290,909 research outputs found
The pre-Lie structure of the time-ordered exponential
The usual time-ordering operation and the corresponding time-ordered
exponential play a fundamental role in physics and applied mathematics. In this
work we study a new approach to the understanding of time-ordering relying on
recent progress made in the context of enveloping algebras of pre-Lie algebras.
Various general formulas for pre-Lie and Rota-Baxter algebras are obtained in
the process. Among others, we recover the noncommutative analog of the
classical Bohnenblust-Spitzer formula, and get explicit formulae for operator
products of time-ordered exponentials
Geometric Intersection Number and analogues of the Curve Complex for free groups
For the free group of finite rank we construct a canonical
Bonahon-type continuous and -invariant \emph{geometric intersection
form}
Here is the closure of unprojectivized Culler-Vogtmann's
Outer space in the equivariant Gromov-Hausdorff convergence topology
(or, equivalently, in the length function topology). It is known that
consists of all \emph{very small} minimal isometric actions of
on -trees. The projectivization of provides a
free group analogue of Thurston's compactification of the Teichm\"uller space.
As an application, using the \emph{intersection graph} determined by the
intersection form, we show that several natural analogues of the curve complex
in the free group context have infinite diameter.Comment: Revised version, to appear in Geometry & Topolog
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
- …