21 research outputs found
Combinatorial Physics, Normal Order and Model Feynman Graphs
The general normal ordering problem for boson strings is a combinatorial
problem. In this note we restrict ourselves to single-mode boson monomials.
This problem leads to elegant generalisations of well-known combinatorial
numbers, such as Bell and Stirling numbers. We explicitly give the generating
functions for some classes of these numbers. Finally we show that a graphical
representation of these combinatorial numbers leads to sets of model field
theories, for which the graphs may be interpreted as Feynman diagrams
corresponding to the bosons of the theory. The generating functions are the
generators of the classes of Feynman diagrams.Comment: 9 pages, 4 figures. 12 references. Presented at the Symposium
'Symmetries in Science XIII', Bregenz, Austria, 200
Hopf Algebras in General and in Combinatorial Physics: a practical introduction
This tutorial is intended to give an accessible introduction to Hopf
algebras. The mathematical context is that of representation theory, and we
also illustrate the structures with examples taken from combinatorics and
quantum physics, showing that in this latter case the axioms of Hopf algebra
arise naturally. The text contains many exercises, some taken from physics,
aimed at expanding and exemplifying the concepts introduced
A product formula and combinatorial field theory
We treat the problem of normally ordering expressions involving the standard boson operators a, ay where [a; ay] = 1. We show that a simple product formula for formal power series | essentially an extension of the Taylor expansion | leads to a double exponential formula which enables a powerful graphical description of the generating functions of the combinatorial sequences associated with such functions | in essence, a combinatorial eld theory. We apply these techniques to some examples related to specic physical Hamiltonians
Dobinski-type relations: Some properties and physical applications
We introduce a generalization of the Dobinski relation through which we
define a family of Bell-type numbers and polynomials. For all these sequences
we find the weight function of the moment problem and give their generating
functions. We provide a physical motivation of this extension in the context of
the boson normal ordering problem and its relation to an extension of the Kerr
Hamiltonian.Comment: 7 pages, 1 figur
Heisenberg-Weyl algebra revisited: Combinatorics of words and paths
The Heisenberg-Weyl algebra, which underlies virtually all physical
representations of Quantum Theory, is considered from the combinatorial point
of view. We provide a concrete model of the algebra in terms of paths on a
lattice with some decomposition rules. We also discuss the rook problem on the
associated Ferrers board; this is related to the calculus in the normally
ordered basis. From this starting point we explore a combinatorial underpinning
of the Heisenberg-Weyl algebra, which offers novel perspectives, methods and
applications.Comment: 5 pages, 3 figure
Hierarchical Dobinski-type relations via substitution and the moment problem
We consider the transformation properties of integer sequences arising from
the normal ordering of exponentiated boson ([a,a*]=1) monomials of the form
exp(x (a*)^r a), r=1,2,..., under the composition of their exponential
generating functions (egf). They turn out to be of Sheffer-type. We demonstrate
that two key properties of these sequences remain preserved under
substitutional composition: (a)the property of being the solution of the
Stieltjes moment problem; and (b) the representation of these sequences through
infinite series (Dobinski-type relations). We present a number of examples of
such composition satisfying properties (a) and (b). We obtain new Dobinski-type
formulas and solve the associated moment problem for several hierarchically
defined combinatorial families of sequences.Comment: 14 pages, 31 reference
Laguerre-type derivatives: Dobinski relations and combinatorial identities
We consider properties of the operators D(r,M)=a^r(a^\dag a)^M (which we call
generalized Laguerre-type derivatives), with r=1,2,..., M=0,1,..., where a and
a^\dag are boson annihilation and creation operators respectively, satisfying
[a,a^\dag]=1. We obtain explicit formulas for the normally ordered form of
arbitrary Taylor-expandable functions of D(r,M) with the help of an operator
relation which generalizes the Dobinski formula. Coherent state expectation
values of certain operator functions of D(r,M) turn out to be generating
functions of combinatorial numbers. In many cases the corresponding
combinatorial structures can be explicitly identified.Comment: 14 pages, 1 figur
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
Hopf algebras: motivations and examples
This paper provides motivation as well as a method of construction for Hopf
algebras, starting from an associative algebra. The dualization technique
involved relies heavily on the use of Sweedler's dual