2012/11/08 by Igor Khavkine, Khavkine, Igor
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Black Holes and Theoretical Physics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Noncommutative and Quantum Gravity Theories
paper · pdf · doi:10.48550/arxiv.1211.1914
openalex publication_date 2012/11/08 · openalex created_date 2022/09/01 · openalex updated_date 2026/07/28
The goal of this work, motivated by the desire to understand causality in\nclassical and quantum gravity, is an in depth investigation of causality in\nclassical field theories with quasilinear equations of motion, of which General\nRelativity is a prominent example. Several modern geometric tools (jet bundle\nformulation of partial differential equations (PDEs), the theory of symmetric\nhyperbolic PDE systems, covariant constructions of symplectic and Poisson\nstructures) and applies them to the construction of the phase space and the\nalgebra of observables of quasilinear classical field theories. This\nconstruction is shown to be diffeomorphism covariant (using auxiliary\nbackground fields if necessary) using categorical tools in a strong parallel\nwith the locally covariant field theory (LCFT) formulation of quantum field\ntheory (QFT) on curved spacetimes. In this context, generalized versions of\nLCFT axioms become theorems of classical field theory, which includes a\ngeneralized Causality property. Considering deformation quantization as the\nconnection to QFT, a plausible conjecture is made about the Causal structure of\nquantum gravity. In the process, conal manifolds are identified as the\ngeneralization of the causal structure of Lorentzian geometry to quasilinear\nPDEs. Several important concepts and results are generalized from Lorentzian to\nconal geometry. Also, the proof of compatibility of the Peierls formula for\nPoisson brackets and the covariant phase space symplectic structure for\nhyperbolic systems is generalized to now encompass systems with constraints and\ngauge invariance.\n