Bullets and Shells, Book Review

Bullet and Shell, the Civil War as the Soldier saw it is an inexpensive book at Amazon and a good read. The mild fighting as the Arab regions shake off old dictators gives me a reason to write a book review for it: it’s a lesson on why civil war should be avoided.

The book was written in 1884 by George F. Williams, an author with a very common name and possibly several other books. The book is a fictionalized memoir of the U.S. Civil War, 1861-1865, and follows the experiences of a young college graduate who enlists in the union side.

“Spin Path Integral” paper proofs sent off.

The “Spin Path Integrals and Generations” paper got accepted at Foundations of Physics. This initiates a series of emails that make you feel like a real researcher. I’m at the stage where they’ve sent the first cut proofs and asked me to make changes.

I screwed up some of the section titles (when I cut out the section on the mixing angles and inserted a section deriving spin-1/2) and so I fixed those things and clicked them into the online proof correction system. And this is the message you get:

Next thing is to finish up the 1st revision to the paper on unitary matrix parameterizations at Phys. Lett. B. I need some more calculations for the CKM and MNS matrices in magic form. I’m hoping this will finish up this week, I’ll get a week of feedback from my peeps, and then send it back to the reviewers there. And then Marni and I are working on improvements to our joint paper.

Progress in Publishing the New Physics.

Doing physics is fun. Writing papers is boring. Publishing them is quite painful. This year I’ve been concentrating on getting stuff published; I haven’t updated this blog. But with March madness receding into the past I’ve got some time and I’ll update what’s going on.

Marni Sheppeard has a new blog, Arcadian Pseudofunctor. She and I have submissions to the FFP10 conference proceedings. I still haven’t heard whether mine was accepted. I would think it’s getting kind of late. Of course Marni and I are writing papers; I expect to see another half dozen between the two of us by the end of the year. Right now I’ve got three more papers in the peer review process:

Spin Path Integrals and Generations

The Spin Path Integrals and Generations paper at Foundations of Physics got 2 of 3 reviews recommending publication with major revisions. I revised it on March 15th. Presumably the editor sent it out for review as it showed “under review” until March 27th. Since then it’s shown “Reviews Completed”. I imagine the editors are arguing over whether or not it should be published. I can understand this; the paper is radical in that it purports to give an explanation of the generation structure of the fermions from first principles. This is a major problem in elementary particles so it’s a serious step to publish it in your journal. It would be embarrassing, especially given the history Foundations of Physics has for publishing junk physics.

The third reviewer on “Spin Path Integrals and Generations” asked that I remove the sections on mixing angles and hadrons. This gave me more room in the paper so I added a section showing how spin-1/2 shows up in the long time limit.

ArXiv and the Wolfenstein Parameterization

A paper appeared on arXiv last week, “On one parametrization of Kobayashi-Maskawa matrix” 0912.0711 by Petre Dita. The abstract:

An analysis of Wolfenstein parametrization for the Kobayashi-Maskawa matrix shows that it has a serious flaw: it depends on three independent parameters instead of four as it should be. Because this approximation is currently used in phenomenological analyzes from the quark sector, the reliability of almost all phenomenological results is called in question. Such an example is the latest PDG fit from \cite{CA}, p. 150. The parametrization cannot be fixed since even when it is brought to an exact form it has the same flaw and its use lead to many inconsistencies.

Among phenomenologists, this is a pretty serious accusation. There are hundreds of papers on arXiv alone that use the Wolfenstein parameterization. It’s the basis for the PDG estimates on the CKM matrix. If it’s true this is really big news in elementary particles.

The Dita paper claims that the Wolfenstein parameterization is defective because its apparent four real degrees of freedom are redundant; instead there are only three. Such a defect would prevent the parameterization from exploring “almost all” of the space of possible 3×3 unitary matrices. Instead of the whole 4-dimensional real manifold of 3×3 unitary matrices (up to multiplication of rows and columns by complex phases), one would obtain only a 3-dimensional submanifold.

In particular, the paper claims that it is impossible to use the Wolfenstein parameterization to obtain a unitary 3×3 matrix with the magnitude of all amplitudes the same (and equal to sqrt(1/3) ). This is the “democratic unitary 3×3 matrix”, a subject Marni Sheppeard and I have explored at length. It took me a few minutes to verify that it is possible to set these parameters (lambda, A, rho, and eta) to obtain a unitary matrix with all magnitudes equal.

