Category Archives: physics

CKM in MNS form! Victory!

The MNS matrix (in tribimaximal form, which is compatible with all experimental measurements) can be written in elegant form as a unitary matrix which is the sum of a real 1-circulant matrix and an imaginary 2-circulant matrix. See Doubly Magic Matrices and the MNS. Now I’ve got the CKM matrix in the same form. And here it is:

The cool thing about the above is that it only involves 6 real numbers. Three reals define the 1-circulant matrix, +0.973313178, -0.008576543, and +0.000466480, while three more reals define the 2-circulant, +0.225761835, +0.040012680, and -0.004273188.

I’ve marked the larger contributions with red to help you see the symmetry. The real matrix has each of its rows shifted one to the right (i.e. 1-circulant), while the imaginary matrix has them shifted two to the right (i.e. 2-circulant). Note that the above matrix is unitary, which you will have to verify by taking dot products of its rows with the complex conjugatges of its rows, and the same for the columns. (You should get 1s and 0s.)

The experimental values I’m using for the CKM matrix come from hep-ph/0706.3588) and are:

If you square the elements of a row or column of the above, and add them up, you get 1, but it is not unitary. To put it into unitary form, we supplement each of the above real numbers by multiplying by a complex phase. The matrix is unitary when the resulting rows and columns are orthonormal.

The experimental measurement has 9 real numbers. Multiplying them by complex phases so as to make the matrix unitary is going to give you 18 real numbers. And yet I’ve been able to do this, put the experimental numbers into unitary form, and ended up with only 6 real numbers, and those are used in an elegant and symmetrical manner. This is by far the most elegant version of the CKM matrix around. It is the number crunched result of an observation by Marni Sheppeard that the CKM matrix is approximately the sum of a 1-circulant and a 2-circulant matrix. Well, now we have it exactly that way. Victory! But more to come.

To get an experimental measurement from the new magic form, compute the magnitude of the complex number. For instance, to get the top right value of 0.0042982, compute

|+0.000466480 – 0.004273188 i| = sqrt(0.000466480^2 + 0.004273188^2)
Uh, I get 0.0042986, which is within rounding errors on the experimental measurement number.

More generally, the nine elements of the experimental CKM matrix can be found by taking one element from the set {+0.973313178, -0.008576543, +0.000466480} along with one element from the set {+0.225761835, +0.040012680, -0.004273188} and computing the RMS value.
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CKM as a magic unitary matrix II

It seems much longer but it was only two months ago that I wrote a post giving the CKM matrix in magic unitary form. With half the Nobel prize in physics going to Kobayashi and Maskawa, the K and M of the CKM matrix, I should include a quick update.

The CKM matrix is usually written in absolute magnitude form. Recent experimental measurements, after correcting to ensure compatibility with unitarity (from hep-ph/0706.3588), is:

If you square the elements of a row or column of the above, and add them up, you get 1, but it is not unitary. To put it into unitary form, we supplement each of the above real numbers by multiplying by a complex phase. The matrix is unitary when the resulting rows and columns are orthonormal.

There are a lot of ways we could choose the complex phases. However, there is only one way (except for the sign of the imaginary unit i) of writing the matrix as a unitary matrix whose rows and columns sum to unity. This is called the “magic” property. The MNS matrix is quite simple when written this way. So I wrote a computer program and found the magic unitary form for the CKM matrix, to see if it would also have a simple form. Here it is:

Note that each row and column sums to 1, the rows and columns are orthonormal, and the absolute value of each element is as given for the experimental measurements at the top of the page.

The above is a slight improvement from the data I gave before. This is due to an error in one of the digits of the input data, that is, in the experimental measurements. Of course it is not accurate to all its digits, but I’ve included them because it’s a complicated nonlinear relationship between the absolute values of a unitary matrix and its magic unitary form; I don’t know how to properly scale the errors. (I could get them by varying the input data to the computer program that finds the magic solution. If someone wants this data, ask for it in the comments.) Below the fold, I’ll include the data in LaTeX format so you can copy it more easily:
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The X(3872) Files: Scanning for new mesons

New mesons! Let’s begin with a picture (from encarta) showing how new particles (states) were discovered back in the good old days:

Work on the big meson paper is continuing. I’m almost done with the isoscalar mesons. So in addition to describing just what these things are, I’m including a few lines on the new J/psi states, the X(3872), X(3940), X(3945), X(4260), and X(4360).

