The kinematical Hilbert space of Loop Quantum Gravity from BF theories

TL;DR

This paper derives the kinematical Hilbert space of Loop Quantum Gravity from BF theories by imposing Hamiltonian constraints and analyzing the structure of states under Lorentz transformations.

Contribution

It shows how to obtain LQG's Hilbert space from BF theories, clarifying the role of constraints and symmetries in the construction.

Findings

Projection to Ashtekar-Barbero representations is effective

Reduction to SU(2) invariant intertwiners is detailed

LQG states' properties under Lorentz transformations are discussed

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.