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Dapor-Liegener formalism of loop quantum cosmology for Bianchi I spacetimes

We discuss the quantization of vacuum Bianchi I spacetimes in the modified formalism of loop quantum cosmology recently proposed by Dapor and Liegener. This modification is based on a regularization procedure where both the Euclidean and Lorentzian terms of the Hamiltonian are treated independently. Whereas the Euclidean part has already been dealt with in the literature for Bianchi I spacetimes, the Lorentzian one has not yet been represented quantum mechanically. After a brief review of the quantum kinematics and the quantization of the Euclidean sector, we represent the Lorentzian part of the Hamiltonian constraint by an operator according to the factor ordering rules of the Martin-Benito--Mena Marugan--Olmedo prescription. We study the general properties of this quantum operator and the superselection rules derived therefrom, resulting in an action similar to that of the Euclidean operator except that the orientation of the densitized triad is not preserved, a fact which leads to a generic enlargement of the superselection sectors. We conclude with an explanation of the mechanism that prevents this enlargement in the isotropic case and a comment on the effect of alternative prescriptions for the implementation of the improved dynamics.