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.

So I wrote up a one page paper giving the solution and sent it up to arXiv: Comment on “On one parametrization of Kobayashi-Maskawa matrix.” Did they accept it? No. And as of 24 hours later, no explanation.

So here’s my view of the situation. ArXiv had no problem whatsoever with publishing a paper that accused hundreds of phenomenology papers of being based on a very basic error. But they refused to publish a paper showing that the standard view of the CKM matrix is correct.

This is not the case of an amateur writing an insane paper claiming that the majority view on physics is wrong. It’s the reverse; an amateur is correcting an obviously defective paper written by a professional. Clearly arXiv cares more about the institution of the authors who contribute papers than they care about the actual content.

I’ve given my first lecture on physics to students at ITT Technical Institute, so I suppose I’m technically again in academia and can no longer claim amateur status. Also, I’ve got the final proofs back from IJMPD on my gravity paper, and I’m now quite convinced it will appear this month, so soon I should have my first published, peer reviewed, paper. This is after spending 25 years in the “real world”.

When I posted the gravity paper to arXiv, the geniuses there put it into the “gen-ph” category instead of gr-qc, hep-th, or astro-ph like every one of the hundreds of other papers which won honorable mentions at the annual Gravity Essay contests. It will look kind of funny when I update the paper with the citation data in a couple weeks; about the only paper published in IJMPD that is stuck in gen-ph is mine. Compare, for example, to IJMPD papers on gr-qc. Has arXiv treated me like crap? Hell yes!

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.

In quantum mechanics, it’s never the case that making steady changes to a state or pair of states causes their transition amplitude to become zero and then remain zero. Instead, upon obtaining zero transition amplitudes by arranging for a parameter to continuously change, we find that continuing to change the parameter causes the transition amplitude to become nonzero. For this reason, from basic quantum mechanical principles, it would be surprising that the forbidden regions of a black hole should have more than an infinitesimal thickness.

For the parameter that continuously changes, so as to create a forbidden region, we can propose the flux density of gravitons. This will be maximal at the central mass concentration, and then decrease, approximately following a 1/r^2 law, as one moves away from the center. Along this line, my paper that won an honorable mention at the annual gravity essay contest, provides calculations showing that this flux has to interact with itself in order to match tests of general relativity, see The force of gravity in Schwarzschild and Gullstrand-Painlevé coordinates, 0907.0660.

My paper on Spin Path Integrals

Now the paper I’m discussing at FFP10, Spin Path Integrals and Generations, proposes that the spin-1/2 elementary fermions have internal structure. The idea is that the left (or right) handed electron oriented with spin in the (1,1,1) direction, has three components, R, G, and B, which we will assume are oriented in the +x, +y, and +z directions. This behavior appears only at extremely short times; we don’t have sufficient energies in our experiments to observe it. However, the three degrees of freedom create three orthogonal particles which we do observe as the three generations. The paper discusses various examples of this. If you do all this, it turns out that spin and position end up with similar Feynman path integral behavior. This is a way of unifying the internal and external degrees of freedom of the elementary fermions.

I chose the +x, +y, and +z directions because they are taken from the three “mutually unbiased bases” of the spin-1/2 algebra, that is, the algebra of the Pauli spin matrices, the Hilbert space of two dimensions. Mutually unbiased bases are the finite dimensional version of complementary variables such as position and momentum. This is why they are used to make spin path integrals and position path integrals have similar behavior; we are taking the ideas behind position and momentum and applying it to spin.

Mutually unbiased bases are defined by requiring that transition probabilities be equal. Thus, if we are to include gravitation in this sort of theory we must assume that the effect of gravitons is to change the transition probabilities.

The paper is about the transition amplitudes and long term propagators for spin moving between orientation in the +x, +y, and +z direction (or any other three perpendicular directions). This is a model of the left or right handed electron; to get a complete model of the electron we have to allow for transitions between them. How can we modify these transition probabilities in order to model the effect of gravity?

From my paper, it’s clear that the (gravity modified) transition probabilities between +x, +y, and +z will have to stay the same. Thus we will assume that gravity modifies the transition probabilities between the states contributing to the left handed electron, say +x, +y, and +z, and the states contributing to the right handed electron, -x, -y, and -z.

If these transition probabilities are modified (but the ones among the + or – states remain unchanged), the requirement that we use a complete set of mutually unbiased bases amounts to our modifying the six states so that, for example, the +x, +y, and +z are moved closer together, and the -x, -y, and -z are moved farther apart.

