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Abstract

A nonpertubative approach to quantum gravity using precanonical field quantization originating from the covariant De Donder-Weyl Hamiltonian formulation which treats space and time variables on equal footing is presented. A generally covariant "multi-temporal" generalized Schrödinger equation on the finite dimensional space of metric and space-time variables is obtained. An important ingredient of the formulation is the "bootstrap condition" which introduces a classical space-time geometry as an approximate concept emerging as the quantum average in a self-consistent with the underlying quantum dynamics manner. An independence of the theory from an arbitrarily fixed background is ensured in this way. The prospects and unsolved problems of precanonical quantization of gravity are outlined. *

Converting classical theories to quantum theories by solutions of the Hamilton-Jacobi equation

“…The De Donder-Weyl equation for gravity of Einstein-Hilbert action has been derived in[39]. The application of De Donder-Weyl theories to canonical quantum gravity has been suggested in[40,41].…”

New multisymplectic approach to the Metric-Affine (Einstein-Palatini) action for gravity

“…Finally, Im ψ L ⊂ S f . Indeed, equations (32) and (33) for ψ H imply that all the points in Im ψ L verify the constraints c µν and m ρσ,µ . The constraints r α βγ,ν and i ρσ,µν are also satisfied because they arise from the tangency condition on the semiholonomic constraints (see Section 3.2.2) and the integrability condition respectively; and then they are satisfied for holonomic sections which are solutions to the Lagrangian field equations.…”

“…Thus, in [3,8,9,10,21,25,26,27,35,36,43,44,46] general aspects of the theory are studied in this way, meanwhile other papers are devoted to consider several particular problems. For instance, in [7,24,41,42] the reduction and projectability of higher-order theories (such as the Hilbert-Einstein model) is analized, in [47] the vielbein models of General Relativity are studied using the multisymplectic formulation and in [32,33,34] interesting contributions to the problem of the precanonical quantization of gravity are done.…”

A New Antisymmetric Bilinear Map for Type-I Gauge Theories

“…The Peierls bracket [6,10,11] has been applied in the modern literature on the manifestly covariant [12,13,14,15,16,17,18,19,20,21] approach to quantization of gauge theories, including gravity, and some authors have even gone so far as to suggest that the Peierls bracket can be used to actually define the functional integral itself [20].…”