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

TL;DR

This paper introduces a new local structural approach to classical field theory focusing on observables as functionals, integrating geometric, analytic, and algebraic methods to better understand dynamics and linearizations, inspired by quantum field theory developments.

Contribution

It presents a novel local framework for classical field theory centered on functionals of field configurations, bridging to quantum theory and refining computational methods.

Findings

A rigorous analytic refinement of classical field theory formulae.

A new pathway for understanding dynamics via linearizations.

Enhanced analysis of nonlinear hyperbolic PDEs.

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.