## ESI programMATHEMATICAL AND PHYSICAL ASPECTS ofPERTURBATIVE APPROACHES toQUANTUM FIELD THEORY
Scientific Aims: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 experimentally. 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 one. Perturbation theory in quantum field theory has been developped as a rigorous mathematical framework in the fifties-sixties thanks to 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 uncovered. 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 mathematics. 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 Hochschild one-cocycles of the Hopf algebra. This Hochschild cohomology of perturbation theory illuminates the 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 program. 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 and Wald 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 metric, 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 holography.
- 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 configuration spaces, 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:Christoph BERGBAUER, David BROADHURST, Francis BROWN, Jacques BROS, Kurusch EBRAHIMI-FARD, Henry EPSTEIN, Bertfried FAUSER, Christopher J. FEWSTER, Herbert GANGL, John GRACEY, Dan GRIGORE, Stefan HOLLANDS, Bernard S. KAY, Pjotr MARECKY, Gerardo MORSELLA, Ugo MOSCHELLA, Heiner OLBERMANN, Nicola PINAMONTI, Karl-Henning REHREN, Abhijnan REJ, Giuseppe RUZZI, Manfred SALMHOFER, Guenther SCHARF, Klaus SIBOLD, Oleg TARASOV, Ivan TODOROV, Fabian VIGNES-TOURNERET, Walter VAN SUIJLEKOM, Rainer VERCH, Stefan WEINZIERL, Eberhard ZEIDLER Workshop:A workshop will be held in the week 26.03 -- 30.03. See the schedule for the talks |