Cantor Sets Rule! (the world of non-integrable dynamical systems)

I just stopped by to quickly profess my sudden and deep love for Cantor sets. So there.

Modest Understanding of Lie Groups Part 0: U(1)

This semester I had the pleasure to take a very little nice course on mathematics, mathematics for physicists that is. What this means is that half the course we dealt with lie groups and the remaining month or so we studied path integrals. Now, why is this interesting? It just happens to be the sexiest mathematics available to me at this point.

In case you didn't know, finding symmetries in physics leads to a deeper understanding of the phenomena at hand. This is obvious to any undergraduate student facing for the first time electromagnetism. The most basic problem of this course is finding the electrical field a distance d above an very long line of uniform density charge. Needless to say, you want to know how much the line would pull (or repel) a charge, should you feel like putting one a distance d above it. Of course you don't need to understand much about physics to eventually see that it doesn't matter where you place it, as long as it is a distance d perpendicular to the cable. Clearly this is because the line of charge is very long and this places are practically the same to the line. From this information you then can guess that the electric field must only depend on the coordinate perpendicular to the line, a trivial conclusion, but proves the point just fine I guess.

Again, why am I talking about this? Turns out our most precious tool (for the moment) allowing us understanding the world, the Standard Model, is based on symmetry groups. Namely it is usually represented by SU(3)xSU(2)xU(1). Let's start by understanding the simplest part of this: U(1). Imagine a circle, or rather, its points:
In this figure, A and B are two points on the circle. All the points on this circle are characterized by some properties. For example, if a point on the circle is represented by the vector

the point

is also on the circle! Having seen this, it's easy to see that the points on a circle with the operation of addition (since each point is characterized by an angle, we can understand it as the sum of their angles) form a group. If we see this circle on the complex plane, a point on the circle can be represented by a complex number

and a rotation about the center of the circle will be given by multiplying this number by the following phase factor

This will take us

that is, another point on the circle (closure). If this is new to you, try and find the identity and inverse elements.

We can see this phase factor as a 1X1 matrix, and call it U. It's clear then that in this case

But in general for bigger matrices

I hope then to have explained how this implies that the complex numbers of norm 1 form a group under the operation of multiplication. This group is the most simple I can think of for now, the U(1) group (unitary matrices of rank 1, which satisfy the last equation).
Tune in next time for a brief explanation on all the other classic groups.

Overblow!

Finally! I can produce overblows! albeit just on the holes 1 and 6. I had to tune up my C harmonica to do so, but its a sweet feeling. Ill try and make some proof of this by the weekend, right now i just have too much work.

Also, I'll speak about Robert B. Laughlin, who came to Mexico city this week, but there's no time!

Linguistic Phase Transition

It seems it would be best if I made future post on plain old English since we're having people coming from Astronomers. In the Wild. Oh well...

And Now for Something Completely Different...

(Nota: A partir de ahora tambien estare participando en Astronomers. In the Wild., asi que transcribo mis aportaciones a dicho blog)

A man with string theory up his nose.

As our gracious host has already pointed out, I'm actually a physics student, so it's an honor to be able to write for this fine astronomy blog. In any case I'm beginning to understand the AdS/CFT correspondence, which as you may already know, deals with the apparent duality that exist between some string theories and quantum field theories.
What we can aspire to do with this approach is to investigate quantum field theories without relying on perturbation theory, which limit our understanding of nature. For example, Quantum Cromodynamics (QCD) enjoys asymptotic freedom at high energies, this means that since interactions are weak at those energy scales it's sane to use perturvative methods. However in cases where energy scales are small, we simply do not know how to perform calculations. As always numerical methods are useful to some extent, but even they have a hard time dealing with some situations (they involve dealing with some dynamical quarks, as far as I know). Other option is to study the AdS/CFT correspondence and gather bits of information about gauge theories in general. It all boils down to this: either, A) Find a string theory dual to QCD to be able to make calculations or B) Understand properties about gauge theories in general so you can make predictions about QCD. The second option is more viable and has had success in the past. But I'll explain that some other time.

Premio Nobel en Física, edición 2007

El próximo martes sabremos quien ha ganado el premio Nobel de física 2007. El año pasado las especulaciones acerca de quien ganaría el premio fueron bastante acertadas, y el estudio de las anisotropías de la radiación cósmica de fondo les otorgaron el Nobel a John C. Mather y George F. Smoot, reconocimiento inevitable por uno de los descubrimientos mas interesantes en cosmología actual.
Este año la situación no es tan clara. Aun cuando el descubrimiento de la expansíon acelerada del universo mediante el estudio de supernovas me parece un descubrimiento extraordinariamente interesante, quizás existan áreas de investigación de la física aplicada o experimental las cuales han sido pasadas por alto por nosotros la gente mal informada.

En menos de un año entrara en funcionamiento el LHC, el cual quizás nos permita ver física mas allá del modelo estándar y quizás en 5 o 10 años podamos tener un nuevo premio Nobel para la física fundamental (tras sacarnos de nuestra peculiar situación actual).

Para algunas predicciones interesantes sobre los ganadores de este año vease
Astronomer in the Wild

Desastre de hoy

Desde lejos el caos se ve bastante ordenado no es asi?
Dvd sin caja esperando la mas mínima excusa para hacerse la victima.
Plato sucio por la izquierda con esperanzas de que reparen el agua algún día para lavarlo.
Somersault por Kenzaburo Oe bajo la fiel laptop.
Contenido de mis bolsillos esparcidos sobre el escritorio (aun estoy tratando de determinar que distribución siguen).
4 harmonicas, de izquierda a derecha:
Golden Melody A, Marine band special 20 C, Marine Band Bb, Big River D
Junto a ellas un vaso que se usa 4 veces por cada vez que se lava.

Let the good times roll

Hey everybody,
Let's have fun
You only live for once
And when you're dead you're done
so let the good times roll

Ensayos

La base de toda actividad creativa es la meditación consciente o inconsciente. Si bien la primera es la base de lo que se puede hacer cotidianamente, la segunda tiene un encanto indudable y en veces lleva a conclusiones sobre la realidad que parecerían haber salido de la nada (incluso podría llamarsele inspiración divina si uno se inclina en esa dirección). Vamos dejando en este blog pedazos de lo que sucede en mi realidad local, para hacer consciente lo inconsciente y para la gente que le interese.

Procurare cada semana al menos dejar por aquí algunas notas sobre lo que estoy estudiando y como va progresando mi habilidad con la armónica. Inevitablemente todo esto en retrospectiva parecerá poco elegante e inclusive bastante verde, pero sin duda es la puerta a estados mas interesantes.