Histories and observables in covariant field theory

Motivated by DeWitt’s viewpoint of covariant field theory, we define a general notion of a non-local classical observable that applies to many physical Lagrangian systems (with bosonic and fermionic variables), by using methods that are now standard in algebraic geometry. We review the methods of local functional calculus, as they are presented by Beilinson and Drinfeld, and relate them to our construction. We partially explain the relation of these with Vinogradov’s secondary calculus. The methods present here are all necessary to understand mathematically properly, and with simple notions, the full renormalization of the standard model, based on functional integral methods. Our approach is close in spirit to non-perturbative methods since we work with actual functions on spaces of fields, and not only formal power series. This article can be seen as an introduction to well-grounded classical physical mathematics, and as a good starting point to study quantum physical mathematics, which make frequent use of non-local functionals, like for example in the computation of Wilson’s effective action. We finish by describing briefly a coordinate-free approach to the classical Batalin–Vilkovisky formalism for general gauge theories, in the language of homotopical geometry.