were solved, then and the products taken, the hermitian singlet state, which is Lorentz invariant, would result.
A final note of speculation is based on the notion that each line of the above singlet disentanglement corresponds to a possible orientation of the two spins, but those two spins can be in only one quadrant at any instance. Therefore, upon separation, it is suggested here that the product state from only one of the four lines exists, and this, when separated, forms the single 2D spin in free space which is described by the 2D Dirac equation.
It does not stop here. The three entangled triplet states (the other Bell states) are similarly decomposed into products, and so on for all entangled states composed of any number of particles. In short, it is easy to modify the Clebsch-Gordan series that couples angular momentum to include the 2D spin. That series contains no entanglement.
This is one of the papers I got up on the quantum archives before the moderators blacklisted me for my ideas. Paper is called Separation of Bell States.
In summary, the 2D spin plays a role for all spin interactions and removes entanglement from quantum theory.
Entanglement is a property of quantum mechanics, but not of Nature.
Here is the simple disentangled proof: