A universe of processes and some of its guises

TL;DR

This paper explores the foundational role of symmetric monoidal categories in quantum theory, highlighting how categorical quantum mechanics reproduces key aspects of quantum physics and connects to various mathematical and physical theories.

Contribution

It provides a conceptual foundation for categorical quantum mechanics without requiring prior category theory knowledge, linking it to broader developments in physics and mathematics.

Findings

Reproduces a large fragment of quantum theory using categorical methods

Provides new insights into quantum foundations and information

Connects categorical quantum mechanics to diverse mathematical areas

Abstract

Our starting point is a particular `canvas' aimed to `draw' theories of physics, which has symmetric monoidal categories as its mathematical backbone. In this paper we consider the conceptual foundations for this canvas, and how these can then be converted into mathematical structure. With very little structural effort (i.e. in very abstract terms) and in a very short time span the categorical quantum mechanics (CQM) research program has reproduced a surprisingly large fragment of quantum theory. It also provides new insights both in quantum foundations and in quantum information, and has even resulted in automated reasoning software called `quantomatic' which exploits the deductive power of CQM. In this paper we complement the available material by not requiring prior knowledge of category theory, and by pointing at connections to previous and current developments in the foundations of physics. This research program is also in close synergy with developments elsewhere, for example in representation theory, quantum algebra, knot theory, topological quantum field theory and several other areas.