The quantization of the Hamiltonian for a scalar field is performed in the framework of Quantum Reduced Loop Gravity. We outline how the regularization can be performed by using the analogous tools adopted in full Loop Quantum Gravity and the matrix elements of the resulting operator between basis states are analytic coefficients. These achievements open the way for a consistent analysis of the Quantum Gravity corrections to the classical dynamics of gravity in the presence of a scalar field in a cosmological setting. *

This article sheds new light on the problem of cosmological reduction in Loop Quantum Gravity. We critically analyze Quantum Reduced Loop Gravity -an attempt to extract the cosmological sector of the full theory. We reconsider the reduction procedure applied to the states of the kinematical Hilbert space, developing a comparative analysis with previous efforts in the literature. We show that the constraints of the model were formerly instantiated in an inconsistent fashion, leading to an overconstrained dynamics and an ill-defined Hilbert space. We then scrutinize alternative implementations of symmetry-reduction. While remaining unaffected by the shortcomings encountered in Quantum Reduced Loop Gravity, these latter procedures bridge the gap between the full theory and former endeavors in Loop Quantum Cosmology. * bilski@zjut.edu.cn † marciano@fudan.edu.cn i The well known non-trivial structure of the vacuum was emphasized for gauge theories, providing a possible framework to account for quantum theories of gravity, by inspections of their instantonic solutions, as argued in Ref.[2].

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