# Rotations, Harmonic Oscillators, Gaussians, Ladders

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: Characterizing the harmonic oscillator creation and annihilation operators in a rotationally invariant way

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

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The first two representations are related by the Bargmann transform, which is a unitary transformation between $L^2( \mathbb{R}, dx)$ and $L^2( \mathbb{C}, e^{-z\bar{z}}dzd\bar{z})$. This transformation expresses the invariance of the quantization under the change of polariztion: real in the first case and Kähler in the second case. (Both cases treat the quantization of $\mathbb{C} = T^{*}\mathbb{R}$ with their canonical symplectic forms).
The third case coresponds to the quantization of $\mathbb{C^{\times}}$ in real polarization, which corresponds basically to performing the canonical transformation:
$dx\wedge dp = dH\wedge d\theta$