A diagrammatic categorification of the fermion algebra

TL;DR

This paper develops a graphical categorification of the fermion algebra, creating a diagrammatic framework that mirrors quantum mechanics, with categorical structures representing Fock states and their inner products.

Contribution

It introduces a novel diagrammatic category for the fermion algebra, linking categorical morphisms to quantum states and reproducing quantum mechanical results.

Findings

Categorical analogues of Fock states are represented as 1-morphisms.

The dimension of 2-morphism spaces matches inner products of Fock states.

Results align exactly with traditional quantum mechanics.

Abstract

In this paper, we study the diagrammatic categorification of the fermion algebra. We construct a graphical category corresponding to the one-dimensional fermion algebra, and we investigate the properties of this category. The categorical analogues of the Fock states are some kind of 1-morphisms in our category, and the dimension of the vector space of 2-morphisms is exactly the inner product of the corresponding Fock states. All the results in our categorical framework coincide exactly with those in normal quantum mechanics.