22,343 research outputs found
Trees
An algebraic formalism, developped with V. Glaser and R. Stora for the study
of the generalized retarded functions of quantum field theory, is used to prove
a factorization theorem which provides a complete description of the
generalized retarded functions associated with any tree graph. Integrating over
the variables associated to internal vertices to obtain the perturbative
generalized retarded functions for interacting fields arising from such graphs
is shown to be possible for a large category of space-times.Comment: minor corrections, references added, no change in result
The Galois Complexity of Graph Drawing: Why Numerical Solutions are Ubiquitous for Force-Directed, Spectral, and Circle Packing Drawings
Many well-known graph drawing techniques, including force directed drawings,
spectral graph layouts, multidimensional scaling, and circle packings, have
algebraic formulations. However, practical methods for producing such drawings
ubiquitously use iterative numerical approximations rather than constructing
and then solving algebraic expressions representing their exact solutions. To
explain this phenomenon, we use Galois theory to show that many variants of
these problems have solutions that cannot be expressed by nested radicals or
nested roots of low-degree polynomials. Hence, such solutions cannot be
computed exactly even in extended computational models that include such
operations.Comment: Graph Drawing 201
Rota-Baxter algebras, singular hypersurfaces, and renormalization on Kausz compactifications
We consider Rota-Baxter algebras of meromorphic forms with poles along a
(singular) hypersurface in a smooth projective variety and the associated
Birkhoff factorization for algebra homomorphisms from a commutative Hopf
algebra. In the case of a normal crossings divisor, the Rota-Baxter structure
simplifies considerably and the factorization becomes a simple pole
subtraction. We apply this formalism to the unrenormalized momentum space
Feynman amplitudes, viewed as (divergent) integrals in the complement of the
determinant hypersurface. We lift the integral to the Kausz compactification of
the general linear group, whose boundary divisor is normal crossings. We show
that the Kausz compactification is a Tate motive and that the boundary divisor
and the divisor that contains the boundary of the chain of integration are
mixed Tate configurations. The regularization of the integrals that we obtain
differs from the usual renormalization of physical Feynman amplitudes, and in
particular it may give mixed Tate periods in some cases that have non-mixed
Tate contributions when computed with other renormalization methods.Comment: 35 pages, LaTe
Feynman integrals and motives
This article gives an overview of recent results on the relation between
quantum field theory and motives, with an emphasis on two different approaches:
a "bottom-up" approach based on the algebraic geometry of varieties associated
to Feynman graphs, and a "top-down" approach based on the comparison of the
properties of associated categorical structures. This survey is mostly based on
joint work of the author with Paolo Aluffi, along the lines of the first
approach, and on previous work of the author with Alain Connes on the second
approach.Comment: 32 pages LaTeX, 3 figures, to appear in the Proceedings of the 5th
European Congress of Mathematic
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