How to Find a New Hadron

Hadrons (mesons and baryons) are found by analyzing large numbers of particle interactions. When you plot the data, the graphs have little, uh, well the technical term is “bump” . Yes, hadrons, like children, begin as bumps. A hadron bump looks like this:

The above is from hep-ph/0510365. This bump happens to be one of the states, the X(3940), with which this post is concerned.
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Scrap Metal and European Banks Collapse

More excitement in the markets: Three days ago we got quoted $0.60 per pound for stainless steel scrap. Today we dropped by with 10 thousand pounds. The manager was apologetic, but they were no longer buying at any price. Scrap metal prices collapsed about in half (at least here on the West Coast), probably due to a lack of buying in China.

And remember my post on the unusual activity at the Federal Reserve? “ For example, it must be noted that business cycles are world-wide” and therefore business cycles cannot be blamed on individuals, political parties, countries, or even continents. The bubbles always burst eventually, and when people in one country see the bubble burst in another, monkey see, monkey do. They act to avoid the consequences of a bubble bursting in their own country and that bursts their bubble. (Hence the famous stock trading advice: Avoid Panic. But, if you absolutely must panic, try to do it before everybody else does.)

Well, as expected, European financial companies are failing and for the same reason US financial companies have had trouble. They did the same thing that US financial companies did. And their equivalents to our Federal Reserve are doing equivalent things.

The human inclination to get over enthusiastic in markets, and to ignore risks, is universal. If you outlawed it one way, they’d find another way to do it. It’s like teenagers and sex. But even worse, humans imitate other humans so the European troubles are identical to our own. Too much debt, and, in this bubble, an inclination to create mortgage backed securities to theoretically decrease risks, part of which is due to physicists. The latest news from Europe:

Wall Street crisis spreads through Europe’s banks
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PDG on road to fixing psi(3770) meson mass

The big paper on the mesons is coming along. Yesterday I was checking the mass formulas for heavy quarkonium, a subject which was discussed here a few months ago. While checking numbers for the paper, I found that the Particle Data Group has changed the values for psi(3770) in our favor! For us it was exciting so Kea says I have to blog it. And in fact I feel that I did an inadquate job the last posting on this.

So first, a little background. Mesons are made of a quark and an anti-quark, plus the color and electric force that binds them together. The usual method of modeling them is to simplify the color force and treat it as if it were just a scalar force like the electric force, and then calculate the binding energies (and therefore the masses of the mesons) by using the methods used to calculate atomic energies.

What I’ve been doing instead is looking at the problem from the point of view of quantum information theory. With this one ignores the spatial and momentum information and looks only at the information content of the particle states. When one does this to a spin-1/2 particle like the electron, one uses qubits. Since I want to model the color force but don’t care about spin, I use qutrits instead with the three states being red, green, and blue.

The usual method of modeling the mesons works best for lowest energy states of the heaviest quarkonium. This is because these states are the least relativistic (because the quarks are so heavy) and the color force isn’t as strong (since these quarks are so heavy, their deepest bound states are smaller than other mesons, and since the quarks are close to each other, the color force is reduced by asymptotic freedom). With my method, the reverse should be true; I should be more accurate at bigger states where color is more important. These states are either higher excitations or have lighter quarks. This is because I treat the color states correctly but don’t work on getting momentum modeled correctly.
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Dark Flow, the Speed of Gravity, and the CMB

Kashlinsky, Atrio-Barandela, Kocevski, and Ebeling have just put out a preprint on the peculiar motions of galactic clusters: A measurement of large-scale peculiar velocities of clusters of galaxies: results and cosmological implications. In short, they claim that all galactic clusters appear to have a motion with respect to the cosmic microwave background (CMB). The motion of a galactic cluster slightly effects the energy of the microwave radiation that travels through it, so they use the temperature map of the CMB to determine the velocity of those galactic clusters.

