Lattice gravity and cosmology

The background-independent quantum gravity is the necessary framework to construct generally relativistic quantum field theory. By assuming the ADM decomposition of spacetime, it is possible to define the metric-independent Fock space for this formulation. This space, known as spin network, is invariant under the SU(2) symmetry and the spatial diffeomorphisms transformations. It is the Fock space for the model called loop quantum gravity in which the canonical operators are the quantized holonomies of the Ashtekar connection and the fluxes of densitized dreibein. I will present an improved construction of the lattice gravity and its gauge-fixed cosmological reduction based on the same lattice variables. The approach is based on the geometric expansion of holonomies into power series up to the quadratic order terms in the regularization parameter. As a result, a more accurate procedure is obtained in which the symmetry of holonomies assigned to links is directly reflected in the related distribution of connections. The application of the procedure to the Hamiltonian constraint regularization provides its lattice analog, the domain of which has a natural structure of elementary cells sum. In consequence, the related scalar constraint operator, which spectrum is independent of intertwiners, can be defined. The cosmological phase space reduction of lattice gravity requires rigorous application of gauge-fixing conditions that reduce the SU(2) symmetry and the spatial diffeomorphisms invariance. The internal symmetry is fixed to the Abelian case and the diffeomorphisms invariance is simultaneously reduced to spatial translations. The obtained Hamiltonian constraint is finite (without any cut-off introduction) and exact (without the holonomy expansion around short links). Furthermore, it has the expected 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.