Monthly Archives: February 2008

Book Review: West With the Night, Beryl Markham

West With the Night is the autobiography of Beryl Markham, a pioneering aviator of pre WW2 Africa. This remarkable autobiography dates to 1942 and is so beautifully written that if you know someone who loves flying you should pick them up a copy. As usual, let me introduce the book by a few quotes:

From the time I arrived in British East Africa at the indifferent age of four and went through the barefoot stage of early youth hunting wild pig with the Nandi, later training race-horses for a living, and still later scouting Tanganyika and the waterless bush country between the Tana and Athi Rivers, by aeroplane, for elephant, I remained so happily provincial I was unable to discuss the boredom of being alive with any intelligence until I had gone to London and lived there a year. Boredom, like hookworm, is endemic.

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Berry ( or Pancharatnam-Berry or quantum ) phase.

Classically, if a vector is parallel transported around a closed circuit on a curved space, it returns with its orientation altered. A drawing modified from the one at the a review of Berry’s Geometric Phase on the web page of Nicola Manini (assistant professor at Milan University) shows the classical effect:
Classical Berry's phase drawing
In the drawing above, when the red vector is parallel transported through the 13 marked points around the sphere (all on a spherical triangle), the vector returns rotated.

If the vector is considered as a 2-d object in the tangent space of the surface of the sphere, the rotation of the vector can be given by an angle. The angle that the vector is rotated by is proportional to the surface area of the spherical triangle. If the spherical triangle is built from three 90 degree angles, it is clear that the vector is rotated by 90 degrees, and, under the (correct) assumption that the rotation angle is proportional to the area of the spherical triangle, this tells us that the ratio of spherical triangle area to rotation angle is given by 1. That is, if the spherical triangle has surface area A, the vector will be rotated by the angle A. A similar effect occurs in quantum mechanics when a spin state is sent through a sequence of orientations. This is kind of interesting in that we are turning a measure of spherical area into an angle.

In the usual spinor representation of quantum mechanics, the phase of a quantum state is arbitrary; one can always multiply a spinor \psi by a complex phase e^{i\kappa} and the new spinor represents the same state.

Density matrices are simpler in that they do not carry arbitrary phase information and consequently we can discuss Berry phase very elegantly and easily in density matrix formalism, which is the subject of this post.
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Test post. Ignore.

Is this coming from the future???

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Koide formulas and Qubit / Qutrit MUBs

Recently we’ve been discussing Mutually Unbiased Bases or MUBs on this blog. This has not been for any particular interest in pure mathematics, but instead in their application to a preon theory of the elementary particles, and the E8 quantum numbers. The relationship is rather long and difficult to explain. All the major pieces are already on the web in various places, but it would be rather difficult for someone to piece together the details as they are spread around, and use various notation. The full explanation is too long for a blog post, but what I can do is give an overview of the explanation, along with links to resources. I will begin with the end, the Koide formula, as it hasn’t been discussed here in a few weeks:

Koide Formulas: The Koide formula for the charged leptons that we will use here is:
\sqrt{m_n} = \mu_1(1+\sqrt{2}\cos(2in\pi/3 + 2/9) )
where \mu_1 is a constant with units of square root of mass, n is the generation number and m_n is the mass of the generation n charged lepton, that is, the masses of the electron, muon, and tau. The formula is accurate to a part in a million.

There is a similar formula for the neutrinos (i.e. the neutral leptons):
\sqrt{m_n} = \mu_0(1+\sqrt{2}\cos(2in\pi/3 + \pi/12 + 2/9 ) ).
The accuracy of the neutrino formula is unknown, it accounts for the differences in masses of the neutrinos, but those masses are rather poorly measured.
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Jay Yablon on Tensors and Symmetry

My approach to elementary particles is to try to build them up from a Clifford algebra assumed to have something to do with spacetime. This is inherently geometric, but it is not the only way to do geometry. A similarity is that both of us end up relating the three spatial dimensions to color. (Forgive me if I read too much into your papers, Jay.)

