If you want to sing out, sing out

Why is there a 1919 photo of a silent movie star on my blog? I was watching TV just now. There was an advertisement for one odd thing or another. It abused a song that caught my attention. It was easy to recognize the singer, Cat Stevens, but I was sure it wasn’t on any album of his, and I knew he had stopped cutting new music long before I quit buying it.

A quick google search for the lyrics found that the song is one that Cat Stevens wrote for the romantic comedy movie Harold and Maude. It was somewhat shocking to see in the theater because that part of the audience that is “in the know” bursts into laughter at the first few scenes, that of a suicide by hanging. The song wasn’t released on any Cat Stevens album until a greatest hits album in 1984. I almost never buy greatest hits albums for artists I like, so I don’t have a copy of the song. The above photo is Ruth Gordon age 29, over 50 years before she played Maude in 1971. If you want to hear it, it’s possible that google will find a version.

And I’ve got a solution for fixing my paper. There will have to be another in the series. Mother nature is a rhymes with witch. Now I understand the mathematical relation between quantum numbers and path integrals much better. Just because an object is primitive it doesn’t always mean that it has unit trace.

Uncertain Spin

I’m releasing two papers that relate Heisenberg’s uncertainty principle, spin-1/2, the generations of elementary fermions, their masses and mixing matrices, and their weak quantum numbers. I haven’t blogged anything about these because I’ve been so busy writing, but I should give a quick introduction to them.

Heisenberg’s uncertainty principle states that certain pairs of physical observables (i.e. things that physicists can measure) cannot both be known exactly. The usual example is position and momentum. If you measure position accurately, then, by the uncertainty principle, the momentum will go all to Hell. That means that if you measure the position again, you’re likely to get a totally different result. Spin (or angular momentum), on the other hand, acts completely differently. If you measure the spin of a particle twice, you’re guaranteed that the second measurement will be the same as the first. It takes some time to learn quantum mechanics and by the time you know enough of it to question why spin and position act so differently you’ve become accustomed to these differences and it doesn’t bother you very much.

If you want to figure out where an electron goes between two consecutive measurements the modern method is to use Feynman’s path integrals. The idea is to consider all possible paths the particle could take to get from point A to point B. The amplitude for the particle is obtained by computing amplitudes for each of those paths and adding them up. The mathematical details are difficult and are typically the subject of first year graduate classes in physics. Spin, on the other hand, couldn’t be simpler. Spin-1/2 amounts to the simplest possible case for a quantum system that exhibits something like angular momentum.

The Proton Spin Puzzle

For 20 years QCD has been unable to guess the structure of the most common stable hadron, the proton. This is exemplified in the “Proton Spin Puzzle.” A recent review article:

The proton spin puzzle: where are we today?
Steven D. Bass Invited Brief Review for Modern Physics Letters A, 17 pages
The proton spin puzzle has challenged our understanding of QCD for the last 20 years. New measurements of polarized glue, valence and sea quark polarization, including strange quark polarization, are available. What is new and exciting in the data, and what might this tell us about the structure of the proton ? The proton spin puzzle seems to be telling us about the interplay of valence quarks with the complex vacuum structure of QCD.
http://arxiv.org/abs/0905.4619
Mod.Phys.Lett.A24:1087-1101,2009

The conclusion ends with the following (my emphasis):

“The spin puzzle appears to be a property of the valence quarks. Given that SU(3) works well, within 20%, in beta decays and the corresponding axial-charges, then the difference between and suggests a finite subtraction in the g1 spin dispersion relation. If there is a finite subtraction constant, polarized high-energy processes are not measuring the full singlet axial-charge: and the partonic contribution can be different. Since the topological subtraction constant term affects just the first moment of g1 and not the higher moments it behaves like polarization at zero energy and zero momentum. The proton spin puzzle seems to be telling us about the interplay of valence quarks with the complex vacuum structure of QCD.”

My theory for quarks involves analyzing the interaction between the valence quarks and the sea in the quantum information theory limit, that is, when position and momentum are ignored. I represent color bound states as 3×3 matrices. (See equation (41) of Spin Path Integrals and Generations). The diagonal entries on the matrix are propagators for color not being changed. For a proton, these are the valence quarks. The off diagonal entries are color changing, these correspond to the activity of gluons.

I end up with three solutions to the bound state problem. In terms of absolute values (i.e. ignoring colors), the solutions are 1-circulant; each row of the 3×3 matrix is the same as the one above. There are six off diagonal entries and three diagonal entries. So naively, the contribution from the valence quarks is about half the contribution from the sea. So as far as back of envelope calculations, I would have the spin contribution from the valence quarks at around 0.33 of the total proton spin.

Equation (6) from the review article:

In the parton model, this is “interpreted as the fraction of the proton’s spin which is carried by the intrinsic spin of its quark and antiquark constituents.” According to the paper, a puzzle is “Why is the quark spin content … so small?” But in my theory, 1/3 is a natural value for the percentage of the proton that is quark as opposed to sea.