In this paper we derive the uncertainty principle for the Loop Quantum Cosmology homogeneous and isotropic FLWR model with the holonomy-flux algebra. The uncertainty principle is between the variables c, with the meaning of connection and µ having the meaning of the physical cell volume to the power 2/3, i.e v 2/3 or a plaquette area. Since both µ and c are not operators, but rather the random variables, the Robertson uncertainty principle derivation that works for hermitian operators, can not be used. Instead we use the Wigner-Moyal-Groenewold phase space formalism. The Wigner-Moyal-Groenewold formalism was originally applied to the Heisenberg algebra of the Quantum Mechanics. One can derive from it both the canonical and path integral QM as well as the uncertainty principle. In this paper we apply it to the holonomy-flux algebra in case of the homogeneous and isotropic space. Another result is the expression for the Wigner function on the space of the cylindrical wave functions defined on R b in c variables rather than in dual space µ variables.

The expansion of a square integrable function on SL(2, C) into the sum of the principal series matrix coefficients with the specially selected representation parameters was recently used in the Loop Quantum Gravity [10], [11]. In this paper we prove that the sum used originally in the Loop Quantum Gravity: ∞ j=0 |m|≤j |n|≤j

The main result of this paper is the 4-dimensional supermetric version of the Wheeler-DeWitt equation, that uses only one time variable for the both roles -as internal time and for the ADM split, as Hamiltonian evolution parameter. We study the ADM split with respect to the scalar massless field serving as internal time. The 4-dimensional hyper-surfaces φ=const span the 5-dimensional space with the scalar field being the fifth coordinate. As a result we obtain the analog of the Wheeler-DeWitt equation for the 4-dimensional supermetric. We compare the ADM action with the non-compactified Kaluza-Klein action for the same physical space and obtain the equation for the extrinsic curvature and the scalar massless field.

We consider general relativity as a limit case of the scalar–tensor theory with Barbero–Immirzi (BI) field when the field tends to a constant. We use Shapiro time delay experimental value of [Formula: see text] provided by the Cassini spacecraft to find the present BI parameter value.