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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MUBs and Symmetry Breaking

One can obtain a Hilbert space from a wide variety of physical situations. Suppose we are to use a Hilbert space to model elementary particles, for instance. One the one hand, we’d like to have our model cover as many particles as possible. But to do this, we may have to assume a symmetry that is not exactly correct. For example, to get a model that includes isospin, we have a natural inclination to assume that isospin is a perfect symmetry when, in reality, it cannot be. Assuming perfect symmetry as an approximation leaves us with the need to later correct for this by including a symmetry breaking.

Recently, we’ve been discussing using Mutual Unbiased Bases (MUBs) to model the elementary particles. If we want to include very disparate objects as states in different MUB bases, we need to also include a symmetry breaking mechanism.

When elementary particle theory is built from symmetry alone, one finds that there are a nearly infinite number of ways of breaking symmetry. The freedom to do this makes it difficult to find our way. In the case of MUBs, however, we can write down some very natural ways to break symmetry if we treat the Hilbert space as the vector space of a Clifford algebra like the Dirac algebra. In this post we will show how the natural symmetry breaking in the Dirac algebra can lead to structures that are reminiscent of the structure of the elementary particles.
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Qutrit Mutually Unbiased Bases (MUBs)

Mutually Unbiased Bases are sets of bases for a Hilbert space that are “unbiased:” the transition probabilities between any two states from different bases are equal. For a Hilbert space of dimension 3 (i.e. qutrits), the transition probability is 1/3. The operator space of a Hilbert space of dimension n is n^2, in this case the operator space has 9 dimensions. Each base consists of 3 quantum states. It turns out that a base uses up 3-1 = 2 degrees of freedom of the operator space, and the scalar part of the operator space is shared by all. So for a 3-dimensional Hilbert space, there are at most four mutually unbiased bases. In this post I will derive a set of four such bases.
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The Generalized Pauli Group

I’ve been watching Perimeter Institute lectures again, and one that dips into the stuff I’m working came up. In addition to the lecture, there is an acrobat file that contains the slides and photos of the blackboard. This is 124 pages long and it is these pages that I will reference.

Applications of the generalized Pauli group in quantum information
Speaker(s): Thomas Durt – Vrije Universiteit Brussel
Abstract: It is known that finite fields with d elements exist only when d is a prime or a prime power. When the dimension d of a finite dimensional Hilbert space is a prime power, we can associate to each basis state of the Hilbert space an element of a finite or Galois field, and construct a finite group of unitary transformations, the generalised Pauli group or discrete Heisenberg-Weyl group. Its elements can be expressed, in terms of the elements of a Galois field. This group presents numerous applications in Quantum Information Science e.g. tomography, dense coding, teleportation, error correction and so on. The aim of our talk is to give a general survey of these properties and to present recently obtained results in connection with three problems: -the so-called ”Mean King’s problem” in prime power dimension, -discrete Wigner distributions, -and quantum tomography . Finally we shall discuss a limitation of the possible dimensions in which the so-called epistemic interpretation can be consistently formulated, in relation with the existence of finite affine planes, Euler’s conjecture and the 36 officers problem.
Date: 10/10/2007 – 4:00 pm

This has been discussed a little over at Kea’s blog and at Matti’s blog as well, and I figure I should write something up here as well, since I am also applying “Mutually Unbiased Bases” (MUBs); what I call snuarks are the Pauli algebra example of a set of MUBs.

I’ve added an explicit calculation in density matrix formalism for a maximal MUB for the Dirac algebra, and I’ve converted the solution into spinor form. This is a wonderful illustration of how much more powerful the density matrix formalism is than the spinor formalism.
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Contemporaneity and Density Matrices

I’m continuing to watch the Perimeter Institute lectures and watching the latest one, Dynamic Time: The “Missing Link” in the Search for a Unified Theory? by Avshalom Elitzur, has influenced me to type up my version of the ontology of quantum mechanics.

As Elitzur states, the central conundrum is to ask the question, “what is the past, the present and the future”. The intuitive answer to this is that only the present is real. The future is just our expectations, and the past is just our memory. The standard physics answer to the question is that physics must be written as a theory on space time. The past and future exist just as surely as left and right, or up and down, and the notion of “now” is either a figment of our imagination, or an arbitrary choice of the time coordinate origin.

My view on this is in the middle. I believe that the past and future do exist, but that the present should be treated as a preferred time. In this, I part ways with relativity which denies preferred things. And analogously, I part ways with gauge theory, which denies a preferred gauge. As a matter of philosophy, I believe in the existence of a unique (i.e. exactly 1) world which must therefore have a unique representation in mathematics. If we are unable to distinguish between two different representations, it is only because of the inadequacy of our understanding and not due to the world being several different things at once.

These ideas are difficult to explain because the word “time” is overloaded in our language. To get the point across, let us consider the 2-slit experiment with a single quantum particle, and examine the target region from two different perspectives. This experiment is simpler than the “2-hole experiment” because the symmetry in the direction of the slit reduces one of the dimensions. Instead of three dimensions of space and one time dimension, the 2-slit experiment has only two space dimensions. Furthermore, we concentrate our attention on the target (which is the only part of the experiment which we can “measure” anyway). This reduces the problem to one spatial dimension. We look at only a single time slice, so our pictures are only 1+0 dimensional.

The wave function shows interference. It might look like this:
2 slit experiment wave function
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Fundamental Bohmian Density Matrices

I’ve been listening to Perimeter Institute lectures and came upon one by Roderich Tumulka that was interesting enough that I looked up his papers. They are voluminous. One that applies more or less directly to the area I play in is this one: On the Role of Density Matrices in Bohmian Mechanics quant-ph/0311127, by Detlef Dürr, Sheldon Goldstein, Roderich Tumulka, and Nino Zanghì and published as Foundations of Physics 35 (2005) 449-467.

A density matrix is relevant in yet another way: in a modified version of Bohmian mechanics in which the particles are guided not by a wave function but by a density matrix. Let us call this W-Bohmian mechanics. Whereas in the conventional version of Bohmian mechanics the wave function (of the universe) is something real, as an objective component of the state of the universe at a given time, in W-Bohmian mechanics instead of a wave function (of the universe) we may have only a density matrix. This density matrix does not arise in any way from an analysis of the theory, but is built into the fundamental postulates of W-Bohmian mechanics. It is a fundamental density matrix, W_{fund}, in contrast to the four other density matrices we have discussed, which were derived objects, derived from \psi and Q.

This is one of only a very few references I’ve seen in the literature to the possibility that the density matrix is fundamental. Since this is one of my assumptions in my view of elementary particles, it is gratifying to see it in print.

They approach the subject very thinly. One can find how one defines a density matrix from a state vector in most quantum mechanics text books, but as far as I know, it is only in my book on density matrices that one can find a demonstration of how one defines a state vector from the density matrix. As part of my NYR to be more professional about physics, I’m rewriting the book to remove the inside jokes, make the cover art actually look like a physics book instead of a heavy equipment operation guide, etc. So if, like some of my friends, you think a physics book with a picture of a ditch witch on the front is cool, you’d better order a copy soon because it is not going to be available for very much longer (and I’m not going to buy you one).

The new book will be in paperback format 8.5″ x 11″ (Misner Thorne & Wheeler inspired my layout for both this and the paperback), and will be cheaper at around $15 to $20.

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