Fourth Week’s seminar

Brandon has requested we change the time so he can make a train, that selfish bastard. So would Wednesday (tomorrow) at 3:30 be ok with people?

UCSB Workshop

Apparently there’s a workshop going on right now at UCSB on gauge theory and Langlands duality. There are lecture notes and even some poorly shot video. I don’t actually know what Langlands duality is unfortunately. Here are some notes from a talk given by Witten right here at Stony on the subject. It appears that everything involved is above my head.

Third week seminar

This Thursday we will meet at 3:30 PM. Brandon will continiue his discussion on bundles.

Elements of Electromagnetism (II)

I would like to continue the discussion I started here about electromagnetism. Last time I stoped right when things were getting interesting: we had reached special relativity. Some words about special relativity follow.

Elements of Electromagnetism (I)

Electromagnetism is really cute. In this post (and its sequel) my aim is to provide a very broad overview of what physicist mean when they talk about the theory of electromagnetism or Maxwell theory. First of all I would like to list some of the references I used to get my information from and some other documents were one can pursue further details.

1. “Gauge theories in particle physics” by Aitchison and Hey – chapter 2 provides most of what I mention here.
2. “Classical Electrodynamics” by JDJ – Just in case people really wanted to learn electromagnetism.
3. “The conceptual basis of quantum field theory” by G’tH – Look here for more details on Yang-Mills theory, a generalization of Maxwell theory.
4. “Special Relativity: from Einstein to strings” by Schwarz and Schwarz – Here are some other topics I wanted to talk about but maybe will not have time to cover.
5. “A first course in string theory” by BZ – Stuff on Kalb-Ramond and Born-Infeld.

So I was requested to talk about electromagnetism as a gauge theory. What is electromagnetism? What is a gauge theory? And how is electromagnetism a gauge theory? For now I will start with the first question. We will work our way through some history and we will reach a point where we will say something like “… in this sense E&M is a gauge theory of local phase transformations… “. This will hopefully answer the third question, leaving the second question (what is a gauge theory) for another occasion.

So what is electromagnetism? My answer is that it depends who you ask.

This week’s seminar

We will meet this Thursday, 3:10 PM at Eitan’s office (Math 2-122). I will be talking about electromagnetism.

Seminar time

So far the arranged time for the seminar is Thursdays @ 3:10 PM. Let your voice be heard if you have issues.

This week Brandon will talk about fiber bundles.

Second Quantization

Since Melvin wrote about how I tried to introduce him to fiber bundles, I’ll try to relay what he taught me about second quantization. (I was totally lost when he was patiently explaining Feynman path integrals and diagrams :-/ ) First we start with the canonical commutation relations for quantum mechanics. Then we promote from an “operator” to a “label” for the field which we’ll assume is a scalar field. After choosing a Lagrangian which makes sense to physicists if not to me, we minimize the action to get the Klein-Gordon equation . Solving the K-G equation in a Fourier-ish way gives . The and are annihilation and creation operators respectively. Then we start with some Hilbert space with a vacuum state and apply the creation operator to get a particle with momentum , . So on and so forthing we can generate a Fock space maybe.

Melvin tried to expound on the difference between first and second quantization. He explained that Quantum Mechanics was when you perform canonical quantization, upgrading observables to operators (maybe) and second quantizing is Quantum Field Theory where you see particles as excitations of the quantized field.

Manifolds and Bundles

Today I met with Eitan for a few hours. After he realized that I did not know what a bundle was, he procedded to define some terms for me.

Eitan started with the concept of a manifold. At the moment it made sense to me and I had seen it before, so I did not bother to write it down. Sadly, I cannot reproduce it off the top of my head. So Eitan if you read this, remind me (again) what is manifold.

Then he moved on to the tangent and cotangent spaces. One can understand the tangent space of a manifold at the point as the set of “velocities” of a “particle” at . Let . Then one think of as a directional derivative and as a derivation, which satisfies:

• a product rule:
• a linear rule:

Meanwhile the cotangent space at a point is a set of linear maps from the tangent space at that point to the real numbers. One thinks of elements of the cotangent space as momenta, although the meaning of this is not very clear to me…

Turning now to bundles. Roughly speaking one has a base space with a space called the fiber attached at each point . Examples of bundles are the tangent bundle and the cotangent bundle.

Finally Eitan concluded with a section, which he defined as a choice of for all . Most of the fields in physics are really sections of some (vector) bundle. For example the gravitational potential (the spacetime metric) is a section of the cotangent bundle tensored with itself.

That was all I fitted to my sheet of paper. Hopefully we can go into more detail during the course of this mini-seminar. After Eitan finished his nice overview of bundles I tried explaining what was the deal with second quantization and the path integral. But honestly I was not prepared so I do not blame him for getting a bit confused.

Gauge theory seminar

Let us have a gauge theory seminar! Here are some of references that might be useful:

• Gauge theories in Particle Physics by Aitchison & Hey. This is for some of the physics basics.
• Geometry, topology and physics by Nakahara. General mathematical background.
• The geometry of physics by Frankel. More general mathematical background.
• The conceptual basis of quantum field theory by ‘t Hooft. Basics of QFT. Can be found here.
• Preparation for Gauge Theory by Svetlichny. More specific mathematical background. Can be found at the arXiv.
• The geometrical setting of gauge theories of the Yang-Mills type by Daniel and Viallet. Yet more specific mathematics. Can be found here.

Suggestions are welcomed.