D. A Dirac equation for mirror states

It is shown here that the different Dirac Algebra leads to a different Dirac Equation and this diplays mirror states which combine to give states of odd and even Parity.

I suppose those who do look at this blog wonder what I am up to because who’s interested in these equations?  Well I am not trying to reach the general public right now, but on the other hand the Dirac equation and spin are something all undergraduate physics students study.  I have felt for a long time a different look into the foundations of physics is needed.  Things have not moved much in areas where entanglement is used as a resource, although papers abound.  I believe people are beginning to think there must be some way out of the problems with physics, from dark matter down to our failure to describe the double slit experiment. We must be missing something.

Then of course there is the absurdity of non-locality, my pet peeve.

I believe the way out of many of these problems is to accept that spin has a 2D structure rather than being a point particle.  Both structured and point particle spin rest on the same firm mathematical foundations and offer a choice between the status quo, and something that answers a lot of questions, at least for me.  So I thought I would write down those ideas in the simplest way, yet detailed enough so that those who have the background can follow.

A different Dirac equation

In the last entry, Part C, “A different Dirac Algebra”, I noted that the small change in the gamma matrix γ² of replacing the Pauli spin operator σ² by iσ² makes a big difference.  First it changes one of the three spatial variables from our usual 3D real space to 2D space.  (I am tempted to say “Welcome to Flatland”).  That spatial variable changes into a new time variable, a rotational time, which is frequency or phase.

This new time is analogous to usual linear time but for angular motion.  So just as different inertial frames have different linear times, so the flat plane of 2D spin can rotate at different relative frequencies.  Spin is a different sort of matter from mass with momentum, having angular momentum rather than linear momentum.  There are two types of angular momentum, orbital (like moon moving around the Earth) and spin (like a spinning top). It is not surprising that angular time arises for structured particles.

Usual spin is considered to a point particle, so angular time cannot exist.

A different Dirac Equation

As soon as we have a Dirac Algebra, we can immediately write down the Dirac Equation.  For the different Dirac Algebra of

the different Dirac equation has two forms

This equation is neither hermitian nor Lorentz invariant* which is usually bad news for any equation, but let’s move on and see where this leads.

The reason it has two forms lies in the fact that space is locally completely isotropic for an isolated particle, say an electron in interstellar space. When a spin interacts with something, like when we try to measure it, then space becomes anisotropic (we turn on a magnetic field (which is a vector) to measure its magnetic moment).

If space is isotropic, then the labels 1 and 3 are indistinguishable and can be permuted without changing the physical meaning of the equation. Therefore if 1 and 3 are interchanged, the 2 term changes sign.

Let P₁₃ be the operator that permutes these two labels, then

And we get the two forms above for the different Dirac equation.

Mirror States

In fact the operator P₁₃ is a reflection operator.

Applied to the Dirac equation above it gives

That is, these two states are reflections of each other, see the figure, The operation of reflection via P₁₃ changes one state into its mirror image.  This is exactly the property sort by Yang and Lee to solve the fact that parity is not conserved for the electro-weak force.  Using their example, if cobalt atoms undergo beta decay, and you watch it in a mirror, then the magnetic moments are not reflected, and so parity is violated.

This does not make much sense but is an experimental reality.

However when beta decay occurs and we do not observe it, then parity is conserved.

Parity States

From the above different Dirac equation, the two mirror states exist simultaneously. Therefore we can define states as sums and differences of the two,

These states have definite parity when reflected,

And so parity is conserved in this different Dirac equation, but destroyed when we try to measure those states.

The next step is to change that non-hermitian, non- Lorentz invariant Dirac equation from mirror states to states of definite parity, .  When this is done, the resulting equation is both hermitian and Lorentz invariant.

And that makes good physical sense.

Summary

Mirror states were postulated to resolve the parity breaking of the electro-weak force and from the different Dirac algebra, they appear naturally.  Since there is no external field, the magnetic moments (that bisect the quadrants of the 13 plane), are reflected faithfully.  In contrast, in the presence of a magnetic field, the 1 and 3 components cannot be permuted because each axis is generally oriented differently with respect to the field, and are no longer indistinguishable.  Hence the mirror states cannot exist when observed, and parity is broken.

I will discuss this different Dirac equation in the next entry.

^(*)  A non-hermitian equation means that things we observe can be complex (z=x±iy), rather than real (z=x).  An equation that is not Lorentz invariant means that physics at different places in the Universe is different.