Dec 31, 2005

Rusty Links

Rusty Chain

I took this last week of the year off to organize my things, cause I will probably move to UK next year to begin a work applying Statistical Physics to Cryptography. I was cleaning my wardrobe, which does not have only clothes but also books, old toys and a lot of papers. Then I found a folder and a notebook covered with some dust and with the title "Internet" (time goes by very fast...). I opened then and found a lot of links that I probably found interesting someday and I decided to put them in my page. I begun this task this morning and they are really interesting links. So, I decided to put them here for everyone who read this also have the chance to take a look at them. Let me list them in the exact order that I found them:

Sodaplay: classic. Everyone must know it by now, but I´m glad to have rediscovered it.

Jim Loy´s Homepage: the amount of information about science-related themes is huge. How this guy manage to write so much is a mistery to me.

2d Curves: a collection of mathematical bidimensional curves.

Igor Nikitin´s Homepage: has an interesting document on String Theory.

Kolmogorov: a very complete page about Kolmogorov.

Wilfrid Hodges´ Homepage

Douglas Arnold´s Homepage: with an interesting page about Some disasters attributable to bad numerical computing.

The Online Books Page

O Cerebro Nosso de Cada Dia: a site about the brain originally in Portuguese, but with an English translation as Our Daily Brain.

That´s it. Probably I will find more as my cleaning proceeds and I´ll put them here.

Happy New Year for everyone!

Picture: Rusty Chain by Hilly Wakeford

Dec 29, 2005

Phononic Crystals

Phononic Crystal

I found an interesting article in Physics Web by Gorishnyy et al. named Sound ideas (the beautiful picture above was taken from this article). The article talks about a kind of crystal, named a phononic crystal, that can be constructed in such a way to create specific "band gaps" for waves travelling in this solid. This means that you can control which frequency cannot propagate in the crystal, creating, for example, materials that become isolators for particular sounds or mechanical waves.

The band gaps are created by carefull design of the crystals allowing a control of the dispersion relation, the relation between frequency and wave number, in phonons, which are quantized modes of vibration in a solid. This quantization of vibrational modes comes from a treatment using the machinery of quantum mechanics and is a very important mechanism that, among other things, influence the heat condictivity of materials. For a short introduction to the theory of elementary excitations in solids see Elementary Excitations in Solids : Lectures on Phonons, Electrons, and Plasmons by David Pines.

Phononic crystals may have a lot of interesting technological applications described in detail inside the article, in the words of the authors

"Phononic crystals will provide researchers in acoustics and ultrasonics with new components that offer the same level of control over sound that mirrors and lenses provide over light."

Over my desk:

Dec 23, 2005

Cosmic Collision

Galaxies NGC 2207 & IC 2163

A friend send me this link named Cosmic Collision this week. It is a subsite of the official Hubble site where it is described how it will look like the collision of our Milky Way with the Andromeda Galaxy. The site shows the story with a narrated video and have a lot of scientific explanations in a simple but precise way.

The Milky way is indeed colliding already with other minor galaxies of our local neighborhood in our trip in the direction of Virgo Cluster named the Local Group, like the Magellanic Clouds, but the collision with Andromeda will be much more espectacular due to the size of Andromeda. Our planetary system probably will not be affected due to its tiny size relative to interstelar distances, but in the site they show how the night sky will look like during the collision time. In the end, both galaxies will merge into a large elliptical galaxy.

The collision will occur in about 5 billion years from now, what remembered me of a story someone told me once (I don´t remember who...): A scientist was giving a lecture about the death of our sun. At some point, a person raised a shaking arm and asked in a trembling voice Excuse-me, professor, when did you say that will occur?. The professor answered In about 5 billion years.. The guy then took a deep breath and said in relief Oh... I thought you have said 5 MILLION...

As a last comment, the Hubble site has a lot of beautiful pictures and interesting explanations. Don´t be in a hurry when navigating there and you will enjoy every mouse click.

Over my desk:

1. Deriving Landauer’s erasure principle from statistical mechanics, Jacobs (quant-ph/0512105).

2. Spin Glasses: a Perspective, Sherrington (cond-mat/0512425).

3. Projective geometry and special relativity, Delphenich (gr-qc/0512125).

4. The Study of the Pioneer Anomaly: New Data and Objectives for New Investigation, Turyshev (gr-qc/0512121).

5. Quantum information and computation, Bub (quant-ph/0512125).

Picture: Colliding galaxies NGC 2207 and IC 2163, NASA.

