Algebraic Structure of Classical Field Theory: Kinematics and Linearized Dynamics for Real Scalar Fields

Abstract

We describe the elements of a novel structural approach to classical field theory, inspired by recent developments in perturbative algebraic quantum field theory. This approach is local and focuses mainly on the observables over field configurations, given by certain spaces of functionals which are studied here in depth. The analysis of such functionals is characterized by a combination of geometric, analytic and algebraic elements which (1) make our approach closer to quantum field theory, (2) allow for a rigorous analytic refinement of many computational formulae from the functional formulation of classical field theory and (3) provide a new pathway towards understanding dynamics. Particular attention will be paid to aspects related to nonlinear hyperbolic partial differential equations and their linearizations.

68 pages, no figures. Version 7: very minor corrections, DOI added. Version published (Online First) in Communications in Mathematical Physics, journal reference will be added when available

References in corpus (7)

• Globally hyperbolic spacetimes can be defined as "causal" instead of "strongly causal"
• Batalin-Vilkovisky formalism in the functional approach to classical field theory
• Removal of violations of the Master Ward Identity in perturbative QFT
• Fermionic fields in the functional approach to classical field theory
• Global hyperbolicity is stable in the interval topology
• Global analysis of quasilinear wave equations on asymptotically Kerr-de Sitter spaces
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Cited by in corpus (7)

• Covariant canonical formulations of classical field theories
• $C^{\ast }$-algebraic approach to interacting quantum field theory: Inclusion of Fermi fields
• The algebra of Wick polynomials of a scalar field on a Riemannian manifold
• Lorentzian 2d CFT from the pAQFT perspective
• Superluminal local operations in quantum field theory: A ping-pong ball test
• Field Theory via Higher Geometry I: Smooth Sets of Fields
• Møller maps for Dirac fields in external backgrounds