Histories and observables in covariant field theory

Abstract

Motivated by DeWitt's viewpoint of covariant field theory, we define a general notion of 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 (standard) 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 the 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. 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, that 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.Comment: 49 page

Comparison theorems for Kan, faintly universal and strongly universal derived functors

“…Indeed, the new geometry in particular provides a convenient method of encoding total derivatives and leads to a covariant description of the classical BV complex which arises as a specific case of general constructions. Further evidence for this standpoint appears in [16,17].…”

A new approach to model categorical homotopy fiber sequences

“…Moreover, we say that a homotopy fiber sequence (resp., a long homotopy fiber sequence) of M is objectwise fibrant if its four vertices are fibrant objects of M (resp., if all homotopy fiber sequences (17) are objectwise fibrant). We also say that a morphism of homotopy fiber sequences (resp., of long homotopy fiber sequences) of M is an objectwise weak equivalence if its four component morphisms are weak equivalences of M (resp., if all morphisms of homotopy fiber sequences (19) are objectwise weak equivalences).…”

Homotopical Poisson reduction of gauge theories

“…For the reader's convenience, we recall shortly the formulation summed-up in [Pau10] and fully described in [Pau11] of general variational problems, and its grounding on functorial geometry. This is certainly useful, but not strictly necessary to understand our final results.…”

“…We refer to the article [Pau10] for detailed acknowledgements and more references on this work, that is its direct continuation (with improvements and simplifications). Special thanks are due to Jim Stasheff for his detailed comments of loc.…”

“…We refer to Schapira's survey [Sch10] for an efficient introduction to the general methods of linear algebraic analysis on varieties and to Beilinson and Drinfeld's book [BD04] for the non-linear setting. This section expands on the article [Pau10], giving more details and explanations. In particular, we use systematically the functor of point approach to spaces of fields, as explained in Section 1 (see also [Pau11] for a complete treatment) without further comments.…”