Dec 20, 2005

Geometric Algebra

The algebra of complex numbers is related to geometry by the Argand plane. Using it, we see that the operation of multiplying by i is equivalent to a 90 degrees rotation in the counterclockwise direction. A little more advanced concept is that of quaternions, that as complex numbers, are a set of numbers that can represent rotations in 3D space. In both these cases, there is a beautiful connection between algebric structures and geometry that can be used to express physical laws in a concise way.

The notorious way to use geometry in physics is by means of Gibbs' vector calculus, which became widespread in physical sciences and engineering. In 1878 Clifford created a structure with the name geometric algebra uniting the dot and the cross products of two vectors into a single entity named the geometric product, which for two vectors a and b is written as
\[ab=a\cdot b +a \wedge b,\]

where the first term is the dot (scalar) product and the second the wedge or exterior product, which generalize the cross product that turns out to be a particular case in 3 dimensions.

Although it has a lot of applications in physics, it was eclipsed by Gibbs' vector calculus and was forgotten untill 1960 when David Hestenes, trying to recover the geometric meaning of the Clifford algebra related to spin discovered that geometric algebra is a "universal language for mathematics, physics and engineering."

There are a complete introductory course as Lecture Notes in the site of the Department of Physics of the University of Cambridge.

The interesting fact, that my former PhD advisor pointed me, is that there is a hope that this structure can lead to a geometric interpretation of the misterious use of complex numbers in Quantum Mechanics. However, I need to read more the lecture notes to talk about that.

Papers over my desk (or in my desktop):

  • Vegetation's Red Edge: A Possible Spectroscopic Biosignature of Extraterrestrial Plants - Seager et al. (astro-ph/0503302)
  • Causal Sets: Discrete Gravity (Notes for the Valdivia Summer School) - Sorkin (gr-qc/0309009)
  • The General Quantum Interference Principle and the Duality Computer - Long (quant-ph/0512120)
  • Entropic Priors - Caticha and Preuss (physics/0312131)
  • On Math, Matter and Mind - Hut et al. (physics/0510188)
Picture: Quantum Notions - Gerard von Harpe

Dec 14, 2005

A Matter of Doubt

Black Hole

In the most fundamental level of nature lies two concepts that are central to physics: energy and matter. Energy is a fundamental entity present everywhere. Even empty space contains energy (what renders the term "empty space" a little inaccurate).

Matter is a concept directly associated with mass. Matter particles are particles with mass. Mass started as two "different" quantities: a measure of inertia, what comes from Newton´s formula
and gravitational charge, again given by Newton as
where the gravitational constant G is so small that renders gravity the weakest of all forces in nature. Although nothing in principle says that gravitational charge and coefficient of inertia should be the same thing, Newton already confirmed by making experiments that both concepts agree with great precision. This point was late clarified by General Relativity, where we learned that gravity is only a deformation in spacetime and what we see as an atractive force is just a geodesic path, but the detailed explanation can be found in, for example, Robert Wald's General Relativitybook and in Sean Carrol's Website under the title Lecture Notes on General Relativity, so I will postpone it for a future post.

Mass is known to be equivalent to energy since Einstein´s Special Relativity. His famous formula
which is valid for a body AT REST, means that even objects with no movement and subject to no forces have some energy that can be extracted from its mass. Indeed, the atomic bomb relied on this formula to produce an amazing amount of energy from a relatively small piece of matter.

The making of the atomic bomb shows that to extract energy from matter is (relatively) easy, but the converse is not so. The main problem is that we still does not know what exactly is the mechanism that converts energy into matter. We have clues, both experimental and theoretical, but a complete explanation is still lacking.

In the first place, we expect that energy can be transformed into matter because we believe that in the beginning there was only energy in the universe and, somehow at some point in the far past, this energy gave birth matter particles. Second, we know that it can happen because there are experimental evidence for a phenomenon called pair creation, where a photon acquires sufficient energy and generates a positron and an electron. However, this is totally random and we cannot predict when and how this will happen.

