Rotations, harmonic oscillators, Gaussians, ladders

Question

I am trying to understand better the quantization of the harmonic oscillator.

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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?

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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 the harmonic oscillator: Representation of double cover of $U(n)$ on eigenspaces of harmonic oscillator

Answer 1

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 $

before quantization. It is a result by Śniatycki, that upon choosing a real polarization along the radius, the quantization space is spanned by "distributional wave functions" having support only on the Bohr-Sommerfeld fibers which are the level sets corresponding to the quantized Harmonic oscillator energies. There are two recent Arxiv articles by Śniatycki (first, second), where he reviews his Bohr-Sommerfeld quantization and treats further applications.