A generic Hopf algebra for quantum statistical mechanics

TL;DR

This paper introduces a Hopf algebra framework for a simple bosonic quantum model, highlighting combinatorial structures like Bell and Stirling numbers, and drawing parallels to complex quantum field theories.

Contribution

It presents a novel Hopf algebra description of a bosonic quantum system using combinatorial elements, simplifying the understanding of algebraic structures in quantum models.

Findings

Hopf algebra structure identified in bosonic operators

Combinatorial elements linked to Bell and Stirling numbers

Graphical representation analogous to Feynman diagrams

Abstract

In this paper, we present a Hopf algebra description of a bosonic quantum model, using the elementary combinatorial elements of Bell and Stirling numbers. Our objective in doing this is as follows. Recent studies have revealed that perturbative quantum field theory (pQFT) displays an astonishing interplay between analysis (Riemann zeta functions), topology (Knot theory), combinatorial graph theory (Feynman diagrams) and algebra (Hopf structure). Since pQFT is an inherently complicated study, so far not exactly solvable and replete with divergences, the essential simplicity of the relationships between these areas can be somewhat obscured. The intention here is to display some of the above-mentioned structures in the context of a simple bosonic quantum theory, i.e. a quantum theory of non-commuting operators that do not depend on space-time. The combinatorial properties of these boson creation and annihilation operators, which is our chosen example, may be described by graphs, analogous to the Feynman diagrams of pQFT, which we show possess a Hopf algebra structure. Our approach is based on the quantum canonical partition function for a boson gas.