---
date: 2021-11-05 11:12
modification date: Friday 5th November 2021 11:12:05
title: quantization
aliases: [quantization]

---

Last modified date: <%+ tp.file.last_modified_date() %>

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- Tags: 
	- #physics 
- Refs:
	- <https://arxiv.org/pdf/1309.1150.pdf#page=48&zoom=140,-146,740> 
- Links:
	- Possibly related to [deformation quantization](Unsorted/deformation%20quantization.md)?

---


# quantization

Idea: replace a symplectic manifold $(M, \omega)$ modeling a system in classical mechanics with a quantum mechanical system $$Q: ( C^\infty(X; \RR), \ts{\wait, \wait}) \to (\Endo \mch, [\wait, \wait]) \qquad\in \liealg\slice\CC$$ where $[Q(f), Q(g)] = ih Q(\ts{f, g})$.

Steps:

- Regard $\omega$ as the curvature of $\mce \to X$ a principal $G\dash$bundle for $G = \U_1$ with connection $\nabla$
- Polarize: split phase space into position and momentum.
	- For $x\in X$ picks out $P_x\in (\TT_p X)_\CC$ isotropic Lagrangians
- Associate $\mch = L^2(H^0(\mcl))$ a Hilbert space of states coming from the sections of an associated Hermitian line bundle $\mcl$ whose covariant derivative along $P_x$ vanishes: $$\Gamma_{\mathcal{P}}(M, \mathcal{L})=\left\{s \in \Gamma(M, \mathcal{L}): \nabla_X s=0, X \in \Gamma(M, \mathcal{P}) \subset \Gamma(M, T M)\right\}$$
- Map Hamiltonians $H\in C^\infty(X; \RR)$ to linear operators $\tilde H\in \Endo \mch$ sending Poisson brackets to commutators.

![](attachments/2023-02-17-quant.png)

![](attachments/Pasted%20image%2020220806163710.png)
![](attachments/Pasted%20image%2020220806164503.png)
![](attachments/Pasted%20image%2020220806164634.png)
![](attachments/Pasted%20image%2020220806164816.png)

Related to [[Fock space]]:
![](attachments/Pasted%20image%2020220806164704.png)

For a Hamiltonian:

![](attachments/Pasted%20image%2020220510182009.png)

![](attachments/Pasted%20image%2020220429214748.png)

![Pasted image 20211105111208.png](Pasted%20image%2020211105111208.png)
![Pasted image 20211105111252.png](Pasted%20image%2020211105111252.png)

![Pasted image 20211105111350.png](Pasted%20image%2020211105111350.png)
![Pasted image 20211105111448.png](Pasted%20image%2020211105111448.png)
![Pasted image 20211105111635.png](Pasted%20image%2020211105111635.png)
![Pasted image 20211105111653.png](Pasted%20image%2020211105111653.png)

# Application to TQFTs

See [TQFT](Unsorted/TQFT.md):

![](attachments/Pasted%20image%2020220408100635.png)
![](attachments/Pasted%20image%2020220408100653.png)