Special generalized densities and propagators: a geometric account

TL;DR

This paper provides a geometric framework for understanding special generalized densities, propagators, and solutions of field equations, linking classical distributions with quantum field propagators in flat spacetime.

Contribution

It offers a systematic geometric description of various densities and propagators, and introduces a novel geometric perspective on free quantum fields and their propagators.

Findings

Geometric description of Leray and principal value densities

Representation of propagators via graded commutators in quantum fields

Connection between classical distributions and quantum propagators

Abstract

Starting from a short review of spaces of generalized sections of vector bundles, we give a concise systematic description, in precise geometric terms, of Leray densities, principal value densities, propagators and elementary solutions of field equations in flat spacetime. We then sketch a partly original geometric presentation of free quantum fields and show how propagators arise from their graded commutators in the boson and fermion cases.