290,909 research outputs found

    The pre-Lie structure of the time-ordered exponential

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    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

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    For the free group FNF_{N} of finite rank N≥2N \geq 2 we construct a canonical Bonahon-type continuous and Out(FN)Out(F_N)-invariant \emph{geometric intersection form} :cvˉ(FN)×Curr(FN)→R≥0. : \bar{cv}(F_N)\times Curr(F_N)\to \mathbb R_{\ge 0}. Here cvˉ(FN)\bar{cv}(F_N) is the closure of unprojectivized Culler-Vogtmann's Outer space cv(FN)cv(F_N) in the equivariant Gromov-Hausdorff convergence topology (or, equivalently, in the length function topology). It is known that cvˉ(FN)\bar{cv}(F_N) consists of all \emph{very small} minimal isometric actions of FNF_N on R\mathbb R-trees. The projectivization of cvˉ(FN)\bar{cv}(F_N) 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

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    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
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