# Questions tagged [quantization]

Questions on various methods and aspects of quantization

71
questions

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### Quantization of normal distribution

For $n\in\mathbb{N}$, denote by $\mathcal{Q}_n$ the set of all probability measures on $\mathbb{R}$ that are supported on at most $n$ points.
Question: Is it known which element in $\mathcal{Q}_n$ is ...

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votes

**1**answer

101 views

### Nonlinear ODE to linear PDE?

I am interested in when and how one can trade a non-liner ODE for a linear PDE. To explain what this could look like here is a physics-inspired discussion.
Consider a classical mechanical system with ...

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182 views

### On prequantization bundles over integral symplectic manifolds

I am trying to clarify certain subtleties regarding prequantization bundles over symplectic manifolds, for which I haven't found any clear explanation so far.
Let me fix some definitions first.
...

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269 views

### Polarizations in algebraic and symplectic geometry

In context of Abelian varieties there are a couple of equivalent ways to
introduce the polarization of a algebraic variety. One way is to
choose a line bundle $\mathcal{L}$ which satisfies certain ...

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### Deformation quantization of infinite dimensional Poisson manifolds

In 1999, Karaali wrote a review of formal deformation quantization for a class she took with Weinstein.
She ends the paper with the following remark:
Another question that remains involves the ...

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**1**answer

78 views

### Hilbert module over $C_0(\Lambda)$ as space of continuous sections of HIlbert bundle

Let $\Lambda$ be a manifold and $p:H\to\Lambda$ a continuous Hilbert bundle with $H(\lambda):=p^{-1}(\lambda)$. Suppose $\Gamma_0^0(\Lambda)$ is the space of continuous sections vanishing at infinity ...

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671 views

### The quantization problem: modern quantization procedures and current status

The quantization problem is one of the most well-known current problems of theoretical and mathematical physics. It is also part of Hilbert's sixth problem (on the axiomatization of physics - see here ...

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### What are Harish-Chandra bimodules used for?

There are many recent papers on classification of Harish-Chandra bimodules for rational Cherednik algebras and, more generally, non-commutative algebras which are quantizations of symplectic ...

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votes

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541 views

### 1-dimensional pure gauge theory

I am learning TQFT from compact Lie groups by Freed, Hopkins, Lurie,
and Teleman: https://arxiv.org/abs/0905.0731 , and got stuck very hard
even in the first section ($n = 1$), which was "trivial but ...

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### The Planck constant for mathematicians

The questions
Q1. What are simple ways to think mathematically about the physical meanings of the Planck constant?
Q2. How does the Planck constant appear in mathematics of quantum mechanics? In ...

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**1**answer

346 views

### Kontsevich Formality sign convention

Since my question is related to sign convention, I want to define everything from the very beginning. $T_{poly}^k(M)=\Gamma(\wedge^{k+1} TM)$ are the multi vector fields with shifted degree and with ...

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114 views

### On different definitions of a prequantization space

Geometric quantization associates to a symplectic manifold $(M,\omega)$ a hermitian line bundle $L \to M$ with connection $\nabla$ whose curvature is $\omega$ (up to some constant).
Without talking ...

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625 views

### Yang-Mills theory with non-compact gauge groups G

Physicists are familiar working with Yang-Mills theory with compact and semi-simple gauge groups $G$ (Lie groups).
However, it is not entirely clear the formulation of Yang-Mills theory with non-...

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**1**answer

119 views

### Reference request for quantum Teichmuller space

I would like to ask for some detailed reference for quantum Teichmuller theory, better in a mathematical taste. I read a little bit on Kashaev's or Chekhov and Fock's, but find that I need to fill ...

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**1**answer

367 views

### Physical intuition behind prequantization spaces

Given a symplectic manifold $(M,\omega)$ with integral symplectic form, that is $$\omega \in \text{Im}(H_2(M,\mathbb{Z}) \to H_2(M,\mathbb{R})),$$ one can form a so-called prequantization space, that ...

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348 views

### Isn't the quantomorphism group really just the “WKB-quantomorphism” group?

Introduction
In his second-most upvoted post, called "Why quantum mechanics?" (second only to his post on fibre bundles & gauge theory) in the physics SE community, Urs Schreiber, in the setting ...

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votes

**1**answer

391 views

### Fedosov vs. Kontsevich deformation quantization : a beginner survey

I'm a condensed matter physicist who tries to understand the details of deformation quantization.
In my self-made training, I've found two huge pieces of work, namely
Fedosov, B. V. (1994). "A ...

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votes

**1**answer

2k views

### What is the relation between BRST quantization and gauge fixing quantization

To quantize gauge field, one usually use gauge-fixing procedure and then plus ghost field, my question is what the relation between BRST quantization and gauge fixing quantization is? Because it seems ...

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127 views

### Could we extend the star product on a Poisson manifold from its ring of smooth functions to its de Rham complex?

Let $M$ be a smooth manifold with a Poisson bracket $\{-,-\}$. Kontsevich proved that there exists a deformation quantization of $M$, i.e. let $C^{\infty}(M)[[\hbar]]=C^{\infty}(M)\otimes_{\mathbb{R}}\...