And the result is that the whole (observable) universe appears to be moving with respect to the CMB. This was not expected because the observable universe is approximately isotropic and so shouldn’t be going anywhere. They write (in the abstract):

This flow is difficult to explain by gravitational evolution within the framework of the concordance LCDM model and may be indicative of the tilt exerted across the entire current horizon by far-away pre-inflationary inhomogeneities.

However, the tilt is easy to explain when you assume that the speed of gravity is larger than C: If gravitational interactions travel faster than light, you will automatically be able to feel the gravitational attraction of matter even if it is too far away for you to see.
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The image of the Sun

Today at Moses Lake the sun went down in cloudless skies. Naturally, my mind drifted to the subjects of religion, time, astronomy, sunspots, optics, history, and climate, and of I ran into the fermentation section of our ethanol plant to see if I could image the sun. The west wall of our plant has a lot of little (maybe 1/8″ = 3mm) holes in it and a good view of the setting sun. The sun shines through each of these holes, and they produce images of the sun on the opposite wall. These are, literally, sunspots in the sense of “spots of sun”:

Images of the sun like these were used by the Catholic Church as a very accurate sun dial. They drew curves on the floor to indicate noon, summer and winter solstice, etc. These are called Meridian lines. When one specifies a time as “AM” or “PM”, this is an abbreviation for “ante meridian” or “post meridian.” The moment at which the disk is perfectly centered on the meridian curve indicates high noon. High noon occurs in slightly different directions depending on the time of year, so the meridian figures are curved in a distorted figure eight, called an analemma. The very interesting history of the meridian lines inside churches is the subject of the book The Sun in the Church: Cathedrals as Solar Observatories by J. L. Heilbron.

With the improvement of time keeping, the necessity of making solar observations by sunshaft decreased and some of the beautiful meridian curves have been removed from cathedral floors. The Catholic Church, however, still supports astronomy. The tradition is carried on at the Vatican Observatory in Arizona.

You can get a better sun image at home if you use a piece of paper. After adjusting contrast and brightness, here’s my image of the sun:

The above is a nice clean image of the sun. You can tell that it is in pretty good focus by noticing how sharp the border of the sun’s disk is. If there were sunspots visible at that time, I would have seen them.
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Quantum Entanglement

Alice and Bob are very anti-social electrons. For financial reasons, they have to share a house (a helium atom). Due to their antisocial nature, they come to a condition where both of them never are in the same spot (state) at the same time. We’re interested in their spin, (political spin). If we ask one of them what their spin is, a question that technically would be phrased like “is your spin aligned with the y-axis?”, the answer we must get is “with” or “against”. That is all the answer they can give. And if we then immediately go and ask the other the same question, we will have to get the opposite answer. If Alice answered “with”, then Bob will answer “against”.

This all assumes that Alice and Bob are in their “ground state”, which is the worst financial condition (lowest energy) possible. If one of they hae a little cash, they could be in an “excited state” and they could end up with the same spin. But in that case, they would still have to be found in different positions (and with different energies). For example, Alice could be downstairs eating while Bob is upstairs sleeping.

The physicists say that these sort of living together difficulties arise because electrons are fermions: anti-social quantum creatures in that two of them are never found in precisely the same quantum state. This is called Fermi-Dirac statistics, or the Pauli exclusion principle. Fermi, Dirac and Pauli are three physicists. “Statistics” from the fact that when you make computations using “statistical mechanics,” you have to count up the number of ways a certain situation can be achieved, and if the particles can’t fit into the same state it reduces the number of ways. Fewer ways makes that situation less likely. Statistical mechanics forms the foundation of thermodynamics, the science of temperature, pressure, volume and all that.

So electrons are fermions. This is a good thing. The reason that gravity doesn’t pull you down to the center of the earth is because the electrons in your shoes can’t fit into the same quantum states as the electrons in the stuff you’re walking on. The same principle keeps neutron stars from collapsing into black holes. Hooray for fermions!!!

