I am trying to understand better the quantization of the Harmonic Oscillator.  
<hr>

Here are three ways of thinking about the Harmonic Oscillator.  

 - Eigenfunctions of the differential operator: $H = -\frac{d^2}{dx^2} + x^2$
 - Eigenfunctions of the oscillator $H = a a^\dagger+ \frac{1}{2}$
 - Special orbits of the $U(1)$ action on the complex plane, level sets of the moment map $H = p^2 + x^2$.

Are there any places that explain all three of these on equal footing?  Items 1 and 2 have a [Wick formula][1]
$$ \langle a b c d\rangle = \langle a b \rangle \langle c d\rangle + \langle a c \rangle \langle b d\rangle + \langle a d \rangle \langle bc \rangle$$
Is there an analogue in the symplectic geometry case (item 3)?  

I want to understand better why this is a duality

$$ {\tt rotation,}\;e^{it}\in U(1)\leftrightarrow {\tt gaussians,}\;e^{-x^2} \leftrightarrow {\tt eigenstates, }\;|n\rangle$$

Something to that effect, mentioned in these [notes][2].  Does any rotation action get quantized this way?
<hr>
This question involves rotation actions, in a different way than this other MO qustion:
https://mathoverflow.net/questions/35054/characterizing-the-harmonic-oscillator-creation-and-annihilation-operators-in-a-r

**EDIT** Here is another MO post where the Bargmann transform arises in quantization of Harmonic Oscillator:
https://mathoverflow.net/questions/94535/representation-of-double-cover-of-un-on-eigenspaces-of-harmonic-oscillator


  [1]: http://www.lptl.jussieu.fr/files/fi.pdf
  [2]: http://www.math.columbia.edu/~woit/notes23.pdf