There is a curious theoretical phenomenon called Unruh-Hawking Radiation, sometimes treated separetely as Unruh Effect and Hawking Radiation, which is related to matter creation too. It is theoretical because we can deduce it from quantum mechanics and relativity, but the effect was not observed experimentally at this moment. Hawking discovered that black holes can induce production of pairs of matter particles around its event horizon and the emnission spectrum of these particles is a black body spectrum with temperature
T=\hbar g/2\pi ck_B
where g is the local gravity acceleration. The equivalence principle of general relativity requires that a gravitational field is equivalent to acceleration and this implies that an accelerated observer can see a background of matter particles where an observer at rest see only the vacuum, and this particles obbey the same spectrum distribution of the particles near the black hole with temperature
T=\hbar a/2\pi ck_B
where a is the acceleration.

The only explanation till now about how particles acquire mass comes from the Higgs mechanism, a kind of symmetry breaking involving a particle called the Higgs boson. But the Higgs boson has not been found experimentally at the moment and there is another problem: we must assume that the Higgs has a mass itself, what only puts the problem in another level: from where comes the mass of the Higgs? A self-interaction, you would say, but it´s just a circular argument, does not help too much.

The matter-energy problem has not been in the first plane of research in the last decades, but there are something very fundamental in this problem that must be understood if we want to go on with our aim of understanding how the universe works and how it appeared.

Dec 6, 2005

Quantum Limitations

Quantum Foam

As I already said in another post, quantum mechanics is a wide confirmed and one of the most successful theories about nature we (humans) ever created. The agreement of predictions with experiments is amazing and there are no known experiments that contradict the theory.

However, this is not the end of the story. QM is successful for its mathematics describes nature with tantalizing precision, but the math was tailored from experiments to fit them. This means that QM, unlike Relativity, is not derived from some fundamental principle. The lack of this principle is what is behind the great numbers of alternative interpretations apart from the ortodox one, which leads to strange situations like the Schroedinger Cat.

The lack of a first principles derivation still is responsible for the existence of alternative theories that try to explain qunatum phenomena, like Bohmian Mechanics, where David Bohm tries to explain quantum behavior by a misterious quantum field that permeates spacetime, and Stochastic Electrodynamics, pioneered by Timothy Boyer, which uses classical mechanics plus a random background field of electric particles and is able to find a lot of good results. But no theory yet has been proven to be exactly equal or superior to QM.

When I talk about the success of QM, I´m not talking yet about quantum field theory (QFT). QFT arises from the merging of QM with special relativity. It has a lot of success, but the way these results are extracted from the body of the theory is very trick and most of scientists have the feeling that this should not be the final answer. Although when supplied with some experimental measurements QFT can give results that agree with experiments by one part in a billion (in QED, for example), the calculations come from infinite series expansions that do not converge. The expansions are truncated and a lot of work on renormalizing (take the infinites away) the theory must be made.

The point is that even giving the correct values for several quantities, QFT begins with a very questionable (in my view) procedure: you simply transform equations from classical physics in equations for fields, solve by expanding in Fourier series and then impose quantum commutation relations between the fields and the canonical conjugate momenta. It is a recipe. We don´t know exactly what we are doing, but we borrow the procedure of imposing these relations from plain QM and go on. Another thing: the conjugate momenta comes from a lagrangean density that is constructed in such a way that it gives the correct equations of motion, again without any fuindamental principle. This procedure works with some tricks for Electrodynamics and for Weak and Strong Forces (giving rise to electroweak theory and to QCD), but fails miserably with gravity.

In my view (and my only view, what means that it is not the current view of scientific community), without a clear understanding of what we are really doing, we can´t even be sure if we should quantize gravity. Critics of string theory say that after so much time without success, maybe string theory is a wrong way, but the endeavour of quantizing gravity is much older and we could not do it till this day. I´m not saying that quantum gravity is not worth pursuing, I´m just saying that maybe there is a tiny possibility that nature did not choose this path. However, today the probability that QG exists probably is higher than that it do not. We have to wait more theoretical results or experiments.

Just to cite a tentative for deriving QM from first principles, it is worth looking at the papers of Ariel Caticha, he is trying to show that QM can be obtained by applying principles of information theory and bayesian inference to physics. The main theory is in Insufficient reason and entropy in quantum theory (quant-ph/9810074). He is also trying to show that general relativity can be obtained from the same principles: The Information Geometry of Space and Time (gr-qc/0508108).

Picture: Quantum Foam - taken from