Abstract
Renormalization in perturbative quantum field theory is based on a Hopf algebra of Feynman diagrams. A precondition for this is locality. Therefore one might suspect that non-local field theories such as matrix or tensor field theories cannot benefit from a similar algebraic understanding. Here I show that, on the contrary, perturbative renormalization of a broad class of such field theories is based in the same way on a Hopf algebra. Their interaction vertices have the structure of graphs. This gives the necessary concept of locality and leads to Feynman diagrams defined as “2-graphs” which generate the Hopf algebra. These results set the stage for a systematic study of perturbative renormalization as well as non-perturbative aspects, e.g. Dyson-Schwinger equations, for a number of combinatorially non-local field theories with possible applications to random geometry and quantum gravity.
Highlights
Locality in QFT allows to perform perturbative renormalization order by order
The renormalization operation subtracting the divergences of this subdiagram
For perturbatively renormalizable field theories, it is the possibility to find for each divergent subgraph a counter term of the same external structure which allows to subtract the divergent part of an amplitude
Summary
Locality in QFT allows to perform perturbative renormalization order by order. Subtracting divergences is described mathematically by a Hopf algebra of Feynman diagrams [1,2,3,4]. The renormalization operation subtracting the divergences of this subdiagram
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