Factorization in Quantum Field Theory: An Exercise in Hopf Algebras and Local Singularities

Abstract

I discuss the role of Hochschild cohomology in Quantum Field Theory with particular emphasis on Dyson-Schwinger equations. Talk given at Frontiers in Number Theory, Physics and Geometry; Les Houches, March 2003 [arXiv:hep-th/0306020].1 Furthermore, the structure of the Dyson-Schwinger equations in gauge theories eliminates overlapping divergences altogether upon use of gauge invariance [13,14].

The residues of quantum field theory - numbers we should know

“…As we are interested in the limit ε → 0, it is consistent to maintain only coefficients which have a pole or finite part in ε as we did above. This gives a second grading which, in accord with quantum field theory [4], is provided by the augmentation degree [4,5]. Note that this is consistent with what we did in the previous section, upon noticing that…”

“…This connection between Hochschild closedness and locality is universal in quantum field theory [5,10], and will be discussed in detail in [1].…”

“…While it is fascinating to import quantum field theory methods into number theory, which suggest to resum perturbation theory amplitudes of a MS scheme making use of the Birkhoff decomposition of [13] combined with progress thanks to Ramis and others in resumation of asymptotic series, as beautifully suggested recently [14], the problem is unfortunately much harder still for a renormalizable quantum field theory. We indeed have almost no handle outside perturbation theory on such schemes, while on the other hand the NP solution of DSE with physical side-constraints like F (α, 1) = 1 is amazingly straightforward and resums perturbation theory naturally once one has recognized the role of the Hochschild closed 1-cocycles [4,11,5]. Such an approach will be exhibited in detail in [1].…”

“…The following discussion was essentially given already in [5] and is repeated here with special emphasis on the analogies pointed out in the previous two sections.…”

“…Altogether, this allows to summarize the structure in QFT as a vast generalization of the introductory study in the previous sections. It turns out that even the quantum structure of gauge theories can be understood along these lines [5]. A full discussion is upcoming [1].…”