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

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

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

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

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

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