Quantization of event horizon radius and Quasar Redshifts

I’m getting ready for the FFP10 meeting later this month. In reading the abstracts of those who will be giving talks or posters, I came upon “Analyses of the 2dF deep field” by Chris Fulton, Halton Arp and John G. Hartnett. The abstract is about the relationship between low redshift and high redshift astronomical objects. The claim is that some quasars have redshifts that do not give their true distance; instead, they are much closer. Looking on arXiv finds: The 2dF Redshift Survey II: UGC 8584 – Redshift Periodicity and Rings by Arp and Fulton.

If these high and low redshift objects actually are related, this places doubt on the Hubble relation. In addition, when low and high redshift objects appear to be related, their redshifts are related by quantum values . From observations, Arp has proposed that quasars evolve from high to low redshift, and finally become regular galaxies.

Now for quasars to have redshifts that differ from their true distances implies that their redshifts are determined gravitationally; that is, what we are seeing is partly the redshift of light climbing out of a gravitational potential. And if these redshifts are quantized, this gives a clue that the structure inside the event horizon of a black hole is not a simple central singularity but instead there must be repetitive structure.

In a classical black hole, the region inside the event horizon can only be temporarily visited by regular matter. Even light cannot be directed so as to increase its radius in this region. Let’s refer to this region as the “forbidden region” of the black hole as it is near the central singularity. For the classical black hole, this includes everything inside the event horizon. We will be considering the possibility that the forbidden regions of a black hole occur as infinitesimally thin shells, and that between these shells, light can still propagate outwards:

Forbidden regions shown in red.

Event Horizons as Quantum amplitudes

If we were looking for a quantum mechanical definition of the inside of a black hole, we could define it as the region where particles have a zero probability of moving outwards. We could say that the transition probability for the particle moving outwards is zero. However, in quantum mechanics probabilities are defined as the squared magnitudes of complex amplitudes. The way we compute transition probabilities is from complex transition amplitudes. If the transition amplitude between two states is zero, we say that they are “orthogonal”. Zero transition amplitudes correspond to points where a sine wave is zero; at these points, deviations to either side give nonzero transition amplitudes:

How to get zero probabilities from nonzero in QM.

My Gravity paper accepted for publication

I’ve just got notice that my gravity paper, titled The force of gravity in Schwarzschild and Gullstrand-Painleve coordinates has been accepted for publication in the International Journal of Modern Physics D, with only a very minor modification.

I’m kind of surprised by this, given that the paper proposes a new theory of gravity. I was expecting to have that portion excised.

And to help make a week more perfect, my paper for Foundations of Physics, titled Spin Path Integrals and Generations, got a good review along with a nasty one (and much good advice from both), and the editor has asked for me to revise the manuscript and resubmit. So I suppose this paper will also eventually be published. I’m a little over half finished with the rewrite. This paper is, if anything, even more radical than the gravity paper.

Finally, the Frontiers of Fundamental and Computational Physics conference organizers have chosen my abstract (based on the Foundations of Physics paper) for a 15 minute talk. The title is Position, Momentum, and the Standard Model Fermions. Marni Sheppeard (my coauthor for a third paper, “The discrete Fourier transform and the particle mixing matrices” which so far is having some difficulty getting published), is giving a related talk, Ternary logic in lepton mass quantum numbers immediately following mine.

So all in all, I am a very lucky amateur physicist

The Young Headhunter Returns

I visited the University of Washington bookstore a week ago and they had a copy of the new translation of Táin_Bó_Cúailnge (The Tain) by Ciaran Carson on sale for $5.98 in hardback, so of course I bought it.

If you’re unaware of this classic, I’ll type in a page. For context, Cu Chulainn is returning to his own village after killing three of the village’s enemies. Perhaps due to post-traumatic stress disorder, or maybe blood-lust, he’s now a bit berserk and needs psychiatric attention.

“There’s a man approaching us in a chariot,” cried the look-out in Emain Macha. “He’s got the bloody heads of his enemies in his chariot, and a flock of wild birds overhead, and a wild stag hitched behind. He’ll spill the blood of every soldier in the fort unless you act quickly and send the naked women out to meet him.”

Cu Chulainn turned the left board of his chariot towards Emain to show his disrespect, and he said:
“I swear by the god of Ulster, that unless a man is sent to fight me, I’ll spill the blood of everybody in the fort.”

