A Diagrammatic Approach to Quantum Dynamics

TL;DR

This paper introduces a diagrammatic, categorical framework for quantum dynamics, linking algebraic structures with physical concepts like clocks and time quantization, and deriving key quantum results from a process-theoretic perspective.

Contribution

It develops a novel diagrammatic approach to quantum dynamics based on categorical algebra, connecting observables, time, and energy through algebraic laws and process theory.

Findings

Quantum dynamical systems as algebras of a dagger Frobenius monad

Derivation of Schrödinger's equation from process theory

Diagrammatic proofs of Stone's and von Neumann's propositions

Abstract

We present a diagrammatic approach to quantum dynamics based on the categorical algebraic structure of strongly complementary observables. We provide physical semantics to our approach in terms of quantum clocks and quantisation of time. We show that quantum dynamical systems arise naturally as the algebras of a certain dagger Frobenius monad, with the morphisms and tensor product of the category of algebras playing the role, respectively, of equivariant transformations and synchronised parallel composition of dynamical systems. We show that the Weyl Canonical Commutation Relations between time and energy are an incarnation of the bialgebra law and we derive Schr\"{o}dinger's equation from a process-theoretic perspective. Finally, we use diagrammatic symmetry-observable duality to prove Stone's proposition and von Neumann's Mean Ergodic proposition, recasting the results as two faces of the very same coin.