I’m going to do a series on trying to get at what exactly a mathematician means when referring to a “quantum field theory” or QFT. This is pretty tough to get at since if you ask a physicist the answer is likely to be completely different than a mathematician. Also, people within disciplines often refer to different things when they mention it.
Problem: A mathematician needs to frame problems about QFT’s in terms that are familiar. A physicist must do the same likewise. When translating, these two frames are precisely the same (at least in theory, physicist have been able to do things that work in which the exact mathematical reason that they work is still a mystery and such…). This post is to try to bring some of these concepts to mathematical language, since even books with titles such as QFT for Mathematicians assume way too much knowledge of physics on the first page and don’t truly get the translation across.
First, let’s start a little more basic. What is a classical field theory? There are two main parts to this. There is the fiber bundle and the Lagrangian. I think we see why this is interesting to me now. I’m quite interested in algebraic topology, so the term fiber bundle gives good indication that we are headed that way.
For those not familiar with the fiber bundle, it really is quite a brilliant construction. It is so natural (although it might take a bit of time of staring at definitions to think so), yet powerful. A fiber bundle consists of a base space B and a total space E (topological of course). We have a “projection” type map that is locally trivial. Now we also have the fiber space F. Here is where that locally trivial condition applies. What we mean is that for any , we can find a neighborhood of B, say U, such that is homeomorphic to and so if we factor through that homeomorphism and then project, it is the same as just the map.
This is probably overwhelming at this point, but let’s look at how simple it really is through some examples. You should easily get the feel for it that way. The trivial bundle is to just taking and the projection really is projecting by . This trivially satisfies the definitions, and the “fibers” just look like “slices” of E in that you get the whole of F but just a point of B ().
Another common example is to look at the tangent bundle (hmm “bundle” is in the name…) of a manifold. This has some more structure to it since it is a vector space as the fiber space and is hence known as a “vector bundle.” So in particular we could let our manifold be none other than . Then our vectors we have in , so . Another vector bundle along the same lines is the normal bundle to a manifold. I don’t want to go too far off topic, but essentially fiber bundles tell us how much a space is “twisted.”
I know that is rather inadequate, but I do want to get to QFT’s, and I’m only on the first part of the definition of a CFT. One more quick thing that will be needed at some point in the future is called the section of a fiber bundle. Sadly global sections are not in general possible, but we can do it locally. Take an open set U in the base space B, then the section is a continuous map such that for every x in U. At this point we can start throwing around terms like sheaf, but we’ll move on.
So let’s sum up and reword with a view towards CFT. We mostly want things to be manifolds, since this is classical physics. So basically our fiber bundle will look locally like . Now we say that a classical field is a section of that fiber bundle. Does this fit with our traditional view?
Classical mechanics is just taking our manifold (base space) to be time, i.e. and the fiber space to be the space of all possible configurations of the mechanical system. So this works. Let’s get the Lagrangian into place now. Another classical field theory is statistical mechanics. We could just let our space be traditional 3-dimensional space and the fiber space is . The field is just a real function on . Here we will have the Lagrangian as a function . Remember that fields in this case are functions on . The standard is .
I just realized that Lagrangians could probably use an entire post on their own. So don’t worry about what is written there. It is just a flavor of next post. I’ll finish up CFT’s next time with basically a post on Lagrangians and some more examples that should give a better feel.