Related papers: Some combinatorial aspects of quantum field theory
We relate renormalization in perturbative quantum field theory to the theory of limiting mixed Hodge structures using parametric representations of Feynman graphs.
We formulate the Hopf algebraic approach of Connes and Kreimer to renormalization in perturbative quantum field theory using triangular matrix representation. We give a Rota-Baxter anti-homomorphism from general regularized functionals on…
Recently, the theory of renormalization in perturbative quantum field theory underwent some exciting new developments. Kreimer discovered an organization of Feynman graphs into combinatorial Hopf algebras. The process of renormalization is…
We review the structures imposed on perturbative QFT by the fact that its Feynman diagrams provide Hopf and Lie algebras. We emphasize the role which the Hopf algebra plays in renormalization by providing the forest formulas. We exhibit how…
Matrix field theory is a combinatorially non-local field theory which has recently been found to be a non-trivial but solvable QFT example. To generalize such non-perturbative structures to other models, a more combinatorial understanding…
We develop a general framework for quantum field theory on noncommutative spaces, i.e., spaces with quantum group symmetry. We use the path integral approach to obtain expressions for $n$-point functions. Perturbation theory leads us to…
In recent years enormous progress has been made in perturbative quantum field theory by applying methods of algebraic geometry to parametric Feynman integrals for scalar theories. The transition to gauge theories is complicated not only by…
We review the combinatorial structure of perturbative quantum field theory with emphasis given to the decomposition of graphs into primitive ones. The consequences in terms of unique factorization of Dyson--Schwinger equations into Euler…
We find new properties of the topological transition polynomial of embedded graphs, $Q(G)$. We use these properties to explain the striking similarities between certain evaluations of Bollob\'as and Riordan's ribbon graph polynomial,…
In this article we are introducing combinatorial spectra of graphs, this is a generalization of $H$-Hamiltonian spectra. The main motivation was to made from $H$-Hamiltonian spectra an operation and develop some algebra in this field. An…
In this contribution to the proceedings of the Corfu Summer Institute 2015, I give an overview over quantum field theories on non-commutative Moyal space and renormalization. In particular, I review the new features and challenges one faces…
In this paper we reformulate in a simpler way the combinatoric core of constructive quantum field theory We define universal rational combinatoric weights for pairs made of a graph and one of its spanning trees. These weights are nothing…
We propose a general formulation of perturbative quantum field theory on (finitely generated) projective modules over noncommutative algebras. This is the analogue of scalar field theories with non-trivial topology in the noncommutative…
A noncommutative Feynman graph is a ribbon graph and can be drawn on a genus $g$ 2-surface with a boundary. We formulate a general convergence theorem for the noncommutative Feynman graphs in topological terms and prove it for some classes…
We consider the number \bar N(q) of points in the projective complement of graph hypersurfaces over \F_q and show that the smallest graphs with non-polynomial \bar N(q) have 14 edges. We give six examples which fall into two classes. One…
We study a combinatorial model of the quantum scalar field with polynomial potential on a graph. In the first quantization formalism, the value of a Feynman graph is given by a sum over maps from the Feynman graph to the spacetime graph…
Recent elegant work on the structure of Perturbative Quantum Field Theory (PQFT) has revealed an astonishing interplay between analysis(Riemann Zeta functions), topology (Knot theory), combinatorial graph theory (Feynman Diagrams) and…
We give a parametric representation of the effective noncommutative field theory derived from a $\kappa$-deformation of the Ponzano-Regge model and define a generalized Kirchhoff polynomial with $\kappa$-correction terms, obtained in a…
We give arguments for the necessity to employ Quantum Clifford Hopf Gebras in quantum field theory. The role of the antipode is examined, Feynman diagrams are re-interpreted as tangles of graphical calculus. Regularization due to the design…
The paper deals with some spectral properties of (mostly infinite) quantum and combinatorial graphs. Quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics, nanotechnology, optics, and…