An Axiomatics and a Combinatorial Model of Creation/Annihilation Operators
Marcelo Fiore
TL;DR
The paper develops a categorical axiomatic framework for bosonic creation/annihilation operators on Fock space and a motivating combinatorial model. It employs a category $\mathcal{S}$ with biproduct $(\mathrm{O},\oplus)$ and symmetric monoidal $(\mathrm{I},\otimes)$, along with a strong monoidal functor $\mathrm{F}$, to axiomatically capture operator commutation relations and coherent states, while a parallel bicategorical model based on profunctors realises Fock space as the free symmetric monoidal completion and derives the same algebraic identities via coends. The two frameworks illuminate the combinatorial content of the relations, with explicit constructions of creation/annihilation operators and coherent states and a detailed coend-based derivation of the canonical commutation relations. The work situates these ideas within connections to linear logic, differential linear logic, and related categorical models, offering a unified language bridging physics, logic, and combinatorics. Overall, the paper provides a rigorous operational foundation for Fock-space constructions in category theory and demonstrates how combinatorial techniques recover essential quantum-algebraic identities.
Abstract
A categorical axiomatic theory of creation/annihilation operators on bosonic Fock space is introduced and the combinatorial model that motivated it is presented. Commutation relations and coherent states are considered in both frameworks.