From quantum mechanics to quantum field theory: The Hopf route

Solomon, A. I.; Duchamp, G. H. E.; Blasiak, P.; Horzela, A. and Penson, K. A. (2011). From quantum mechanics to quantum field theory: The Hopf route. Journal of Physics: Conference Series, 284 012055.

URL: http://dx.doi.org/10.1088/1742-6596/284/1/012055

Abstract

We show that the combinatorial numbers known as Bell numbers are generic in quantum physics. This is because they arise in the procedure known as 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, 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 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.

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