Canonical Lagrangian Dynamics and General Relativity

TL;DR

This paper introduces a covariant Lagrangian-based approach to constrained dynamics in general relativity, emphasizing the Lagrangian phase space and demonstrating closure of constraints without gauge fixing, with implications for quantum gravity.

Contribution

It presents a novel covariant formalism using a Lagrangian one-form for constrained systems, applicable to generally covariant theories like Einstein-Cartan gravity.

Findings

Constraint algebra closes without gauge fixing.

Applicable to degenerate symplectic forms in gravity.

Brief discussion on implications for quantum theory.

Abstract

Building towards a more covariant approach to canonical classical and quantum gravity we outline an approach to constrained dynamics that de-emphasizes the role of the Hamiltonian phase space and highlights the role of the Lagrangian phase space. We identify a "Lagrangian one-form" to replace the standard symplectic one-form, which we use to construct the canonical constraints and an associated constraint algebra. The method is particularly useful for generally covariant systems and systems with a degenerate canonical symplectic form, such as Einstein Cartan gravity, to which we apply the method explicitly. We find that one can demonstrate the closure of the constraints without gauge fixing the Lorentz group or introducing primary constraints on the phase space variables. Finally, using geometric quantization techniques, we briefly discuss implications of the formalism for the quantum theory.