Loop Quantization of Shape Dynamics

Loop Quantum Gravity (LQG) is a promising approach to quantum gravity, in particular because it is based on a rigorous quantization of the kinematics of gravity. A difficult and still open problem in the LQG program is the construction of the physical Hilbert space for pure quantum gravity. This is due to the complicated nature of the Hamilton constraints. The Shape Dynamics description of General Relativity (GR) replaces the Hamilton constraints with spatial Weyl constraints, so the problem of finding the physical Hilbert space reduces to the problem of quantizing the Weyl constraints. Unfortunately, it turns out that a loop quantization of Weyl constraints is far from trivial despite their intuitive physical interpretation. A tentative quantization proposal and interpretation proposal is given in this contribution.