Effect of gravity is to warp the MUBs.

The paper doesn’t discuss this, but the natural interpretation of the +x, +y, and +z states is that the particle is moving at some speed in these various directions. Since the maximum particle speed is c, that speed must be c sqrt(3). This result is similar to that of Feynman’s checkerboard model of the Dirac equation in 3+1 dimensions. See around equation (42) of Peter Plavchan’s informal paper Feynman’s Checkerboard, the Dirac equation, and spin.

With this interpretation, the effect of warping the mutually unbiased bases is to effect the maximum speeds of the particle in the +(1,1,1) and -(1,1,1) directions. For the illustration above, the particle speed in the -(1,1,1) direction has been slightly increased while the speed in the +(1,1,1) direction has slightly decreased. For a black hole, this would correspond to different speeds for the radial inward -(1,1,1) direction as opposed to the radial outward +(1,1,1) direction.

These calculations are easy to do with the particle’s spin oriented parallel or antiparallel to the gravitational force. To obtain more arbitrary directions, one uses linear superposition in a manner similar to how spin-1/2 can be written in terms of spin-up and spin-down.

Gravity as changes to velocity
These sorts of ideas about gravity, that it should be interpreted as a modification of the natural velocities of particles with differences between inbound and outbound particles, is used in the important paper by Andrew J. S. Hamilton and Jason P. Lisle, The River Model of Black Holes, Am.J.Phys.76:519-532,2008. This paper models rotating and non rotating black holes as a river of “space” that is sucked into the black hole. The river defines a velocity at each point in space and from this one can derive the various properties of black holes.

The non rotating coordinates used in Hamilton and Lisle’s papers are Gullstrand-Painleve, the same used in my paper on flux gravity. These coordinates, along with their rotating (and charged) generalizations, are unique in that they allow general relativity to be rewritten in terms of David Hestenes’ geometric algebra. This amounts to getting rid of the tensors general relativity is usually defined with, and replacing them with functions of Dirac’s gamma matrices. This is a particularly useful version of general relativity because Dirac’s gamma matrices are used to model the elementary fermions. The Cambridge geometry (geometric algebra) group has many papers giving calculations of electrons in black hole coordinates using these methods.

Redshifts are sort of multiplicative; to compare them we look at ratios of (1+z). For example, if an object has an intrinsic redshift (due to gravitation) of (1+y), and it is at a distance or moving with a velocity that gives a redshift of (1+x), then its total redshift is (1+x)(1+y).

The claim of the redshift quantization folks is that redshifts are quantized according to a factor of (1+z) = (1 + 0.23) . The factor 0.23 is suspiciously close to 2/9, the factor (prominent in my spin path integral paper above) which Marni Sheppeard and I call “that damned number”, which I’ve assumed comes from a sum over infrared divergences. These arise when considering particles with very small energies or very long distances.

Quasar redshift quantization

Quasar redshifts are observed by emission and absorption lines. In order for these to be gravitational rather than due to distance, the light we see, and the atoms responsible for the emission and absorption, have to originate deep inside event horizons of black holes. As far as arranging for the light to escape, this is not too difficult: Since the forbidden regions are thin, light need only tunnel through it in order to carry the imprint of the inner layer.

Inside the most central forbidden region, matter is not constrained to fall to the singularity so this region will consist of normal matter. Provided temperatures are sufficiently low, normal matter can exist in this region. This matter will emit light, providing emission lines, and imprint light with absorption lines. The light then works it way out of the quasar.

For all this to happen, the temperature of the black hole has to become sufficiently low that normal matter can exist deep with the forbidden regions. This can only happen if the black hole is sufficiently cool. Arp’s observation is that new quasars have high redshift so the implication is that they are cool as they are ejected, and then heat up as they age.

A black hole is heated by matter falling into it, so a cold black hole would be a good candidate for eventually heating up and running through the quasar sequence of quantized redshifts. When the black hole attracts sufficient matter to form a new galaxy, it has been sufficiently heated that its quantum structure becomes hidden.

As far as evidence, it might be possible to find a quasar with two sets of emission / absorption lines. This would happen when a black hole has cooled just sufficiently to expose two different forbidden regions. This would be observed as a single quasar with two different redshifts.

