Spacetime Covariant Form of Ashtekar's Constraints

TL;DR

This paper develops a covariant, spacetime formulation of Ashtekar's constraints in general relativity using multisymplectic techniques, providing a boundary-aware, Hamiltonian analysis compatible with non-perturbative quantization.

Contribution

It introduces a fully covariant, multisymplectic approach to derive Ashtekar's constraints, connecting Hamiltonian and covariant formalisms in gravity theories.

Findings

Constraint equations are derived with boundary terms considered.

Hamiltonian constraint is linear in multimomenta.

Covariant analysis is valid on any spacelike hypersurface.

Abstract

The Lagrangian formulation of classical field theories and in particular general relativity leads to a coordinate-free, fully covariant analysis of these constrained systems. This paper applies multisymplectic techniques to obtain the analysis of Palatini and self-dual gravity theories as constrained systems, which have been studied so far in the Hamiltonian formalism. The constraint equations are derived while paying attention to boundary terms, and the Hamiltonian constraint turns out to be linear in the multimomenta. The equivalence with Ashtekar's formalism is also established. The whole constraint analysis, however, remains covariant in that the multimomentum map is evaluated on {\it any} spacelike hypersurface. This study is motivated by the non-perturbative quantization program of general relativity.