I keep finding similarities with what Jay R. Yablon is doing. While my approach is algebraic, he uses tensors. Unlike the usual approach to elementary particles, he puts tensors at the root and more or less derives the symmetries from there. For example, with the elementary particles, he gets that the 4x3x2 structure of the fermions arises from the 4! = 4x3x2x1 permutations of the four spacetime indices. From his recent paper, the fermions are organized as follows (with L=leptons, R, G, B = colors of quarks):
Jay's 4x3x2x1 = 4! split of the fermions

As I do, Jay relates the structure of the baryons to the structure of quarks. Further, he goes into looking at the hydrogen atom as a composition of three quarks and a lepton. This is fun stuff.

Interesting? Go visit Jay’s obscure blog and read the tensor theory behind all this.

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MUBs, Preons, and Lisi’s E8 model

If a Hilbert space is d-dimensional, we expect that the number of elementary particles we can describe with it is d. For example, a Pauli spinor is 2-dimensional and electrons come in two states, spin-up and spin-down. The Dirac spinors are 4-dimensional and the related theory describes electrons plus their antiparticles, and therefore 4 states.

So long as we think of elementary particles as the things that can only be represented by spinors (i.e. the 2×1 or 4×1 vectors of the Pauli or Dirac theory, respectively), each Hilbert space can only provide a home for a number of particles given by the dimension of the vectors of that space. There are d complex degrees of freedom for each particle and therefore, since quantum mechanics is more elegant with complex wave functions, there is room for only d particles represented by that spinor. That’s all there is, and that’s how elementary particles has been done for many decades.

In the usual theory, the Clifford algebra acts on the states. When we discuss a “basis” here it will be in the context of the d-dimensional Hilbert space, that is, we will mean a basis for the states that the Clifford algebra acts upon. A base defines the quantum states that we can consider to be different particles, or different aspects of the same particle. For the Dirac algebra, one basis has the four particles: {spin up electron, spin down electron, spin up positron, spin down positron}. But this is not the only possible basis for the Dirac algebra quantum states. We could instead pick {right handed electron, left handed electron, right handed positron, left handed positron}.

From a quantum information point of view, the splitting of the Hilbert space into a particular basis is a somewhat inadequate description of the information contained in a quantum state on the space. Suppose we have a large number of identical states and we wish to determine what state it is. If we make measurements with respect to just one basis, we will get the right answer if the state is entirely within that basis, but this won’t necessarily be all the information about the state.

For example, suppose the state is a spin-1/2 state. If it is polarized in the y-direction, then anytime we measure it in the z-direction we will get the value + or – with equal probabilities. This will not allow us to distinguish between, an +y oriented state and a -y oriented state. And vice versa. To determine what state we really have we need to measure it with respect to more than just the usual single direction. And this is where MUBs come in. It turns out that if you measure an arbitrary quantum state (even a mixed density matrix state) with respect to a “complete set of MUBs”, you will get just barely enough information to completely determine the quantum state.

The idea we will discuss here is to use MUBs to describe elementary particles, a subject of current research. We will briefly discuss the motivation, the history, and some of the advantages of this idea, and finally show how it can be used to derive some of the features of Garrett Lisi’s E8 model
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Love = Negative Energy

Perhaps due to a lack of details regarding his or their martyrdom, the Catholic Church pulled Saint Valentine from its liturgical veneration in 1969. Since that time, the holiday has expanded world-wide to areas that have never heard of early Roman martyrs. What a descent. From ecstatic religious devotion to a crude worshipping at the altars of sex and money. The now unholy day is coming up soon, and I thought that the following exchange would be appropriate for the occasion:

.

I am sorry for bothering you with this question.

But my high school students are asking this questions and I am not able to answer. Could you help me?

You wrote in Physics forum.

“When energy is released, it means that the binding is increased. The number of nucleons in gamma decay (emission of a photon, if I recall) stays constant. Therefore, the binding energy per nucleon increases.”

I am not able to find any textbooks or website explaining this. Could you indicate where I can find it so that I can explain iit.

Regards,
xxx

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