MATHEMATICAL AND PHYSICAL ASPECTS
QUANTUM FIELD THEORY
is taking place from 05.03 to 30.04 2007
at the Erwin Schrödinger Institute
) in Vienna.
Quantum Field Theory aims at a unifying
description of nature on the basis of the principles of
quantum physics and field theory. Its main success is the
development of a standard model for the theory of
elementary particles which describes physics between the
atomic scale and the highest energies which can be reached in
present experiments. It has, however, turned
out to be also very important in other branches of physics,
in particular for solid state physics. Its mathematical
complexity is enormous and has induced many new developments
in pure mathematics. In its original formulation it was plagued
by divergencies whose removal by renormalization lead to
fantastically precise predictions which could be verified
A full construction of quantum field theories was possible up
to now only for unphysical models. For realistic models one
still has rely on uncontrollable approximations under which
perturbation theory, which constructs the models as formal
power series in the coupling constants, is the most important
Perturbation theory in quantum field theory has been developped as
a rigorous mathematical framework in the fifties-sixties
the work of Hepp, Lehmann, Symanzik, Zimmermann, Steinmann,
Epstein, Glaser, Bogoliubov, Stückelberg
and several others.
These authors found a mathematically consistent method to
construct the perturbation series of quantum field theory at
all orders, thereby making mathematical
sense of the recipes for renormalizations suggested before.
More recently, we experienced a renewed interest in the
foundations of perturbation theory.
Two independent directions were traced. The first took place
around 1996, due to Brunetti
and Fredenhagen, and was centered around the problem of
constructing quantum field theories on curved spacetimes, and
the other started around the end of the
nineties and is due to Connes
and Kreimer and deals with structural insights
into the combinatorics of Feynman
graphs via Hopf algebras. In both cases there arise
direct connections to the
application of quantum field theory
to physics problems. The two settings gave a lot of striking
results and applications that were unforeseen before. In
particular, new aspects of the renormalization group were
List of Topics and Main Open Problems:
- An important progress in the connection to
mathematics has been obtained
recently by Connes and Kreimer. Their idea of
using Hopf algebras in perturbation theory
has led to a mathematical understanding of the forest
formula in momentum space.
Kreimer's original insight originated from a
study of number-theoretic properties
of Feynman integrals and related the amplitudes term by
term in the perturbative expansion to
polylogarithms and motivic theory as well as,
ultimately, to arithmetic geometry.
It turns out that Feynman graphs carry a pre-Lie
algebra structure in a natural manner.
Antysymmetrizing this pre-Lie algebra delivers a Lie
algebra, which provides a universal enveloping
algebra whose dual is a graded commutative Hopf
algebra. It has a recursive coproduct which agrees
with the Bogoliubov recursion in renormalization
theory. While this gives a mathematical framework to
perturbation theory in momentum space Feynman
integrals, it also suggests to incorporate notions
of perturbative quantum field theory into
Indeed, very similar Hopf algebras have emerged in
mathematics in the study of motivic theory and the
polylogarithm through the works of Spencer Bloch,
Pierre Deligne, Sasha Goncharov and Don Zagier. We
ultimately hope that a link can be established
between number theory and quantum field theory in
studying the relevant Hopf algebras and their
relation in detail.
A major problem here is the understanding of the quantum
equations of motion, which are governed by the closed
one-cocycles of the Hopf algebra.
This Hochschild cohomology of perturbation theory
role of locality in momentum and coordinate space approaches.
At the same time, it provides a crucial input
into the function theory of the polylog,
and certainly into a yet to be developed function theory
of quantum field theory amplitudes.
Extensions of these ideas to gauge
theories are under active investigation. Furthermore,
the connection to motivic theory is now under
consideration in a collaboration with Spencer
Bloch, and we expect to present results at the time of the
At the same time, Connes and Marcolli are
incorporating the techniques of arithmetic geometry
into quantum field theories, which utilize again the
underlying Hopf structure in the context of
Tannakian categories, intimately connected again to
the theory of the polylogarithm.
- Another important direction of recent research
has been put forward by
Brunetti and Fredenhagen and refined by Hollands
in a series of papers. The local point of
view is emphasized
and allows a description of perturbation theory on any
background spacetime. Basic to this
approach is the connection with the field of microlocal
analysis pioneered by Radzikowski.
These methods allowed the cited authors to prove
for the first
time, that up to possible additional invariant terms of the
the classification of renormalization in a general
spacetime follows the same rules as
that on Minkowski spacetime.
Actually the theory suggests further
possibilities, the most important of which is
a conceptually new approach to quantum gravity, at least in
the perturbative sense. We expect to deliver results at the
time of the program.
Another direction is that taken by Dütsch and Rehren for
perturbation theory on AdS and
connections with the quantum field theory perspectives on
- Renormalization group ideas seem to be
crucial in both approaches.
Other groups have pioneered different ideas, for
instance by making rigorous the work
of Polchinski and Wilson. However,
a connection between all these
seemingly different perspectives is lacking and an important
issue would be a comparison and attempt to find
a possible unification. A first step in this
direction was done by T. Krajewski and collaborators.
He showed how to use tree-like expansions and the
universal Hopf algebra of rooted trees to
reformulate the Wilson-Polchinski approach.
In the local approach this connection is under investigation
by Brunetti, Dütsch and Fredenhagen.
- An important issue which does not seem to
be fully explored is in the connection between
the Epstein and Glaser method of constructing renormalized
perturbative quantum field theory which is crucial for the
implementation of the locality principle, and the Hopf
algebra combinatorics of
BPHZ renormalization. Initial work has been done
motivated by Fulton and MacPherson's work on
leading to a satisfactory answer in the absence of gravity.
The basic idea is to dualize the celebrated result
of Fulton MacPherson, that the compactification of
configuration spaces is stratified by rooted trees,
to a statement on the continuation of distributions
to the diagonals in configuration space. This is
evidently possible in flat space using again the
Hopf algebra structure of rooted trees.
To understand the connection to the above approach
by Brunetti, Fredenhagen and others
is an important future goal.
Among the participants:
Christopher J. FEWSTER,
Bernard S. KAY,
Walter VAN SUIJLEKOM,
A workshop will be held in the week 26.03 -- 30.03. See the schedule for the talks