Continuous formulation of the Loop Quantum Gravity phase space
Laurent Freidel, Marc Geiller, Jonathan Ziprick
TL;DR
The paper shows that the discrete holonomy-flux phase space on a graph $\Gamma$ is isomorphic to the symplectic reduction $\mathcal{P}_{\Gamma,\Gamma^*}$ of the continuous GR phase space by a flatness constraint outside the dual graph $\Gamma^*$, thereby identifying $P_\Gamma$ with a particular gauge-equivalence class of continuous geometries. It reveals that flux variables encode both intrinsic and extrinsic geometry and label gauge-equivalence classes, explaining their noncommutativity and the non-uniqueness of the geometry they define. By introducing two covariant gauge choices—the singular (LQG) gauge and the flat (spin-foam) gauge—the work reconciles the loop-geometry and spin-foam pictures, linking twisted geometries to Regge-type piecewise-flat geometries. It also develops cylindrical consistency for extending discrete observables to the full continuum, and demonstrates that, in the flat gauge, the reduced phase space is the cotangent bundle $T^*\mathcal{M}_{\Gamma^*}$ of the moduli space of flat connections, suggesting a flat-connection viewpoint for quantization. Overall, the results provide a classical bridge between LQG and spin-foam formalisms, offer a precise interpretation of holonomy-flux data, and outline a path toward a classical dynamics formulation of gravity in this truncated, graph-based setting.
Abstract
In this paper, we study the discrete classical phase space of loop gravity, which is expressed in terms of the holonomy-flux variables, and show how it is related to the continuous phase space of general relativity. In particular, we prove an isomorphism between the loop gravity discrete phase space and the symplectic reduction of the continuous phase space with respect to a flatness constraint. This gives for the first time a precise relationship between the continuum and holonomy-flux variables. Our construction shows that the fluxes depend on the three-geometry, but also explicitly on the connection, explaining their non commutativity. It also clearly shows that the flux variables do not label a unique geometry, but rather a class of gauge-equivalent geometries. This allows us to resolve the tension between the loop gravity geometrical interpretation in terms of singular geometry, and the spin foam interpretation in terms of piecewise flat geometry, since we establish that both geometries belong to the same equivalence class. This finally gives us a clear understanding of the relationship between the piecewise flat spin foam geometries and Regge geometries, which are only piecewise-linear flat: While Regge geometry corresponds to metrics whose curvature is concentrated around straight edges, the loop gravity geometry correspond to metrics whose curvature is concentrated around not necessarily straight edges.