TITLE: Why is the harmonic oscillator so important? (pure viewpoint sought). How to motivate its role in Getzler's work on Atiyah-Singer? 
QUESTION [17 upvotes]: 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.

REPLY [4 votes]: 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.