Graphs in Perturbation Theory

Abstract

This thesis provides an extension of the work of Dirk Kreimer and Alain Connes on the Hopf algebra structure of Feynman graphs and renormalization to general graphs. Additionally, an algebraic structure of the asymptotics of formal power series with factorial growth, which is compatible with the Hopf algebraic structure, will be introduced. The Hopf algebraic structure on graphs permits the explicit enumeration of graphs with constraints for the allowed subgraphs. In the case of Feynman diagrams a lattice structure, which will be introduced, exposes additional unique properties for physical quantum field theories. The differential ring of factorially divergent power series allows the extraction of asymptotic results of implicitly defined power series with vanishing radius of convergence. Together both structures provide an algebraic formulation of large graphs with constraints on the allowed subgraphs. These structures are motivated by and used to analyze renormalized zero-dimensional quantum field theory at high orders in perturbation theory. As a pure application of the Hopf algebra structure, an Hopf algebraic interpretation of the Legendre transformation in quantum field theory is given. The differential ring of factorially divergent power series will be used to solve two asymptotic counting problems in combinatorics: The asymptotic number of connected chord diagrams and the number of simple permutations. For both asymptotic solutions, all order asymptotic expansions are provided as generating functions in closed form. Both structures are combined in an application to zero-dimensional quantum field theory. Various quantities are explicitly given asymptotically in the zero-dimensional version of ϕ 3 , ϕ 4 , QED, quenched QED and Yukawa theory with their all order asymptotic expansions.

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

“…In this work I show that the Hopf-algebraic structure underlying renormalization in cNLFT is very general and independent of any specific theory, along a similar logic as for local field theory [8]. Renormalizablity of various cNLFTs is known as for example Grosse/Wulkenhaar's non-commutative field theory [16,17] related to Kontsevich's matrix model [15,32], tensor-field models [20,22] and group field theories [23].…”

“…For perturbative field theory it is convenient to define graphs in terms of half-edges associated to vertices which are then pairwise combined into edges [8][9][10]. This captures nicely the appearance of self-loops, multi-edges and external legs in Feynman diagrams.…”

“…The two definitions in terms of either property (4a) or (4b) are equivalent [8] since an involution (4a) partitions H into sets of either one or two elements where the latter are in one-to-one correspondence to edge sets (4b).…”

“…Algebraic structure can be defined on 2-graphs in the same way as for 1-graphs [8].…”

“…It is well known that this works in many cases for local, i.e. point-like, interactions [1][2][3][4][5][6][7][8][9][10]. But there are also examples of renormalizable field theories with certain "non-local" interactions for which a Connes-Kreimer type Hopf algebra can be found [11][12][13][14].…”