Phase space for gravity with boundaries
Alberto S. Cattaneo
TL;DR
This work presents a geometric formulation—the Kijowski–Tulczyjew (KT) method—of constructing the reduced phase space for Lagrangian field theories on manifolds with boundary, by treating solutions as isotropic relations between boundary data endowed with a closed 2-form. It develops a staged reduction: starting from the bulk action, extracting boundary data via preboundary spaces, and then performing coisotropic and split-Lagrangian reductions to obtain a true symplectic boundary phase space; the evolution is encoded as relations rather than flows. The Palatini–Cartan (coframe) formulation of gravity in four dimensions serves as the central, nontrivial example, where the fields are a coframe $e$ and a connection $\omega$, the action $S_M[e,\omega]$ yields EL equations $e F_\omega + \frac{\Lambda}{6} e^3 = 0$ and $e d_\omega e = 0$, and a boundary analysis produces a coisotropic Cauchy data space with gauge and diffeomorphism constraints generated by explicit Hamiltonians. The framework also clarifies how boundary conditions and gluing work via evolution relations and their compositions, and it connects to BV–BFV formalisms for manifolds with boundary, providing a robust route toward quantization. Overall, the paper offers a flexible, geometry-driven route to obtain consistent reduced phase spaces for gravity and other field theories with boundaries.
Abstract
This explanatory note, based on the geometrical method by Kijovski and Tulczyjew, describes the construction of the reduced phase space of Lagrangian field theories, i.e., the correct space of initial conditions with its symplectic structure. Several examples and, in particular, the case of four-dimensional gravity in the coframe formalism (Palatini--Cartan theory) are analyzed.