Cosmology

The cosmology implied by this model is one where the effects of the big bang and dark energy are due to changes in the background level of graviton flux. The effect of gravitons on matter is to change the probability amplitudes in such a way that, at low levels, the probability of a transition is increased. The transitions influenced (raised) are those that cause the particle to move in the direction from which the graviton flux arrived.

So over cosmological time, the internal clocks on particles is sped up. In a flat universe, the frequency of light is not changed with time; there is no stretching of space to redden it. However, the clocks of the observer do change and an observer will see ancient light reddened according to how much time has gone by.

It should be noted that when using Schwarzschild coordinates, the clocks of particles are slowed by the presence of a gravitating body. But in the velocity model, this is due to the velocity of those particles (near the speed of light) rather than the gravitons per se. To see particle clocks slowed by gravity, one can instead use Gullstrand-Painleve (GP) coordinates.

In GP coordinates, the particle speeds depend on the direction. For particles on a radial axis, the modified particle speeds are modified in such a way that the difference in velocity between an outgoing particle and an incoming one is still 2c (as in free space). In such a case, a particle moving back and forth between two points will be slowed down, relative to free space. The analogous example given in freshman physics classes is that when you travel one direction at 60+v mph, and travel back at 60-v, you will arrive later than someone who travelled at speed 60 both directions.

At the foundation of quantum mechanics particles with mass m have a de Broglie frequency proportional to its mass. If we are to increase the frequency over cosmological time, this is the same effect as increasing the mass. So a cosmological theory compatible with the evidence given in this post would be one where the masses of the elementary particles increase over time.

As it turns out, just such a cosmology theory is propounded by none other than Halton Arp. It’s called the “variable mass theory” and is mentioned in the paper I referenced above, Evolution of Quasars into Galaxies and its Implications for the Birth and Evolution of Matter.

The variable mass theory was originated by Jayant Narlikar. With Hoyle, he developed what is also called “Hoyle-Narlikar” theory. This was sufficiently long ago that it is difficult to find review articles on it. A more recent version is the quasi-steady state cosmological model (QSSC) Cosmology and Cosmogony in a Cyclic Universe, J.Astrophys.Astron.28:67-99,2007 (0801.2965), by Narlikar, Geoffrey Burbidge and R.G. Vishwakarma.

In theories where the internal clocks of particles changes with the background graviton flux, it is also the case that the speed of light (as measured by the particles for propagation between two fixed points) will decrease with time. This is the subject of Louise Riofrio’s cosmology.

Weird Gravitational effects

Finally, I would be remiss without mentioning that theories of gravity that involve graviton flux imply that when planets are aligned, gravity should be a little stronger. This effect, however, can be canceled by absorption of gravitons. It’s not clear what the net effect should be, however, strange behavior of pendulums and gravity measuring devices during total eclipses of the sun have been seen, and not seen. See my blog post, The Moon’s Subtle Influence.

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

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.

Among Cu Chulainn’s many many feats was destroying the Lia Fáil (Stone of destiny) by splitting it with his sword. Since then, the stone has failed to sing out in joy when a rightful king of Ireland places his feet on it, uh, with exceptions for Conn of the Hundred Battles and Brian Boru. The stone as it appears today:

An illustration of Cu Chulainn, in his chariot, from wikipedia, in his pretty, non-fighting, mode. Note the absence of beard:

His secret weapon is the Gáe Bulga, a sort of javelin that enters with a single wound but then spreads into barbs that fill the body. He uses this to kill his long time friend, Fer Diad.

A note in the book mentions the 1973 rock album by Horslips titled The Tain. Multiple bad copies of most of its songs are on you-tube, if you want to listen to them. My favorite cut is Dearg Doom, or Cu Chulainn’s lament. I’m not sure who did this version, it doesn’t quite sound like the original to me, but it’s not bad; a sweet sad song. Someone listening to you sing this will suppose that it is a song of lost love, rather than one suitable for use after you’ve just killed your best friend.

Life was a game.
Now I miss your name;
Your golden hair.
No more in your eyes is the blue of skies – Only shame.
A lonely soul, betrayed by love,
You walked into the stream.
With tears of love upon my cheeks
I heard your final scream.

More recently, The Decemberists cut an EP titled The Tain. And they put out a somewhat creepy music video for the track using silhouette crepe paper stop motion animation. A mild example of the creepiness is the aftermath of the final battle:

In the video, you will note that Cu Chulainn, when in fighting mode, has one of his eyes bulged out onto his cheek. This follows the original text and is actually fairly mild compared to the written description.