“Bring on the naked women!” said Conchobar.

The women of Emain came out to meet him, led by Mugain, the wife of Conchobar Mac Nessa, and they bared their breasts at him.

“These are the warriors you must take on today,” said Mugain.

He hid his face. The warriors of Emain grabbed him and threw him into a barrel of cold water. The barrel burst to bits about him. They threw him into another barrel and the water boiled up till it seemed it was boiling with fists. By the the time they’d put him into a third barrel, he’d cooled down enough just to warm the water through. Then he got out and Mugain the queen wrapped him in a blue cloak with a silver broach in it, and a hooded tunic. She brought him to sit on Conchobar’s knee, and that was where he sat from then on.

“Is it any wonder,” said Fiacha Mac Fir Febe, “that someone [Cu Chulainn] who did all this when he was seven should triumph against all odds and beat all comers in fair fight, now that he’s reached seventeen?”

Some notes
Cu Chulainn is a national hero to both sides of the border in Ireland. The famous statue of his death appears on the 1966 10 shilling coin:

Chariots appear prominently in The Tain but there’s at least some argument about whether these existed in Ireland. I have little doubt, see this page for a discussion and illustration. For the wealthy, imagine a chariot hooded with expensive colorful cloth. For the very wealthy, imagine the cloth covered further with bird feathers. The warrior is right handed and stands on the right side of a chariot, so the left side is the unarmed side. Hence the disrespect. Perhaps this has something to do with why the British drive on the left side of the road; they’re showing respect from the days of chariots.

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.

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 looked like this:

Mohe eclipse data

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

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.

Matrix Decomposition by Discrete Fourier Transform

Given a 3-vector of complex numbers, (A,B,C), define its discrete Fourier transform as

where . 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 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 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 where is the matrix:

Discrete Fourier transform matrix

where . With this definition, the discrete Fourier transform of the democratic matrix D, is:

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:

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.

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

New Paper on Hadrons and Koide’s mass formula

I’ve got a paper on the hadrons ready to submit to Phys Math Central. This is a fairly new peer reviewed open access journal for which I have a “pass” that allows me to avoid having to pay the $1500 submission fee, so long as I submit before January 31. This is a big deal and I want to do it right, so I’m looking for advice from readers.

The paper as it stands is here:
Koide mass formulas for the hadrons, 49 pages, LaTeX.

The subject is the extension of Koide’s lepton mass formula to the neutrinos and then to the hadrons. I’ve written the background section so it should be accessible to typical grad students in physics.

I’ve put this together as an example of applying quantum information theory to the practical problem of the hadron masses. This all is fairly simple stuff and it uses very basic ideas in quantum mechanics.

More arXiv papers, Dec 15, 2008

Various papers which I may not yet have read, but want to take a look at:

Geodesic stability, Lyapunov exponents and quasinormal modes by Vitor Cardoso, Alex S. Miranda, Emanuele Berti, Helvi Witek, and Vilson T. Zanchin.

Categorical Foundation of Quantum Mechanics and String Theory by A. Nicolaidis.

A Finite Electroweak Model Without a Higgs Particle by J. W. Moffat and V. T. Toth.

A list of astrophysical paradoxes, and on the idea of “paradox” in general:Astrophysical Paradoxes by Dragoljub A. Cucic.

A particularly interesting paper for me:
Infinite Statistics, Symmetry Breaking and Combinatorial Hierarchy by V.Shevchenko:

The physics of symmetry breaking in theories with strongly interacting quanta obeying infinite (quantum Boltzmann) statistics known as quons is discussed. The picture of Bose/Fermi particles as low energy excitations over nontrivial quon condensate is advocated. Using induced gravity arguments it is demonstrated that the Planck mass in such low energy effective theory can be factorially (in number of degrees of freedom) larger than its true ultraviolet cutoff. Thus, the assumption that statistics of relevant high energy excitations is neither Bose nor Fermi but infinite can remove the hierarchy problem without necessity to introduce any artificially large numbers. Quantum mechanical model illustrating this scenario is presented.

This week’s arXiv haul (Dec 5, 2008)

More than the usual number of interesting articles at arXiv caught my eye this week. I’m thinking about making this a weekly habit.

