Questions tagged [quantization]
Questions on various methods and aspects of quantization
85 questions
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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 \...
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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-...
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The equivalence of stochastic quantization and path integral quantization
I am looking for a reference in which the equivalence of stochastic quantization and
path integral quantization has been shown. It would be great if I can see such a relation for a Euclidean quantum ...
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0
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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 ...
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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)} \...
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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 ...
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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....
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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},$
$(...
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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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SL(2,C) Chern-Simons theory in genus 1
In Link, Witten claims (p. 54) that to quantize the moduli space of flat $SL(2,\mathbb{C})$ connections on a torus, one can simply quantize the cotangent bundle of a real torus and take the part ...
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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-...
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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 ...
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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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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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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$-...
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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-...
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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 $\{-,-\}...
CommunityBot
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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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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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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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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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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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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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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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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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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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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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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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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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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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Quantization of a classical system (e.g. the case of a billiard)
There have been already several questions asking for an introduction to quantum mechanics
for a mathematician, but this one is slightly different, and more restrictive.
I know (some)
quantum ...
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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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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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basic questions on quantum integrable systems
I have been learning about (classical) integrable systems lately, e.g. in the examples of a Lax pair etc. I frequently run into the term 'quantum integrable system'. May I ask a few questions:
What ...
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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 ...
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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 ...
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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 $\pi:G\...
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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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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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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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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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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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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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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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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 ...
CommunityBot
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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 ...
CommunityBot
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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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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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