Renormalization in Combinatorially Non-Local Field Theories: The Hopf Algebra of 2-Graphs

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 sh...

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Bibliographic Details
Published in:Mathematical physics, analysis, and geometry Vol. 24; no. 2
Main Author: Thürigen, Johannes
Format: Journal Article
Language:English
Published: Dordrecht Springer Netherlands 01.06.2021
Springer Nature B.V
Subjects:
Algebra
Analysis
Apexes
Applications of Mathematics
Feynman diagrams
Field theory
Geometry
Group Theory and Generalizations
Mathematical and Computational Physics
Matrix algebra
Physics
Physics and Astronomy
Quantum field theory
Quantum gravity
Quantum theory
Tensors
Theoretical
16T30
05C10
81T15
Non-local field theory
MSC2020: 81T18
81T32
Random geometry
16T05
Renormalization
Hopf algebra
81V17
ISSN:1385-0172, 1572-9656
Online Access:Get full text
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Summary: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.
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ISSN:1385-0172
1572-9656
DOI:10.1007/s11040-021-09390-6