249 research outputs found
Dynkin operators and renormalization group actions in pQFT
Renormalization techniques in perturbative quantum field theory were known,
from their inception, to have a strong combinatorial content emphasized, among
others, by Zimmermann's celebrated forest formula. The present article reports
on recent advances on the subject, featuring the role played by the Dynkin
operators (actually their extension to the Hopf algebraic setting) at two
crucial levels of renormalization, namely the Bogolioubov recursion and the
renormalization group (RG) equations. For that purpose, an iterated integrals
toy model is introduced to emphasize how the operators appear naturally in the
setting of renormalization group analysis. The toy model, in spite of its
simplicity, captures many key features of recent approaches to RG equations in
pQFT, including the construction of a universal Galois group for quantum field
theories
Right-handed Hopf algebras and the preLie forest formula
Three equivalent methods allow to compute the antipode of the Hopf algebras
of Feynman diagrams in perturbative quantum field theory (QFT): the Dyson-Salam
formula, the Bogoliubov formula, and the Zimmermann forest formula. Whereas the
first two hold generally for arbitrary connected graded Hopf algebras, the
third one requires extra structure properties of the underlying Hopf algebra
but has the nice property to reduce drastically the number of terms in the
expression of the antipode (it is optimal in that sense).The present article is
concerned with the forest formula: we show that it generalizes to arbitrary
right-handed polynomial Hopf algebras. These Hopf algebras are dual to the
enveloping algebras of preLie algebras -a structure common to many
combinatorial Hopf algebras which is carried in particular by the Hopf algebras
of Feynman diagrams
Nonlocal, noncommutative diagrammatics and the linked cluster Theorems
Recent developments in quantum chemistry, perturbative quantum field theory,
statistical physics or stochastic differential equations require the
introduction of new families of Feynman-type diagrams. These new families arise
in various ways. In some generalizations of the classical diagrams, the notion
of Feynman propagator is extended to generalized propagators connecting more
than two vertices of the graphs. In some others (introduced in the present
article), the diagrams, associated to noncommuting product of operators inherit
from the noncommutativity of the products extra graphical properties. The
purpose of the present article is to introduce a general way of dealing with
such diagrams. We prove in particular a "universal" linked cluster theorem and
introduce, in the process, a Feynman-type "diagrammatics" that allows to handle
simultaneously nonlocal (Coulomb-type) interactions, the generalized diagrams
arising from the study of interacting systems (such as the ones where the
ground state is not the vacuum but e.g. a vacuum perturbed by a magnetic or
electric field, by impurities...) or Wightman fields (that is, expectation
values of products of interacting fields). Our diagrammatics seems to be the
first attempt to encode in a unified algebraic framework such a wide variety of
situations. In the process, we promote two ideas. First, Feynman-type
diagrammatics belong mathematically to the theory of linear forms on
combinatorial Hopf algebras. Second, linked cluster-type theorems rely
ultimately on M\"obius inversion on the partition lattice. The two theories
should therefore be introduced and presented accordingl
Tree expansion in time-dependent perturbation theory
The computational complexity of time-dependent perturbation theory is
well-known to be largely combinatorial whatever the chosen expansion method and
family of parameters (combinatorial sequences, Goldstone and other Feynman-type
diagrams...). We show that a very efficient perturbative expansion, both for
theoretical and numerical purposes, can be obtained through an original
parametrization by trees and generalized iterated integrals. We emphasize above
all the simplicity and naturality of the new approach that links perturbation
theory with classical and recent results in enumerative and algebraic
combinatorics. These tools are applied to the adiabatic approximation and the
effective Hamiltonian. We prove perturbatively and non-perturbatively the
convergence of Morita's generalization of the Gell-Mann and Low wavefunction.
We show that summing all the terms associated to the same tree leads to an
utter simplification where the sum is simpler than any of its terms. Finally,
we recover the time-independent equation for the wave operator and we give an
explicit non-recursive expression for the term corresponding to an arbitrary
tree.Comment: 22 pages, 2 figure
Enveloping algebras of preLie algebras, Solomon idempotents and the Magnus formula
We study the internal structure of enveloping algebras of preLie algebras. We
show in particular that the canonical projections arising from the
Poincar\'e-Birkhoff-Witt theorem can be computed explicitely. They happen to be
closely related to the Magnus formula for matrix differential equations.
Indeed, we show that the Magnus formula provides a way to compute the canonical
projection on the preLie algebra. Conversely, our results provide new insights
on classical problems in the theory of differential equations and on recent
advances in their combinatorial understanding
A Fast Algorithm for the Construction of Integrity Bases Associated to Symmetry-Adapted Polynomial Representations. Application to Tetrahedral XY4 Molecules
Invariant theory provides more efficient tools, such as Molien generating
functions and integrity bases, than basic group theory, that relies on
projector techniques for the construction of symmetry--adapted polynomials in
the symmetry coordinates of a molecular system, because it is based on a finer
description of the mathematical structure of the latter. The present article
extends its use to the construction of polynomial bases which span possibly,
non--totally symmetric irreducible representations of a molecular symmetry
group. Electric or magnetic observables can carry such irreducible
representations, a common example is given by the electric dipole moment
surface. The elementary generating functions and their corresponding integrity
bases, where both the initial and the final representations are irreducible,
are the building blocks of the algorithm presented in this article, which is
faster than algorithms based on projection operators only. The generating
functions for the full initial representation of interest are built recursively
from the elementary generating functions. Integrity bases which can be used to
generate in the most economical way symmetry--adapted polynomial bases are
constructed alongside in the same fashion. The method is illustrated in detail
on XY4 type of molecules. Explicit integrity bases for all five possible final
irreducible representations of the tetrahedral group have been calculated and
are given in the supplemental material associated with this paper
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