Quantum group intertwiner space from quantum curved tetrahedron
Han M, Hsiao CH, Pan Q (2024)
Publication Type: Journal article
Publication year: 2024
Journal
Book Volume: 41
Article Number: 165008
Journal Issue: 16
Abstract
In this paper, we develop a quantum theory of homogeneously curved tetrahedron geometry, by applying the combinatorial quantization to the phase space of tetrahedron shapes defined in Haggard et al (2016 Ann. Henri Poincaré 17 2001-48). Our method is based on the relation between this phase space and the moduli space of SU(2) flat connections on a 4-punctured sphere. The quantization results in the physical Hilbert space as the solution of the quantum closure constraint, which quantizes the classical closure condition M 4 M 3 M 2 M 1 = 1 , M ν ∈ SU ( 2 ) , for the homogeneously curved tetrahedron. The quantum group U q ( su ( 2 ) ) emerges as the gauge symmetry of a quantum tetrahedron. The physical Hilbert space of the quantum tetrahedron coincides with the Hilbert space of 4-valent intertwiners of U q ( su ( 2 ) ) . In addition, we define the area operators quantizing the face areas of the tetrahedron and compute the spectrum. The resulting spectrum is consistent with the usual Loop-Quantum-Gravity area spectrum in the large spin regime but is different for small spins. This work closely relates to 3+1 dimensional Loop Quantum Gravity in presence of cosmological constant and provides a justification for the emergence of quantum group in the theory.
Involved external institutions
United States (USA) (US)How to cite
APA:
Han, M., Hsiao, C.H., & Pan, Q. (2024). Quantum group intertwiner space from quantum curved tetrahedron. Classical and Quantum Gravity, 41(16). https://doi.org/10.1088/1361-6382/ad5f71
MLA:
Han, Muxin, Chen Hung Hsiao, and Qiaoyin Pan. "Quantum group intertwiner space from quantum curved tetrahedron." Classical and Quantum Gravity 41.16 (2024).
BibTeX: Download