09 October 2006

A new class of entries

I think I might like to spend some time thinking about definitions in mathematical physics. What is a quantum field, for instance? Physicists usually give a slightly incoherent answer: a quantum field is a quantum particle at every point, just like a field is a number at every point. You ask them to unpack this a bit, and some might remember that there may be global — what the physicists call "topological" — issues with such a definition, but for now let's only be concerned with the local definition, where a field is a function. So what should a quantum field be?

Conveniently, I'm taking three classes right now on related questions: Differential Geometry, Geometric Methods to ODEs, and Quantum Field Theory. I would like to start a series of entries blogging those classes, and relating it back to such foundational questions. I hope to get to answers involving infinitesimals: Robinson's "Non-standard Analysis", or Kock's "Synthetic Geometry". I don't have the answers yet.

What's most important about fields is their geometric nature. Like the physicists and the classical differential geometers, I may from time to time refer to coordinates, but ultimately I'd like a coordinate-invariant picture — indeed, one without coordinates at all. I also hope to ask and answer issues about how to regularize our fields, by which I mean "how continuous should they be?" This is an extremely non-trivial question: not only is it extremely unclear how to demand that two
"nearby" "quantum particles" be "similar" (we can demand as much of classical fields: for any epsilon, there should be a delta at each point so that within the delta ball at that point the fields don't vary more than epsilon; perhaps we should find the right metric on Schrodinger-quantized particles?), but the physicists don't even want to be stuck with, say, C^\infty fields. They want \delta functions to work within their formalism. And yet they adamantly refuse to consider "pathological" fields that are too "wildly varying".

Eventually, it would be nice also to understand the Lagrangian and Hamiltonian, and this almost-symmetry between position and momentum. For now, I'd like to end this entry with some basic definitions.

Manifolds: There are many equivalent definitions of a manifold. Since the physicists and classical geometers like to work with coordinates (replacing geometry-defined, invariant objects with coordinate-defined, covariant objects), I'll use the definition that mentions coordinates explicitly. A manifold is a (metrizable) topological space M with a maximal atlas — to each "small" open set U in M we assign a module of "coordinate patches" \phi:U\to\R^n, which should be homeomorphisms, subject to some regularity condition: if \phi:U\to\R^n and \psi:V\to\R^n, then \phi\psi^{-1} should be, say, smooth wherever it's defined. In general, modifying the word manifold modifies the condition on \phi\psi^{-1}: a C^\infty manifold has that all the \phi\psi^{-1}'s are C^\infty, for example. I will generally be interested only in C^\infty (aka "smooth") manifolds, although once we understand what kinds of functions the physicists are ok with, we may change that restriction. For a manifold, I demand that the atlas be maximal in the sense that it list all possible coordinatizations consistent with the smoothness condition. It is, of course, sufficient to simply cover our space with (coherent) patches, defining the rest as all other possibilities.

So that we can generalize this definition if we need to, it would be nice to reword this definition in the language of sheaves. The god-given structure on a smooth manifold is exactly enough to tell which functions are differentiable: a sheaf is a topological space along with a ring of "smooth functions" on each open set, so that the function rings align coherently (in full glory, a sheaf is a (contravariant) functor from the category of open sets in the space to the category of commutative \R-algebras whatever your sheaf is of, along with some "local" axioms, which ultimately say that to know a function I need exactly to know it on an open cover). I probably won't use this description, largely because I don't know what other conditions I would want to put on my sheaf in order to make it into something like a smooth manifold. Clearly every manifold generates a sheaf, and I have it on good authority that if two manifolds have the same sheaf, then they are the same manifold.

So what about our most-important of objects: a field? A field is a "section" of a "bundle".

Let's start with the latter of those undefined words. To each point p\in M, we associate a (for now) vector space V_p, called the "fiber at p". And let's (for now) demand some isotropy: V_p should be isomorphic to V_q for any given p and q in M, although not necessarily canonically so. (When we move to the realm of infinite-dimensional fibers, we may demand only that the fibers be somehow "smoothly varying" — I'm not sure yet how to define this. So long as everything is finite-dimensional, the isomorphism class of a fiber is determined by an integer, and integers cannot smoothly vary, so it suffices to consider bundles where the dimension of the fibers is constant.)

There should be some sort of association between nearby fibers: locally (on small neighborhoods U) the bundle should look like U\times V. So I ought to demand that the bundle be equipped with a manifold structure, which aligns coherently with M: a bundle E is a manifold along with a projection map \pi_E : E\to M, such that the inverse image of each point is a vector space. This is the same as saying that among the coordinate patches in E's atlas, there are some of the form \Phi: \pi^{-1}(U) \to \R^(n+k), (where, of course, n is the dimension of M and k is the dimension of each fiber) so that \Phi = (\phi,\alpha), where \phi is a coordinate patch on M and \alpha is linear on each fiber. We can naturally embed M\into E by identifying each point p\in M with (p,0) in E (where 0 is the origin of the fiber at p).

I will soon make like a physicist and forget about global issues, but I do want to provide one example of why global issues are important: the cylinder and the mobius strip are both one-dimensional ("line") bundles over the circle. The latter has a "twist" in it: as you go around the circle, you come back with an extra factor of -1.

So what's a section of a bundle? A (global) section is a map s:M\to E so that \pi s:M\to M is the identity, i.e. a section picks out one vector from each fiber. We will for now think of our sections as being C^\infty.

The most important kinds of fields are "scalar" fields, by which I exactly mean a function, i.e. a number at every point. I want to do this because I want to consider other spaces of fields as modules over the ring of scalar fields, so I need to be able to multiply. Of course, there are many times when I don't want a full-fledged scalar field. The potential energy, for instance, is only defined up to a constant: I will eventually need my formalism to accommodate objects that have fields as derivatives, but aren't fields themselves. Since potentials don't care about constants, we could imagine that after going around a circle we measure a different potential energy than we had to begin with, but that we never picked up any force. The string theorists, in fact, need similar objects: locally, string theory looks like (conformal) field theory on the string's worldsheet. But perhaps the string wraps around a small extra dimension? This is why in the previous paragraph I refer to "global" sections: I really ought to allow myself a whole sheaf of fields, understanding that sometimes I want to work with fields that are only defined in a local area. But the physicists are generally clever about this type of problem, so, at the risk of saying things that we might think generalize but actually don't, I'm going to restrict my attention to scalar fields.

In which case, yes, by "scalar field" I mean "function from M \to \R". "A section of M\times\R". "A number at each point". Those who prefer to start with the sheaf of scalar fields will be happy to know that, when I define tangent vectors and their relatives in the next entry, I will start with these scalar fields.