On the "spin connection foam" picture of quantum gravity from precanonical quantization

Precanonical quantization is based on a generalization of the Hamiltonian formalism to field theory, the so-called De Donder-Weyl (DW) theory, which does not require a spacetime splitting and treats the space-time variables on an equal footing. Quantum dynamics is described by a precanonical wave function on the finite dimensional space of field coordinates and space-time coordinates, which satisfies a partial derivative precanonical Schr\"odinger equation. The standard QFT in the functional Schr\"odinger representation can be derived from the precanonical quantization in a limiting case. An analysis of the constraints within the DW Hamiltonian formulation of the Einstein-Palatini vielbein formulation of GR and quantization of the generalized Dirac brackets defined on differential forms lead to the covariant precanonical Schr\"odinger equation for quantum gravity. The resulting dynamics of quantum gravity is described by the wave function or transition amplitudes on the total space of the bundle of spin connections over space-time. Thus, precanonical quantization leads to the "spin connection foam" picture of quantum geometry represented by a generally non-Gaussian random field of spin connection coefficients, whose probability distribution is given by the precanonical wave function. The normalizability of precanonical wave functions is argued to lead to the quantum-gravitational avoidance of curvature singularities. Possible connections with LQG are briefly discussed.