Symmetry-reduced loop quantum gravity: plane waves, flat space
and the Hamiltonian constraint

J 2025

Symmetry-reduced loop quantum gravity: plane waves, flat space and the Hamiltonian constraint

HINTERLEITNER, Franz

Basic information

Original name

Symmetry-reduced loop quantum gravity: plane waves, flat space and the Hamiltonian constraint

Name in Czech

Symetricky redukovaná smyčková kvantová gravitace: rovinné vlny, plochý prostor a hamiltonovská vazba

Authors

HINTERLEITNER, Franz (40 Austria, guarantor, belonging to the institution)

Edition

Classical and Quantum Gravity, UK, IOP Publishing Ltd, 2025, 1361-6382

Other information

Language

English

Type of outcome

Article in a journal

Field of Study

10300 1.3 Physical sciences

Country of publisher

United Kingdom of Great Britain and Northern Ireland

Confidentiality degree

is not subject to a state or trade secret

References:

URL

Impact factor

Impact factor: 3.700 in 2024

Organization unit

Faculty of Science

DOI

https://doi.org/10.1088/1361-6382/adc5d4

UT WoS

001468553500001

EID Scopus

2-s2.0-105002782657

Keywords in English

loop quantum gravity; gravitational wave dispersion; Hamiltonian constraint

Abstract

In the original language

Loop quantum gravity (LQG) methods are applied to a symmetry-reduced model with homogeneity in two dimensions, derived from a Gowdy model. The conditions for the propagation of unidirectional plane gravitational waves at exactly the speed of light are set up in the form of two null Killing equations in terms of Ashtekar variables and imposed as operators on the quantum states of the system. Owing to symmetry reduction the gauge group of the system reduces formally from SU(2) to U(1). Under the assumption of equal spacing of the holonomy eigenvalues, the solutions are not normalizable in the sense of the usual inner product on U(1). Taking over the inner product from the genuine gauge group SU(2) of LQG renders the obtained states normalizable, nevertheless fluctuations of geometrical quantities remain divergent. In consequence, the solutions of the (non-commuting) Killing conditions have to be renormalized. Two kinds of renormalization are presented. The combination of the occurrence of non-commuting Killing operators and the necessity of renormalization indicates fluctuations of the propagation speed, i. e. dispersion of gravitational waves. Finally the same methods are applied to the Hamiltonian constraint with the same result concerning normalizability. After renormalization the constraint is not exactly satisfied any more, which suggests the presence of some kind of interacting matter.

Cite

HINTERLEITNER, Franz. Symmetry-reduced loop quantum gravity: plane waves, flat space and the Hamiltonian constraint. Classical and Quantum Gravity. UK: IOP Publishing Ltd, 2025, vol. 42, No 8, p. 1-35. ISSN 1361-6382. Available from: https://doi.org/10.1088/1361-6382/adc5d4.

@article{2510477,
   author = {Hinterleitner, Franz},
   article_location = {UK},
   article_number = {8},
   doi = {https://doi.org/10.1088/1361-6382/adc5d4},
   keywords = {loop quantum gravity; gravitational wave dispersion; Hamiltonian constraint},
   language = {eng},
   issn = {1361-6382},
   journal = {Classical and Quantum Gravity},
   title = {Symmetry-reduced loop quantum gravity: plane waves, flat space and the Hamiltonian constraint},
   url = {https://doi.org/10.48550/arXiv.2403.11864},
   volume = {42},
   year = {2025}
}

TY  - JOUR
ID  - 2510477
AU  - Hinterleitner, Franz
PY  - 2025
TI  - Symmetry-reduced loop quantum gravity: plane waves, flat space and the Hamiltonian constraint
JF  - Classical and Quantum Gravity
VL  - 42
IS  - 8
SP  - 1-35
EP  - 1-35
PB  - IOP Publishing Ltd
SN  - 13616382
KW  - loop quantum gravity
KW  - gravitational wave dispersion
KW  - Hamiltonian constraint
UR  - https://doi.org/10.48550/arXiv.2403.11864
N2  - Loop quantum gravity (LQG) methods are applied to a symmetry-reduced model with homogeneity in two dimensions, derived from a Gowdy model. The conditions for the propagation of unidirectional plane gravitational waves at exactly the speed of light are set up in the form of two null Killing equations in terms of Ashtekar variables and imposed as operators on the quantum states of the system. Owing to symmetry reduction the gauge group of the system reduces formally from SU(2) to U(1). Under the assumption of equal spacing of the holonomy eigenvalues, the solutions are not normalizable in the sense of the usual inner product on U(1). Taking over the inner product from the genuine gauge group SU(2) of LQG renders the obtained states normalizable, nevertheless fluctuations of geometrical quantities remain divergent. In consequence, the solutions of the (non-commuting) Killing conditions have to be renormalized. Two kinds of renormalization are presented. The combination of the occurrence of non-commuting Killing operators and the necessity of renormalization indicates fluctuations of the propagation speed, i. e. dispersion of gravitational waves. Finally the same methods are applied to the Hamiltonian constraint with the same result concerning normalizability. After renormalization the constraint is not exactly satisfied any more, which suggests the presence of some kind of interacting matter.
ER  -

HINTERLEITNER, Franz. Symmetry-reduced loop quantum gravity: plane waves, flat space and the Hamiltonian constraint. \textit{Classical and Quantum Gravity}. UK: IOP Publishing Ltd, 2025, vol.~42, No~8, p.~1-35. ISSN~1361-6382. Available from: https://doi.org/10.1088/1361-6382/adc5d4.

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