Quantum-reduced loop gravity: Cosmology

We introduce a new framework for loop quantum gravity: mimicking the spin foam quantization procedure we propose to study the symmetric sectors of the theory imposing the reduction weakly on the full kinematical Hilbert space of the canonical theory. As a first application of quantum-reduced loop gravity we study the inhomogeneous extension of the Bianchi I model. The emerging quantum cosmological model represents a simplified arena on which the complete canonical quantization program can be tested. The achievements of this analysis could elucidate the relationship between loop quantum cosmology and the full theory.

Figure 1

An elementary tetrahedron $\Delta \in T$ constructed by adapting it to a graph $\Gamma$ which underlies a cylindrical function.

Figure 2

The representation of a vertex in reduced quantization: the quantum numbers $n_1$, $n_2$, $n_3$ are conserved along the directions $i=1$, 2, 3, respectively.

Figure 3

The action of the operator associated with the curvature changes $U(1{)}_i$ quantum numbers such that it maps the state out of the gauge-invariant Hilbert space (we did not draw the edges along the third direction).

Figure 4

Six-valent vertex.