Quantum spin dynamics (QSD): IV. Euclidean quantum gravity as a model to test Lorentzian quantum gravity
Open accessAbstract
The quantization of Lorentzian or Euclidean 2 + 1 gravity by canonical methods is a well studied problem. However, the constraints of 2 + 1 gravity are those of a topological field theory and therefore resemble very little those of the corresponding Lorentzian 3 + 1 constraints. In this paper we canonically quantize Euclidean 2 + 1 gravity for an arbitrary genus of the spacelike hypersurface with new, classically equivalent constraints that maximally probe the Lorentzian 3 + 1 situation. We choose the signature to be Euclidean because this implies that the gauge group is, as in the 3 + 1 case, SU (2) rather than . We employ, and carry out to full completion, the new quantization method introduced in preceding papers of this series which resulted in a finite 3 + 1 Lorentzian quantum field theory for gravity. The space of solutions to all constraints turns out to be much larger than that obtained by traditional approaches, however, it is fully included. Thus, by a suitable restriction of the solution space, we can recover all former results which gives confidence in the new quantization methods. The meaning of the remaining `spurious solutions' is discussed.
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A similar problem occurs in the loop quantum gravity entropy calculation because of the Immirzi parameter.
“A similar problem arises in the loop quantum gravity calculation [10] because of the so-called Immirzi parameter [29].”
From Symmetry reduction for quantized diffeomorphism-invariant theories of connections (2000) · cites this paper as [29]
Physical states of BF theory solve the scalar constraint.
“The fact that physical states of BF theory solve the scalar constraint is well known [353], since roughly F(A) = 0 implies EEF(A) = 0.”
From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [353]
The anomaly that [p^I(B),p^J(B)] is not expressible in terms of a p^I(B') is due to not properly smearing the φ's.
“The reason for this 'anomaly' is due to the fact that we did not properly smear the φ 's as explained in [26] :”
From Kinematical Hilbert spaces for fermionic and Higgs quantum field theories (1998) · cites this paper as [26]
The correspondence {.,.}→1/(iħ)[.,.] holds everywhere only if p's and φ's are smeared in three directions.
“we can expect the correspondence { ., . } → 1 i ¯ h [ ., . ] to hold everywhere only if p 's and φ 's are smeared in three directions.”
From Kinematical Hilbert spaces for fermionic and Higgs quantum field theories (1998) · cites this paper as [26]
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Cites
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Ashtekar, A., & Lewandowski, J. (1997) · arXiv (Cornell University) · doi:10.48550/arxiv.gr-qc/9711031 · Cited by 11 · see citing works
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Research works printed but not linked: 8
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The Spin-Foam Approach to Quantum Gravity
Perez, A. (2013) · Living Reviews in Relativity · doi:10.12942/lrr-2013-3 · Cited by 677 · see citing works
Thiemann, T. (1998) · Classical and Quantum Gravity · doi:10.1088/0264-9381/15/5/012 · Cited by 416 · see citing works
Self-citationKinematical Hilbert spaces for fermionic and Higgs quantum field theories
Thiemann, T. (1998) · Classical and Quantum Gravity · doi:10.1088/0264-9381/15/6/006 · Cited by 184 · see citing works
Self-citationThree-dimensional loop quantum gravity: physical scalar product and spin-foam models
Noui, K., & Perez, A. (2005) · Classical and Quantum Gravity · doi:10.1088/0264-9381/22/9/017 · Cited by 181 · see citing works
Symmetry reduction for quantized diffeomorphism-invariant theories of connections
Bojowald, M., & Kastrup, H. A. (2000) · Classical and Quantum Gravity · doi:10.1088/0264-9381/17/15/311 · Cited by 168 · see citing works
Quantum spin dynamics: VIII. The master constraint
Thiemann, T. (2006) · Classical and Quantum Gravity · doi:10.1088/0264-9381/23/7/003 · Cited by 99 · see citing works
Self-citationAre the spectra of geometrical operators in loop quantum gravity really discrete?
Dittrich, B., & Thiemann, T. (2009) · Journal of Mathematical Physics · doi:10.1063/1.3054277 · Cited by 96 · see citing works
Self-citationPerez, A., & Pranzetti, D. (2010) · Classical and Quantum Gravity · doi:10.1088/0264-9381/27/14/145009 · Cited by 53 · see citing works
Eleven-dimensional supergravity as a constrained topological field theory
Ling, Y., & Smolin, L. (2001) · Nuclear Physics B · doi:10.1016/s0550-3213(01)00063-3 · Cited by 29 · see citing works
Finite spin-foam-based theory of three- and four-dimensional quantum gravity
Gambini, R., & Pullin, J. (2002) · Physical Review D · doi:10.1103/physrevd.66.024020 · Cited by 10 · see citing works
Hyperinvariant spin network states — an AdS/CFT model from first principles
Otto, F., Mansuroglu, R., Schuch, N., Gühne, O., & Sahlmann, H. (2026) · Journal of High Energy Physics · doi:10.1007/jhep06(2026)145 · Cited by 1 · see citing works
- Authors
- Thiemann, T.
- Affiliations
- Harvard University (US)
- Corresponding author
- Thiemann, T
- ORCID
- Thiemann, T
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 1998-05-01
- Volume
- 15
- Issue
- 5
- Pages
- 1249-1280
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W1479721997
- Cited by
- 128
- References
- 4
- Citations by year
- 2026: 1 · 2024: 5 · 2023: 4 · 2021: 4 · 2020: 2 · 2019: 1 · 2018: 2 · 2016: 3 · 2015: 4 · 2014: 1 · 2013: 8 · 2012: 4
- FWCI
- 8.26
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Funders
- U.S. Department of Energy (DE‐FG02‐)
- Harvard University
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Keywords
- Lorentzian quantum gravity, canonical quantization, topological field theory, SU(2) gauge group, spurious solutions, constraint surface, quantum spin dynamics
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