  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.