The helium atom has two electrons. While two electrons cannot be in the same state, all electrons are identical in that they all have negative charge and are attracted to positively charged things. In the case of the helium atom, the nucleus is positively charged and the two electrons are attracted to it. They can’t escape, but they can’t be in the same state either.
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Quantum Cloning

It’s time I blogged some physics instead of filler like mouse transportation. There’s a lot of physics stuff going on around here but right now it’s kind of hush-hush and I can’t tell you about it. Which reminds me, I found an older version (perhaps a reader will disavow me of the notion that it is the oldest) of the line used in Top Gun, “I could tell you, but then I’d have to kill you“: Alexandre Dumas, in The Man in the Iron Mask aka The Vicomte de Bragelonne, writes:
“It is a state secret,” replied d’Artagnan, bluntly; “and as you know that according to the King’s orders it is under the penalty of death that any one should penetrate it, I will, if you like, allow you to read it and have you shot immediately afterwards.”

“The man in the iron mask” was a mysterious 17th century prisoner of the reign of Louis IV in France. Will I spoil the book if I tell you that in it, the state secret is that the man in the iron mask is the exact twin of the King of France? I hope not. It’s germane; in this post we will discuss what one would have to do to make the twin (or clone) of a quantum object, a [state] secret that evaded science until quite recently.

I will explain why this is of interest, and how this comes about in the language of quantum mechanics. For us, the quantum object will be an electron, and it’s state will be its spin.
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Quantum Field Mouse Cube Theory

I’ve been staying at our ethanol plant in Moses Lake, and I’ve got a bit of a rodent problem.

The locals tell me that these are “field mice”, and that if you have a place near grain fields these will invade your home. Wikipedia says “field mice” actually are meadow voles. Reading further, one can tell the difference by doing things like counting toes and looking at ear shapes and the like. I’ll more carefully examine the next prisoner and likely extract more information from him.

So far I’ve caught 3 with the Mouse Cube. Walmart sells these for $1.50. They catch the mouse because they can push a door open from the outside, but can’t open it again from the inside. Here’s prisoner #3 checking to see if that door is going to open just one more time:

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The Physics of David W. Talmage

Professor David W. Talmage (ret) has kindly allowed me to put a copy of his latest physics paper on the web:
Relativity with a Quantum Field

A sharp distinction has been made between the confirmed observations that were predicted and form the essential core of the theory of relativity and the untestable explanations that have become the lore of the theory and its vision of reality. The possibility is explored that it is these explanations, not the observations, that are incompatible with quantum mechanics. The explanation of the gravitational red shift, that photons lose energy as they climb out of a gravitational gradient, is the keystone to this lore. Once this keystone is removed the remaining explanations lose their coherence. Alternate explanations are presented that are not only compatible with quantum mechanics but require the existence of a quantum field.

This is only David’s latest in a fairly long list of papers supporting and exploring Lorentzian Relativity which the excellent Wikipedia article calls “Lorentz Ether Theory” . These were the aether theories that they don’t tell you about in school, the ones that are fully compatible with experiment. For them, the preferred reference frame is undetectable; what Einstein did was remove that reference frame.
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FQXi Essay Contest: The Nature of Time

The Foundational Questions Institute, FQXi, is running an essay contest, with the subject of the Nature of Time. Up to 21 prizes are to be awarded, with amounts ranging from $1000 to $10,000 per prize. But probably more important are the bragging rights. You have until December 1, 2008 to submit an essay. Your essay, assuming it’s “serious”, will be displayed on their website from soon after you turn it in until the contest concludes in mid December.

Right now, the contributions look fairly weak so who knows, maybe a little typing now will get you a little money, and some bragging rights, in early 2009. Of course I’ve typed up a contribution, and of course it has to do with the nature of time, as is suggested by the density matrix formulation of quantum mechanics.

I included some short comments on the subject of non Hermitian density matrices. These have something to do with raising and lowering operators and it’s worth typing up a quick blog post on the subject. Surely somebody is going to learn something.
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New Evidence Against Einstein: Si-32 and Ra-226

A post at physics forums has just pointed out to me a fascinating new article on arXiv that one shows that the radioactive decay rate of certain isotopes depends on what time of year one makes the measurement.

