33 research outputs found
Renormalization automated by Hopf algebra
It was recently shown that the renormalization of quantum field theory is
organized by the Hopf algebra of decorated rooted trees, whose coproduct
identifies the divergences requiring subtraction and whose antipode achieves
this. We automate this process in a few lines of recursive symbolic code, which
deliver a finite renormalized expression for any Feynman diagram. We thus
verify a representation of the operator product expansion, which generalizes
Chen's lemma for iterated integrals. The subset of diagrams whose forest
structure entails a unique primitive subdivergence provides a representation of
the Hopf algebra of undecorated rooted trees. Our undecorated Hopf
algebra program is designed to process the 24,213,878 BPHZ contributions to the
renormalization of 7,813 diagrams, with up to 12 loops. We consider 10 models,
each in 9 renormalization schemes. The two simplest models reveal a notable
feature of the subalgebra of Connes and Moscovici, corresponding to the
commutative part of the Hopf algebra of the diffeomorphism group:
it assigns to Feynman diagrams those weights which remove zeta values from the
counterterms of the minimal subtraction scheme. We devise a fast algorithm for
these weights, whose squares are summed with a permutation factor, to give
rational counterterms.Comment: 22 pages, latex, epsf for figure
Lessons from Quantum Field Theory - Hopf Algebras and Spacetime Geometries
We discuss the prominence of Hopf algebras in recent progress in Quantum
Field Theory. In particular, we will consider the Hopf algebra of
renormalization, whose antipode turned out to be the key to a conceptual
understanding of the subtraction procedure. We shall then describe several
occurences of this or closely related Hopf algebras in other mathematical
domains, such as foliations, Runge Kutta methods, iterated integrals and
multiple zeta values. We emphasize the unifying role which the Butcher group,
discovered in the study of numerical integration of ordinary differential
equations, plays in QFT.Comment: Survey paper, 12 pages, epsf for figures, dedicated to Mosh\'e Flato,
minor corrections, to appear in Lett.Math.Phys.4
The Hopf algebra of Feynman graphs in QED
We report on the Hopf algebraic description of renormalization theory of
quantum electrodynamics. The Ward-Takahashi identities are implemented as
linear relations on the (commutative) Hopf algebra of Feynman graphs of QED.
Compatibility of these relations with the Hopf algebra structure is the
mathematical formulation of the physical fact that WT-identities are compatible
with renormalization. As a result, the counterterms and the renormalized
Feynman amplitudes automatically satisfy the WT-identities, which leads in
particular to the well-known identity .Comment: 13 pages. Latex, uses feynmp. Minor corrections; to appear in LM
Simply Generated Trees, B-series and Wigner Processes
We consider simply generated trees and study multiplicative functions on
rooted plane trees. We show that the associated generating functions satisfy
differential equations or difference equations. Our approach considers B-series
from Butcher's theory, the generating functions are seen as generalized
Runge-Kutta methodsComment: 19 pages, 1 figur
Combinatorics of Rooted Trees and Hopf Algebras
We begin by considering the graded vector space with a basis consisting of
rooted trees, graded by the count of non-root vertices. We define two linear
operators on this vector space, the growth and pruning operators, which
respectively raise and lower grading; their commutator is the operator that
multiplies a rooted tree by its number of vertices. We define an inner product
with respect to which the growth and pruning operators are adjoint, and obtain
several results about the multiplicities associated with each operator.
The symmetric algebra on the vector space of rooted trees (after a degree
shift) can be endowed with a coproduct to make a Hopf algebra; this was defined
by Kreimer in connection with renormalization. We extend the growth and pruning
operators, as well as the inner product mentioned above, to Kreimer's Hopf
algebra. On the other hand, the vector space of rooted trees itself can be
given a noncommutative multiplication: with an appropriate coproduct, this
gives the Hopf algebra of Grossman and Larson. We show the inner product on
rooted trees leads to an isomorphism of the Grossman-Larson Hopf algebra with
the graded dual of Kreimer's Hopf algebra, correcting an earlier result of
Panaite.Comment: 19 pages; final revision has minor corrections, slightly expanded
sect. 4 and additional reference