From Quantum Mechanics to Quantum Field Theory: The Hopf route

Allan I. Solomon (1, 2) , Gérard Henry Edmond Duchamp (3) , Pawel Blasiak (4) , Andrzej Horzela (4) , Karol A. Penson (2)

1 Department of Physics and Astronomy [Milton Keynes]
2 LPTMC - Laboratoire de Physique Théorique de la Matière Condensée
3 LIPN - Laboratoire d'Informatique de Paris-Nord
4 H. Niewodniczanski Institute of Nuclear Physics

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.

Mots clés: quantum mechanics; combinatorics; Hopf algebra

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Combinatoire [math.CO]; Calcul formel [cs.SC]

Allan I. Solomon, Gérard Henry Edmond Duchamp, Pawel Blasiak, Andrzej Horzela, Karol A. Penson. From Quantum Mechanics to Quantum Field Theory: The Hopf route. 2010. ⟨hal-00531102⟩