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232 views

### Is there any work on quantization of distributions?

Let $G$ be a Lie group and consider the space $C_c^\infty(G)$ of compactly supported complex-valued smooth functions on $G$ and $D'(G) = (C_c^\infty(G))'$ the topological dual linear space of $C_c^\...

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158 views

### Analogue of Kontsevich's formality theorem for quantization of Courant algebroids

In his 1997 preprint, M. Kontsevich proved the formality of the differential graded algebra controlling deformations of the associative and commutative algebra of functions on a manifold, seen as an ...

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126 views

### Reference request: Prequantization of canonical transformations and Lie group action

Hello to MathoverFlow community
I have some seemingly technical questions on applications of geometric quantisation to Lie group representation theory.
We shall start by giving background definitions....

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514 views

### From classical to quantum mechanics

Let ($X,\omega$) be a symplectic manifold (phase space of some physical system). Consider the algebra $\mathcal{C}^{\infty}(X,\mathbb{R})$ of smooth functions on $X$ and $[\omega]\in \textrm{H}^{2}_{\...

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395 views

### Kontsevich weights in the complex algebraic setting

In Kontsevich's Deformation quantization of Poisson manifolds, he gives an explicit formula for the star product:
$$
f \star g = fg + \sum_{n=1}^\infty \hbar^n \sum_{\Gamma \in G_n} w_\Gamma B_{\Gamma}...

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394 views

### Geometric quantization of Teichmuller space

The quantizations of Teichmuller space I have seen are via special coordinates (e.g. the paper of Chekhov and Fock) or conformal blocks. Does one get an equivalent quantization by geometric ...

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256 views

### Deformation quantization of Poisson bracket without star-product

Kontsevich's formality theorem implies in particular that star-products on a $C^\infty$-manifold $M$,
$$f\star g = fg + \sum_{k\geq1} \hbar^k B_k(f,g),\qquad f,g\in C^\infty(M),$$ where $B_k$ are ...

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268 views

### Geometric quantization: why are the prequantum operators self-adjoint?

I'm reading a bit about geometric quantization and, among the axioms of this construction, is one requiring that the operator $\hat f = -\textrm i \hbar \nabla _{X_f} + f$ associated to the classical ...

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148 views

### Distinguishing the Duflo star product

$\newcommand{\g}{\mathfrak g}\newcommand{\h}{\hbar}$
For a finite dimensional Lie algebra $\g$, he Duflo isomorphism is a complicated algebra isomorphism between the $\g$-invariant part $S(\g)^\g$ of ...

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260 views

### Quantization of $S^2$ as $C^*$-algebra?

The general context for the question - is belief that quantization of compact symplectic manifolds can be endowed with the structure of $C^*$-algebra (see MO230695).
The particular question is about ...

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214 views

### Existence/characterization/properties of $C^*$-algebras which “are” quantization of compact symplectic manifolds?

Understanding of "quantization" achieved much progress recent years, especially after Kontsevich breakthrough on deformation quantization, where he proved one-to-one correspondence between Poisson ...

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402 views

### Squeezing physics out of formal deformation quantizations

I am reading various texts concerning the concept of "quantization". I am interested in quantization on Riemannian manifolds (as opposed to just on $\Bbb R ^n$); for absolute clarity, I am interested ...

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### Does quantum mechanics ever really quantize classical mechanics?

I was curious about a physics question which I thought might be suitable for mathoverflow. I looked at the answer to this question, but it's not what I'm looking for.
Basically, classical mechanics ...

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302 views

### From symplectic manifold to Hilbert spaces [closed]

What could be a mathematical model of such physical wish? I'm looking for something sending a symplectic manifold $(M,\omega)$ to a Hilbert space $H_{M}$ with the following properties:
1- We should ...

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### Ambidexterity and Quantization

Here the nlab says about Hopkins-Lurie's ambidexterity paper:
"The discussion in the article is apparently motivated as part of what it takes to make precise the discussion of quantization in ...

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**2**answers

334 views

### Hamiltonian group actions in the context of holomorphic line bundles

When studying Hamiltonian group actions, a very nice set up might be to take the following:
Let $M$ be a compact Kähler manifold with (integral) Kähler form $\omega$, endowed with a Hamiltonian $G$ ...

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146 views

### Reference of $\hbar$-differential operator from symplectic geometry perspective

I am reading Bates and Weinstein's book 'Lectures on the Geometry of Quantization'. In Chapter 6, they defined the $\hbar$-differential operator, and showed (Theorem 6.7) that the Lagrangian ...

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375 views

### 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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**2**answers

260 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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votes

**1**answer

855 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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203 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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136 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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**1**answer

104 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 Calabi-...

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667 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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202 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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244 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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537 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 $(\mathcal{C}^{\infty}(M)[[t]],\...

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482 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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331 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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220 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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206 views

### higher order Noether identities

Noether's second variational theorem gives a correspondence between symmetries of a Lagrangian and Noether identities, which are relations among the Euler–Lagrange equations.
How about relations ...