Questions tagged [quantization]
Questions on various methods and aspects of quantization
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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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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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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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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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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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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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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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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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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 g=fg+\{f,g\}\...
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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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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
$$
f\star g = \sum_{n=0}^\infty \...
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What are the implications of torsion in H^2 for geometric quantization?
Given a real manifold $M$ with symplectic $2$-form $\omega$,
one can ask whether the cohomology class $[\omega] \in H^2(M;{\mathbb R})$ lies in the image of
$H^2(M;{\mathbb Z})$. If so, one can ask ...
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Kontsevich's formality theorem from an explicit homotopy
Suppose that $X$ is a smooth manifold, whose $C^{\infty}$-functions we denote by $A$. Let $D_{poly}^*(A):=\bigoplus_{n\geq -1}Hom(A^{\otimes n+1},A)$ be the Lie algebra of polydifferential operators ...
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Yang-Mills theory with non-compact gauge groups G
Physicists are familiar working with Yang-Mills theory with compact and semi-simple gauge groups $G$ (Lie groups).
However, it is not entirely clear the formulation of Yang-Mills theory with non-...
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basic questions on quantum integrable systems
I have been learning about (classical) integrable systems lately, e.g. in the examples of a Lax pair etc. I frequently run into the term 'quantum integrable system'. May I ask a few questions:
What ...
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Quantization of conjugacy classes in a Lie group
Let $G$ be a Lie group (and to be safe, let's assume it is semisimple). Consider the action of $G$ on itself by conjugation, and form the GIT (algebro-geometric) quotient $G/\!/G$. Then let $\pi:G\...
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Physical intuition behind prequantization spaces
Given a symplectic manifold $(M,\omega)$ with integral symplectic form, that is $$\omega \in \text{Im}(H_2(M,\mathbb{Z}) \to H_2(M,\mathbb{R})),$$ one can form a so-called prequantization space, that ...
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Fedosov vs. Kontsevich deformation quantization : a beginner survey
I'm a condensed matter physicist who tries to understand the details of deformation quantization.
In my self-made training, I've found two huge pieces of work, namely
Fedosov, B. V. (1994). "A ...
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Geometric quantization: why are the prequantum operators self-adjoint?
I'm reading a bit about geometric quantization and, among the axioms of this construction, is one requiring that the operator $\hat f = -\textrm i \hbar \nabla _{X_f} + f$ associated to the classical ...
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From classical to quantum mechanics
Let ($X,\omega$) be a symplectic manifold (phase space of some physical system). Consider the algebra $\mathcal{C}^{\infty}(X,\mathbb{R})$ of smooth functions on $X$ and $[\omega]\in \textrm{H}^{2}_{\...
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Existence/characterization/properties of $C^*$-algebras which "are" quantization of compact symplectic manifolds?
Understanding of "quantization" achieved much progress recent years, especially after Kontsevich breakthrough on deformation quantization, where he proved one-to-one correspondence between Poisson ...
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Kontsevich weights in the complex algebraic setting
In Kontsevich's Deformation quantization of Poisson manifolds, he gives an explicit formula for the star product:
$$
f \star g = fg + \sum_{n=1}^\infty \hbar^n \sum_{\Gamma \in G_n} w_\Gamma B_{\Gamma}...
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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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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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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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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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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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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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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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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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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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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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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 $C^{\infty}(M)[[\hbar]]=C^{\infty}(M)\otimes_{\mathbb{R}}\...
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Higgs mechanism from a deformation quantization point of view
Is it possible to describe the Higgs mechanism from a deformation quantization point of view? How would one do it? Are there aspects of the Higgs mechanism and Higgs particle which one may see clearer ...
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Ambidexterity and Quantization
Here the nlab says about Hopkins-Lurie's ambidexterity paper:
"The discussion in the article is apparently motivated as part of what it takes to make precise the discussion of quantization in ...
user62675
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1-dimensional pure gauge theory
I am learning TQFT from compact Lie groups by Freed, Hopkins, Lurie,
and Teleman: https://arxiv.org/abs/0905.0731 , and got stuck very hard
even in the first section ($n = 1$), which was "trivial but ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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