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.

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 simplest possible case for something like Heisenberg’s uncertainty principle occurs in spin-1/2. This is a 2-dimensional Hilbert space so calculations are a lot simpler than the infinite dimensional case needed for position and momentum. In these finite dimensional Hilbert spaces the quantum information theorists refer to two observables as “mutually unbiased” if they are related the way that position and momentum are. That is, knowing one of the observables exactly means that you can know nothing at all about the other observable. In terms of quantum mechanics that means that the transition probabilities are all equal, or unbiased.

A paper by George Svetlichny describes Feynman’s path integrals using the concept of MUBs:

Feynman’s Integral is About Mutually Unbiased Bases, George Svetlichny, 2007:
The Feynman [position path] integral can be seen as an attempt to relate, under certain circumstances, the quantum-information-theoretic separateness of mutually unbiased bases to causal proximity of the measuring processes.

My two latest papers apply the idea of path integrals being about mutually unbiased bases to spin-1/2. Under this assumption, spin is no longer an observable that stays constant after you measure it. Instead, it jumps around just like position. So the papers are about applying the idea behind Heisenberg’s uncertainty principle to the simplest quantum mechanical system possible, the qubit (or spin-1/2). The two papers are for Foundations of Physics. The first is:

Spin Path Integrals and Generations, Carl Brannen, 2009:
Two consecutive measurements of the position of a non relativistic free particle will give entirely unrelated results. Recent quantum information research by G. Svetlichny, J. Tolar, and G. Chadzitaskos have explained this property of position observables as a result of a path in the Feynman integral being mathematically defined as a product of incompatible states; that is, as a product of mutually unbiased bases (MUBs). On the other hand, two consecutive measurements of a particle’s spin give identical results. This raises the question “what happens when spin path integrals are computed over products of MUBs?” We show that the usual Pauli spin is obtained in the long time limit along with three generations of particles. We propose applications to the masses and mixing matrices of the elementary fermions.

The above paper is in the “under review” stage at Foundations of Physics. This seems to be the middle stage for a paper, the first stage is “editor assigned” or something like that. The paper buries deeply into the shaky foundations of physics. Those who are working on the periphery of physics imagine that the foundations are more stable than they are and will find the paper a bit shocking. For laughs, I’ve linked in the copy that I uploaded to arXiv. Some moronic Cornell grad student moderated it off. Maybe I’ll resubmit it when the paper is accepted for publishing, maybe not. The second paper is:

Path Integrals and the Weak Force, Carl Brannen, 2009:
In a previous paper, we showed that spin-1/2 can arise from a more primitive form of spin called “tripled Pauli spin”, along with three generations of elementary fermions. This gave a possible explanation for various coincidences in the fermion masses and mixing matrices. In this paper we continue the analysis. We show that the weak hypercharge (t_0), and weak isospin (t_3) quantum numbers can be derived from the long term propagators of three tripled Pauli spin particles. This completes a derivation of the standard model elementary fermions.

The second paper should be ready to submit to Foundations of Physics around the end of the month. The above is a first draft. Of course I’d appreciate comments and corrections. Right now I’m thinking that my next paper will be about what all this says about the geometry of spacetime. Or maybe I’ll write an introduction to Clifford algebra.

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.

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.

Elevation of the Sun
If the Mohe anomaly is due to gravitons emitted from the sun and or moon, then we need to adjust for how far off the horizon these bodies are. The eclipse was observed at latitude 53°29′20″ N and longitude 122°20′30″ E, on March 9, 1997. Sunrise was at 06:20:00 local time, with first contact at 08:03:29, totality from 09:08:18 to 09:11:04, and fourth (last) contact at 10:19:50. The first anomaly peaked at around at around 7:30 while the second anomaly goes to at 10:20.

To see how far off the horizon the sun was at 7:30 and 10:20, we go to John Walker’s convenient applet Your Sky, and find the sky map for that spot on the earth at that time. Whoops, he doesn’t quite manage to get the numbers exactly right. Here’s his picture for the center of totality, showing the sun erroneously peeking out underneath the moon:

John Walker's eclipse image

Okay, so let’s go to Chris Obyrne’s Javacript eclipse calculator and select the 1997 data at the given longitude and latitude. We find that first contact has the sun 14 degrees of the horizon while fourth contact is 28 degrees. This is compatible with Walker’s program having the sun 21 degrees over the horizon at the center of totality.