A new article, Evidence for Correlations Between Nuclear Decay Rates and Earth-Sun Distance by Jere H. Jenkins, Ephraim Fischbach, John B. Buncher, John T. Gruenwald, Dennis E. Krause, and Joshua J. Mattes, August 25, 2008, suggests that the different decay rates, which have been observed by laboratories in the US and Germany, on Si-32 and Ra-226, are correlated with the distance to the sun, and therefore, to the flux of neutrinos. However, in both sets of data there is a lag which suggests an effect due to the gravitational potential of the sun (which would be modified by the gravitational potential of the local galaxy stars), or perhaps local motion relative to a preferred reference frame. Either way, this is not good news for Einstein.

Relativity as an Anthropic Theory

As creatures of incredibly complicated biochemistry, which involves very large numbers of very carefully constructed molecules, our lives depend on the laws of physics as much as they depend on the earth not getting too close or too far away from the sun.
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Bit From Trit, and Lubos’ Booboo

Black holes and the standard model is a bit out of whack, as far as symmetry goes. I’ll discuss the issues, and then discuss how I think the problem is solved.

Quasinormal modes of vibration of black holes.
One of the curiousities of attempts to unify general relativity and the standard model is the quasinormal modes of vibration of black holes. This requires a little explanation.

It’s long been known that black holes exponentially approach a condition where the only numbers that characterize them is their mass, their spin, and their electric charge. This is sometimes called the “black holes have no hair” theorem, also known as Price’s Theorem, but named the “no hair theorem” by John Wheeler.

Suppose we begin with a black hole that has just a little hair, that is, we begin with a slightly perturbed black hole, one that is not quite symmetric. Over time, this hair will disappear. In the process of disappearing, the perturbation will change. As the perturbation dies out, it is possible that the perturbation will act as a sine wave multiplied by an exponential decay: \exp(-at) \;\sin(bt) , where a and b are constants with units of 1/time. If so, the constant b defines a characteristic frequency of this particular perturbation. And this is a fundamental characteristic of the black hole.
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Unitary Circulant Matrices and Idempotentcy

Let A_k be a complete set of N annihilating primitive idempotents. That is, \Sigma_k A_k = 1 , and A_jA_k = 0 when j is not k, and A_k A_k = A_k for all k. Also suppose that each A_k is Hermitian.

Let \alpha_k be a set of N real numbers. Then:
\Sigma_k e^{i\alpha_k}A_k is unitary.
I’m guessing that the reader will find the proof immediately. If not, ask in the comments and I’ll give the short proof.

I’ve been working on the CKM matrix recently, which is a 3×3 unitary matrix. For 3×3 matrices, the simplest complete set of annihilating primitive idempotents is the diagonal primitive idempotents, that is, the matrices that are zero except for a single one somewhere on the diagonal:
Diagonal 3x3 matrix complete set of annihilating primitive idempotents
The unitary matrices generated by this set are simply the diagonal matrices with complex phases down the diagonal.

The 1-Circulant Primitive Idempotents

Around here, our favorite complete set of mutually annihilating primitive idempotents for the 3×3 matrices are the 1-circulant ones:
Complete set of annihilating circulant primitive idempotent 3x3 matrices
Label the above three 3×3 matrices as P_I, P_J, P_K .

Interpreted as density matrix states, these are generated from the bra / ket states as shown in the right side of the above. Just as all other pure density matrix states created from state vectors, they are Hermitian. In addition, they are all magic. The rows and columns of P_I sum to unity, while the rows and columns of the P_J and P_K sum to zero.

So these are just what we need to write down an elegant parameterization of the 1-circulant unitary matrices:
e^{i\theta_I}P_I + e^{i\theta_J}P_J + e^{i\theta_K}P_K.
This is a unitary matrix whose rows and columns sum to e^{i\theta_I} .

