I am trying to understand better the quantization of the Harmonic Oscillator.
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 $$ \langle a b c d\rangle = \langle a b \rangle \langle c d\rangle + \langle a c \rangle \langle b d\rangle + \langle a d \rangle \langle bc \rangle$$ 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. Does any rotation action get quantized this way?
This question involves rotation actions, in a different way than this other MO qustion: http://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: Representation of double cover of $U(n)$ on eigenspaces of harmonic oscillator