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Dirk Kreimer

Publications and source records attributed to Dirk Kreimer.

At least 73 records · Page 4Linked to original sources
arXiv2002

New mathematical structures in renormalizable quantum field theories

Computations in renormalizable perturbative quantum field theories reveal mathematical structures which go way beyond the formal structure which is usually taken as underlying quantum field theory. We review these new structures and the role they can play in future developments.

hep-th↗
arXiv2002

Structures in Feynman Graphs -Hopf Algebras and Symmetries

We review the combinatorial structure of perturbative quantum field theory with emphasis given to the decomposition of graphs into primitive ones. The consequences in terms of unique factorization of Dyson--Schwinger equations into Euler products are discussed.

hep-th↗
arXiv2002

Insertion and Elimination: the doubly infinite Lie algebra of Feynman graphs

The Lie algebra of Feynman graphs gives rise to two natural representations, acting as derivations on the commutative Hopf algebra of Feynman graphs, by creating or eliminating subgraphs. Insertions and eliminations do not commute, but rather establish a larger Lie algebra of derivations which we here determine.

hep-th↗
arXiv2001

On the Invariance of Residues of Feynman Graphs

We use simple iterated one-loop graphs in massless Yukawa theory and QED to pose the following question: what are the symmetries of the residues of a graph under a permutation of places to insert subdivergences. The investigation confirms partial invariance of the residue under such permutations: the highest weight transcendental is invariant under such a permutation. For QED this result is gauge invariant, ie the permutation invariance holds for any gauge. Computations are done making use of the Hopf algebra structure of graphs and employing GiNaC to automate the calculations.

hep-th↗
arXiv2000

Combinatorics of (perturbative) quantum field theory

We review the structures imposed on perturbative QFT by the fact that its Feynman diagrams provide Hopf and Lie algebras. We emphasize the role which the Hopf algebra plays in renormalization by providing the forest formulas. We exhibit how the associated Lie algebra originates from an operadic operation of graph insertions. Particular emphasis is given to the connection with the Riemann--Hilbert problem. Finally, we outline how these structures relate to the numbers which we see in Feynman diagrams.

hep-th↗
arXiv2000

Shuffling quantum field theory

We discuss shuffle identities between Feynman graphs using the Hopf algebra structure of perturbative quantum field theory. For concrete exposition, we discuss vertex function in massless Yukawa theory.

hep-th↗
arXiv2000

Renormalization in quantum field theory and the Riemann-Hilbert problem II: the $β$-function, diffeomorphisms and the renormalization group

We showed in part I (hep-th/9912092) that the Hopf algebra ${\cal H}$ of Feynman graphs in a given QFT is the algebra of coordinates on a complex infinite dimensional Lie group $G$ and that the renormalized theory is obtained from the unrenormalized one by evaluating at $\ve=0$ the holomorphic part $γ_+(\ve)$ of the Riemann-Hilbert decomposition $γ_-(\ve)^{-1}γ_+(\ve)$ of the loop $γ(\ve)\in G$ provided by dimensional regularization. We show in this paper that the group $G$ acts naturally on the complex space $X$ of dimensionless coupling constants of the theory. More precisely, the formula $g_0=gZ_1Z_3^{-3/2}$ for the effective coupling constant, when viewed as a formal power series, does define a Hopf algebra homomorphism between the Hopf algebra of coordinates on the group of formal diffeomorphisms to the Hopf algebra ${\cal H}$. This allows first of all to read off directly, without using the group $G$, the bare coupling constant and the renormalized one from the Riemann-Hilbert decomposition of the unrenormalized effective coupling constant viewed as a loop of formal diffeomorphisms. This shows that renormalization is intimately related with the theory of non-linear complex bundles on the Riemann sphere of the dimensional regularization parameter $\ve$. It also allows to lift both the renormalization group and the $β$-function as the asymptotic scaling in the group $G$. This exploits the full power of the Riemann-Hilbert decomposition together with the invariance of $γ_-(\ve)$ under a change of unit of mass. This not only gives a conceptual proof of the existence of the renormalization group but also delivers a scattering formula in the group $G$ for the full higher pole structure of minimal subtracted counterterms in terms of the residue.

