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Combinatorics and Boson normal ordering: A gentle introduction

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 Added by Pawel Blasiak
 Publication date 2007
  fields Physics
and research's language is English
 Authors P. Blasiak




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We discuss a general combinatorial framework for operator ordering problems by applying it to the normal ordering of the powers and exponential of the boson number operator. The solution of the problem is given in terms of Bell and Stirling numbers enumerating partitions of a set. This framework reveals several inherent relations between ordering problems and combinatorial objects, and displays the analytical background to Wicks theorem. The methodology can be straightforwardly generalized from the simple example given herein to a wide class of operators.



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We solve the normal ordering problem for (A* A)^n where A* (resp. A) are one mode deformed bosonic creation (resp. annihilation) operators satisfying [A,A*]=[N+1]-[N]. The solution generalizes results known for canonical and q-bosons. It involves combinatorial polynomials in the number operator N for which the generating functions and explicit expressions are found. Simple deformations provide examples of the method.
106 - H. Cheballah 2008
In this paper, we show that the infinite generalised Stirling matrices associated with boson strings with one annihilation operator are projective limits of approximate substitutions, the latter being characterised by a finite set of algebraic equations.
84 - P Blasiak 2005
We solve the boson normal ordering problem for (q(a*)a + v(a*))^n with arbitrary functions q and v and integer n, where a and a* are boson annihilation and creation operators, satisfying [a,a*]=1. This leads to exponential operators generalizing the shift operator and we show that their action can be expressed in terms of substitutions. Our solution is naturally related through the coherent state representation to the exponential generating functions of Sheffer-type polynomials. This in turn opens a vast arena of combinatorial methodology which is applied to boson normal ordering and illustrated by a few examples.
240 - Pawel Blasiak 2010
Non-commutativity is ubiquitous in mathematical modeling of reality and in many cases same algebraic structures are implemented in different situations. Here we consider the canonical commutation relation of quantum theory and discuss a simple urn model of the latter. It is shown that enumeration of urn histories provides a faithful realization of the Heisenberg-Weyl algebra. Drawing on this analogy we demonstrate how the operator normal forms facilitate counting of histories via generating functions, which in turn yields an intuitive combinatorial picture of the ordering procedure itself.
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The recent COSY-11 collaboration measurement of the two-proton correlation function in the pp -> ppeta reaction, reported at this meeting [1], arouse some interest in a simple theoretical description of the correlation function. In these notes we present a pedagogical introduction to the practical methods that can be used for calculating the correlation function.
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