De Donder-Weyl Hamiltonian formulation and precanonical quantization of vielbein gravity

TL;DR

This paper develops a covariant Hamiltonian and precanonical quantization framework for vielbein gravity, avoiding space-time splitting, and introduces a quantum description of space-time geometry using a Dirac-like Schrödinger equation.

Contribution

It presents the first covariant De Donder-Weyl Hamiltonian formulation and precanonical quantization of vielbein gravity, including a novel treatment of constraints and quantum geometric interpretation.

Findings

Representation of vielbeins as differential operators

Derivation of a Dirac-like precanonical Schrödinger equation

Discussion of quantum space-time geometry and singularity avoidance

Abstract

The De Donder-Weyl (DW) covariant Hamiltonian formulation of Palatini first-order Lagrangian of vielbein (tetrad) gravity and its precanonical quantization are presented. No splitting into the space and time is required in this formulation. Our recent generalization of Dirac brackets is used to treat the second class primary constraints appearing in the DW Hamiltonian formulation and to find the fundamental brackets. Quantization of the latter yields the representation of vielbeins as differential operators with respect to the spin connection coefficients, and the Dirac-like precanonical Schr\"odinger equation on the space of spin connection coefficients and space-time variables. The transition amplitudes on this space describe the quantum geometry of space-time. We also discuss the Hilbert space of the theory, the invariant measure on the spin connection coefficients, and point to the possible quantum singularity avoidance built in in the natural choice of the boundary conditions of the wave functions on the space of spin connection coefficients.