This volume comprises review papers presented at the Conference on Antidifferentiation and the Calculation of Feynman Amplitudes, held in Zeuthen, Germany, in October 2020, and a few additional invited reviews. The book aims at comprehensive surveys and new innovative results of the analytic integration methods of Feynman integrals in quantum field theory. These methods are closely related to the field of special functions and their function spaces, the theory of differential equations and summation theory. Almost all of these algorithms have a strong basis in computer algebra. The solution of the corresponding problems are connected to the analytic management of large data in the range of Giga- to Terabytes. The methods are widely applicable to quite a series of other branches of mathematics and theoretical physics.

This textbook on Feynman integrals starts from the basics, requiring only knowledge of special relativity and undergraduate mathematics. Feynman integrals are indispensable for precision calculations in quantum field theory. At the same time, they are also fascinating from a mathematical point of view. Topics from quantum field theory and advanced mathematics are introduced as needed. The book covers modern developments in the field of Feynman integrals. Topics included are: representations of Feynman integrals, integration-by-parts, differential equations, intersection theory, multiple polylogarithms, Gelfand-Kapranov-Zelevinsky systems, coactions and symbols, cluster algebras, elliptic Feynman integrals, and motives associated with Feynman integrals. This volume is aimed at a) students at the master's level in physics or mathematics, b) physicists who want to learn how to calculate Feynman integrals (for whom state-of-the-art techniques and computations are provided), and c) mathematicians who are interested in the mathematical aspects underlying Feynman integrals. It is, indeed, the interwoven nature of their physical and mathematical aspects that make Feynman integrals so enthralling.

This book contains a valuable discussion of renormalization through the addition of counterterms to the Lagrangian, giving the first complete proof of the cancellation of all divergences in an arbitrary interaction. The author also introduces a new method of renormalizing an arbitrary Feynman amplitude, a method that is simpler than previous approaches and can be used to study the renormalized perturbation series in quantum field theory.

An Introduction to Quantum Field Theory is a textbook intended for the graduate physics course covering relativistic quantum mechanics, quantum electrodynamics, and Feynman diagrams. The authors make these subjects accessible through carefully worked examples illustrating the technical aspects of the subject, and intuitive explanations of what is going on behind the mathematics. After presenting the basics of quantum electrodynamics, the authors discuss the theory of renormalization and its relation to statistical mechanics, and introduce the renormalization group. This discussion sets the stage for a discussion of the physical principles that underlie the fundamental interactions of elementary particle physics and their description by gauge field theories.

This book presents recent and ongoing research work aimed at understanding the mysterious relation between the computations of Feynman integrals in perturbative quantum field theory and the theory of motives of algebraic varieties and their periods. One of the main questions in the field is understanding when the residues of Feynman integrals in perturbative quantum field theory evaluate to periods of mixed Tate motives. The question originates from the occurrence of multiple zeta values in Feynman integrals calculations observed by Broadhurst and Kreimer. Two different approaches to the subject are described. The first, a OC bottom-upOCO approach, constructs explicit algebraic varieties and periods from Feynman graphs and parametric Feynman integrals. This approach, which grew out of work of BlochOCoEsnaultOCoKreimer and was more recently developed in joint work of Paolo Aluffi and the author, leads to algebro-geometric and motivic versions of the Feynman rules of quantum field theory and concentrates on explicit constructions of motives and classes in the Grothendieck ring of varieties associated to Feynman integrals. While the varieties obtained in this way can be arbitrarily complicated as motives, the part of the cohomology that is involved in the Feynman integral computation might still be of the special mixed Tate kind. A second, OC top-downOCO approach to the problem, developed in the work of Alain Connes and the author, consists of comparing a Tannakian category constructed out of the data of renormalization of perturbative scalar field theories, obtained in the form of a RiemannOCoHilbert correspondence, with Tannakian categories of mixed Tate motives. The book draws connections between these two approaches and gives an overview of other ongoing directions of research in the field, outlining the many connections of perturbative quantum field theory and renormalization to motives, singularity theory, Hodge structures, arithmetic geometry, supermanifolds, algebraic and non-commutative geometry. The text is aimed at researchers in mathematical physics, high energy physics, number theory and algebraic geometry. Partly based on lecture notes for a graduate course given by the author at Caltech in the fall of 2008, it can also be used by graduate students interested in working in this area. Sample Chapter(s). Chapter 1: Perturbative quantum field theory and Feynman diagrams (350 KB). Contents: Perturbative Quantum Field Theory and Feynman Diagrams; Motives and Periods; Feynman Integrals and Algebraic Varieties; Feynman Integrals and GelfandOCoLeray Forms; ConnesOCoKreimer Theory in a Nutshell; The RiemannOCoHilbert Correspondence; The Geometry of DimReg; Renormalization, Singularities, and Hodge Structures; Beyond Scalar Theories. Readership: Graduate students and researchers in mathematical physics and theoretical physics.

This advanced, accessible textbook on effective field theories uses worked examples to bring this important topic to a wider audience.