Preprints of our group
Publications (including arXiv)
The list of publications of Dirk Kreimer can be found here .
A quantum field theory
is specified by the one-particle irreducible (1PI) Green functions of the theory.
For a renormalizable theory, a finite number of amplitudes needs to be fixed by experiment
to determine the theory completely.
The mathematical structure of the corresponding set of 1PI Green functions
upon the study of the underlying Hopf algebra structure
of a perturbative expansion in the coupling.
Before we describe the main results following this approach, we point out some introductory material.
A good source reviewing the combinatorial approach to renormalization is
Dominique Manchon's review of my work
Having digested that, you might consider my paper with Christoph Bergbauer in
This clarifies the structure of equations of motion originating from Hochschild cohomology
or Lie algebra cohomology, for that matter.
This has consequences for quantum field theory beyond perturbation theory.
Look at Karen Yeats thesis
for an introduction.
Apart from this mainly combinatorical aspects,
there are hard analytic questions in quantum field theory,
which connect it to beautiful problems in algebraic geometry.
On the interface of these two aspects lives my recent work with
on a connection between limiting mixed Hodge structures and renormalization
There, you will find also the necessary details on graph polynomials
and the core Hopf algebra.
An account of the connection between periods and values of Feynman integrals from a mathematicians perspective,
which is actually quite illuminating to a physicist, is given by Francis Brown
in his paper in Commun.Math.Phys.287 (2009) DOI 10.1007/s00220-009-0740-5
So highlights of this approach then include:
i) We have learned recently how to make progress with quantum field theory beyond perturbation theory
in such an approach, see beta function QED
and references there,
a collaboration with Karen Yeats
Guillaume van Baalen
and David Uminsky
Slavnov--Taylor identities find a natural explanation
in this approach, see for example Anatomy of a gauge theory
and Walter van Suijlekom
. From here, nice connections to the core Hopf algebra, to unitarity of the
S-matrix and recursive relations between multi-leg tree-level ampltudes emerge
My very recent paper
with Walter is a first step in this direction.
iii) As mentioned above, the renormalization process is captured as a limiting mixed Hodge
in collaboration with Spencer Bloch
Much work remains to connect this to
iv) Amplitudes in beta-functions and anomalous dimensions typically deliver
multiple zeta values
found in collaboration with David Broadhurst
and hence should be of
, as indicated by
and Hélène Esnault
v) quantum gravity has some very interesting features with regard to these Hopf algebra structures,
see this remark
and a short recent review
It suggests recursive relations between n-point and (n-1)-point amplitudes
which need better understanding,
and which should have bearing on gauge theories as well. The above-quoted paper
is a first result illuminating this connection.
vi) a field theoretic prescription of a Brownian gas can lead to a
Poissonian ground state
relating to similar algebraic structures,
a collaboration with Andrea Velenich
and Letitia Cugliandolo
vii) Minimal subtraction schemes in the context of dimensional regularization connect the perturbative expansion
to a Birkhoff decomposition, as worked out in collaboration with Alain Connes
in several papers,
NCG and renormalization
See the work
and Matilde Marcolli
for mathematical consequences.
viii) Matt Szczesny
and Kobi Kremnizer
developed a connection
to Ringel-Hall algebras.
ix) Kurusch Ebrahimi-Fard
and collaborators Dominique Manchon
clarified largely the
of all this.
x) Loic Foissy
Hochschild cohomological classification
of Dyson--Schwinger equations, a connection
established together with
xi) Christoph Bergbauer
, who continues to work on
between Epstein-Glaser renormalization, Hopf algebras and Fulton-MacPherson compactifications.
xii) Other work which relates to this approach includes
aspects of the calcul moulien
of the works of Li Guo
, and also
in collaboration with
demand further clarification.