The conformal factor in the parameter-free construction of spin-gauge variables for gravity

The newly found conformal decomposition in canonical general relativity is applied to drastically simplify the recently formulated parameter-free construction of spin-gauge variables for gravity. The resulting framework preserves many of the main structures of the existing canonical framework for loop quantum gravity related to the spin network and Thiemann’s regularization. However, the Barbero-Immirzi parameter is now converted into the conformal factor as a canonical variable. It behaves like a scalar field but is somehow non-dynamical since the effective Hamiltonian constraint does not depend on its momentum. The essential steps of the mathematical derivation of this parameter-free framework for the spin-gauge variables of gravity are spelled out. The implications for the loop quantum gravity programme are briefly discussed.