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Reduced Phase Space Quantization for Loop Quantum Gravity and Algebraic Quantum Gravity

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In this talk we propose a Reduced Phase Space Quantization approach to Loop Quantum Gravity. The idea is to combine the relational formalism introduced by Rovelli in the extended form developed by Dittrich and the Brown-Kuchar-Mechanism. The relational formalism can be used to construct gauge invariant observables for constrained systems such as General Relativity, while the Brown-Kuchar-Mechanism is a particular application of the relational formalism in which pressureless dust is taken as the clock of the system. By combining these two we obtain a framework in which the constraints of General Relativity deparametrize such that the algebra of observables has a very simple structure and furthermore we obtain a so called physical Hamiltonian generating the evolution of those observables. The quantization of the reduced phase space and the physical Hamiltonian can be obtained by using standard LQG techniques and gives a direct access to the physical Hilbert space, which is much harder to achieve in the standard Dirac quantization.
Additionally we will analyze the quantization in the Algebraic Quantum Gravity context and discuss the differences that occur. Finally we will present recent results where this framework has been applied to cosmological perturbation theory.