Algebraic quantum gravity (AQG): IV. Reduced phase space quantization of loop quantum gravity

2007/11/01 by Kristina Giesel, K. Giesel, T. Thiemann +1 · 149 citations
Mathematics · Physics and Astronomy · #Algorithm #Black Holes and Theoretical Physics #Canonical quantization #Canonical quantum gravity #Classical mechanics #Cosmology and Gravitation Theories #General relativity #Hamiltonian (control theory) #Hilbert space #Linear-quadratic-Gaussian control #Loop quantum gravity #Mathematical physics #Mathematics #Noncommutative and Quantum Gravity Theories #Observable #Optimal control #Phase space #Physics #Quantization (signal processing) #Quantum #Quantum gravity #Quantum mechanics #Theoretical physics #gr-qc #hep-th

paper · pdf · doi:10.1088/0264-9381/27/17/175009

published in Classical and Quantum Gravity 27(17), 175009 (IOP Publishing) · 31 pages, no figures

arxiv created 2007/11/01 · openalex publication_date 2010/07/19 · arxiv updated 2014/11/18 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/06

Abstract

We perform a canonical, reduced phase space quantization of general relativity by loop quantum gravity (LQG) methods. The explicit construction of the reduced phase space is made possible by the combination of (a) the Brown–Kuchař mechanism in the presence of pressure-free dust fields which allows to deparametrize the theory and (b) Rovelli's relational formalism in the extended version developed by Dittrich to construct the algebra of gauge-invariant observables. Since the resulting algebra of observables is very simple, one can quantize it using the methods of LQG. Basically, the kinematical Hilbert space of non-reduced LQG now becomes a physical Hilbert space and the kinematical results of LQG such as discreteness of spectra of geometrical operators now have physical meaning. The constraints have disappeared; however, the dynamics of the observables is driven by a physical Hamiltonian which is related to the Hamiltonian of the standard model (without dust) and which we quantize in this paper.

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