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Coherent State Transforms for Spaces of Connections

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Abhay Ashtekar, Jerzy Lewandowski, Donald Marolf, José Mourão, Thomas Thiemann

General Relativity and Quantum Cosmology · Submitted 5 Dec 1994

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The authors' main claim The resulting ``coherent state transform'' provides a holomorphic representation of the holonomy $C^\star$ algebra of real gauge fields.

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arXiv IDgr-qc/9412014 (version 1)
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CitationAbhay Ashtekar, Donald Marolf, Jerzy lewandowski, Jos\'e Mour\~ao, Thomas Thiemann. "Coherent State Transforms for Spaces of Connections." arXiv:gr-qc/9412014 (1994). https://pith.science/paper/MCORADJ7
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What the abstract itself states, as probabilities: claims first 0.09 · states limitation 0.16 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.82 · mentions code or data availability 0.01.

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arXiv listing abstract v1. Results and the main claim are underlined; .

The Segal-Bargmann transform plays an important role in quantum theories of linear fields. Recently, Hall obtained a non-linear analog of this transform for quantum mechanics on Lie groups. Given a compact, connected Lie group $G$ with its normalized Haar measure $\mu_H$, the Hall transform is an isometric isomorphism from $L^2(G, \mu_H)$ to ${\cal H}(G^{\Co})\cap L^2(G^{\Co}, \nu)$, where $G^{\Co}$ the complexification of $G$, ${\cal H}(G^{\Co})$ the space of holomorphic functions on $G^{\Co}$, and $\nu$ an appropriate heat-kernel measure on $G^{\Co}$. We extend the Hall transform to the infinite dimensional context of non-Abelian gauge theories by replacing the Lie group $G$ by (a certain extension of) the space ${\cal A}/{\cal G}$ of connections modulo gauge transformations. The resulting ``coherent state transform'' provides a holomorphic representation of the holonomy $C^\star$ algebra of real gauge fields. This representation is expected to play a key role in a non-perturbative, canonical approach to quantum gravity in 4-dimensions.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 The Segal-Bargmann transform plays an important role in quantum theories of linear fields. background 1.00 0.04 0.02 asserted 1.00
s1 Recently, Hall obtained a non-linear analog of this transform for quantum mechanics on Lie groups. background 1.00 0.03 0.03 asserted 1.00
s2 Given a compact, connected Lie group $G$ with its normalized Haar measure $\mu_H$, the Hall transform is an isometric isomorphism from $L^2(G, \mu_H)$ to ${\cal H}(G^{\Co})\cap L^2(G^{\Co}, \nu)$, where $G^{\Co}$ the complexification of $G$, ${\cal H}(G^{\Co})$ the space of holomorphic functions on $G^{\Co}$, and $\nu$ an appropriate heat-kernel measure on $G^{\Co}$. background 1.00 0.07 0.06 asserted 1.00
s3 We extend the Hall transform to the infinite dimensional context of non-Abelian gauge theories by replacing the Lie group $G$ by (a certain extension of) the space ${\cal A}/{\cal G}$ of connections modulo gauge transformations. contribution 0.71 0.92 0.26 asserted 1.00
s4 The resulting ``coherent state transform'' provides a holomorphic representation of the holonomy $C^\star$ algebra of real gauge fields. result 0.88 0.89 0.55 asserted 1.00
s5 This representation is expected to play a key role in a non-perturbative, canonical approach to quantum gravity in 4-dimensions. interpretation 0.96 0.10 0.28 hedged possibility 0.90

What kind of paper the classifier reports this is: theoretical result 0.57 · evidence from proof or derivation 0.96.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

The Segal-Bargmann transform plays an important role in quantum theories of linear fields. Recently, Hall obtained a non-linear analog of this transform for quantum mechanics on Lie groups. Given a compact, connected Lie group $G$ with its normalized Haar measure $\mu_H$, the Hall transform is an isometric isomorphism from $L^2(G, \mu_H)$ to ${\cal H}(G^{\Co})\cap L^2(G^{\Co}, \nu)$, where $G^{\Co}$ the complexification of $G$, ${\cal H}(G^{\Co})$ the space of holomorphic functions on $G^{\Co}$, and $\nu$ an appropriate heat-kernel measure on $G^{\Co}$. We extend the Hall transform to the infinite dimensional context of non-Abelian gauge theories by replacing the Lie group $G$ by (a certain extension of) the space ${\cal A}/{\cal G}$ of connections modulo gauge transformations. The resulting ``coherent state transform'' provides a holomorphic representation of the holonomy $C^\star$ algebra of real gauge fields. This representation is expected to play a key role in a non-perturbative, canonical approach to quantum gravity in 4-dimensions.

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15 arXiv papers cite this in 26 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 16 sentences without a confident label

Uses its method or tool 2 papers
  • Applying the heat-kernel technique to several copies of $SU(2)$ allows to build coherent spin-network states for LQG [ref] .

    Coherent states for FLRW space-times in loop quantum gravity · Coherent spin-networks · 0.87

  • Using this map, the vertex amplitude in the holonomy representation can be written as [equation] The integrals on $Spin(4)$ impose the Euclidean invariance of the vertex, and the integrand is given by a product of `face' terms $P$ given by the following distribution over $L^2(SU(2))⊗L^2(Spin(4))≃L^2(SU(2)^3)$: [equation] Expression ([ref]) can be easily put in the holomorphic representation using the Segal-Bargamann transform [ref] : [equation] where [equation] Notice that the last expression can be seen as a distribution on $ℋL^2(SL(2,ℂ))⊗L^2(Spin(4))$.

    Spinfoams in the holomorphic representation · III. 4d Euclidean vertex amplitude in the holomorphic representation · 0.83

Cites it as background 8 papers

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