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Cited by 84Noncommutative and Quantum Gravity TheoriesRelativity and Gravitational TheoryBlack Holes and Theoretical Physics

Lorentz covariance of loop quantum gravity

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Rovelli, Carlo, Speziale, Simone·2011·Physical Review D
DOI 10.1103/physrevd.83.104029
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Abstract

The kinematics of loop gravity can be given a manifestly Lorentz-covariant formulation: the conventional $SU(2)$-spin-network Hilbert space can be mapped to a space $\mathcal{K}$ of $SL(2,\mathbb{C})$ functions, where Lorentz covariance is manifest. $\mathcal{K}$ can be described in terms of a certain subset of the projected spin networks studied by Livine, Alexandrov and Dupuis. It is formed by $SL(2,\mathbb{C})$ functions completely determined by their restriction on $SU(2)$. These are square-integrable in the $SU(2)$ scalar product, but not in the $SL(2,\mathbb{C})$ one. Thus, $SU(2)$-spin-network states can be represented by Lorentz-covariant $SL(2,\mathbb{C})$ functions, as two-component photons can be described in the Lorentz-covariant Gupta-Bleuler formalism. As shown by Wolfgang Wieland in a related paper, this manifestly Lorentz-covariant formulation can also be directly obtained from canonical quantization. We show that the spinfoam dynamics of loop quantum gravity is locally $SL(2,\mathbb{C})$-invariant in the bulk, and yields states that are precisely in $\mathcal{K}$ on the boundary. This clarifies how the $SL(2,\mathbb{C})$ spinfoam formalism yields an $SU(2)$ theory on the boundary. These structures define a tidy Lorentz-covariant formalism for loop gravity.

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  • A criterion equivalent to the Master-constraint criterion can be written in a manifestly Spin(4)-invariant way.

    “Nevertheless, as shown in [333], one can also write down a criterion (equivalent to the Master-constraint one) in a manifestly Spin(4) invariant way.”

    From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [333]

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Authors
Rovelli, C., & Speziale, S.
Venue
Physical Review D
Publisher
American Physical Society (APS)
Published
2011-05-17
Volume
83
Issue
10
Pages
104029
ISSN
  • 0556-2821 (ISSN-L)
  • 1550-7998
  • 1550-2368
DOI
10.1103/physrevd.83.104029
OpenAlex
W2162020092
Cited by
84
References
30
Citations by year
2026: 1 · 2024: 2 · 2023: 1 · 2022: 3 · 2021: 2 · 2020: 2 · 2019: 3 · 2018: 4 · 2017: 5 · 2016: 6 · 2015: 3 · 2014: 10 · 2013: 14 · 2012: 17
FWCI
12.73
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • HAL (submittedVersion, other-oa)
  • arXiv (Cornell 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
loop quantum gravity, Lorentz covariance, canonical quantization

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