Questions tagged [geometric-quantization]

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Inner product on global sections of positive line bundle

Let $\Sigma = S^2$ be thought of as a Riemann surface, and let $L$ be a Hermitian line bundle on $\Sigma$ with curvature $2$-form $-2 \pi i \Omega \in \Omega^2(\Sigma, \mathbb{R})$. Then $L$ is a ...
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Classification of square roots of line bundles and metalinear/metaplectic structures

Reading some books and articles about geometric quantization I got confused about the classification of square roots of complex line bundles over a manifold. Consider the group of isomorphism classes ...
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On the construction of principal $S^1$-bundles with prescribed characteristic form

I am trying to understand an argument made by Kobayashi (https://projecteuclid.org/download/pdf_1/euclid.tmj/1178245006, page 35) in his construction of a principal $S^1$-bundle with connection $1$-...
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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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polarization of symplectic manifold

The geometric quantization can be considered as an approach the formalize the way of associating a quantum theory corresponding to a given classical theory. Suppose we start with a sympetic manifold $(...
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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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Does symplectic morphism after geometric quantization induce Hilbert spaces morphism?

Let us have two symplectic manifolds $(M, \,\omega)$ and $(N, \,\omega')$ and morphism between them: $$ \varphi \ :\ M \to N.$$ Then we geometrically quantize these systems: we add a prequantum line ...
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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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On the existence and classification of prequantization spaces over a closed symplectic manifold

Let $(M,\omega)$ be a closed symplectic manifold. If the cohomology class $[\omega]$ is rational, that is if it lies in the image of the natural homomorphism $H^2(M,\mathbb{Z}) \to H^2(M,\mathbb{R})$, ...
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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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Non-diffeomorphic smooth structures on the quotient of a manifold by an integrable distribution

In geometric quantization, one of the important ingredients is an integrable distribution $D$ (let's say real) on some manifold $M$ (symplectic, but this is not important). The resulting object is $M/...
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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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A representation similar to coadjoint representation?

In a project of quantization, I come up with a finite dimensional representation of $so(d)$ that I wish to find some decent references for it. I guess it could have been studied thoroughly in ...
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Second-order term of the Fedosov quantised product

In Fedosov's version of quantisation of functions on a symplectic manifold, the product is given in terms of a symplectic connection. I have looked through Fedosov's book in deformation quantisation, ...
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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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138 views

Complex structure on the set of prequantization line bundles

For geometric quantization, the set of equivalence classes of prequantization line bundles of a quantizable symplectic manifold $(M, ω)$ is parametrized by $H^1(M, S^1)$ which represents the ...
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pre-quantization of Jet bundle

We know that the notion of Jet bundle $J^kM×\mathbb{R}$, is generalization of cotangent bundle. What is the prequantization of $J^kM×\mathbb{R}$?
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3 votes
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a question about geometric quantization background

I don't understand why for geometric description of a regular system, we take always the classical phase space as a symplectic manifold?
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