The previous post showed a parameterization of the magic unitary 3×3 matrices. These included 2-circulant parts as well as 1-circulant. In the above, the P_I primitive idempotent is both 1-circulant and 2-circulant, while the other two matrices are 2-circulant. Can we get an elegant parameterization of the magic matrices this way?
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CKM as a Magic Unitary Matrix

Since writing the MNS as a magic unitary matrix, of course I’ve been working on writing the CKM matrix the same way. This involved learning a lot more about 3×3 unitary and 3×3 magic matrices, and writing a Java program to do the heavy lifting.

The first thing one must do to deal with magic unitary matrices is to define a parameterization of these matrices. A full parameterization of all unitary matrices requires 9 real variables. Five of these define the arbitrary complex phases that can be applied to any row or column (yes there are 3 rows and 3 columns, but one of them is redundant). The remaining 4 variables are usually written in the Wolfenstein parameterization. In this parameterization, three variables are mixing angles and the fourth variable defines the CP violation.

If one is given a magic unitary matrix, the effect of multiplying any row or column by a complex phase would be to destroy the magic. Consequently, putting a unitary matrix into magic form (if this can be done) amounts to choosing a set of unique phases. Therefore, the largest number of free parameters we can expect to need to define magic unitary matrices is the 4 used in the Wolfenstein parameterization (which also amounts to choosing a unique set of phases). In fact, I’ve found a parameterization of the unitary magic matrices on 4 real parameters so one supposes that any unitary matrix can be put into a (more or less unique) magic form.

First, let’s write some reduced notation. We will abbreviate the 1-circ and 2-circ matrices as follows:
Abbreviated circulant matrices
Note that that “E” and “F” are reversed in the above 2-circ definition. This is to make the Fourier transforms easier to deal with. In our abbreviated notation, we are to solve:
1-circulant + 2-circulant matrix as unitary
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Unitarity and the CKM Matrix

I’m busily working on writing the CKM matrix in Kea’s form, that is, as a unitary matrix that is the sum of 1-circulant and 2-circulant matrices. With the MNS matrix this was easier because I could begin with a unitary matrix, but the CKM matrix is usually given in absolute value form.

Looking through the literature, I’ve found a beautiful paper that digs to the core of the unitarity problem for the CKM matrix: A new type fit for the CKM matrix elements, Petre Dita, hep-ph/arXiv:0706.3588:

Abstract: The aim of the paper is to propose a new type of fits in terms of invariant quantities for finding the entries of the CKM matrix from the quark sector, by using the mathematical solution to the reconstruction problem of 3 x 3 unitary matrices from experimental data, recently found. The necessity of this type of fit comes from the compatibility conditions between the data and the theoretical model formalised by the CKM matrix, which imply many strong nonlinear conditions on moduli which all have to be satisfied in order to obtain a unitary matrix.

Dita’s CKM matrix estimate is:
Petre Dita\'s estimate for the CKM matrix
This needs a little explaining, probably.
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Pop quiz! Spin Powers, and latest MNS news

I’ve been working on a derivation of the tribimaximal MNS matrix from first principles. It looks like it will require an assumption that the underlying particles follow a set of mutually unbiased bases (MUB). The calculation involves spin products. These are messy to compute unless you know some tricks I’ve discussed here previously.

So, did I waste my breath? Is teaching how to compute these things like shouting down a well? Let’s have a contest. First person to solve the following problem gets a $50 prize:
Spin product
“Solve” means to write the above in reduced closed form in the comments section. The solution must be exact, for example, it must use \pi rather than some numeric approximation such as 3.14159. Now I’ve chosen the values so that it will be easy to do with the techniques shown here, but will cause problems with mathematica or other automatic assistants. So go for the glory! Of course the winner can also ask that the prize be donated to a needy physicist such as Marni Sheppeard.

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Electroweak unification, Quarks

To recap the previous post we began by combining the SU(2) spin-1/2 and U(1) operators into 2×2 matrices. We then showed that the leptons were solutions of the idempotency equation UU = U for 2×2 matrices subject to the additional requirement that the solutions be eigenstates of electric charge Q. For pure density matrix formalism, individual particle states are represented by primitive idempotents (with trace = 1), so we then converted these idempotents into primtive form by embedding them into 4×4 matrices. In doing this, we found that the idempotents given by the 2×2 matrices were composite, each being composed of two sub particles.

