Research Interests

A quantum field theory is specified by the one-particle irreducible (1PI) Green functions of the theory. Knowledge of the underlying free quantum field theory then allows to reconstruct the connected Wightman functions, and hence the quantum fields.

For a renormalizable theory, a finite number of amplitudes needs to be fixed by experiment to determine the theory completely. We thus need to determine the 1PI-Green functions corresponding to the monomials appearing in any corresponding renormalizable Lagrange density.

The mathematical structure of these corresponding set of 1PI Green functions reveals itself upon the study of the underlying Hopf algebra structure of a perturbative expansion in the coupling, which allows for an arithmetic analysis of terms in the perturbative expansion, and the
study of fix-point equations underlying non-perturbative physics.

A good source reviewing the combinatorial approach to renormalization is Dominique Manchon’s review of my work. Arithmetic aspects are nicely summarized in a talk by Francis Brown: Quantum Field Theory and Arithmetic.

Here are some selected research papers of mine:

Quantization of gauge fields, graph polynomials and graph cohomology
Dirk Kreimer, Matthias Sars, Walter D. van Suijlekom, Annals Phys. 336 (2013) 180-222
We review quantization of gauge fields using algebraic properties of 3-regular graphs. We derive the Feynman integrand at n loops for a non-abelian gauge theory quantized in a covariant gauge from scalar integrands for connected 3-regular graphs, obtained from the two Symanzik polynomials. The transition to the full gauge theory amplitude is obtained by the use of a third, new, graph polynomial, the corolla polynomial. This implies effectively a covariant quantization without ghosts, where all the relevant signs of the ghost sector are incorporated in a double complex furnished by the corolla polynomial -we call it cycle homology- and by graph homology.

Angles, scales and parametric renormalization
Francis Brown, Dirk Kreimer, Lett.Math.Phys. 103 (2013) 933-1007
We decompose renormalized Feynman rules according to the scale and angle dependence of amplitudes. We use parametric representations such that the resulting amplitudes can be studied in algebraic geometry.

Properties of the corolla polynomial of a 3-regular graph
Dirk Kreimer, Karen Yeats, Electr.J.Comb., Volume 20, Issue 1 (2013) P41
We investigate combinatorial properties of a graph polynomial indexed by half-edges of a graph which was introduced recently to understand the connection between Feynman rules for scalar field theory and Feynman rules for gauge theory. We investigate the new graph polynomial as a stand-alone object.

Feynman amplitudes and Landau singularities for 1-loop graphs
Spencer Bloch, Dirk Kreimer, Commun.Num.Theor.Phys. 4 (2010) 709-753.
We use mixed Hodge structures to investigate Feynman amplitudes as functions of external momenta and masses.

The QCD beta-function from global solutions to Dyson-Schwinger equations
Guillaume van Baalen, Dirk Kreimer, David Uminsky, Karen Yeats, Annals Phys.325:300-324,2010
We study quantum chromodynamics from the viewpoint of untruncated Dyson-Schwinger equations turned to an ordinary differential equation for the gluon anomalous dimension. This nonlinear equation is parameterized by a function P(x) which is unknown beyond perturbation theory. Still, very mild assumptions on P(x) lead to stringent restrictions for possible solutions to Dyson-Schwinger equations. We establish that the theory must have asymptotic freedom beyond perturbation theory and also investigate the low energy regime and the possibility for a mass gap in the asymptotically free theory.

A remark on quantum gravity
Dirk Kreimer, Annals Phys.323:49-60,2008
We discuss the structure of Dyson–Schwinger equations in quantum gravity and conclude in particular that all relevant skeletons are of first order in the loop number. There is an accompanying sub Hopf algebra on gravity amplitudes equivalent to identities between n-graviton scattering amplitudes which generalize the Slavnov Taylor identities. These identities map the infinite number of charges and finite numbers of skeletons in gravity to an infinite number of skeletons and a finite number of charges needing renormalization. Our analysis suggests that gravity, regarded as a probability conserving but perturbatively non-renormalizable theory, is renormalizable after all, thanks to the structure of its Dyson–Schwinger equations.

Mixed Hodge Structures and Renormalization in Physics

Spencer Bloch, Dirk Kreimer, Commun.Num.Theor.Phys.2:637-718,2008.
We relate renormalization in perturbative quantum field theory to the theory of limiting mixed Hodge structures using parametric representations of Feynman graphs.

On Motives Associated to Graph Polynomials
Spencer Bloch, Hélène Esnault, Dirk Kreimer, Commun.Math.Phys. 267 (2006) 181-225
The appearance of multiple zeta values in anomalous dimensions and β-functions of renormalizable quantum field theories has given evidence towards a motivic interpretation of these renormalization group functions. In this paper we start to hunt the motive, restricting our attention to a subclass of graphs in four dimensional scalar field theory which give scheme independent contributions to the above functions.

Renormalization in quantum field theory and the Riemann-Hilbert problem II: the β-function, diffeomorphisms and the renormalization group
Alain Connes, Dirk Kreimer, Commun.Math.Phys. 216 (2001) 215-241
We showed in part I (hep-th/9912092) that the Hopf algebra 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 $\gamma_+(\ve)$ of the Riemann-Hilbert decomposition $\gamma_-(\ve)^{-1}\gamma_+(\ve)$ of the loop $\gamma(\ve)\in G$ 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 g0=gZ1Z−3/23 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 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 $\gamma_-(\ve)$ 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.

Exact solutions of Dyson-Schwinger equations for iterated one-loop integrals and propagator-coupling duality
D.J.Broadhurst, D.Kreimer, Nucl.Phys. B600 (2001) 403-422
The Hopf algebra of undecorated rooted trees has tamed the combinatorics of perturbative contributions, to anomalous dimensions in Yukawa theory and scalar ϕ3 theory, from all nestings and chainings of a primitive self-energy subdivergence. Here we formulate the nonperturbative problems which these resummations approximate. For Yukawa theory, at spacetime dimension d=4, we obtain an integrodifferential Dyson-Schwinger equation and solve it parametrically in terms of the complementary error function. For the scalar theory, at d=6, the nonperturbative problem is more severe; we transform it to a nonlinear fourth-order differential equation. After intensive use of symbolic computation we find an algorithm that extends both perturbation series to 500 loops in 7 minutes. Finally, we establish the propagator-coupling duality underlying these achievements making use of the Hopf structure of Feynman diagrams.

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