The kinematical Hilbert space of loop quantum gravity from BF theories

Abstract

Abstract. In this work, it is demonstrated how the kinematical Hilbert space of Loop Quantum Gravity (LQG) can be inferred from the configuration space of BF theories via the imposition of the Hamiltonian constraints. In particular, it is outlined how the projection to the representations associated with Ashtekar-Barbero connections provides the correct procedure to implement second-class constraints and the corresponding nontrivial induced symplectic structure. Then, the reduction to SU(2) invariant intertwiners is analyzed and the properties of LQG states under Lorentz transformations is discussed.

Perturbative BF theory

“…The gauge structure group of the BF theory naturally suggests possible connections with (2+1) gravity [6,7,8,9,10,11,12,13,14,15,16,17,18,19], and applications of the BF formalism in the context of loop quantum gravity have also been studied [20,21,22,23,24,25,26]. Generalizations of the BF models in higher dimensions have been considered [27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42].…”

2-connections, a Lattice Point of View

“…This is reminiscent of what happens in some gauge theories where symmetries between symmetries appear due to the presence of second class constraints. It is worth noting that in BF theory, which is known to have this kind of metasymmetries, and which has close relations to quantum gravity [4], the gauge fields defining the theory form a 2-connection [6]. All this tends to prove that higher category theory is a fertile ground where theories can be enriched, by systematically extending basic structures underlying them using the two main tools of higher category theory: internalization and enrichment [3].…”

Physical states in Quantum Einstein-Cartan Gravity

“…τ k,ρ IJ being Lorentz generators in the irreducible representation (k, ρ). Here, we refer to Naimark classification of Lorentz irreducible representations [22] (a brief introduction is also given in [23]), in which k is a integer or semi-integer non-negative number and ρ ∈ C. Each representation is constructed as a tower of SU (2) irreducible representations with spin numbers j from k up to infinity, i.e.…”