Canonical Lagrangian dynamics and general relativity

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 it explicitly. We find that one can compute the constraint algebra and demonstrate the closure of the constraints without gauge fixing the Lorentz group or introducing primary constraints on the phase space variables. Ultimately our aim is towards a more covariant approach for canonical quantum gravity, and we discuss a possible route to quantization. Applying these techniques and using methods from geometric quantization, we find a new representation of the pre-quantum operator corresponding to the Hamiltonian constraint, which in contrast to the standard representation, is a kinematical operator that simply generates timelike diffeomorphisms on functionals of the Lagrangian phase space. This opens the possibility for a kinematical spacetime-diffeomorphism invariant Hilbert space.