Questions on various methods and aspects of quantization

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### The function algebra $C^{\infty}(M\#N)$ of the connected sum of two spaces

Operations such as taking union or Cartesian products of spaces have direct analogues in term of algebra of functions on them (direct sum and tensor product, respectively),
my question is:
Is there ...

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vote

**2**answers

126 views

### Deformation quantization of a closed Riemann surface with genus >1

Quantization of of an elliptic curve can be done in different ways.
In C^*-algebraic version,
one can start with the C^*-algebra ...

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**0**answers

81 views

### The normalization axiom of a quantization

Guillemin, Ginzburg and Karshon explain a quantization in their book [Chap 6,MR1929136] as follows.
The quantization is a process which associates to a symplectic manifold $M$ a Hilbert space ...

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**0**answers

196 views

### Why is geometric quantization (esp. Berezin-Toeplitz quantization) interesting for a symplectic geometer/topologist?

I know that many symplectic geometers are interested in quantization as well.
From what I understood, quantization isn't expected to be used as a tool to answer symplectic questions (as in ...

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**3**answers

117 views

### graded generalization of the Moyal–Weyl product

Has anyone written about the graded generalization of the Moyal–Weyl product/star product, that is, where the original algebra is already graded? Is it just a matter of signs?

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**1**answer

87 views

### Equivalence of star products on two differents Poisson algebras?

Let $A$, $B$ be two commutative and associative $\mathbb k$-algebras and let $A_\hbar:=A[[\hbar]]$, $B_\hbar:=B[[\hbar]]$ be the corresponding ring of formal series. Of sense [Deformation theory and ...

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vote

**1**answer

53 views

### Projective volume form

Upon reading K. Costello's paper on Witten genus, I wonder when, on a smooth (quasi-)projective variety $X$, the canonical bundle $\omega_X$ admits a left $D$-module structure (other than the ...

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**2**answers

383 views

### Quantization by cohomology

Ok, so I have heard some cool stuff here and there about how to Quantize Yang-Mills via cohomology, can anyone refer any texts in the literature that have shed some light on this, I mean I have some ...

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**0**answers

146 views

### Fractional Derivatives [closed]

How far these Theories of "Fractional Derivatives" be rigorized ? I have few books and references on Fractional Differential Equations etc (mainly they stress on Applied Mathematics parts and similar ...

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**1**answer

123 views

### Absent 2nd order terms in deformation quantization of Poisson manifolds

I am reading Kontsevich' famous paper on deformation quantization of Poisson manifolds. In section 1.4.2 on page 4 he gives the general formula for the star product associated to a Poisson structure ...

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**1**answer

262 views

### Formal series convergence in deformation quantization and $C^*$-condition

A link between formal series convergence in deformation quantization (strict deformation quantization) and producing $C^*$-algebras instead of mere $*$-algebras (which ...

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votes

**2**answers

323 views

### Physical meaning of the integral cohomology condition in Souriau-Kostant pre-quantization?

The question is in the title. The form of the condition looks like the Bohr-Sommerfeld quantization formula of angular momentum, is there a link between the two formulas?

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**0**answers

209 views

### Bohr topos and quantization

Bohrification is a natural way to construct a quantum "phase space" (with some nice insights on foundational problems like non-contextuality through Kochen-Specker etc). I was wondering, since we get ...

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**1**answer

201 views

### choices of connection in prequantization

In the definition of pre-quantization of representation $f\to \hat{f}$, (here $\hat{f}$ is Hermitian operator)of $C^{\infty}(M)$ on $L^2(M,L,\mu)$ where $\mu$ is Hermitian form, suppose that there ...

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**1**answer

227 views

### Does the vanishing of the Poisson bracket on $S(\mathfrak{g})^{\mathfrak{g}}$ inspire the disover of Duflo's isomorphism theorem?

For any finite dimensional Lie algebra $\mathfrak{g}$, we know that the universal enveloping algebra $U(\mathfrak{g})$ is a deformation of the symmetric algebra $S(\mathfrak{g})$. In fact let's define
...

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**1**answer

308 views

### Reconciling two notions of geometric quantization.

Let $(M,\omega)$ be a compact symplectic manifold and $(L,\nabla)$ a prequantum line bundle. There are two schemes to quantize this data:
Choose a polarization $P$ of $M$ and define the quantum ...

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**0**answers

233 views

### D-modules as quantization of modules on cotangent bundle

If we have filtration on D-module compatible with some good filtration on differential operators sheaf then adjoint graded module is module over functions on cotangent bundle. So morally D-module is ...

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**2**answers

454 views

### Is the quantum algebra unique (up to isomorphism) in deformation quantization ?

Consider a Poisson algebra A (i.e. commutative algebra with Poisson bracket).
Let $\hat A$ be a deformation quantization of the algebra A. We know that construction of deformation quantization and ...

