28 research outputs found
Groups of tree-expanded series
We describe the proalgebraic groups represented by three Hopf algebras on
planar binary trees previously introduced by the author and Christian Brouder
in relation with the renormalization of quantum electrodynamics. Using two
monoidal structures and a set-operad structure on planar binary trees, we show
that these groups can be realized on formal series expanded over trees, and
that the group laws are generalization of the multiplication and the
composition of usual series in one variable. All the constructions are done in
a general operad-theoretic setting, and then applied to the duplicial operad on
trees.Comment: 30 pages; added references, to appear in J.Algebr
Renormalization of QED with planar binary trees
The renormalized photon and electron propagators are expanded over planar
binary trees. Explicit recurrence solutions are given for the terms of these
expansions. In the case of massless Quantum Electrodynamics (QED), the relation
between renormalized and bare expansions is given in terms of a Hopf algebra
structure. For massive quenched QED, the relation between renormalized and bare
expansions is given explicitly.Comment: Uses feynmf package. 20 page
From quantum electrodynamics to posets of planar binary trees
This paper is a brief mathematical excursion which starts from quantum
electrodynamics and leads to the Moebius function of the Tamari lattice of
planar binary trees, within the framework of groups of tree-expanded series.
First we recall Brouder's expansion of the photon and the electron Green's
functions on planar binary trees, before and after the renormalization. Then we
recall the structure of Connes and Kreimer's Hopf algebra of renormalization in
the context of planar binary trees, and of their dual group of tree-expanded
series. Finally we show that the Moebius function of the Tamari posets of
planar binary trees gives rise to a particular series in this group.Comment: 13 page
Noncommutative version of Borcherds' approach to quantum field theory
Richard Borcherds proposed an elegant geometric version of renormalized
perturbative quantum field theory in curved spacetimes, where Lagrangians are
sections of a Hopf algebra bundle over a smooth manifold. However, this
framework looses its geometric meaning when Borcherds introduces a (graded)
commutative normal product. We present a fully geometric version of Borcherds'
quantization where the (external) tensor product plays the role of the normal
product. We construct a noncommutative many-body Hopf algebra and a module over
it which contains all the terms of the perturbative expansion and we quantize
it to recover the expectation values of standard quantum field theory when the
Hopf algebra fiber is (graded) cocommutative. This construction enables to the
second quantize any theory described by a cocommutative Hopf algebra bundle.Comment: Frontiers of Fundamental Physics 14, Jul 2014, Marseille, Franc
Five interpretations of FaĂ di Bruno's formula
International audienceIn these lectures we present five interpretations of the Fa' di Bruno formula which computes the n-th derivative of the composition of two functions of one variable: in terms of groups, Lie algebras and Hopf algebras, in combinatorics and within operads