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.
Ruth_Gordon_1919

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.

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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.
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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 g_a^{(0)}|_{pDIS} and g_a^{(8)} 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: g_a^{(0)} and the partonic contribution g_a^{(0)}|_{pDIS}= g_a^{(0)}-C_\infty 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:
g_a^{(0)}|_{pDIS,Q^2\to\infty} = 0.33 \pm 0.03(stat.) \pm 0.05(syst.)
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.

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The Moon’s Subtle Influence

Science or fiction, sometimes it is hard to tell. In 1997, a group of Chinese scientists hooked up a sensitive gravimeter, to automatically record the earth’s gravitational field (or more accurately, the local acceleration of the earth’s crust) in the obscure northeast China town of Mohe, Heilongjiang (Black Dragon River) province. They chose this town because it was near the center of the 1997 solar eclipse, achieving totality for about 2 minutes. They chose the most accurate unit available, it can detect the reduction in gravitation when it is raised 1cm.

After the eclipse they examined their data. They found the usual tidal effects and slow drifts but they also found an interesting signal at the beginning and end of the eclipse, a signal that indicated that the earth’s gravitation field weakened slightly, or that the location was lifted into the air a few cm, or, perhaps, the gravitational field of the sun or moon had increased slightly. Their data, published in Phys Rev D 62, 041101, in units of 10^{-6}cm/s^2 looked like this:

Mohe gravimeter eclipse data

Mohe eclipse data

Let’s look at the data. Our first step will be to look at the elevation of the sun.
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The Force of Gravity

Six weeks ago I submitted a paper, “The Force of Gravity in Schwarzschild and Gullstrand-Painleve Coordinates” to the annual Gravity Essay Contest at the Gravity Research Foundation.

The Gravity Research Foundation
The Gravity Research Foundation (see the informative wikipedia article) was started in 1948 by a wealthy businessman, Roger Babson, who also started Babson College, a private business college. Babson’s motivation was to help physicists discover antigravity. Physicists soon convinced him to instead fund new research into gravitation (and who knows, maybe the antigrav equipment will appear later). And so this has become a mainstream annual essay contest, with many winners with Nobel Prize winners recognizable in the list of winners.

The results are in today. I got an “honorable mention”. The email comes with a sentence: “Please expect an invitation from Dr. D. V. Ahluwalia regarding possible publication in a special issue of IJMPD.” This is the International Journal of Modern Physics D, a peer reviewed physics journal (impact factor of 1.87) which specializes in gravitation, astrophysics, and cosmology.

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Matrix Decomposition by Discrete Fourier Transform

Given a 3-vector of complex numbers, (A,B,C), define its discrete Fourier transform as
(a,b,c) = (A+B+C,A+wB+w^*C,A+w^*B+wC)
where w = \exp(2i\pi/3) . That is, I’ll use lower case letters to denote the discrete Fourier transforms of UPPER case letters. The above leaves off a factor of \sqrt{1/3} but it will do.

Of interest today will be vectors (A,B,C) which happen to satisfy A+B+C = 0. These are eigenvectors of the Democratic D matrix

Democratic matrix

Democratic matrix with all entries D


that is, the matrix all of whose entries are equal to the complex number D. Of course their eigenvalues are zero. None of this is particularly interesting until we move from linearity to bilinearity and work with the discrete Fourier transforms of 3×3 matrices.

Define the Fourier transform of a 3×3 matrix U as u = F^{-1}UF/3 where F is the matrix:

Discrete Fourier transform matrix

Discrete Fourier transform matrix


where w = \exp(2i\pi/3) . With this definition, the discrete Fourier transform of the democratic matrix D, is:
Fourier transform of democratic matrix

Fourier transform of democratic matrix


This is a nice simplification.

Now let A+B+C=0 and compute some discrete Fourier transforms of four kinds of matrices, 1-circulant, 2-circulant, and two new types I will call “bra” and “ket” for obvious reasons. Untransformed matrices on the left, their transforms on the right, note that they fit together like the pieces of a jigsaw puzzle:
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An Immorality Tale

She was born with given name Johanna Maria Magdalena and a last name of either Behrend or Ritschel, my sources disagree. Her parents were unmarried, did she receive the last name of her father, Oskar Ritschel, or her mother, Auguste Behrend? In either case it was November 11, 1901. She was one of the most fascinating personalities of her time.

Bild 183-R22014

Youth
Her mother worked as a servant in Berlin and her father was an engineer who worked in various places around Europe. Soon after her birth, they married, but only for 3 years. Until she was 5, she stayed with her mother. Then she went to Belgium to visit her father who, after a delay of two years and insistent requests from the mother, finally told her that he had sent their child to be educated by the nuns at a convent (Catholic) boarding school in Brussels.

Her mother met and married a Jewish businessman, Richard Friedländer. When, the couple saw the conditions at the convent her mother decided to transfer her daughter to another convent, one that was less strict, in Vilvoorde, Belgium. Her parents moved to Schaerbeek, near Brussels (Belgium), and now she was able to come home to visit. With the marriage, she became Johanna Maria Magdalena Friedländer, and from the age of 7 she was raised in a household that observed both Catholic and Jewish customs.

In 1914, the world descended into the horror of the first world war. As German aliens living in Belgium, overnight the Friedländers became refugees. Eventually they made it to the German border, probably feeling fortunate that there was space available on a cattle car for them. As the modern world is one of passenger jets, the railroad was the transportation mode of the first half of the 20th century. Transport by livestock car is not a pleasant thing. Later, in the second world war, many thousands would be transported this way to the concentration camp at Buchenwald, where Richard Friedländer died. But let us return to her story.

Survivors at Buchenwald, April 16, 1945

Survivors at Buchenwald, April 16, 1945


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