Related papers: Periods and Feynman amplitudes
The role of hyperlogarithms and multiple zeta values (and their generalizations) in Feynman amplitudes is being gradually recognized since the mid 1990's. The present lecture provides a concise introduction to a fast developing subject that…
In this talk we discuss mathematical structures associated to Feynman graphs. Feynman graphs are the backbone of calculations in perturbative quantum field theory. The mathematical structures -- apart from being of interest in their own…
Periods are numbers represented as integrals of rational functions over algebraic domains. A survey of their elementary properties is provided. Examples of periods includes Feynman Integrals from Quantum Physics and Multiple Zeta Values…
Perturbative scattering amplitudes in gauge theories have remarkable simplicity and hidden infinite dimensional symmetries that are completely obscured in the conventional formulation of field theory using Feynman diagrams. This suggests…
I analyze the algebraic patterns underlying the structure of scattering amplitudes in quantum field theory. I focus on the decomposition of amplitudes in terms of independent functions and the systems of differential equations the latter…
We investigate from a mathematical perspective how Feynman amplitudes appear in the low-energy limit of string amplitudes. In this paper, we prove the convergence of the integrands. We derive this from results describing the asymptotic…
There is a fruitful interplay between algebraic geometry on the one side and perturbative quantum field theory on the other side. I review the main relevant mathematical concepts of periods, Hodge structures and Picard-Fuchs equations and…
Feynman diagrams are the foremost tool in the perturbative study of quantum field theory. In gauge theories, the full potential of this tool is revealed when it is combined with the Slavanov-Taylor identities associated with the local gauge…
This article explores an operational model for transition amplitudes between measurements proposed by Goyal et al. within the quantum reconstruction program. To classify suitable amplitude algebras, we distinguish mathematical axioms,…
We discuss a remarkable new approach initiated by Cachazo, Svrcek and Witten for calculating gauge theory amplitudes. The formalism amounts to an effective scalar perturbation theory which in many cases offers a much simpler alternative to…
These lecture notes provide a self-contained introduction to Euler integrals, which are frequently encountered in applications. In particle physics, they arise as Feynman integrals or string amplitudes. Our four selected topics demonstrate…
We use mixed Hodge structures to investigate Feynman amplitudes as functions of external momenta and masses.
Hardy's paradox is analysed within Feynman's formulation of quantum mechanics. A transition amplitude is represented as a sum over virtual paths which different intermediate measurements convert into different sets of real pathways.…
This article introduces moduli spaces of coloured graphs on which Feynman amplitudes can be viewed as 'discrete' volume densities. The basic idea behind this construction is that these moduli spaces decompose into disjoint unions of open…
Physical properties of scattering amplitudes are mapped to the Riemann zeta function. Specifically, a closed-form amplitude is constructed, describing the tree-level exchange of a tower with masses $m_n^2 = \mu_n^2$, where…
We describe the origins of recurrence relations between field theory amplitudes in terms of the construction of Feynman diagrams. In application we derive recurrence relations for the amplitudes of QED which hold to all loop orders and for…
Single-scale Feynman diagrams yield integrals that are periods, namely projective integrals of rational functions of Schwinger parameters. Algebraic geometry may therefore inform us of the types of number to which these integrals evaluate.…
In these lectures I will give an introduction to Feynman integrals. In the first part of the course I review the basics of the perturbative expansion in quantum field theories. In the second part of the course I will discuss more advanced…
In these lectures we discuss some of the mathematical structures that appear when computing multi-loop Feynman integrals. We focus on a specific class of special functions, the so-called multiple polylogarithms, and discuss introduce their…
A survey is given on the present status of analytic calculation methods and the mathematical structures of zero- and single scale Feynman amplitudes which emerge in higher order perturbative calculations in the Standard Model of elementary…