We're all familiar with space and maybe somewhat comfortable with spacetime. We can give ourselves coordinates, labeling our position in space (and time). But math and physics routinely uses other spaces to do calculations or simply as a framework which things act. They are often very similar to "real" space that we are familiar with.
I want to mention a couple of spaces that are of relevance in physics as well as math. One is the path space. This is a great deal more complicated than ordinary space. For one, it is infinite dimensional. That gets mighty difficult to imagine. But it is a necessary space when thinking about the first formulation of the path integral: our first re-formulation of quantum mechanics.
How do we think about path space, then? First, a path is simply some way of getting from one point to another. So we need what we'll call the "base space." It is just like some ordinary space that we're familiar with. We can run around and all that jazz. We consider two points in this base space. A path is any line (or curvy line or even fractal) that connects these two points. For the case of quantum, we'll keep the less nice looking paths, the ones with kinks and jagged points and things in the path, the ones that go all the way to the moon and back. But these curves must be continuous, the path can't abruptly end in the middle and start back up somewhere else.
Ok we know what a path is. Technically now I am going to have to restrict paths into equivalence classes defined by thin homotopy: simple reparameterizations of the path. If we change how fast we move along this path, that is a thin homotopy, and the two paths, which differ only by the speed along which we travel it are said to be the same path. The two paths aren't the exact same paths (we more along it differently (magnitude of the derivative is different) but we call them the same because they take us across the same places.
We can set up the space of paths now. It is tough to imagine. But if we arbitrarily choose a point in this space, we imagine having chosen a particular path between our two points. Ok, so far so good: a single point in this abstract path space forces us to imagine an entire curve in our ordinary space between two points. Choose a neighboring point in path space, and we are choosing a path which is almost like our original path. This new path differs by just a small amount. Fine. If we choose a point in path space very far from the original point, this path and the original path will look very different. For now we'll continue to assume that every path in this path space has the same endpoints.
We can imaging connecting two points in path space via some curve (still in path space, remember). What does this give us? At each point on the curve in path space, we have a slightly different curve in space. A curve in path space, then, defines a surface that connects two curves in ordinary space. This is called a homotopy between the two curves. There are cases where two curves, with the same start/end points and everything, cannot be taken smoothly from one to the other curve, so there isn't a nice smooth surface with the two curves as boundaries. This will be due to the topology of ordinary space (whether ordinary space is finite & periodic or if there are holes).
We could go on about this. Let's just point out one thing: the word "thin homotopy" was used before and now we've seen "homotopy." Yes they are related, yes thin homotopy is a specific type of homotopy. What type? Imagine this 2-D surface connecting the two curves being the surface with the minimum possible surface area. Two curves are called "thin homotopic" if this particular surface has zero surface area. Think about it. (We can define a metric on path space (talk about "distances" between points) using this definition, too).
Anyway, what good is this structure? Remember we have included all sorts of nasty paths with corners and the like. We can define quantum mechanics using this path space. We consider a starting point and an ending point for a particle. The particle goes from start to finish at some known velocity, which can of course change during the course of the trip.
We will look at path space now. Assign a number to each path. This number will end up being:
\[ \mathrm{exp}\Big\{ -i\int_{t_1}^{t_2} dt \; (T - V)\Big\} \]
Where $T$ is the kinetic energy of the particle and $V$ is the potential energy the particle feels. The term inside the exponential is $\sqrt{-1}$ times the "action." For those who have taken statistical mechanics, this might remind you of the Boltzmann probability factor, and that is exactly what we have here.
We have this number assigned to EVERY point in path space. Now we add up all of these numbers. If the result is finite, then we can normalize the thing to define a probability distribution on path space. It'll turn out that the larges probability point in path space (highest probability path in ordinary space) will be the classical path, the path we expect from Newton. Why? The classical path is where the action is a minimum, which means that exponential is a maximum. So the largest contribution to the sum of all the numbers (the path integral) will be from the classical trajectory.
That was fun. Should we generalize this path integral to be able to consider fields next so we can start describing quantum field theory?
