2015/09/29 by Kurusch Ebrahimi‐Fard, Kurusch Ebrahimi-Fard, Frédéric Patras +1
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Bialgebra #Class (philosophy) #Computer science #Coproduct #Epistemology #Exposition (narrative) #Field (mathematics) #Hopf algebra #Mathematical physics #Mathematics #Object (grammar) #Physics #Planar #Pure mathematics #Quantum #Quantum field theory #Quantum mechanics #Simple (philosophy) #math-ph #math.CO #math.MP
paper · pdf · doi:10.1007/s11467-016-0585-2
published as Springer's Frontiers of Physics, 11(6), 110310 (2016)
arxiv created 2015/09/29 · openalex publication_date 2016/08/12 · arxiv updated 2016/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The aim of this exposition is to provide a detailed description of the use of combinatorial algebra in quantum field theory in the planar setting. Particular emphasis is placed on the relations between different types of planar Green’s functions. The primary object is a Hopf algebra that is naturally defined on variables representing non-commuting sources, and whose coproduct splits into two half-coproducts. The latter give rise to the notion of an unshuffle bialgebra. This setting allows a description of the relation between full and connected planar Green’s functions to be given by solving a simple linear fixed point equation. We also include a brief outline of the consequences of our approach in the framework of ordinary quantum field theory.