"Fields" in classical and quantum field theories

TL;DR

This paper explores the foundational concept of fields in classical and quantum theories using a categorical, groupoid-based approach inspired by Schwinger's quantum mechanics, offering a new interpretation of physical fields.

Contribution

It introduces a novel framework interpreting physical fields as functors between groupoids, providing a categorical perspective on the notion of fields in both classical and quantum contexts.

Findings

Fields can be modeled as functors between groupoids.

The approach offers a new interpretation of space-time and physical states.

Illustrative examples clarify the categorical structures involved.

Abstract

The challenges posed by the development of field theories, both classical and quantum, force us to question their most basic and foundational ideas like the role and origin of space-time, the meaning of physical states, etc. Among them the notion of ``field'' itself is notoriously difficult to address. These notes aim to analyze such notion from the perspective offered by the groupoid description of quantum mechanics inspired by Schwinger's picture of quantum mechanics. Then, a natural interpretation of the notion of physical fields as functors among appropriate groupoids will emerge. The domain of a field in this new picture is a groupoid that describes ``test particles'', and its codomain is a groupoid that describes the intrinsic nature of the system being probed. Such a space of functors carries some natural structures, which are best described in a categorical language. Some illustrative examples will be presented that could help clarify the various abstract notions discussed in the text.