New mathematical structures in renormalizable quantum field theories

Abstract

Computations in renormalizable perturbative quantum field theories reveal mathematical structures which go way beyond the formal structure which is usually taken as underlying quantum field theory. We review these new structures and the role they can play in future developments.

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…”

“…There is a strong analogy here to the Bogoliubov R operation in renormalization theory [4,7], thanks to the fact that Li and L have matching asymptotic behaviour for | z |→ ∞. Indeed, if we let R be defined to map the character Li to the character L, R(Li) = L, and P the projector into the augmentation ideal of H, then…”

“…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].…”

“…In general, this motivates an approach to quantum field theory which is utterly based on the Hopf and Lie algebra structures of graphs [4].…”

Hopf Algebra Structure of Renormalizable Quantum Field Theory

“…The unrenormalized but regularized Feynman rules φ assign to a graph a function (Γ [0] and Γ [1]int being the set of vertices v and internal edges e of Γ)…”