Ladder operators and endomorphisms in combinatorial Physics

Starting with the Heisenberg-Weyl algebra, fundamental to quantum physics, we first show how the ordering of the non-commuting operators intrinsic to that algebra gives rise to generalizations of the classical Stirling Numbers of Combinatorics. These may be expressed in terms of infinite, but row-finite, matrices, which may also be considered as endomorphisms of C[x]. This leads us to consider endomorphisms in more general spaces, and these in turn may be expressed in terms of generalizations of the ladder-operators familiar in physics

Overview of the Heisenberg--Weyl Algebra and Subsets of Riordan Subgroups

In a first part, we are concerned with the relationships between polynomials in the two generators of the algebra of Heisenberg--Weyl, its Bargmann--Fock representation with differential operators and the associated one-parameter group.Upon this basis, the paper is then devoted to the groups of Riordan matrices associated to the related transformations of matrices (i.e. substitutions with prefunctions). Thereby, various properties are studied arising in Riordan arrays, in the Riordan group and, more specifically, in the `striped' Riordan subgroups; further, a striped quasigroup and a semigroup are also examined. A few applications to combinatorial structures are also briefly addressed in the Appendix.Comment: Version 3 of the paper entitled `On subsets of Riordan subgroups and Heisenberg--Weyl algebra' in [hal-00974929v2]The present article is published in The Electronic Journal of Combinatorics, Volume 22, Issue 4, 40 pages (Oct. 2015), pp.Id: 1

Fock Spaces, Landau Operators and the Regular Solutions of time-harmonic Maxwell equations

We investigate the representations of the solutions to Maxwell's equations based on the combination of hypercomplex function-theoretical methods with quantum mechanical methods. Our approach provides us with a characterization for the solutions to the time-harmonic Maxwell system in terms of series expansions involving spherical harmonics resp. spherical monogenics. Also, a thorough investigation for the series representation of the solutions in terms of eigenfunctions of Landau operators that encode $n-$dimensional spinless electrons is given. This new insight should lead to important investigations in the study of regularity and hypo-ellipticity of the solutions to Schr\"odinger equations with natural applications in relativistic quantum mechanics concerning massive spinor fields.Comment: Exposition improved; Some typos corrected; Accepted for publication in J.Phys.A (February 2011). http://www.mat.uc.pt/preprints/ps/p1047.pd

Renormalization: a quasi-shuffle approach

In recent years, the usual BPHZ algorithm for renormalization in perturbative quantum field theory has been interpreted, after dimensional regularization, as a Birkhoff decomposition of characters on the Hopf algebra of Feynman graphs, with values in a Rota-Baxter algebra of amplitudes. We associate in this paper to any such algebra a universal semi-group (different in nature from the Connes-Marcolli "cosmical Galois group"). Its action on the physical amplitudes associated to Feynman graphs produces the expected operations: Bogoliubov's preparation map, extraction of divergences, renormalization. In this process a key role is played by commutative and noncommutative quasi-shuffle bialgebras whose universal properties are instrumental in encoding the renormalization process

The uses of Connes and Kreimer's algebraic formulation of renormalization theory

We show how, modulo the distinction between the antipode and the "twisted" or "renormalized" antipode, Connes and Kreimer's algebraic paradigm trivializes the proofs of equivalence of the (corrected) Dyson-Salam, Bogoliubov-Parasiuk-Hepp and Zimmermann procedures for renormalizing Feynman amplitudes. We discuss the outlook for a parallel simplification of computations in quantum field theory, stemming from the same algebraic approach.Comment: 15 pages, Latex. Minor changes, typos fixed, 2 references adde

Free Rota-Baxter algebras and rooted trees

A Rota-Baxter algebra, also known as a Baxter algebra, is an algebra with a linear operator satisfying a relation, called the Rota-Baxter relation, that generalizes the integration by parts formula. Most of the studies on Rota-Baxter algebras have been for commutative algebras. Two constructions of free commutative Rota-Baxter algebras were obtained by Rota and Cartier in the 1970s and a third one by Keigher and one of the authors in the 1990s in terms of mixable shuffles. Recently, noncommutative Rota-Baxter algebras have appeared both in physics in connection with the work of Connes and Kreimer on renormalization in perturbative quantum field theory, and in mathematics related to the work of Loday and Ronco on dendriform dialgebras and trialgebras. This paper uses rooted trees and forests to give explicit constructions of free noncommutative Rota--Baxter algebras on modules and sets. This highlights the combinatorial nature of Rota--Baxter algebras and facilitates their further study. As an application, we obtain the unitarization of Rota-Baxter algebras.Comment: 23 page

Combinatorial Hopf algebraic description of the multiscale renormalization in quantum field theory

We define in this paper several Hopf algebras describing the combinatorics of the so-called multi-scale renormalization in quantum field theory. After a brief recall of the main mathematical features of multi-scale renormalization, we define assigned graphs, that are graphs with appropriate decorations for the multi-scale framework. We then define Hopf algebras on these assigned graphs and on the Gallavotti-Nicol\`o trees, particular class of trees encoding the supplementary informations of the assigned graphs. Several morphisms between these combinatorial Hopf algebras and the Connes-Kreimer algebra are given. Finally, scale dependent couplings are analyzed via this combinatorial algebraic setting.Comment: 26 pages, 3 figures; the presentation of the results has been reorganized. Several details of various proofs are given and some references have been adde