Lattice classical cosmology

TL;DR

This paper develops a lattice-based reduction of loop quantum gravity's phase space to a cosmological model, resulting in a finite, well-defined Hamiltonian suitable for quantum cosmological evolution analysis.

Contribution

It introduces a novel gauge fixing and reduction method that simplifies loop quantum gravity to a cosmological setting with a finite Hamiltonian without cut-offs.

Findings

Hamiltonian is finite and expressed as a sum over cuboidal cells.

Reduced constraints are globally satisfied and vanish identically.

The model enables quantum cosmological evolution via transition amplitudes.

Abstract

This article presents the lattice-smeared gravity phase space reduction defined by the cosmological gauge-fixing conditions. These conditions are specified to reduce the SU(2) symmetry and the spatial diffeomorphism invariance of the loop quantum gravity's Fock space, known as the spin network. The internal symmetry is fixed to the Abelian case and the diffeomorphism invariance is simultaneously reduced to spatial translations. The unification of the results of the related gauge fixing conditions leads to the gauge generators correlation. Consequently, these conditions become solvable by constant variables; hence the reduced constraints become globally satisfied and vanish identically. By rigorously satisfying the reduced gauge symmetries, the resulting cosmological model is precisely the limit of the gravitational theory expressed in terms of holonomies and fluxes. Moreover, the obtained Hamiltonian constraint is finite (without any cut-off introduction) and as rigorous as an approximation of a Lie group by its representation. Furthermore, it has the form of the sum over elementary cuboidal cells. Finally, the simple structure of its homogeneities and anisotropies should allow to describe the quantum cosmological evolution of the Universe in terms of transition amplitudes, instead of using perturbative approximations.