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Similar to Ashtekar's original formulation, in precanonical quantization of gravity the vielbeins are represented as differential operators with respect to the spin connection coefficients. However, precanonical quantization uses a different generalization of Hamiltonian formalism to field theory, the so-called De Donder--Weyl theory, which does not require a space-time decomposition and treats the space-time variables on the equal footing. Correspondingly, the dynamics of quantum gravity is encoded in the wave function on the space of spin-connection coefficients and space-time variables. Based on the analysis of constraints within the De Donder--Weyl Hamiltonian formulation of Einstein-Palatini vielbein gravity and quantization of generalized Dirac brackets defined on differential forms we derive the analogue of the Schroedinger equation for precanonical wave function. The definition of the scalar product in this formulation lead to the operator valued measure on the space of spin connection coefficients. We argue that the finiteness of the corresponding norm of precanonical wave function leads to the quantum avoidance of curvature singularities. We also point out that the theory leads to the picture of quantum gravity as a non-Gaussian random field of spin connection, which can be viewed as a "spin-connection foam" picture of quantum gravity suggested by precanonical quantization. We will also discuss the validity of the Ehrenfest theorem within the precanonical quantization approach. References: [1] I.V. Kanatchikov, On precanonical quantization of gravity, arXiv:1407.3101 [2] I.V. Kanatchikov, De Donder-Weyl Hamiltonian formulation and precanonical quantization of vielbein gravity, arXiv:1302.2610 [3] I.V. Kanatchikov, Ehrenfest Theorem in Precanonical Quantization arXiv:1501.00480 [4] I.V. Kanatchikov, On the Ehrenfest Theorem in quantum general relativity, work in progress

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We outline the main conceptual and technical ingredients of the precanonical quantization of fields, which is based on the De Donder-Weyl (polysymplectic) Hamiltonian formulation in classical field theory. This approach does not distinguish space and time variables and it describes fields in terms of the wave functions on the finite dimensional space of field variables and space-time variables. We show that the standard functional Schroedinger representation in QFT can be viewed as a limiting case of precanonical quantization. We also show how the classical field equations are obtained from precanonical quantization as equations on expectation values of corresponding operators obtained from precanonical quantization (Ehrenfest theorem). We consider the Ehrenfest theorem both in flat and curved space-time. We also show how the approach works when applied to the problem of quantization of gravity and how it is consistent with the Einstein equations obtained as the equations for the expectation values calculated according to the principles of precanonical quantization. References: [1] I.V. Kanatchikov, Ehrenfest Theorem in Precanonical Quantization arXiv:1501.00480 [2] I.V. Kanatchikov, On precanonical quantization of gravity, arXiv:1407.3101 [3] I.V. Kanatchikov, De Donder-Weyl Hamiltonian formulation and precanonical quantization of vielbein gravity, arXiv:1302.2610 [4] I.V. Kanatchikov, On the Ehrenfest Theorem in quantum general relativity. work in progress [5] I.V. Kanatchikov, Precanonical Quantization and the Schroedinger Wave Functional Revisited, Adv. Theor. Math. Phys. 18 (2014) 1249-1265.

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The approach to quantization of General Relativity, which is based on the De Donder-Weyl (DW) Hamiltonian formulation rather than the canonical Hamiltonian formalism (a relation and differences between both will be briefly discussed) leads to the description of quantum geometry in terms of the Clifford algebra valued transition amplitudes between different values of spin-connections at difference points of space-time. We derive the covariant Schroedinger equation for those amplitudes. Surprisingly, all the physical constants (G,h,c, \Lambda and the constant of "elementary volume" introduced when the classical differential forms are quantized according to the Poisson-Gerstenhaber brackets of the above mentioned DW formalism) are absorbed in the single dimensionless parameter, which depends on the ordering of operators in the expression of the DW Hamiltonian operator. Thus, when the effective cosmological constant is set to be zero, there is no Planck constant in the remaining part of the covariant Schroedinger equation for quantum general relativity, that points to the purely mathematical meaning of the quantum geometry it describes. We discuss a possible interpretation of the result in terms of the parallel transport of Clifford valued wave functions on the spin-connection bundle. References: [1] I.V. Kanatchikov, Precanonical Quantization and the Schroedinger Wave Functional Revisited, Adv. Theor. Math. Phys. 18 (2014) 1249-1265. [2] I.V. Kanatchikov, On precanonical quantization of gravity, arXiv:1407.3101

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We will outline precanonical quantization of vielbein gravity [1] based on the De Donder-Weyl version of the Hamiltonian formalism [2] and apply it to simple cosmological models. One of the outcomes is that precanonical quantization of gravity predicts a non-Gaussian distribution of random cosmological spin-connection field of quantum origin whose interaction with matter may have potentially observable imprints on the large scale distribution of matter in the universe. References. [1] I.V. Kanatchikov, On precanonical quantization of gravity, arXiv:1407.3101 [2] I.V. Kanatchikov, De Donder-Weyl Hamiltonian formulation and precanonical quantization of vielbein gravity, arXiv:1302.2610

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