arXiv subjects

Alessandra Frabetti

Publications and source records attributed to Alessandra Frabetti.

14 recordsLinked to original sources

arXiv2024

Poisson bundles over unordered configurations

In this paper we construct a Poisson algebra bundle whose distributional sections are suitable to represent multilocal observables in classical field theory. To do this, we work with vector bundles over the unordered configuration space of a manifold $M$ and consider the structure of a $2$-monoidal category given by the usual (Hadamard) tensor product of bundles and a new (Cauchy) tensor product which provides a symmetrized version of the usual external tensor product of vector bundles on $M$. We use the symmetric algebras with respect to both products to obtain a Poisson 2-algebra bundle mimicking the construction of Peierls bracket from the causal propagator in field theory. The explicit description of observables from this Poisson algebra bundle will be carried out in a forthcoming paper.

math-ph↗

arXiv2019

Loop of formal diffeomorphisms and F{à}a di Bruno coloop bialgebra

We consider a generalization of (pro)algebraic loops defined on general categories of algebras and the dual notion of a coloop bialgebra suitable to represent them as functors. Our main result is the proof that the natural loop of formal diffeomorphisms with associative coefficients is proalgebraic, and give a full description of the codivisions on its coloop bialgebra.This result provides a generalization of the Lagrange inversion formula to series with non-commutative coefficients, and a loop-theoretic explanation to the existence of the non-commutative F{à}a di Bruno Hopf algebra. MSC: 20N05, 14L17, 18D35, 16T30

math.RA↗

arXiv2015

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.

math-ph↗

arXiv2014

Five interpretations of Faà di Bruno's formula

In 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.

math.CO↗

arXiv2010

Combinatorial Hopf algebras from renormalization

In this paper we describe the right-sided combinatorial Hopf structure of three Hopf algebras appearing in the context of renormalization in quantum field theory: the non-commutative version of the Faà di Bruno Hopf algebra, the non-commutative version of the charge renormalization Hopf algebra on planar binary trees for quantum electrodynamics, and the non-commutative version of the Pinter renormalization Hopf algebra on any bosonic field. We also describe two general ways to define the associative product in such Hopf algebras, the first one by recursion, and the second one by grafting and shuffling some decorated rooted trees.

math.QA↗

arXiv2009

Renormalization Hopf algebras and combinatorial groups

These are the notes of five lectures given at the Summer School {\em Geometric and Topological Methods for Quantum Field Theory}, held in Villa de Leyva (Colombia), July 2--20, 2007. The lectures are meant for graduate or almost graduate students in physics or mathematics. They include references, many examples and some exercices. The content is the following. The first lecture is a short introduction to algebraic and proalgebraic groups, based on some examples of groups of matrices and groups of formal series, and their Hopf algebras of coordinate functions. The second lecture presents a very condensed review of classical and quantum field theory, from the Lagrangian formalism to the Euler-Lagrange equation and the Dyson-Schwinger equation for Green's functions. It poses the main problem of solving some non-linear differential equations for interacting fields. In the third lecture we explain the perturbative solution of the previous equations, expanded on Feynman graphs, in the simplest case of the scalar $ϕ^3$ theory. The forth lecture introduces the problem of divergent integrals appearing in quantum field theory, the renormalization procedure for the graphs, and how the renormalization affects the Lagrangian and the Green's functions given as perturbative series. The last lecture presents the Connes-Kreimer Hopf algebra of renormalization for the scalar theory and its associated proalgebraic group of formal series.

math-ph↗

arXiv2008

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.

math.QA↗

arXiv2007

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.

math.GR↗

arXiv2004

Non-commutative Hopf algebra of formal diffeomorphisms

The subject of this paper are two Hopf algebras which are the non-commutative analogues of two different groups of formal power series. The first group is the set of invertible series with the multiplication, while the second group is the set of formal diffeomorphisms with the composition. The motivation to introduce these Hopf algebras comes from the study of formal series with non-commutative coefficients. Invertible series with non-commutative coefficients still form a group, and we interpret the corresponding new non-commutative Hopf algebra as an alternative to the natural Hopf algebra given by the co-ordinate ring of the group, which has the advantage of being functorial in the algebra of coefficients. For the formal diffeomorphisms with non-commutative coefficients, this interpretation fails, because in this case the composition is not associative anymore. However, we show that for the dual non-commutative algebra there exists a natural co-associative co-product defining a non-commutative Hopf algebra. Moreover, we give an explicit formula for the antipode, which represents a non-commutative version of the Lagrange inversion formula, and we show that its coefficients are related to planar binary trees. Then we extend these results to the semi-direct co-product of the previous Hopf algebras, and to series in several variables. Finally, we show how the non-commutative Hopf algebras of formal series are related to some renormalization Hopf algebras, which are combinatorial Hopf algebras motivated by the renormalization in quantum field theory, and to the renormalization functor given by the double tensor algebra on a bi-algebra.

math.QA↗

arXiv2004

Quantum field theory and Hopf algebra cohomology

We exhibit a Hopf superalgebra structure of the algebra of field operators of quantum field theory (QFT) with the normal product. Based on this we construct the operator product and the time-ordered product as a twist deformation in the sense of Drinfeld. Our approach yields formulas for (perturbative) products and expectation values that allow for a significant enhancement in computational efficiency as compared to traditional methods. Employing Hopf algebra cohomology sheds new light on the structure of QFT and allows the extension to interacting (not necessarily perturbative) QFT. We give a reconstruction theorem for time-ordered products in the spirit of Streater and Wightman and recover the distinction between free and interacting theory from a property of the underlying cocycle. We also demonstrate how non-trivial vacua are described in our approach solving a problem in quantum chemistry.

hep-th↗

arXiv2001

QED Hopf algebras on planar binary trees

In this paper we describe the Hopf algebras on planar binary trees used to renormalize the Feynman propagators of quantum electrodynamics, and the coaction which describes the renormalization procedure. Both structures are related to some semi-direct coproduct of Hopf algebras.

math.QA↗

arXiv2000

Noncommutative renormalization for massless QED

We study the renormalization of massless QED from the point of view of the Hopf algebra discovered by D. Kreimer. For QED, we describe a Hopf algebra of renormalization which is neither commutative nor cocommutative. We obtain explicit renormalization formulas for the electron and photon propagators, for the vacuum polarization and the electron self-energy, which are equivalent to Zimmermann's forest formula for the sum of all Feynman diagrams at a given order of interaction. Then we extend to QED the Connes-Kreimer map defined by the coupling constant of the theory (i.e. the homomorphism between some formal diffeomorphisms and the Hopf algebra of renormalization) by defining a noncommutative Hopf algebra of diffeomorphisms, and then showing that the renormalization of the electric charge defines a homomorphism between this Hopf algebra and the Hopf algebra of renormalization of QED. Finally we show that Dyson's formulas for the renormalization of the electron and photon propagators can be given in a noncommutative (e.g. matrix-valued) form.

hep-th↗

arXiv2000

On the Leibniz cohomology of vector fields

I. M. Gelfand and D. B. Fuks have studied the cohomology of the Lie algebra of vector fields on a manifold. In this article, we generalize their main tools to compute the Leibniz cohomology, by extending the two spectral sequences associated to the diagonal and the order filtration. In particular, we determine some new generators for the diagonal Leibniz cohomology of the Lie algebra of vector fields on the circle.

math.KT↗

arXiv2000

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.

hep-th↗