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**7**answers

669 views

### Quantization of a classical system (e.g. the case of a billard)

There has been already several questions asking for an introduction to quantum mechanics
for a mathematician, but this ons is slightly different, and more restrictive. I know (some)
quantum mechanics, ...

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votes

**2**answers

190 views

### On the definition of ‘smooth vectors’ in Rieffel's “Deformation Quantization for Actions of $ \mathbb{R}^{d} $”.

On the first page of Chapter 1 of Rieffel's Deformation Quantization for Actions of $ \mathbb{R}^{d} $, Rieffel defines a family of seminorms on the space $ A^{\infty} $ of smooth vectors of a Fréchet ...

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**4**answers

534 views

### Higgs mechanism from a deformation quantization point of view

Is it possible to describe the Higgs mechanism from a deformation quantization point of view? How would one do it? Are there aspects of the Higgs mechanism and Higgs particle which one may see clearer ...

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votes

**2**answers

526 views

### Problem of quantization: state of the art

The "problem of quantization":
Find a vector space $Obs$ (as large as possible) of real-valued functions $f(p, q)$ on $R^{2n}$, containing the coordinate ...

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**1**answer

544 views

### Problem of quantization: state of the art [closed]

As the title suggests, I'm interested in finding out the state-of-the-art in the problem of quantization.
Any suggestions and/or feedback would be greatly appreciated.
Regards.

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votes

**1**answer

181 views

### What is the definition of “the $L_\infty$ part of a $G_\infty$ morphism”?

We know that in Tamarkin's proof of Kontsevich's formality theorem, he defined the $G_\infty$ structure on the Hochschild cochain complex $C^\cdot(A,A)$ and constructed a $G_\infty$ morphism from ...

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672 views

### Coherent states vs quantization of Lagrangian submanifold

Coherent states http://en.wikipedia.org/wiki/Coherent_states
are vectors in the Hilbert space which in certain sense are strongly localized
and "corresponds" to points in classical phase space (see ...

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votes

**1**answer

138 views

### BKS pairing for distributional sections

I am trying to understand the Blattner-Kostant-Sternberg pairing as it applies to geometric quantization in real polarizations whose integral manifolds are, for simplicity, compact. I have been trying ...

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**1**answer

208 views

### Implementation of the bounded-distance decoder of Leech-lattice?

Hi all,
I am wondering, anybody can help me how can I find an implemented version of Leech-Lattice quantizer/decoder, i.e., "Matlab", "C++" or "Python" code, using the approach proposed by Ofer ...

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votes

**1**answer

564 views

### SL(2,C) Chern-Simons theory in genus 1

In http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1104202513, Witten claims (p. 54) that to quantize the moduli space of flat $SL(2,\mathbb{C})$ ...

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**2**answers

508 views

### Deriving the Hilbert spaces for Chern-Simons TQFTs with complex gauge group

One method for finding the Hilbert spaces corresponding to surfaces in Chern-Simons TQFT is by geometrically quantizing the phase space, which is just the character variety of the surface. I know that ...

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709 views

### Quantization of conjugacy classes in a Lie group

Let $G$ be a Lie group (and to be safe, let's assume it is semisimple). Consider the action of $G$ on itself by conjugation, and form the GIT (algebro-geometric) quotient $G/\\!/G$. Then let ...

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votes

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869 views

### Lagrangian Submanifolds in Deformation Quantization

Suppose I have a symplectic manifold $M$, and have a deformation quantization of it, i.e. an associative product $\ast:C(M)[[\hbar]]\otimes C(M)[[\hbar]]\to C(M)[[\hbar]]$ so that $f\ast ...

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**1**answer

845 views

### Kontsevich's formality theorem from an explicit homotopy

Suppose that $X$ is a smooth manifold, whose $C^{\infty}$-functions we denote by $A$. Let $D_{poly}^*(A):=\bigoplus_{n\geq -1}Hom(A^{\otimes n+1},A)$ be the Lie algebra of polydifferential operators ...

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**2**answers

599 views

### What are the implications of torsion in H^2 for geometric quantization?

Given a real manifold $M$ with symplectic $2$-form $\omega$,
one can ask whether the cohomology class $[\omega] \in H^2(M;{\mathbb R})$ lies in the image of
$H^2(M;{\mathbb Z})$. If so, one can ask ...

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**4**answers

2k views

### How to see the Phase Space of a Physical System as the Cotangent Bundle

Two things today motivated this question.
First, the professor said that in a lecture Thurston mentioned
Any manifold can be seen as the configuration space of some physical system.
Clearly we ...