---
created: 2022-03-27 00:54
updated: 2023-03-31T16:25
title: "deformation quantization"
aliases: [deformation quantization, star product]
---

---

- Tags 
	- #todo/untagged
- Refs:
	- <https://webspace.maths.qmul.ac.uk/a.ghobadi/Part%20III%20Essay.pdf#page=27&zoom=160,-144,798>
	- Doubek, Deformation theory lecture notes: <https://dml.cz/bitstream/handle/10338.dmlcz/108078/ArchMathRetro_043-2007-5_4.pdf#page=2&zoom=160,-202,592>
	- Deformations of algebra in noncommutative geometry, <https://arxiv.org/pdf/1212.0914.pdf#page=1&zoom=auto,-200,792>
- Links:
	- #todo/create-links 

---

# deformation quantization

![](2023-04-02-13.png)
![](2023-04-02-14.png)

Yields [Frechet spaces](Frechet%20space):
![](2023-04-02-17.png)


![](attachments/Pasted%20image%2020220514173622.png)

![](attachments/Pasted%20image%2020220327005410.png)
![](attachments/Pasted%20image%2020220424234650.png)
Relation to [Weyl algebra](Weyl%20algebra.md)
![](attachments/Pasted%20image%2020220424234718.png)
![](attachments/Pasted%20image%2020220424234822.png)

# Example, deforming an algebra of functions

![](2023-03-31-73.png)

# Star products

Idea: [formal deformations](deformation%20theory%20is%20controlled%20by%20dg%20Lie%20algebras.md#Defining%20formal%20deformations%20of%20an%20algebra) whose coefficients are differential operators.

![](Pasted%20image%2020230403094731.png)

![](2023-03-31-76.png)

Relation to [HH](HH.md):
![](2023-03-31-77.png)

Relation to [Poisson structure](Poisson%20structure.md): ![](2023-03-31-78.png)

Star quantization: ![](2023-04-02-15.png)
Yields a [Poisson structure](Poisson%20structure.md):
![](2023-04-02-16.png)