Representations of the holonomy algebras of gravity and nonAbelian gauge theories
Open accessAbstract
Holonomy algebras arise naturally in the classical description of Yang-Mills fields and gravity, and it has been suggested, at a heuristic level, that they may also play an important role in a nonperturbative treatment of the quantum theory. The aim of this paper is to provide a mathematical basis for this proposal. The quantum holonomy algebra is constructed, and, in the case of real connections, given the structure of a certain C*-algebra. A proper representation theory is then provided using the Gel'fand spectral theory. A corollary of these general results is a precise formulation of the 'loop transform' proposed by Rovelli and Smolin (1990). Several explicit representations of the holonomy algebra are constructed. The general theory developed here implies that the domain space of quantum states can always be taken to be the space of maximal ideals of the C*-algebra. The structure of this space is investigated and it is shown how observables labelled by 'strips' arise naturally.
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The quantum configuration space A/G is a certain completion of the classical configuration space A/G.
“The quantum configuration space A / G is a certain completion of A / G [4,5].”
From Quantum theory of geometry: I. Area operators (1997) · cites this paper as [4]
Section 2.2 (p. 15) identifies the quantum-state space Δ as compact and says the classical configuration space A/G is naturally embedded in it. But the Proof (p. 26) explicitly asks whether A/G is dense in Δ, so the paper does not establish that Δ is a completion of A/G in the usual sense. (LLM, conf=0.87)
Experimental — accuracy not yet measuredOver the past three years, the non-perturbative canonical approach to quantum gravity based on connections has been put on a firm mathematical footing through the development of a new functional calculus on the space of gauge equivalent connections.
“Over the past three years, this approach has been put on a firm mathematical footing through the development of a new functional calculus on the space of gauge equivalent connections [4-11].”
From Quantum theory of geometry: I. Area operators (1997) · cites this paper as [4]
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Experimental — accuracy not yet measuredThe new functional calculus does not use any background fields and is therefore well-suited to a fully nonperturbative treatment.
“This calculus does not use any background fields (such as a metric) and is therefore well-suited to a fully nonperturbative treatment.”
From Quantum theory of geometry: I. Area operators (1997) · cites this paper as [4]
The Motivation (pp. 2–3) presents the formalism as a nonperturbative approach to quantum gravity, and the Conclusion (p. 39) describes its loop-based holonomy operators. Our copy does not explicitly say that the calculus uses no background fields, or that this absence is why it is suited to a fully nonperturbative treatment. (LLM, conf=0.78)
Experimental — accuracy not yet measuredA, G and A/G are all compact spaces in their natural Gel'fand topologies.
“In contrast with the usual Minkowskian situation, however, A , G and A / G are all compact spaces in their natural (Gel'fand) topologies [4-8].”
From Quantum theory of geometry: I. Area operators (1997) · cites this paper as [4]
The Outline (p. 5) says the Gel'fand spectrum Δ is compact, not that A, G, or A/G are compact. The Proof (p. 26) treats the topology induced on A/G by Δ as an open question; its statement that A/G is noncompact in typical metric topologies (p. 27) does not settle the claim about Gel'fand topologies. (LLM, conf=0.76)
Experimental — accuracy not yet measuredThe resulting quantum algebra is exactly the one used in the earlier papers [1-14].
“Furthermore, the resulting quantum algebra is precisely the one used in [1][2][3][4][5][6][7][8][9][10][11][12][13][14].”
From Quantum theory of geometry: III. Non-commutativity of Riemannian structures (1998) · cites this paper as [3]
The Hilbert space used to define the operators is built from gauge invariant cylindrical functions on the space of connections A.
“The Hilbert space in which we will define the operators is constructed from the gauge invariant cylindrical functions on A [6,7,12,19,20].”
From Volume and quantizations (1997) · cites this paper as [6]
The Higgs field framework in the diffeomorphism invariant context is developed by analogy between path holonomy and point holonomy, translating the gauge-field framework to arrive at the Higgs field Hilbert space.
“We develop the framework for the Higgs field, in the diffeomorphism invariant context, by pushing the analogy between a usual path holonomy and a 'point holonomy' constructed from a Higgs Field (see below) and can therefore translate step by step the framework [4, 5, 6, 7, 8] developed for gauge fields to arrive at the Higgs field Hilbert space.”
From Kinematical Hilbert spaces for fermionic and Higgs quantum field theories (1998) · cites this paper as [4]
In [3], diffeomorphism invariant theories of connections for a compact gauge group were canonically quantized, and the representation of the canonical commutation relations chosen is generated by the Abelian C⋆ algebra W of Wilson loop observables via the GNS construction.
“In [3] such theories were canonically quantized and the representation of the canonical commutation relations that was chosen is generated by the Abelian C ⋆ algebra W of Wilson loop observables via the GNS construction [4].”
From Kinematical Hilbert spaces for fermionic and Higgs quantum field theories (1998) · cites this paper as [4]
If one does not work with half densities, the gravitational connection becomes complex valued, so the gravitational gauge group would become SL(2,C), which is non-compact and would make the techniques developed in [4,5,6,7,8,3] inapplicable.
“If one does not work with half densities then it turns out that a) The gravitational connection becomes complex valued. Thus, the gravitational gauge group would become become SL (2 , C) which is non-compact and would make the arsenal of techniques developed in [4, 5, 6, 7, 8, 3] inapplicable.”
From Kinematical Hilbert spaces for fermionic and Higgs quantum field theories (1998) · cites this paper as [4]
If π := i√det(q) η is defined as momentum conjugate to η, integration by parts makes the symplectic structure equal to ∫ d^3x [π ̇η + 1/4 πη E^i_a ̇E^a_i], implying the real-valued gravitational connection A^i_a acquires a purely imaginary correction term -1/4 πψ E^i_a = -i/4 ̄ηη e^i_a, which leads to difficulties because techniques break down for complex valued connections.
“Assume now that we would define π := i √ det( q ) η to be the momentum conjugate to η . Upon an integration by parts we see that the symplectic structure becomes up to a total time derivative equal to ∫ Σ d 3 x [ π ˙ η + 1 4 πηE i a ˙ E a i ] where E i a denotes the inverse of E a i . But since E a i is the momentum conjugate to A i a this implies that the real-valued gravitational connection A i”
From Kinematical Hilbert spaces for fermionic and Higgs quantum field theories (1998) · cites this paper as [4]
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- Authors
- Ashtekar, A., & Isham, C. J.
- Affiliations
- Syracuse University (US)
- ORCID
- Ashtekar, A
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 1992-06-01
- Volume
- 9
- Issue
- 6
- Pages
- 1433-1467
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W2144505517
- Cited by
- 310
- References
- 24
- Citations by year
- 2026: 1 · 2025: 1 · 2024: 7 · 2023: 5 · 2022: 2 · 2021: 8 · 2020: 6 · 2019: 5 · 2018: 10 · 2017: 11 · 2016: 9 · 2015: 6 · 2014: 4 · 2013: 14 · 2012: 8
- FWCI
- 8.98
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Syracuse University
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Mathematics › Mathematical Physics › Advanced Operator Algebra Research
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Keywords
- holonomy algebras, C*-algebras, representation theory, non-perturbative quantum gravity, Yang-Mills theory, Yang-Mills fields, maximal ideals, quantum states, loop quantum gravity
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
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