You can often get the gist of a mathematical subject via an informal explanation involving infinitesimals. But I often find that questions arise from that informal explanation that I’d like resolved, but I don’t want to jump all the way to the full definitions. Without a rigorous basis for reasoning about infinitesimals, it can be tricky to dig any deeper, and I think allowing that slightly deeper digging is a nice benefit that an understanding of nonstandard analysis can provide.
One example of this for me is the Lie bracket of vector fields: this is supposed to be a commutator of vector fields, but isn’t addition of vector fields supposed to be commutative? How is the Lie bracket ever non-zero, given that intuition? The answer may be obvious to most mathematicians, but it wasn’t for me. Fortunately, nonstandard analysis provides a nice way to push an informal, infinitesimal-based understanding of vector fields and Lie brackets far enough to to answer this to my intuitive satisfaction.
This is based on this paper, which provides a development of differential geometry in nonstandard analysis. I’ll simplify the presentation in two ways: One is by assuming that the given manifold we’re considering is compact. The other, which you can ignore if you’re not familiar with these issues, is by working in an internal set theory; this essentially means that we work in a framework where already has infinitesimals in it, rather than having to pass to some extension field containing infinitesimals.
The basic intuition for treating tangent vectors nonstandardly is to think of them as things which tell you how to flow an infinitesimal amount from a given point. To flesh this out, let’s start with a few definitions:
Given two points and an infinitesimal with , let’s say that is if is infinitesimal in some chart (equivalently in every chart). Similarly, let’s say that is if isn’t infinitesimal in some chart (equivalently in any chart).
Now we can build up a nonstandard notions of tangent vectors and vector fields. The paper above makes the choice to fix a single infinitesimal and use it as a sort of global length scale throughout. Given that, we have:
- A prevector at a point is a pair where such that is .
- Two prevectors , are equivalent if is .
- A tangent vector at is a prevector , considered up to equivalence.
Similarly we can define:
- A prevector field on is a function from to such that is for all .
- Two prevector fields , are equivalent if is for all .
- A vector field is a prevector field considered up to equivalence.
Now, we can transfer classical notions over to the nonstandard case. For example, we can add two prevectors based at the same point by doing normal vector addition in a chart: this does depend on what chart you use but only up to . Similarly, it turns out that we get a well-defined vector addition based off of this notion of prevector addition.
Similarly, we can define addition of vector fields pointwise. Furthermore, if there are underlying prevector fields and that satisfy a certain regularity condition (corresponding to classically), then addition of the vector fields is equivalent to the composition of the prevector fields (which, recall, are just functions).
We can also easily define a flow (or integral curve) of a prevector field : Starting at a point , the flow of along for time is ; that is, composed with itself many times.
There’s a sort of bonus “paradox” resolution here: I used to wonder intuitively why differential equations could have non-unique solutions: Isn’t it the case that the differential equation always tells you exactly where to move in the next infinitesimal time step? The answer is no, it doesn’t: for an infinitesimal timestep , it only tells you where to move up to , so you might be able to make different steps that “add up” to appreciably different solutions.
Moving on to the Lie bracket, consider prevector fields and ; as the “commutator” intuition about the Lie Bracket suggests, let’s consider the prevector field . As the “vector fields commute with each other” intuition suggests, this prevector field is everywhere equivalent to the zero prevector field.
But we still want to study it. In order to make it “appreciable”, the paper defines the Lie bracket prevector field of and to be , that is, composed with itself many times. This is sufficiently many times that this prevector field is now distinct from the zero prevector field.
The paper proves that (under a regularity condition corresponding to classically), this does indeed correspond to the classical Lie bracket. I found this to be a very satisfactory resolution of the conflicting intuitions I mentioned at the start of the post.