Characteristics, Conal Geometry and Causality in Locally Covariant Field Theory

The goal of this work, motivated by the desire to understand causality in classical and quantum gravity, is an in depth investigation of causality in classical field theories with quasilinear equations of motion, of which General Relativity is a prominent example. Several modern geometric tools (jet bundle formulation of partial differential equations (PDEs), the theory of symmetric hyperbolic PDE systems, covariant constructions of symplectic and Poisson structures) and applies them to the construction of the phase space and the algebra of observables of quasilinear classical field theories. This construction is shown to be diffeomorphism covariant (using auxiliary background fields if necessary) using categorical tools in a strong parallel with the locally covariant field theory (LCFT) formulation of quantum field theory (QFT) on curved spacetimes. In this context, generalized versions of LCFT axioms become theorems of classical field theory, which includes a generalized Causality property. Considering deformation quantization as the connection to QFT, a plausible conjecture is made about the Causal structure of quantum gravity. In the process, conal manifolds are identified as the generalization of the causal structure of Lorentzian geometry to quasilinear PDEs. Several important concepts and results are generalized from Lorentzian to conal geometry. Also, the proof of compatibility of the Peierls formula for Poisson brackets and the covariant phase space symplectic structure for hyperbolic systems is generalized to now encompass systems with constraints and gauge invariance.