Why is the harmonic oscillator so important? (pure viewpoint sought). How to motivate its role in Getzler's work on Atiyah-Singer?

Question

I'm in the process of understanding the heat equation proof of the Atiyah-Singer Index Theorem for Dirac Operators on a spin manifold using Getzler scaling. I'm attending a masters-level course on it and using Berline, Getzler Vergne.

While I think I could bash my way through the details of the scaling trick known as `Getzler scaling', I have little to no intuition for it.

As I understand it, one is computing the trace of the heat kernel of the ("generalized") Laplacian associated to a Dirac operator. The scaling trick reduces the problem to one about the ("supersymmetric" or "generalized") harmonic oscillator, whose heat kernel is given by Mehler's formula. I am repeatedly assured that the harmonic oscillator is a very natural and fundamental object in physics, but, being a `pure' analyst, I still can't sleep at night.

What reasons are there for describing the harmonic oscillator as being so important in physics?

Why/how might Getzler have thought of his trick? (Perhaps the answer to this lies in the older proofs?)

Is there a good way I could motivate an attempt to reduce to the harmonic oscillator from a pure perspective?

(i.e. "It's a common method from physics" is no good). I'm looking for: "Oh it's simplest operator one could hope to reduce down to such that crucial property X still holds since Y,Z"...or..."It's just like the method of continuity in PDE but a bit different because..."

Thanks.

Answer 1

What reasons are there for describing the harmonic oscillator as being so important in physics?

The harmonic oscillator tends to show up when you're expanding a potential function around non-degenerate critical points.

The simplest example is a physical system described by a map $t \mapsto \phi(t) \in \mathbb{R}$. If the energy function for this system has the form $E(\phi) = \frac{1}{2}|\dot{\phi}|^2 + V(\phi)$, with $V$ bounded below, then the lowest energy states are going to be of the form $\phi_0(t) = \phi_0$, where the constant $\phi_0$ is a minimum of $V$, hence a critical point. So, if your map $\phi$ never deviates too much from $\phi_0$ and $\phi_0$ is a non-degenerage critical point, you can approximate the energy function by $E(\phi) = |\frac{1}{2}\dot{\phi}|^2 + V(\phi_0)+\frac{1}{2}V''(\phi_0)(\phi-\phi_0)^2$.

In other words, the harmonic oscillator potential describes small disturbances around "generic" minima of an energy function. This situation comes up all the time in physics. For example: it shows up in Witten's Supersymmetry & Morse Theory paper, which I think would have been well-known to people working on topology and analysis in the 1980s.

Answer 2

I too am but a mere graduate student trying to sort through some of these same issues, but I might have some helpful insight. I'll let you be the judge.

The basic idea behind the heat equation proof of the index theorem is to extract the right term in the asymptotic expansion for the heat kernel and then appeal to the McKean - Singer formula. According to my understanding the original strategy for doing this was to realize that the index is a cobordism invariant and thus it would suffice to do enough explicit calculations on generators for the cobordism group until all free parameters are fixed; as it turns out the complex projective spaces are a good choice. That's exactly what was done. I believe - and I really hope someone will correct me if I'm wrong since I haven't gotten my hands dirty myself - that the required calculation really boils down to dealing with the quantum mechanical harmonic oscillator when you work it out for $CP_2$. If this is correct, then the first hint that the quantum mechanical harmonic oscillator is important came from a very fundamental example.

But I think a more analytic answer is also possible. Let's say instead of working with a Dirac operator acting on smooth sections of the spinor bundle you instead just consider the usual scalar Laplacian acting on functions. What happens if you imitate the heat kernel proof in this much less subtle context? You wind up reproving Weyl's asymptotic formula for the eigenvalues of the Laplacian. In essence this calculation amounts to rescaling the spacial variable so that your operator is deformed into the constant coefficient operator obtained by freezing coefficients. The basic idea of the Getzler calculus is to rescale both the spacial variable and the Riemannian metric in a compatible way - this rescaling deforms the Clifford algebra into the exterior algebra (thereby making Clifford multiplication act like an order one operator) and hence the Dirac operator into a polynomial coefficient operator. What polynomial coefficient operator is it? We have reached the limit of my ability to motivate things any further, but the answer is the quantum mechanical harmonic oscillator operator. I of course have no idea whether or not the physical significance of this operator can be accounted for according to a similar rescaling argument.

I should also mention that the quantum mechanical harmonic oscillator makes no obvious appearance in the original global proofs of the index theorem. It does, however, make a non-obvious appearance via Bott Periodicity which can be proven essentially using Mehler's formula. Nigel Higson and Eric Guenter wrote a very readable paper explaining most of the details of this proof entitled something like "K-Theory and Group C* Algebras". You can find it on Nigel's website, www.math.psu.edu/higson.

The last thing I will say is that I found Getzler, Berligne, and Verne to be a pretty tough way to penetrate this material. The style pays off in some of the later material, but I think I would have had a lot of trouble learning the heat kernel proof of the index theorem for the first time from that book. You might try John Roe's book "Elliptic Operators, Topology, and Asymptotic Methods" instead of or as a supplement.

I hope this has been helpful!

Answer 3

Here is a fairly physicsy point of view. You are calculating the supertrace of the heat operator, and because this is magically time independent, you can calculate it in the small $t$ limit. The heat operator is the time-evolution operator (propagator) for a certain theory (supersymmetric quantum mechanics) and the supertrace amounts to considering it only on loops (I am skipping some details here!). The small $t$ approximation means that you can consider loops that are nearly constant.

The small $t$ limit is similar to the semiclassical (small $\hbar$) limit, so we will borrow our intuition from there. In the semiclassical approximation you consider small perturbations from a classical solution and expand the action in powers of the perturbation. Because classical solutions are critical points of the action, you get no linear term, so the first interesting term in quadratic in the perturbation. So in the small $t$ limit you can ignore everything but the quadratic part of the action. A quadratic action is basically just a harmonic oscillator.

I believe (and here I am expressing an intuition I have no concrete argument for!) that Getzler's rescaling amounts to converting between the small $t$ limit and the small $\hbar$ limit, since these are not quite the same thing. I also think that you are actually getting a magnetic term, not a quadratic potential (which is the harmonic oscillator), but there is a standard physics trick for going from one to the other.

More generally, if you are in any physics situation where you have a minimum/critical point of a potential/action, it is probably going to make sense to expand it in powers of distance from that point, and the first term of the expansion will be quadratic and therefore a harmonic oscillator. So you can always model systems sufficiently close to their equilibrium by a system of harmonic oscillators. In quantum theory it is only a little bit of a lie to say that the only things we can solve are quadratic theories (harmonic oscillator) so nearly all understanding starts from there and builds out. The magic of the Index Theorem, and most places where QFT give you wonderful results in mathematics, is that this approximation is somehow exact.