Archive for the ‘Differential Geometry’ Category

Loose language

August 10, 2010 3 comments

Most of my work in physics is done from the perspective of a theoretical physicist. For some reason or other, when I was younger I found the clarity and precision of theoretical physics to be much more attractive than the rough and ready approach often found in experimental physics.  Not only was I afflicted with the unfortunate ability to render an experiment senseless simply by being in the same room as it, I also found experimental physics to be too subjective, too open to interpretation for my liking. Thus, at some point I decided to stick to theory and I’ve been there ever since.

As wonderful as I find this particular approach to physics, it’s not without its drawbacks. For one thing, in order to be a good theorist, you’ve got to have a good understanding not only of physics but also of the language in which a theorist speaks about physics. This invariably means that you’ve got to be a good mathematician. Early on, I started to think of myself as mediocre — at best — at mathematics. This lasted for several years until I figured out why I was mediocre: I was attempting to understand mathematics largely by reading papers written by mathematicians. Now, I love mathematicians as much as the next man but, by God, they’re awful at writing in a manner that’s clear to a non-expert. At some point during the last century, the Bourbaki crowd decided it would be a great idea to present mathematics in as dense and impenetrable a style as possible; this caught on and we ended up being left with several generations of mathematicians who believed that it’s fine to write about mathematics in a manner that makes it almost impossible for students to tell what’s going on. I’m all for rigour when presenting and proving results, but rigour is generally of little use in a pedagogical sense.

Of course, one can also go too far the other way. Physicists, for instance, tend to speak about mathematics in a manner that horrifies most mathematicians. In addition, many of the “intuitive” ways in which physicists present ideas really aren’t intuitive at all since they often don’t stand up to even the merest scrutiny or immediately run into difficulties once one attempts to extend intuitive pictures beyond simple cases. Perhaps the canonical examples of this sort of problem are to be found in differential geometry, a field which you really must understand if you want to consider yourself a decent theoretical physicist.

One of the things that used to irritate me considerably when I first attempted to learn differential geometry was the fact that I often found the mathematician’s perspective on the subject to be too abstract and needlessly encumbered by Bourbakization. On the other hand, physicists tend to speak about differential geometry using analogies that are often too loose to be of much help. A good example of this lies in the example of a tangent space to a differentiable manifold. If you’re taught the subject by a physicist or happen to learn about it from one of the many books written by physicists, you’ll often first be presented with a pictorial representation of what a physicist thinks a tangent space “looks like”. For instance, here’s a commonly encountered way of representing the tangent space to S^2, a two-sphere:

This is a nice picture to keep in your head if you find such things helpful. However, it’s not at all representative of what a tangent space actually is. For one thing, representing the tangent space to a point X\in S^2 like this really presupposes that you have embedded S^2 in some higher-dimensional flat space; the tangent space at X is then regarded as a hyperplane in this space. However, differential geometry is largely concerned with the intrinsic geometry of manifolds, that is, the study of their geometry without assuming that they are embedded in a higher-dimensional manifold. The picture above makes sense only if we are talking about the extrinsic geometry; we regard S^2 as being embedded in, say, \mathbb{R}^3 and consider the tangent space at X to be a two-dimensional hyperplane since this is the only real way in which the linearity of the tangent space (it is, after all, just some sort of vector space) can be related to the “flatness” of the plane in the picture.

A further problem with the way physicists speak about differential geometry is that the notation one uses when performing practical calculations isn’t necessarily the best notation to use when teaching the subject. To get an idea of what I mean by this, it’s worth bearing in mind that physicists tend as a rule to play fast and loose with notation; mathematicians, on the other hand, tend to be so precise about their notation that it sometimes induces seizures. For instance, if you check out the book I linked to earlier, you’ll find that the author freely admits that he constantly abuses notation when introducing the concept of a tangent vector as a directional derivative of a function along a curve in a manifold. Typically, this means that physicists often begin by saying that f:M\to\mathbb{R} is a real-valued map on a manifold M but later, when it comes to actually performing interesting calculations, they’ll happily use f to denote what is actually the coordinate presentation f\circ\varphi^{-1}_U of the map in some chart (U,\varphi_U). This irritates me no end and causes huge problems for students when they’re initially learning this stuff.

One of the unhappy consequences of looking at differential geometry through the lens of these loose analogies is that it sometimes convinces people to think that the best way to introduce a topic is to rely heavily on what they consider to be the intuitive aspects. This, however, is often a mistake. The concept of a tangent space, for instance, is a great example of how leaning too heavily on intuitive pictures can actually result in you overlooking the much more important algebraic structure that underlies everything. Let me say something slightly controversial:

Think about a tangent space first and foremost as something algebraic in nature. More specifically, tangent spaces are what you find when you play around with the linear spaces of germs of smooth functions in a neighbourhood of a point p\in M.

Once you’ve got the algebraic nature of tangent spaces under your belt, you can then consider tangent spaces as collections of equivalence classes of smooth curves through p. You’ll then understand not only the geometrical picture of what a tangent space is, but also the algebraic reasons that underlie why it is. I’ll deal with this more in a future post.