Alessandra Frabetti

QED Hopf algebras on planar binary trees

Abstract 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.

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 formulae 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.

Dialgebra homology of associative algebras

Abstract Let A be an associative algebra, and M a bimodule over A. Consider A as a dialgebra (see [5]), with left and right products equal to the given associative multiplication. We define the dialgebra homology HY*(A, M) of A with coefficients in M, and show, when A is unital, that it is isomorphic to the Hochschild homology H*(A, M). It is a consequence of the combinatorial properties of the set of binary trees.

Leibniz homilogy of dialgebras of matrices

Abstract A dialgebra D is a vector space with two associative operations ÷, ⊢ satisfying three more relations. By setting [ x , y ] : = x ÷ y − y ⊢ x , any dialgebra gives rise to a Leibniz algebra. Here we compute the Leibniz homology of the dialgebra of matrices gl ( D ) with entries in a given dialgebra D . We show that HL ( gl ( D )) is isomorphic to the tensor module over HHS ( D ), which is a variation of the natural dialgebra homology HHY ( D ).

Simplicial Properties of the Set of Planar Binary Trees

Planar binary trees appear as the the main ingredient of a new homology theory related to dialgebras, cf.(J.-L. Loday, C.R. Acad. Sci. Paris321 (1995), 141–146.) Here I investigate the simplicial properties of the set of these trees, which are independent of the dialgebra context though they are reflected in the dialgebra homology.The set of planar binary trees is endowed with a natural (almost) simplicial structure which gives rise to a chain complex. The main new idea consists in decomposing the set of trees into classes, by exploiting the orientation of their leaves. (This trick has subsequently found an application in quantum electrodynamics, c.f. (C. Brouder, “On the Trees of Quantum Fields,” Eur. Phys. J. C12, 535–549 (2000).) This decomposition yields a chain bicomplex whose total chain complex is that of binary trees. The main theorem of the paper concerns a further decomposition of this bicomplex. Each vertical complex is the direct sum of subcomplexes which are in bijection with the planar binary trees. This decomposition is used in the computation of dialgebra homology as a derived functor, cf. (A. Frabetti, “Dialgebra (co) Homology with Coefficients,” Springer L.N.M., to appear).

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.