71 research outputs found
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 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
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