Monthly Archives: August 2008

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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