Denis Kochan’s new arXiv article: Does path integral really need a Lagrangian/Hamiltonian?, 0812.0678

Path integral formulation of quantum mechanics is strongly dependent on a given Lagrangian and/or Hamiltonian function. In the paper a simple rearrangement of the path integral to a surface functional integral is proposed. It is shown that the surface integral formulation of a transition probability amplitude is free of any particular choices and requires just the underlying classical equations of motion. A simple example examining functionality of the proposed method is considered.

FQXi Nature of Time Voting Begins

December 1st was the last day to submit an essay on The Nature of Time to FQXi. The contest was open for essays way back on August 4th. I submitted an essay titled Density Operators and Time, back on September 2nd. As of today, there are 127 essays so far. There could be more. There are 48 entries dated December 1st or, interestingly, 2nd. Three of my favorite theoreticians (uh, other than myself) have submitted papers:

Marni Sheppeard wrote Measurement processes and cosmological emergence. This is the only essay that manages to get a mention in for mutually unbiased bases.

Louise Riofrio writes on The Riddle of Time: R = t. This is a revisit of her stuff on R=ct, but with c suppressed, I suppose, so that it doesn’t count as previously published.

David Hestenes writes on the electron Zitterbewegung, Electron time, mass and zitter. This is basically an abbreviation and rewrite of his arXiv article, which, somewhat hilariously, got classified by Cornell as “general physics”: 0802.3227.

Riofrio and Sheppeard got their papers in just before the deadline and may have been a bit rushed. Nevertheless, since these things basically amount to popularity contests, I’ve voted for them. Hopefully, having at least one restricted vote will distinguish them enough that people will read them.

The leading entry for restricted votes is that of Carlo Rovelli, “Forget time” . He argues that we should look for quantum gravity in a form where time plays no role at all.

The Battle of Campeche

Matt Springer, of Texas A&M, via his blog Built On Facts is sending me large numbers of visitors so I thought I would share a small part of the history of Texas, and the Texas navy.

Texas was a part of Mexico when the colony of Spain obtained independence as a result of the Mexican War of Independence, 1810-1821. There followed a brief empire under Agustín de Iturbide followed by the first Mexican Republic in 1824 with Guadalupe Victoria (an assumed name) as President. The election to succeed Guadalupe Victoria was one by the founder of the Partido Moderador (Moderate Party), Manuel Gómez Pedraza, however, before he could take office, the now infamous Antonio López de Santa Anna forced him out and annulled the election. Antonio López de Santa Anna installed the first arguably African-American president of a major North American country, Vicente Guerrero, into power. One of Vicente Guerrero’s most important acts of his brief (1 April 1829 – 17 December 1829) term in power was to ban slavery and emancipate all slaves. The Presidency of Vicente Guerrero ended when his Vice President, Anastasio Bustamante lead a coup against him (and had him executed). More unrest followed, but many must have thought that Mexico was slowly becoming a democracy.

Liberal changes may be good (especially applied to conditions of the early 19th century), but Valentín Gómez Farías changed things too much and too fast for the Catholic and military parts of Mexico which revolted against him. The resulting conflict saw Antonio López de Santa Anna in power as President with the followers of Valentín Gómez Farías forced to hide or leave (mostly to the United States). Enough with democracy; Antonio López de Santa Anna reformed Mexico into a Catholic dictatorship in 1836 and tore up the Constitution of 1824. The new Constitution of 1835 eliminated the loose confederation of states and created a powerful federal government.

New preprint on the weak quantum numbers

I’ve just submitted a paper, Density Matrices and the Weak Quantum Numbers to Foundations of Physics. There are things about the paper that I didn’t include, things that I didn’t think were appropriate to a journal submission and I thought I’d talk about them here, and explain what the paper is talking about to a more general (but still math/physics) audience.

The paper is on the subject of the weak quantum numbers of the left and right handed elementary fermions and anti-fermions. Ignoring color and generation, there are 16 of these quantum objects. I provide a method of defining these quantum numbers by an idempotency equation, that is, by solving an equation of the form . Since pure density matrices satisfy this equation, the calculation is a density matrix calculation based on the permutation group on 3 elements.

The usual method of elementary particles is to assume that a symmetry relates the quantum states. In this calculation, the quantum states themselves are assumed to be composed of group elements of the symmtry. This can be done in density matrix formalism because density matrices can operate on themselves. Also of interest are what happens when different density matrices operate on each other. Particularly when the density matrices are chosen from the basis states of a complete set of mutually unbiased bases. But that’s another paper (mostly written).