hep-th↗
arXiv1999

Renormalization in quantum field theory and the Riemann-Hilbert problem I: the Hopf algebra structure of graphs and the main theorem

This paper gives a complete selfcontained proof of our result announced in hep-th/9909126 showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the Riemann-Hilbert problem. We shall first show that for any quantum field theory, the combinatorics of Feynman graphs gives rise to a Hopf algebra $\Hc$ which is commutative as an algebra. It is the dual Hopf algebra of the envelopping algebra of a Lie algebra $\ud G$ whose basis is labelled by the one particle irreducible Feynman graphs. The Lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa. The corresponding Lie group $G$ is the group of characters of $\Hc$. We shall then show that, using dimensional regularization, the bare (unrenormalized) theory gives rise to a loop $$ \g (z) \in G \qquad z \in C $$ where $C$ is a small circle of complex dimensions around the integer dimension $D$ of space-time. Our main result is that the renormalized theory is just the evaluation at $z = D$ of the holomorphic part $\g_+$ of the Birkhoff decomposition of $\g$. We begin to analyse the group $G$ and show that it is a semi-direct product of an easily understood abelian group by a highly non-trivial group closely tied up with groups of diffeomorphisms. The analysis of this latter group as well as the interpretation of the renormalization group and of anomalous dimensions are the content of our second paper with the same overall title.

hep-th↗
arXiv1999

Renormalization in quantum field theory and the Riemann-Hilbert problem

We show that renormalization in quantum field theory is a special instance of a general mathematical procedure of multiplicative extraction of finite values based on the Riemann-Hilbert problem. Given a loop $γ(z), | z |=1$ of elements of a complex Lie group G the general procedure is given by evaluation of $ γ_{+}(z)$ at z=0 after performing the Birkhoff decomposition $ γ(z)=γ_{-}(z)^{-1} γ_{+}(z)$ where $ γ_{\pm}(z) \in G$ are loops holomorphic in the inner and outer domains of the Riemann sphere (with $γ_{-}(\infty)=1$). We show that, using dimensional regularization, the bare data in quantum field theory delivers a loop (where z is now the deviation from 4 of the complex dimension) of elements of the decorated Butcher group (obtained using the Milnor-Moore theorem from the Kreimer Hopf algebra of renormalization) and that the above general procedure delivers the renormalized physical theory in the minimal substraction scheme.

hep-th↗
arXiv1999

Chen's Iterated Integral represents the Operator Product Expansion

The recently discovered formalism underlying renormalization theory, the Hopf algebra of rooted trees, allows to generalize Chen's lemma. In its generalized form it describes the change of a scale in Green functions, and hence relates to the operator product expansion. Hand in hand with this generalization goes the generalization of the ordinary factorial $n!$ to the tree factorial $t^!$. Various identities on tree-factorials are derived which clarify the relation between Connes-Moscovici weights and Quantum Field Theory.

hep-th↗
arXiv1999

Comment on: Topological Invariants, instantons, and the chiral anomaly on spaces with torsion

In Riemann-Cartan spacetimes with torsion only its axial covector piece $A$ couples to massive Dirac fields. Using renormalization group arguments, we show that besides the familiar Riemannian term only the Pontrjagin type four-form $dA\wedge dA$ does arise additionally in the chiral anomaly, but not the Nieh-Yan term $d^\star A$, as has been claimed in a recent paper [PRD 55, 7580 (1997)].

gr-qc↗
arXiv1999

On Overlapping Divergences

Using set-theoretic considerations, we show that the forest formula for overlapping divergences comes from the Hopf algebra of rooted trees.

hep-th↗
arXiv1998

Locality, QED and Classical Electrodynamics

We report on some conceptual changes in our present understanding of Quantum Field Theory and muse about possible consequences for the understanding of $v>c$ signals.

physics.class-ph↗