Physics – Mathematical Physics
Scientific paper
2010-11-02
J.Phys.Conf.Ser.284:012055,2011
Physics
Mathematical Physics
Scientific paper
10.1088/1742-6596/284/1/012055
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.
Blasiak Pawel
Edmond Duchamp Gérard Henry
Horzela Andrzej
Penson Karol A.
Solomon Allan I.
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