Precanonical Quantum Gravity: Quantization Without the Space–Time Decomposition

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 an equal footing, is presented. A generally covariant “multitemporal” 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 self-consistent with the underlying quantum dynamics. This ensures the independence of the theory from an arbitrarily fixed background. The prospects and unsolved problems of precanonical quantization of gravity are outlined.