Schreiber
Classical field theory via Cohesive homotopy types

An article that has become part of section 1.2 of differential cohomology in a cohesive topos:

on classical mechanics formulated via prequantized Lagrangian correspondences in higher differential geometry, using Higher geometric prequantum theory formulated in terms of differential cohomology in a cohesive topos.

This serves as handout for the talk

and as motivating example for the discussion at

This is a companion of

An account more comprehensive in some ways is at

Contents

Abstract

We discuss here how the refined formulation of classical mechanics/classical field theory (Hamiltonian mechanics, Lagrangian mechanics) that systematically takes all global effects properly into account – such as notably non-perturbative effects, classical anomalies and the definition of and the descent to reduced phase spaces – naturally is a formulation in “higher differential geometry”. This is the context where smooth manifolds are allowed to be generalized first to smooth orbifolds and then further to Lie groupoids, then further to smooth groupoids and smooth moduli stacks and finally to smooth higher groupoids forming a higher topos for higher differential geometry. We introduce and explain this higher differential geometry as we go along. At the same time as we go along, we explain how the classical concepts of classical mechanics all follow naturally from just the abstract theory of “correspondences in higher slice toposes”.

This text is meant to serve the triple purpose of being an exposition of classical mechanics for homotopy type theorists, being an exposition of geometric homotopy theory for physicists, and finally to serve as the canonical example for and seamlessley lead over to the formulation of a local prequantum field theory which supports a localized quantization to local quantum field theory in the sense of the cobordism hypothesis.

Last revised on October 11, 2016 at 03:07:09. See the history of this page for a list of all contributions to it.