Now, if the Mohe measurements are to be interpreted as modifications of the total gravity strength due to (a change in) the sun and moon, then, since the earth’s gravity dominates, we need to divide by the sine of the elevation of the sun. At first contact, this is a factor of 4.13, and at fourth contact 2.13; adjusting the anomaly peaks, this gives the first contact peak at around and the fourth contact goes to . These numbers are somewhat closer to symmetric than the originals.

Tides and Gravimeters

But we can’t interpret the data as measurements of the force of gravity. It is impossible, in a certain sense, to measure the force of gravity at a single point in space. What a gravimeter actually measures, at best, is an acceleration. Installed properly, this acceleration is the force applied to the base of the gravimeter. Consistent with Einstein’s Equivalence principle, a gravimeter in free-fall will measure no gravitational force.

Consequently, interpreting the anomaly peaks as corresponding to changes in the flux of gravitons is simplistic and wrong. But before we discuss what gravimeters actually do measure, let’s discuss the timing of the anomalies.

Gravitational Timing

A significant difference between Einstein’s theory of gravity and Newton’s is that in Newton’s theory, the force is instantaneous at a distance. For theories based on superluminal gravitons, the timing of the pulses needs to be adjusted.

It takes light from the sun about 8.3 minutes to reach the earth. Consequently, while we detect the center of the total eclipse at 09:09:41, the sun actually emitted that light at 09:01:23. This was the time at which the bodies were in alignment.

Thus, if gravitons travel at very high rates, we would expect that the eclipse’s modification of gravity would be symmetric not around the time of deepest eclipse, 09:09:41, but instead around the actual time of alignment, 09:01:23. Thus those who believe in faster than light gravitons expect the gravitonal eclipse effects to appear around 8 minutes before the photon eclipse effects.

Returning to the Mohe gravity data, we see that the first pulse is almost entirely over by the time of first contact, but that the second pulse is close to coincident with fourth contact. To make these two pulses symmetric with respect to the photon measurements of the time of the eclipse, we have to shift the gravity measurements later by around 7 or 8 minutes.

The Acceleration of the Sun and Moon
In preparation for discussing tides and gravimeters, it’s useful to calculate the acceleration, on the earth, of the sun and moon. Since I am a (hopefully soon to be published) general relativity theorist, I will make these calculations in the professional way, using units of centimeters only. To convert from cm to those ugly pedestrian units involving grams and seconds, we will be setting to unity, the gravitational constant G, and the speed of light c:

and so

Looking in wikipedia’s collection of Solar System values, the masses of the sun and moon, and their distances from the earth, are:

Mass Sun: ,
Distance: ,

Mass Moon: ,
Distance:

In theoretical general relativity, the formula for (Newtonian) acceleration is very simple; A = M/r^2. Applying this formula to the above, we get the force that the sun and moon apply to the earth, in units of 1/cm. We can convert these into cm/s^2 by multiplying by c^2. We find:

Sun accel: .
Moon accel: .

Tides

Sailors know that the moon makes stronger tides than the sun; so it is a little surprising that the gravitational acceleration of the sun is stronger in its effect on the earth. The tidal effects of the moon are stronger because the moon is so close.

Tidal forces are caused not by gravitational accelerations, but instead by differences in gravitational accelerations. The strong tides of the moon are caused by the difference in gravitational acceleration between the side of the earth closest to the moon and the side farthest away. As far as tides go, gravity is a 1/r^3 force, not a 1/r^2.

Gravimeters show tidal forces rather handily. They cannot measure actual gravitational forces. Consequently, when we use a gravimeter to look for gravitational anomalies, we need to interpret what we measure not as a gravitational force, but instead as a tidal force.

Since eclipses last several hours, the data for these sorts of experiments has to be corrected for the usual tidal forces. This means that there are lots of opportunities for getting the wrong answer. Later experiments, particularly with the European eclipse of 2003,

Like I said, I don’t know if this is significant. At least it is interesting. For a recent review of conventional explanations for these sorts of observations, see A review of conventional explanations of anomalous observations during solar eclipses by Chris P. Duif, and Review on Possible Gravitational Anomalies by Xavier E. Amador.

Corpuscular Gravity Shielding

The corpuscular graviton shielding theory is that gravity is a shielding effect; two masses are attracted to each other by their shielding of each other from the effect of an isotropic and universal sea of gravitons. This sort of theory expects to see a reduction in the effect of the combined sun and moon gravitational attraction during an eclipse.

