Combinatorial Solutions to Normal Ordering of Bosons

We present a combinatorial method of constructing solutions to the normal ordering of boson operators. Generalizations of standard combinatorial notions - the Stirling and Bell numbers, Bell polynomials and Dobinski relations - lead to calculational tools which allow to find explicitly normally ordered forms for a large class of operator functions.Comment: Presented at 14th Int. Colloquium on Integrable Systems, Prague, Czech Republic, 16-18 June 2005. 6 pages, 11 reference

From Quantum Mechanics to Quantum Field Theory: The Hopf route

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

A generic Hopf algebra for quantum statistical mechanics

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.Comment: 8 pages/(4 pages published version), 1 Figure. arXiv admin note: text overlap with arXiv:1011.052

Euclidean asymptotic expansions of Green functions of quantum fields (II) Combinatorics of the asymptotic operation

The results of part I (hep-ph/9612284) are used to obtain full asymptotic expansions of Feynman diagrams renormalized within the MS-scheme in the regimes when some of the masses and external momenta are large with respect to the others. The large momenta are Euclidean, and the expanded diagrams are regarded as distributions with respect to them. The small masses may be equal to zero. The asymptotic operation for integrals is defined and a simple combinatorial techniques is developed to study its exponentiation. The asymptotic operation is used to obtain the corresponding expansions of arbitrary Green functions. Such expansions generalize and improve upon the well-known short-distance operator-product expansions, the decoupling theorem etc.; e.g. the low-energy effective Lagrangians are obtained to all orders of the inverse heavy mass. The obtained expansions possess the property of perfect factorization of large and small parameters, which is essential for meaningful applications to phenomenology. As an auxiliary tool, the inversion of the R-operation is constructed. The results are valid for arbitrary QFT models.Comment: one .sty + one .tex (LaTeX 2.09) + one .ps (GSview) = 46 pp. Many fewer misprints than the journal versio

Exponential Operators, Dobinski Relations and Summability

We investigate properties of exponential operators preserving the particle number using combinatorial methods developed in order to solve the boson normal ordering problem. In particular, we apply generalized Dobinski relations and methods of multivariate Bell polynomials which enable us to understand the meaning of perturbation-like expansions of exponential operators. Such expansions, obtained as formal power series, are everywhere divergent but the Pade summation method is shown to give results which very well agree with exact solutions got for simplified quantum models of the one mode bosonic systems.Comment: Presented at XIIth Central European Workshop on Quantum Optics, Bilkent University, Ankara, Turkey, 6-10 June 2005. 4 figures, 6 pages, 10 reference