A combinatorial Hopf algebra for the boson normal ordering problem

@article{Bousbaa2015ACH,
  title={A combinatorial Hopf algebra for the boson normal ordering problem},
  author={Imad Eddine Bousbaa and Ali Chouria and Jean-Gabriel Luque},
  journal={arXiv: Combinatorics},
  year={2015}
}
In the aim to understand the generalization of Stirling numbers occurring in the bosonic normal ordering problem, several combinatorial models have been proposed. In particular, Blasiak \emph{et al.} defined combinatorial objects allowing to interpret the number of $S_{\bf{r,s}}(k)$ appearing in the identity $(a^\dag)^{r_n}a^{s_n}\cdots(a^\dag)^{r_1}a^{s_1}=(a^\dag)^\alpha\displaystyle\sum S_{\bf{r,s}}(k)(a^\dag)^k a^k$, where $\alpha$ is assumed to be non-negative. These objects are used to… 

Recent developments in combinatorial aspects of normal ordering

  • M. Schork
  • Mathematics
    Enumerative Combinatorics and Applications
  • 2021
In this paper, we report on recent progress concerning combinatorial aspects of normal ordering. After giving a short introduction to the history and motivation of normal ordering, we present some

References

SHOWING 1-10 OF 26 REFERENCES

The general boson normal ordering problem

On the Structure of Hopf Algebras

induced by the product M x M e M. The structure theorem of Hopf concerning such algebras has been generalized by Borel, Leray, and others. This paper gives a comprehensive treatment of Hopf algebras

Combinatorial algebra for second-quantized Quantum Theory

We describe an algebra G of diagrams that faithfully gives a diagrammatic representation of the structures of both the Heisenberg–Weyl algebra H – the associative algebra of the creation and

Combinatorics and Boson normal ordering: A gentle introduction

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

Combinatorics of boson normal ordering and some applications

We provide the solution to the normal ordering problem for powers and exponentials of two classes of operators. The first one consists of boson strings and more generally homogeneous polynomials,

Commutative combinatorial Hopf algebras

Abstract We propose several constructions of commutative or cocommutative Hopf algebras based on various combinatorial structures and investigate the relations between them. A commutative Hopf

Commutation Relations, Normal Ordering, and Stirling Numbers

Introduction Set Partitions, Stirling, and Bell Numbers Commutation Relations and Operator Ordering Normal Ordering in the Weyl Algebra and Relatives Content of the Book Basic Tools Sequences Solving

Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables

Abstract We introduce a natural Hopf algebra structure on the space of noncommutative symmetric functions. The bases for this algebra are indexed by set partitions. We show that there exists a

THE HOPF ALGEBRAS OF SYMMETRIC FUNCTIONS AND QUASI-SYMMETRIC FUNCTIONS IN NON-COMMUTATIVE VARIABLES ARE FREE AND CO-FREE

We uncover the structure of the space of symmetric functions in non-commutative variables by showing that the underlined Hopf algebra is both free and co-free. We also introduce the Hopf algebra of

On the normal ordering of multi-mode boson operators

Combinatorial aspects of normal ordering for annihilation and creation operators of a multi-mode boson system are discussed. The modes are assumed to be coupled, since otherwise the problem of normal