Such a reduction would mean that the earth’s gravity field would become stronger. This is the opposite of the observation above, but there is evidence for a decrease in other gravimeter measurements. See Un Résultat Gravimétrique pour la Renaissance de la Théorie Corpusculaire “An Experimental Gravimetric Result for the Revival of Corpuscular Theory “, Maurice Duval, (in French). These are difficult measurements and difficult interpretation.

For how corpuscular theories of the graviton fit with these observations, see Un Résultat Gravimétrique pour la Renaissance de la Théorie Corpusculaire “An Experimental Gravimetric Result for the Revival of Corpuscular Theory “, Maurice Duval, (in French).

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.

My initial feeling was to submit the paper instead to Foundations of Physics as that is how I formatted the longer version of the paper originally. The essay contest paper has been cut down to approximate the 1500 word limit for the contest. But Foundations of Physics has an impact factor of 0.951, and is a more general journal, so overall, the paper belongs in IJMPD, if they are willing to publish it.

The Kepler Problem
My paper is about the orbits of test particles around the simplest sort of black holes (or “Schwarzschild metric” ; i.e. non rotating and uncharged. This is a fairly simple problem that was solved analytically a very long time ago, the wikipedia article is called the Kepler problem in general relativity. Of course the original Kepler problem was for Newton’s gravity.

The Kepler problem is to figure out the motion of very small (test) masses in the presence of a large mass. Newton’s gravitation is defined as a force and for this problem it is very simple:

where For simplicity, this uses units where the large mass M = 1, and the gravitational constant G=1. With the relativistic form, we will also put the speed of light c=1. Mathematically, the above is three coupled differential equations in t, one for x, one for y, and one for z.

I wanted to compare these differential equations with those for the corresponding problem in general relativity. This is not so easy to do. The motion of test masses in general relativity is written using geodesic equations. To make a long story short one ends up with four coupled differential equations in the “affine parameter” s, three for x, y, and z, and one (extra one) for t. So the problem is to convert these 4 differential equations to just 3 by eliminating s.

In converting general relativity geodesics into Newtonian forces, one must choose the coordinate system that one will assume to be equivalent to Newtonian. The usual method is to use Schwarzschild coordinates; but I had theoretical reasons (explained briefly in the paper) for preferring Gullstrand-Painleve (GP) coordinates. These coordinates are best explained by the paper, The River Model of Black Holes.

Post Newtonian Expansions
The usual technique for converting from geodesic equations to Newtonian force is to use the “post Newtonian expansion”. This is only an approximation. If one needs higher accuracy, one works with post post Newtonian, etc.

I didn’t want to deal with ugly approximations, and of course post^n Newtonian expansions in the literature are only for Schwarzschild coordinates so I decided to solve the problem exactly. Various people told me that it was impossible or unreasonable. In fact the calculation was horrendous, with hundreds and hundreds of terms. I used a symbolic calculus calculation engine, MAXIMA, which has the essential advantage over its competition that it is free.

Eventually I did manage to solve the geodesic problem exactly, both for Schwarzschild and Painleve coordinates. This is the first result in the paper; these are (6) and (7). The second result was that, in GP coordinates, the exact form can be written as only a few terms, each as powers of the radius.

At long distances r>>1, and low velocities v<<1, the Einstein result approximates the Newtonian. Writing Einstein’s result as a sum over powers of the radius is an invitation to figure out which of these terms have been tested by various tests of general relativity. So the next natural step is to figure out which terms are needed for compatibility with these experimental results. My paper does this, but only for the deflection of light. I’d have done more but I was only aware of the essay contest a few days before the deadline.

Gravity as a Flux of Gravitons
Finally, Newton’s gravity can be explained as being due to a flux of gravitons: Each massive object emits these in all directions and they shoot off in all various directions at an infinite speed. At a distance r, they are spread out over the surface , of the sphere with radius r. This accounts for the decrease in strength of the gravitational field proportional to 1/r^2. But Einstein’s theory gives a correction to this law; what does this say about how the flux of gravitons interacts with itself?

The first thing to note is that in Einstein’s theory, gravity waves move at speed c. In a graviton theory of Einstein’s gravity, one would naturally suppose that the speed of the gravitons is also c. But this is a local speed limit; in practice it will depend on the choice of coordinates. For GP coordinates, the (maximum) speed of light near the gravitating source is always higher than it is far away. (Note that for Schwarzschild coordinates, light slow down near the event horizon and never exceeds the speed of light far away.)

