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Questions tagged [quantization]

Questions on various methods and aspects of quantization

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Does this distribution exist?

Assume there is a distribution in two variables $\mathcal{W}\in\mathcal{S}'(\mathbb{R}^2)$ with Fourier transform $\hat{\mathcal{W}}(\alpha,\beta)\equiv \int_{-\infty}^\infty e^{i(\alpha x+\beta y)} \...
Nicolas Medina Sanchez's user avatar
1 vote
1 answer
65 views

Lipschitz approximation of a probability measure with finite $1$-st moment by the ones with finite $p$-th moment

For $p \in [1, \infty)$, let $\mathcal P_p (\mathbb{R^d})$ be the space of Borel probability measures on $\mathbb R^d$ with finite $p$-th moment. We endow $\mathcal P_p (\mathbb{R^d})$ with the ...
Akira's user avatar
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6 votes
1 answer
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Question about a remark on quantization of Coulomb branches

I will follow the definition of Coulomb branches of $3d$ $\mathcal{N}=4$ gauge theories from the paper by Braverman, Finkelberg and Nakajima, Towards a mathematical definition of Coulomb branches of 3-...
jg1896's user avatar
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2 votes
0 answers
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Efficient decoding of the E8/Leech lattice

Background: Our goal is to quantize a sequence of floating point numbers generated i.i.d. from a standard Gaussian source and minimize the MSE reconstruction error. We can use two bits for each sample....
nalzok's user avatar
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1 vote
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How does $R \equiv 1\ (\text {mod}\ h)\ $?

Definition $:$ Let $H$ be a Hopf algebra. An invertible element $R \in H \otimes H$ is called a coboundary structure on $H$ if $(1)$ $\Delta^{\text {op}} = R \Delta R^{-1},$ $(2)$ $R_{21} = R^{-1},$ $(...
Anacardium's user avatar
1 vote
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69 views

Problem in understanding Theorem $6.2.9$ from Chari and Pressley

The theorem I am referring to here says that if we start with a Lie bialgebra $\mathfrak g$ determined by some skew-symmetric element $r \in \mathfrak g \otimes \mathfrak g$ satisfying classical Yang-...
Anil Bagchi.'s user avatar
3 votes
1 answer
205 views

Understanding definition of quantization of a Poisson-Hopf algebra

I am going through the chapter Quantization of Lie bialgebras from the book A Guide to Quantum Groups by Chari and Pressley. There I found a notion called Quantization which deals with deformations of ...
Anil Bagchi.'s user avatar
9 votes
2 answers
533 views

Physical intuition behind Kontsevich's deformation quantization formula

Kontsevich gives a construction that produces deformation quantization of $C^\infty(M)$ for general Poisson manifolds $M$. The resulting formula (on $\mathbb{R}^n$) is $$ f\star g = \sum_{n=0}^\infty \...
lw h's user avatar
  • 161
2 votes
0 answers
75 views

Most general space for the Wigner-Weyl transform

The Wigner-Weyl transform $\mathfrak{W}$ is a bijective mapping between functions on a phase space and Hilbert space operators in order to map quantum mechanics into a phase-space formulation. Then ...
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Obstruction to deforming vector bundles

Let $X$ be a complex algebraic variety and let $D$ denote any $\mathbb C[[h]]$-deformation of $\mathcal O_X$. Suppose that $D$ is trivial. Then it is well-known that obstructions to deforming any $X$-...
Batabat Volkov's user avatar
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Neural network quantization and arithmetic on affine functions

I'm trying to understand the basics of quantization in Neural networks. Quantization tries to convert a neural network that uses floating point arithmetic to one that uses a lower precision integer ...
anirudh's user avatar
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1 vote
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Is there a version of Nest-Tsygan theorem for smooth variety

Let $M$ be a smooth Poisson manifold and $\mathcal{O}_\hbar (M)$ be a deformation quantisation of $\mathcal{O} (M)$. Nest-Tsygan theorem says that $$HH_i(\mathcal{O}_\hbar (M)[\hbar^{-1}])\cong H^{2d-...
FunctionOfX's user avatar
3 votes
0 answers
363 views

Bi-differential operators in the definition of star product in deformation quantisation

Let $X$ be an (affine) Poisson variety (not necessarily smooth) over an algebraically closed field of characteristic 0 (such as $\mathbb{C}$), denote $\mathcal{O}(X)$ its ring of functions and $\{-,-\}...
FunctionOfX's user avatar
5 votes
1 answer
1k views

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 ...
Steve's user avatar
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4 votes
1 answer
304 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 ...
Weather Report's user avatar
1 vote
0 answers
542 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. ...
BrianT's user avatar
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2 votes
0 answers
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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 ...
user267839's user avatar
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6 votes
0 answers
201 views

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 ...
Daniel Teixeira's user avatar
2 votes
1 answer
143 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 ...
Tan1278's user avatar
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14 votes
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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 ...
Math-Phys-Cat Group's user avatar
13 votes
1 answer
958 views

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 ...
Yellow Pig's user avatar
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6 votes
2 answers
795 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 ...
Student's user avatar
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70 votes
10 answers
11k views

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 ...
Gil Kalai's user avatar
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7 votes
1 answer
404 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 ...
Heinz Doofenschmirtz's user avatar
1 vote
0 answers
165 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 ...
BrianT's user avatar
  • 1,197
10 votes
0 answers
1k 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-...
wonderich's user avatar
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3 votes
1 answer
149 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 ...
Peter Liu's user avatar
  • 253
8 votes
1 answer
606 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 ...
BrianT's user avatar
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5 votes
0 answers
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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 ...
David Roberts's user avatar
9 votes
1 answer
907 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 ...
FraSchelle's user avatar
7 votes
1 answer
3k 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 ...
Hao Yu's user avatar
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7 votes
0 answers
137 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}}\...
Zhaoting Wei's user avatar
  • 8,707
5 votes
1 answer
254 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^\...
Henrique Tyrrell's user avatar
5 votes
0 answers
195 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 ...
thingsthatmighthavebeen's user avatar
4 votes
0 answers
140 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....
Rauan Akylzhanov's user avatar
9 votes
0 answers
537 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}_{\...
mathphys's user avatar
  • 1,619
8 votes
1 answer
418 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}...
Earthliŋ's user avatar
  • 1,181
5 votes
1 answer
452 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 ...
Eric O. Korman's user avatar
5 votes
0 answers
270 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 ...
issoroloap's user avatar
9 votes
1 answer
356 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 ...
Alex M.'s user avatar
  • 5,282
5 votes
0 answers
165 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 ...
Adrien's user avatar
  • 8,369
4 votes
0 answers
319 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 ...
Alexander Chervov's user avatar
9 votes
0 answers
256 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 ...
Alexander Chervov's user avatar
15 votes
0 answers
459 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 ...
Alex M.'s user avatar
  • 5,282
35 votes
5 answers
5k views

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 ...
Dmitry Vaintrob's user avatar
3 votes
0 answers
393 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 ...
Max's user avatar
  • 1,607
6 votes
3 answers
2k views

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 ...
user avatar
3 votes
2 answers
487 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 $G$ ...
Tyler Holden's user avatar
3 votes
1 answer
159 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 ...
Qijun Tan's user avatar
  • 577
6 votes
1 answer
388 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 ...
Ali Fathi's user avatar
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