781 research outputs found
Lessons from Quantum Field Theory - Hopf Algebras and Spacetime Geometries
We discuss the prominence of Hopf algebras in recent progress in Quantum
Field Theory. In particular, we will consider the Hopf algebra of
renormalization, whose antipode turned out to be the key to a conceptual
understanding of the subtraction procedure. We shall then describe several
occurences of this or closely related Hopf algebras in other mathematical
domains, such as foliations, Runge Kutta methods, iterated integrals and
multiple zeta values. We emphasize the unifying role which the Butcher group,
discovered in the study of numerical integration of ordinary differential
equations, plays in QFT.Comment: Survey paper, 12 pages, epsf for figures, dedicated to Mosh\'e Flato,
minor corrections, to appear in Lett.Math.Phys.4
Shuffle relations for regularised integrals of symbols
We prove shuffle relations which relate a product of regularised integrals of
classical symbols to regularised nested (Chen) iterated integrals, which hold
if all the symbols involved have non-vanishing residue. This is true in
particular for non-integer order symbols. In general the shuffle relations hold
up to finite parts of corrective terms arising from renormalisation on tensor
products of classical symbols, a procedure adapted from renormalisation
procedures on Feynman diagrams familiar to physicists. We relate the shuffle
relations for regularised integrals of symbols with shuffle relations for
multizeta functions adapting the above constructions to the case of symbols on
the unit circle.Comment: 40 pages,latex. Changes concern sections 4 and 5 : an error in
section 4 has been corrected, and the link between section 5 and the previous
ones has been precise
Higher loop renormalization of a supersymmetric field theory
Using Dyson--Schwinger equations within an approach developed by Broadhurst
and Kreimer and the renormalization group, we show how high loop order of the
renormalization group coefficients can be efficiently computed in a
supersymmetric model.Comment: 8 pages, 2 figure
Field diffeomorphisms and the algebraic structure of perturbative expansions
We consider field diffeomorphisms in the context of real scalar field
theories. Starting from free field theories we apply non-linear field
diffeomorphisms to the fields and study the perturbative expansion for the
transformed theories. We find that tree level amplitudes for the transformed
fields must satisfy BCFW type recursion relations for the S-matrix to remain
trivial. For the massless field theory these relations continue to hold in loop
computations. In the massive field theory the situation is more subtle. A
necessary condition for the Feynman rules to respect the maximal ideal and
co-ideal defined by the core Hopf algebra of the transformed theory is that
upon renormalization all massive tadpole integrals (defined as all integrals
independent of the kinematics of external momenta) are mapped to zero.Comment: 8 pages, 2 figure
Using the Hopf Algebra Structure of QFT in Calculations
We employ the recently discovered Hopf algebra structure underlying
perturbative Quantum Field Theory to derive iterated integral representations
for Feynman diagrams. We give two applications: to massless Yukawa theory and
quantum electrodynamics in four dimensions.Comment: 28 p, Revtex, epsf for figures, minor changes, to appear in
Phys.Rev.
On the structure and representations of the insertion-elimination Lie algebra
We examine the structure of the insertion-elimination Lie algebra on rooted
trees introduced in \cite{CK}. It possesses a triangular structure \g = \n_+
\oplus \mathbb{C}.d \oplus \n_-, like the Heisenberg, Virasoro, and affine
algebras. We show in particular that it is simple, which in turn implies that
it has no finite-dimensional representations. We consider a category of
lowest-weight representations, and show that irreducible representations are
uniquely determined by a "lowest weight" . We show that
each irreducible representation is a quotient of a Verma-type object, which is
generically irreducible
Spitzer's Identity and the Algebraic Birkhoff Decomposition in pQFT
In this article we continue to explore the notion of Rota-Baxter algebras in
the context of the Hopf algebraic approach to renormalization theory in
perturbative quantum field theory. We show in very simple algebraic terms that
the solutions of the recursively defined formulae for the Birkhoff
factorization of regularized Hopf algebra characters, i.e. Feynman rules,
naturally give a non-commutative generalization of the well-known Spitzer's
identity. The underlying abstract algebraic structure is analyzed in terms of
complete filtered Rota-Baxter algebras.Comment: 19 pages, 2 figure
The Hopf algebra of Feynman graphs in QED
We report on the Hopf algebraic description of renormalization theory of
quantum electrodynamics. The Ward-Takahashi identities are implemented as
linear relations on the (commutative) Hopf algebra of Feynman graphs of QED.
Compatibility of these relations with the Hopf algebra structure is the
mathematical formulation of the physical fact that WT-identities are compatible
with renormalization. As a result, the counterterms and the renormalized
Feynman amplitudes automatically satisfy the WT-identities, which leads in
particular to the well-known identity .Comment: 13 pages. Latex, uses feynmp. Minor corrections; to appear in LM
Calculation of Infrared-Divergent Feynman Diagrams with Zero Mass Threshold
Two-loop vertex Feynman diagrams with infrared and collinear divergences are
investigated by two independent methods. On the one hand, a method of
calculating Feynman diagrams from their small momentum expansion extended to
diagrams with zero mass thresholds is applied. On the other hand, a numerical
method based on a two-fold integral representation is used. The application of
the latter method is possible by using lightcone coordinates in the parallel
space. The numerical data obtained with the two methods are in impressive
agreement.Comment: 20 pages, Latex with epsf-figures, References updated, to appear in
Z.Phys.
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