E. 2D Dirac Equation and Spin

In my previous blog, D. A  Dirac equation for mirror states, it was shown that the two dimensional Dirac algebra leads to mirror states, ψ^±

This can be re-written by combining the two mirror states,

Upon reflection of the 1 and 3 axes the mirror states are interchanged

See the figure,

It follows that the above superpostion gives odd and even parity states,

All this, however, is destroyed by the application of a vector field needed for measurement.

I have a more prosaic view of mirrors states and am not quite able to grasp the notion that there exists a whole mirror world all around us which we cannot see. Rather I just see one state being described in a RH coordinate system and its mirror image in a LH one. From my understanding, Nature does not need a coordinate system, but we do.  Therefore the state, and its mirror image are identical to Nature.

The Dirac equation obtained from the 2D Dirac algebra is non-hermitian and not Lorentz invariant, so cannot have solutions that are physical states. They describe a state, and its mirror image.  You cannot have one without the other, so they occur together. However simply take the sum and difference of the first equation to obtain the following two equations in terms of states of definite parity,

Now we come to a critical part.  The two mirror states are shown in the figure and if you add them, the even parity state is independent of the  “2” axis, and if you subtract them, then the odd parity state is independent of the “1” and “3” axes.  Looking at the second equation above, clearly the first operator does not depend on the 2 axis and the second operator does not depend upon the 1 and 3 axes.  Hence the second equation has two terms that vanish.

2D Dirac and Phase Time Equations

This leaves the first equation, and since the even and odd parity states are orthogonal, the two separate into two independent equations.  One describes a constant precession,

which gives the rotational or phase time.  The second equation is what I am after, which is the two dimensional Dirac Equation,

This equation is both hermitian and Lorentz invariant.

It is important to keep in mind that this is one spin oriented relative to some coordinate frame by angles r=(Θ,Φ) (the body-fixed frame which are Local Hidden Variables LHV) and refers to this particular spin.  In general every spin is oriented differently, and so the spin operator above differs for each spin.

Two Lorentz Invariants

There are two Lorentz invariants, One bisects the 3,1 plane and the other bisects the 3,-1  plane (i.e the even and odd quadrants of the body fixed frame),

We now have another LHV.  In addition to the two angles that orient the structured spin in 3D space, r=(Θ,Φ), the integer, n¹ =±1 indicates which quadrant the spin is oriented. See the figure,

The two possible spin orientations in the spin's body fixed frame, either in the even or odd quadrants.

Since the 1 and 3 axes are assumed to have a magnetic moment of μ the magnitude of the two Lorentz invariants is √2μ. That means that when this spin is not observed, it has a magnitude of √2 larger than observed. This must have some interesting consequences.

Spin Operator

It is straight forward to find the spin states from the above Pauli spin operator,  These states depend upon the LHV (r and and n¹ )

which is the same form as the usual Pauli spin operator, and so it is possible to take over the usual treatment of spin and define the spin operator,

Finally the magnetic moment is root 2 larger than is observed, and it lies along the bisectors of the body fixed frame in the 13 plane.

Summing up

The purpose of this blog is to show that using a different Dirac Algebra, a 2D Dirac equation is obtained and the spin from this bisects the quadrants of the spin’s body fixed frame. The treatment is as mathematically sound as the usual treatment of spin, and therefore puts the structured spin on the same firm footing as usual spin 1/2.

One therefore has a choice.  Accept usual spin that leads to entangled states and a non-local and indeterministic foundation of Nature. Alternately, you can choose the 2D structured spin which gives both a local and realistic view of Nature. Experimentally, the two cannot be distinguished and so the treatment here is not inconsistent with any experimental results.