Monadic Dynamics

TL;DR

This paper introduces a monadic framework for formalising dynamics in physical theories, linking time, change, and system evolution through category theory, and demonstrating its equivalence to path space approaches.

Contribution

It develops a novel monadic approach to dynamics, characterising time and change categorically, and connects it with existing path space methods in physics.

Findings

Dynamics define a canonical notion of time respecting compositional structure.

Framework applies to quantum theory, classical mechanics, and network theory.

Establishes equivalence with path space approaches to dynamics.

Abstract

We develop a monadic framework formalising an operational notion of dynamics, seen as the setting and evolution of initial value problems, in general physical theories. We identify in the Eilenberg-Moore category the natural environment for dynamical systems and characterise Cauchy surfaces abstractly as automorphisms in the Kleisli category. Our main results formally vindicates the Aristotelian view that time and change are defined by one another. We show that dynamics which respect the compositional structure of physical systems always define a canonical notion of time, and give the conditions under which they can be faithfully seen as actions of time on physical systems. Finally, we construct state spaces and path spaces, and show that our framework to be equivalent to the path space approaches to dynamics. The monadic standpoint is thus as strong as the established paradigms, but the shift from histories to dynamics helps shed new light on the nature of time in physics. In the appendix we present some additional structures of wide applicability, introduce prop- agators and draft applications to quantum theory, classical mechanics and network theory.