Questions tagged [quantization]
Questions on various methods and aspects of quantization
85 questions
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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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2
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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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answer
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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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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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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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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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2
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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 ...
user39066
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0
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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 ...
user20510
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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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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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5
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2
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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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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
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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 ...
user21574
3
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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 ...
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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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13
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1
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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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0
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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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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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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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4
votes
2
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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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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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6
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4
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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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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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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.
- 279
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1
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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 $HH^\...
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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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1
vote
1
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290
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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 ...
- 1,025
2
votes
1
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479
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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 ...
- 197
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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 ...
- 1,025
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2
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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 ...
- 1,025
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2
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1k
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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\...
- 18.7k
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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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1
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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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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 ...
- 27.9k
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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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