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.
Best Answer
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.