K. Filtering the 2D spin-cannot filter coherences
In the last entry I showed that the two dimensional spin cannot be fully characterized by measurement because of the Heisenberg Uncertainty Principle. This results because when measured, half the states are randomized away. In fact it was concluded that this 2D spin is capable of measurement along each orthogonal axis, but only one can be measured in an experiment. This means that each such spin is theoretically capable of twice as many clicks as usual spin, even though only half can be detected.
Stated otherwise, the same number of events can be detected from usual spin as the 2D spin, but that number corresponds to only half the events that are possible from the 2D spin.
This point is important for resolving EPR.
Laboratory versus body fixed frame
In this entry I want to make only one small point and contrast the usual 3D Dirac equation and the new 2D equation. I write down both Dirac equations for a free particle spin in the two cases:
The lower case labels, x,y,z denote the body fixed frame and the upper case labels, X, Y, Z denote the laboratory frame. To see the difference, consider a Stern-Gerlach filter,
On the Right Hand Side, the spins are displayed as in two states: up and down, |±>Z. These are the states observed in experiment and are the usual states described nicely on the Bloch Sphere. However just like the double slit experiment, these two states, as described by quantum mechanics, are found only after a large number of spins have passed the inhomogeneous magnetic field. Passing one spin does not give the quantum state.
In other words, consistent with the way quantum mechanics is interpreted, we cannot predict if one spin will be deflected up or down. Only after a statistically large number of spins have passed does the quantum result follow (called Malus’ Law).
Conclusion: the usual three dimensional Dirac equation leads to the usual point particle spin and, when filtered gives the observed states |±>. These quantum states are observed only after a statistical ensemble of spins pass the filter. Many different experiments are done with similarly prepared spins passing the same filter. These accumulate to give the usual observed probabilities of measuring spin up and spin down.
What happens to one spin?
In contrast, and a critical point, the 2D Dirac equation describes one spin that makes up the statistical ensemble. That is, the 2D equation describes a spin before it enters the filter. Before filtering the free spin displays the √2 spin with four possible states. As it reaches the filter, the spin deterministically is deflected up or down thereby showing that Nature is deterministic. As the spin feels the deflection, one of the body fixed axes lines up with the laboratory field and the other precesses perpendicular to that axis and is randomized away as discussed in Blog J of this series.
In short, the 2D Dirac equation describes the state of a single spin before filtering and the usual 3D Dirac equation describes the statistical state that is measured after a large number of 2D spins have passed the filter.
It is that simple.
No spin density matrix for one spin
Now when a bunch of spins are polarized after being filtered, they are organized into pure state of |±>Z. These are polarized along the laboratory Z axis and in general, going back to U. Fano in the Reviews of Modern Physics in 1957, the way to handle a statistical ensemble of spins is with the spin density operator which is given by the well-known formula
Where Pz is the polarization of the ensemble of spins (how well do they point up or down). If the state is pure up |+>Z, then Pz = 1, and if the state is pure down, |->Z, then Pz = -1. Any other state is a superposition of the up and down states. All this is so well known. But notice, there are only two states. The density operator ignores the quantum coherences (generally) because they phase randomize away for macroscopic observation. (just like we saw in blog J.)
What is the equivalent treatment for the spin before it enters the Stern-Gerlach filter?
Recall that Pz is the polarization of the ensemble.
Polarizations and coherences
In the treatment here it is believed that before entering the polarizaing field, the spin is a free particle and displays the √2 states. It is this spin that starts off in the superposed states of the two orthogonal axes, that depend upon the LHV, |±,r=q,f>n1=±1. Note here there are four pure states: two associated with n1=+1 and two with n1=-1, which cannot be simultaneously measured. We measure either n1=+1 states or n1=-1, but not both simultaneously. Half are averaged away when measured.
For this situation we want to display the quantum coherences, so we cannot use a density matrix since the quantum coherences will be lost. Rather it is better to resolve the identity in the basis of the 2D spin. See the figure,
Here is depicted a filter set at angle “a” relative to the laboratory Z axis. The 2D spin approaches the filter in one of its √2 states. These states display both quantum polarizations, diagonal elements, and quantum coherences, off-diagonal elements, and resolving the identity in the 2D spin representation gives,
The matrix elements are (after a lot of work)
Where now pa± is the polarization of one spin (compare with the density matrix above.)
Here “a” is the polarization vector, and “b” and “c” are orthogonal vectors which correspond to the representation (projection) of the 2D spin in the laboratory frame rather than the body fixed frame.
Can only detect polarizations
In an experiment only the diagonal elements can be detected. The quantum coherences are lost. However there is nothing special about the laboratory “Z” axis. We could just have easily used the laboratory “X” axis. If we make that change of representation, then all that happens is the quantum polarization and quantum coherences are flipped, to give (note “+” and ”-“ are interchanged)
Now the tables have turned. The polarization is the Z rep become coherences in the X rep and vice versa. In either rep, half the polarization is unavailable for measurement. Each rep requires a different experimental set up.
This gives some food for thought. The 2D spin completely agrees with the results of experiment. However not everything can be measured in one experiment, so that only half the polarization available in the system can be detected.
It is the inability to detect the polarization from both axes of the spin that Bell’s inequalities appear to be violated. This will be discussed later.
The main point here is that the set-up of the experiment limits our ability to measure all the polarization and what is missed is just the amount needed to resolve the EPR paradox and avoid entanglement.