From Quantum Mechanics to Quantum Field Theory: The Hopf route

TL;DR

This paper reveals the fundamental role of Bell numbers in quantum physics, especially in normal ordering of bosons, and introduces a Hopf algebra framework to extend these ideas to more complex systems.

Contribution

It establishes a connection between Bell numbers and quantum partition functions, introduces a Hopf algebra structure, and discusses the necessity of renormalization in simple quantum models.

Findings

Bell numbers appear naturally in bosonic normal ordering

Partition functions relate to exponential generating functions of Bell numbers

A Hopf algebra framework is developed for quantum systems

Abstract

We show that the combinatorial numbers known as {\em Bell numbers} are generic in quantum physics. This is because they arise in the procedure known as {\em Normal ordering} of bosons, a procedure which is involved in the evaluation of quantum functions such as the canonical partition function of quantum statistical physics, {\it inter alia}. In fact, we shall show that an evaluation of the non-interacting partition function for a single boson system is identical to integrating the {\em exponential generating function} of the Bell numbers, which is a device for encapsulating a combinatorial sequence in a single function. We then introduce a remarkable equality, the Dobinski relation, and use it to indicate why renormalisation is necessary in even the simplest of perturbation expansions for a partition function. Finally we introduce a global algebraic description of this simple model, giving a Hopf algebra, which provides a starting point for extensions to more complex physical systems.