1-Circulant and 2-circulant matrices

In this post, we add the quarks to the picture. To do this, we need to use the 1-circulant and 2-circulant 3×3 matrices Kea talks about. We will write the general 1-circulant and 2-circulant matrices as follows:
generic 1-circulant and 2-circulant 3x3 matrices
Where I, J, K, R, G, and B are complex numbers. Note that there are only 6 complex degrees of freedom in the 1-circulant and 2-circulant matrices, one cannot create an arbitraray 3×3 matrix, with 9 complex degrees of freedom, from 1-circulant and 2-circulant matrices. In addition, setting R=G=B=1 gives a matrix of 1s, the same as setting I=J=K=1. Consequently, the 1-circulant and 2-circulant matrices together, have only 5 complex degrees of freedom, about half that of the 3×3 matrices in general. Writing a 3×3 matrix as a sum of a 1-circulant and a 2-circulant matrix is very restrictive; to write it as just a 1-circulant is even more so.

One obtains the basis for these matrices by setting one of the elements to 1 and the rest to zero. For example, putting I=1, J=0, K=0 gives the unit matrix. These 3×3 basis matrices correspond to permutations on three elements. We will think of the three elements being permuted as red, green, and blue, hence the labels R, G, and B for the 2-circulant matrices (i.e. R labels the permutation that leaves red unchanged and swaps green and blue, etc.). Similarly, “I” labels the permutation that leaves nothing changed, while J and K are the non trivial even permutations.
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Electroweak Unification, Leptons

Quarks will require 1-circulant and 2-circulant matrices. Before adding that complication, let’s unify the lepton quantum numbers. Two quantum numbers (other than generation) distinguish quarks and leptons, weak hypercharge t_0 and weak isospin t_3 . Weak hypercharge is a U(1) quantum number; it has only one generator and therefore is commutative. Weak isospin arrives in two representations, singlets and doublets. The singlets have weak isospin quantum number of 0 and so we can represent them with any sort of 0. The doublets have spin-1/2, which we represent with the Pauli spin matrices:
The Pauli spin matrices
There are three Pauli spin matrices, and they are linearly independent, so complex multilpes of them give 3 complex degrees of freedom. Since complex 2×2 matrices have 4 complex degrees of freedom, there is 1 complex degree of freedom left, the unit matrix:
Unit matrix, what the Pauli matrices don\'t get.

The unit matrix is a natural basis for a U(1) symmetry, so we can combine weak hypercharge with weak isospin into 2×2 matrices. The number of degrees of freedom in the 2×2 matrices is just sufficient to support an SU(2) spin-1/2 and a U(1). Given a representation of an SU(2) operator, and a U(1) operator, the 2×2 matrix representation is simply the sum of the Pauli matrix representation of the SU(2) operator, and the U(1) value times the unit matrix. Similarly, given an arbitrary 2×2 matrix, we can split it into a U(1) portion and the SU(2) portions:
2x2 matrix split into U(1) and SU(2) parts
The above allows the weak hypercharge and weak isospin operators to share a 2×2 matrix representation. To unify the states, we have to pass to a density matrix representation.

Given a normalized quantum state vector (a,b), the density matrix representation of the state is a 2×2 matrix:
Density matrix representation of state (a,b).
Similarly, given a U(1) state \psi , we can convert it into a 2×2 matrix. For qubits, such a (pure) density matrix would be boring, it would only be the unit matrix. But for wave functions that depend on position, the density matrix is not trivial and contains the relative phase information of the quantum state (which is the only phase information that is physical). But for this post, simply note that 2×2 matrices are rich enough to contain both types of quantum numbers. A density matrix is partially characterized by the fact that it is idempotent, that is, \rho^2 = \rho . This characterization is not complete in that the equation has other solutions, in this case 0^2 = 0 , and 1^2 = 1 . These other solutions have trace 0 and 2, the usual pure density matrix has trace 1. It turns out we need these other solutions so their having the wrong trace is an issue. Further down we will show how to convert the traces to 1, but for now let us postpone the discussion.
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