The reason the speed of the gravitons is important is that if the test body is moving with respect to the graviton flux, the speed of the graviton flux will change how much flux is seen by the test mass. For example, if the flux is moving at speed v, then the amount of flux seen by a test particle moving this speed in the same direction as the flux will be reduced to zero. A way to avoid the issue completely is to look at situations where the speed of the gravitons doesn’t matter. This happens if the test particle is not moving; the “gravitostatic” situation.

The fourth point of the paper is a computation of the gravitostatic attraction of gravity in Schwarzschild and GP coordinates. The result shows that if gravity is interpreted as due to a flux of gravitons, then that flux becomes stronger with distance. (That is, when integrated over the surface area of the sphere, the amount of flux increases with the radius.) So in the final part of the paper I showed that the amount of increase in flux is proportional to the square of the flux density. This is compatible with a theory of gravity where the graviton flux interacts with itself. Think “dark energy.”

Anyway, I had hopes that the paper would win a prize, but I guess I can understand why it has to go through peer review; it has a lot of calculations that need verifying.

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

Define the Fourier transform of a 3×3 matrix U as where is the matrix:

Discrete Fourier transform matrix

Fourier transform of democratic matrix

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:

3x3 matrix by Discrete Fourier Transform jigsaw puzzle

The discrete Fourier transform for 3×3 matrices is 1-1 and onto, so the above gives a proof that we can always write a 3×3 matrix uniquely as the sum of a democratic, 1-circulant, 2-circulant, bra, and a ket matrix, each of which, other than the democratic, is subject to the constraint that A+B+C = 0. This gives a decomposition of a 3×3 matrix into five parts, with the democratic part giving one and the others two each degrees of freedom, for a total of nine (complex) degrees of freedom.

Unitary Matrices

Some time ago, I gave a parameterization that seems to be a complete parameterization of the magic unitary 3×3 matrices. This uses 4 real parameters. We can add five more parameters by multiplying rows and columns by arbitrary complex phases. This gives a total of nine parameters, just what are needed for an arbitrary unitary matrix. So I’m wondering how to prove that this parameterization is complete or not.

Of the five components of a 3×3 matrix as given above, three are magic, that is, three have rows and columns sum to the same value. The bra and ket forms have rows or columns that sum to the same value but not both. Consequently, we know that any magic matrix can be written uniquely as the sum of a democratic, a 1-circulant, and a 2-circulant and this must also apply to the magic unitary matrices. Because the decomposition is 1-1 and onto, we know that the magic matrices have 1+2+2 = 5 complex degrees of freedom.

The 1-circulant, 2-circulant, and democratic matrices are closed under multiplication and addition, thus the magic matrices are a subalgebra in the algebra of 3×3 matrices. The unitary matrices are also a subalgebra of the 3×3 matrices and so are the 3×3 magic unitary matrices. Now the 4 parameter parameterization I gave for them was under the additional restriction that the sum of the rows and columns, in addition to being equal, was required to be equal to 1. We can generalize with a 5th parameter to multiply the whole matrix with a complex phase and therefore make the sum of each row and column be an arbitrary complex phase. And this gives 5 parameters that define the magic unitary matrices completely.

This is kind of funny. Magic 3×3 matrices have five complex degrees of freedom, while magic 3×3 unitary matrices have five real degrees of freedom. Could there be a relationship between them?

All this discussion has to do with the democratic, 1-circulant, and 2-circulant portions of the matrices. What about the bra and ket portions?

The method I’ve been using to get a general unitary matrix into magic form is by multiplying the rows and columns by arbitrary complex constants. Ignoring the overall complex phase, we get to pick two rows and two columns for this treatment:

Multiplying rows and columns by phases

Comparing with the table of discrete Fourier transforms, the matrices of complex phases in the above equation can be seen to be the discrete Fourier transforms of 1-circulant matrices, that is, they are of the form

where, as before, the complex numbers A,B, and C are related by A+B+C = 0. This gives two complex degrees of freedom or four real degrees of freedom, again just twice what we need to arrange for multiplication by two complex phases.

So I think I’m getting fairly close to cobbling together a proof that the new parameterization does cover all the unitary matrices, when it is extended by multiplying columns and rows by complex phases.

The funny thing about all this is that it has the feeling of complete triviality, but I’ve never seen the unitary matrices parameterized so nicely.