Quantum Field Theory

Math - Fall 09

Monday, 14 September 2009 00:52 | Author: Anton

Quantum Field Theory: A Tourist Guide for Mathematicians, by Folland.

While in Germany, I talked to Herr Professor Doctor Zeppenfield, and he told me that quantum field theory accounts for much of modern physics. So when I saw Folland's book at the library (near a book I was looking for), I was intrigued. Folland says "[the subtitle] is meant to free me and my readers from guilt about omitting various important but technical topics, viewing others from a point of view that physicists may find perverse, ..., and skipping the gruesome details of certain necessary but boring calculations". What can be better?

I haven't read much so far, but I really like the approach: it is clean, succinct, and informative. The author assumes familiarity with: Fourier analysis, Hilbert spaces, Lie theory, differential geometry, etc. So just reading through bits of this book may be quite an educational adventure.

A fun quote: "Mathematicians look at physical laws such as F=ma with the expectation that they will contain some universal truth about the world. They do, but not in the absolute way that the formula $A=\pi r^2$ is a universal truth about circles; they are always burdened with a certain amount of fine print."

Anyway, a few days ago I posted something on facebook about being "amazed by the full glory of F=ma and how it involves tangent bundles to contangent bundles". A few people asked for an explanation. Here it comes. At least one asker was a non-mathematician, so hopefully I can explain cotangent bundles quickly and convincingly...

The basic idea is that you have some number of particles that move around in some space. For example, you could have 5 particles moving in normal Euclidean 3-space. To store the locations, we need 3 parameters for each one: $(x_1, y_1, y_1)$ for the first one, and likewise for the 4 others. We might as well store them in one list that says where they are all located: $(x_1, y_1, z_1, x_2, y_2, z_2, \ldots, x_5, y_5, z_5)$ . There are 15 numbers all together, so we are in 15-dimensional space, called the configuration space. In general, this could be something more complicated. For example, if the particles had to stay on the surface on the sphere, the configuration space would be 5 copies of the sphere. All laws of physics will be rules concerning the configuration space and how it changes with time. That is, they tell us how the points move.

Now, we need some more information. Many laws of physics depend on the velocity of an object. So for each particle, we throw in three more numbers telling us which way and how fast it's moving. In the Euclidean situation, this just means we throw in another 15 dimensions, 3 for each particle. In general, we don't get two copies of the configuration space, but rather the tangent bundle: a collection of both the possible points and the possible vectors sticking out of them.

Just to complicate the picture some more, Folland switches over to the cotangent bundle, which roughly corresponds to ways of picking coordinates on the tangent bundle. As is, if I pick a point in space and a direction to move in, I don't have any specific coordinates: just a point and a direction, but no numbers to work with. A cotangent bundle is (roughly) all the ways of picking coordinates. At the end of the day, the tangent bundle and the cotangent bundle are very similar: in one case I'm saying "the particle is moving in such and such direction", in the other "let's make that direction, the one in which the particle is moving, the x-axis". The cool thing is, the cotangent bundle has some extra properties, mainly being able to multiply things. I can multiply functions by just multiplying the values they give me (e.g. $x^2 * x^4 = x^6$ ), so now I can somehow multiply directions too. (I'm lying a little bit, but not too much.) It turns out that this ability is very useful for describing physical laws.

In the Euclidean example, switching to the cotangent space does nothing: we still have 30-dimensional space. However, we now call it the phase space. What about physical interpretations? What do the extra 15 dimensions correspond to? Since they live in the cotangent space, they are covectors, not vectors. So they can't be the velocity of the object. Instead, they are the momentum (denoted $p$ ): mass times velocity. And, it turns out, mass is what enables us to move back and forth between the two: in fancier terms, it's an inner product that identifies vectors and covectors so we have $p=mv$ , or in more standard notation for covectors: $p(\cdot)=m(\cdot,v)$ . Quite fancy.

Now, the phase space has natural coordinates: in the example we have the 15 space coordinates as well as 15 momentum coordinates. Forgetting about the fact that we're dealing with 5 particles, let's just number them: $(x_1, \ldots, x_15, p_1, \ldots, x_15)$ . Now, even if we forget about the particles, the directions still match up: $p_1$ is a choice of direction, and I want it to be the choice of the direction $x_1$ . There's a fancy way of writing this down: we have a 2-form $dx_1 \wedge dp_1 + dx_2 \wedge dp_2 + \ldots + dx_5 \wedge dp_5$ . Picking this symplectic form just means matching up the directions with the corresponding momenta. Don't worry about all the d's and wedges if you haven't seen them before. However, I should say that the symplectic form is an element of the second exterior product on the cotangent bundle of the phase space. That means, in a very technical sense, that it eats pairs of vectors and spits out numbers. 30-dimensional vectors in our case, since the tangent bundle to the phase space is 30-dimensional, just like the phase space itself.

Now, in this fun setting is where $F=ma$ happens. Recall that there is kinetic energy: the total energy of all the particles moving, and the potential energy: the energy that the particle would gain by falling to some pre-determined location. For a given point in the phase space, we can compute the total energy by looking at each of the particles and summing up their energies. So there is a function on the phase space, called the Hamiltonian, that gives each point the corresponding energy. The basic rule is that, as time starts ticking, a point can only move to another point with the same Hamiltonian: energy is conserved. Newton's law $F=ma$ can then be rewritten as:

$\nabla_p H = \frac{dx}{dt} = v = m^{-1}p, -\nabla_x H = \frac{dp}{dt} = ma = -V$

where $\nabla$ is the gradient with respect to some variables, and $V$ is the potential energy function. The mass $m$ may be thought of as a matrix, in our example 15 by 15 - dimensional.

If you made it this far (or if you skipped), please let me know whether any of that makes sense. And definitely let me know if you want to read some of the book together. I'll probably be reading either this book or a similar one with a couple of people, and the more the better.

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