The goal of this book is to describe the most powerful methods for evaluating multiloop Feynman integrals that are currently used in practice. This book supersedes the author’s previous Springer book “Evaluating Feynman Integrals” and its textbook version “Feynman Integral Calculus.” Since the publication of these two books, powerful new methods have arisen and conventional methods have been improved on in essential ways. A further qualitative change is the fact that most of the methods and the corresponding algorithms have now been implemented in computer codes which are often public. In comparison to the two previous books, three new chapters have been added: One is on sector decomposition, while the second describes a new method by Lee. The third new chapter concerns the asymptotic expansions of Feynman integrals in momenta and masses, which were described in detail in another Springer book, “Applied Asymptotic Expansions in Momenta and Masses,” by the author. This chapter describes, on the basis of papers that appeared after the publication of said book, how to algorithmically discover the regions relevant to a given limit within the strategy of expansion by regions. In addition, the chapters on the method of Mellin-Barnes representation and on the method of integration by parts have been substantially rewritten, with an emphasis on the corresponding algorithms and computer codes.

The goal of this book is to describe the most powerful methods for evaluating multiloop Feynman integrals that are currently used in practice. This book supersedes the author’s previous Springer book “Evaluating Feynman Integrals” and its textbook version “Feynman Integral Calculus.” Since the publication of these two books, powerful new methods have arisen and conventional methods have been improved on in essential ways. A further qualitative change is the fact that most of the methods and the corresponding algorithms have now been implemented in computer codes which are often public. In comparison to the two previous books, three new chapters have been added: One is on sector decomposition, while the second describes a new method by Lee. The third new chapter concerns the asymptotic expansions of Feynman integrals in momenta and masses, which were described in detail in another Springer book, “Applied Asymptotic Expansions in Momenta and Masses,” by the author. This chapter describes, on the basis of papers that appeared after the publication of said book, how to algorithmically discover the regions relevant to a given limit within the strategy of expansion by regions. In addition, the chapters on the method of Mellin-Barnes representation and on the method of integration by parts have been substantially rewritten, with an emphasis on the corresponding algorithms and computer codes.

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.

The problem of evaluating Feynman integrals over loop momenta has existed from the early days of perturbative quantum field theory. Although a great variety of methods for evaluating Feynman integrals has been developed over a span of more than fifty years, this book is a first attempt to summarize them. Evaluating Feynman Integrals characterizes the most powerful methods, in particular those used for recent, quite sophisticated calculations, and then illustrates them with numerous examples, starting from very simple ones and progressing to nontrivial examples.

Applications of quantum field theoretical methods to gravitational physics, both in the semiclassical and the full quantum frameworks, require a careful formulation of the fundamental basis of quantum theory, with special attention to such important issues as renormalization, quantum theory of gauge theories, and especially effective action formalism. The first part of this graduate textbook provides both a conceptual and technical introduction to the theory of quantum fields. The presentation is consistent, starting from elements of group theory, classical fields, and moving on to the effective action formalism in general gauge theories. Compared to other existing books, the general formalism of renormalization in described in more detail, and special attention paid to gauge theories. This part can serve as a textbook for a one-semester introductory course in quantum field theory. In the second part, we discuss basic aspects of quantum field theory in curved space, and perturbative quantum gravity. More than half of Part II is written with a full exposition of details, and includes elaborated examples of simplest calculations. All chapters include exercises ranging from very simple ones to those requiring small original investigations. The selection of material of the second part is done using the “must-know” principle. This means we included detailed expositions of relatively simple techniques and calculations, expecting that the interested reader will be able to learn more advanced issues independently after working through the basic material, and completing the exercises.

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.

At the fundamental level, the interactions of elementary particles are described by quantum gauge field theory. The quantitative implications of these interactions are captured by scattering amplitudes, traditionally computed using Feynman diagrams. In the past decade tremendous progress has been made in our understanding of and computational abilities with regard to scattering amplitudes in gauge theories, going beyond the traditional textbook approach. These advances build upon on-shell methods that focus on the analytic structure of the amplitudes, as well as on their recently discovered hidden symmetries. In fact, when expressed in suitable variables the amplitudes are much simpler than anticipated and hidden patterns emerge. These modern methods are of increasing importance in phenomenological applications arising from the need for high-precision predictions for the experiments carried out at the Large Hadron Collider, as well as in foundational mathematical physics studies on the S-matrix in quantum field theory. Bridging the gap between introductory courses on quantum field theory and state-of-the-art research, these concise yet self-contained and course-tested lecture notes are well-suited for a one-semester graduate level course or as a self-study guide for anyone interested in fundamental aspects of quantum field theory and its applications. The numerous exercises and solutions included will help readers to embrace and apply the material present ed in the main text.

This open access book bridges a gap between introductory Quantum Field Theory (QFT) courses and state-of-the-art research in scattering amplitudes. It covers the path from basic definitions of QFT to amplitudes, which are relevant for processes in the Standard Model of particle physics. The book begins with a concise yet self-contained introduction to QFT, including perturbative quantum gravity. It then presents modern methods for calculating scattering amplitudes, focusing on tree-level amplitudes, loop-level integrands and loop integration techniques. These methods help to reveal intriguing relations between gauge and gravity amplitudes and are of increasing importance for obtaining high-precision predictions for collider experiments, such as those at the Large Hadron Collider, as well as for foundational mathematical physics studies in QFT, including recent applications to gravitational wave physics.These course-tested lecture notes include numerous exercises with solutions. Requiring only minimal knowledge of QFT, they are well-suited for MSc and PhD students as a preparation for research projects in theoretical particle physics. They can be used as a one-semester graduate level course, or as a self-study guide for researchers interested in fundamental aspects of quantum field theory.

This book introduces readers to a variety of topics surrounding quantum field theory, notably its role in bound states, laser physics, and the gravitational coupling of Dirac particles. It discusses some rather sophisticated concepts based on detailed derivations which cannot be found elsewhere in the literature.It is suitable for undergraduates, graduates, and researchers working on general relativity, relativistic atomic physics, quantum electrodynamics, as well as theoretical laser physics.

Exploring common themes in modern art, mathematics, and science, including the concept of space, the notion of randomness, and the shape of the cosmos. This is a book about art—and a book about mathematics and physics. In Lumen Naturae (the title refers to a purely immanent, non-supernatural form of enlightenment), mathematical physicist Matilde Marcolli explores common themes in modern art and modern science—the concept of space, the notion of randomness, the shape of the cosmos, and other puzzles of the universe—while mapping convergences with the work of such artists as Paul Cezanne, Mark Rothko, Sol LeWitt, and Lee Krasner. Her account, focusing on questions she has investigated in her own scientific work, is illustrated by more than two hundred color images of artworks by modern and contemporary artists. Thus Marcolli finds in still life paintings broad and deep philosophical reflections on space and time, and connects notions of space in mathematics to works by Paul Klee, Salvador Dalí, and others. She considers the relation of entropy and art and how notions of entropy have been expressed by such artists as Hans Arp and Fernand Léger; and traces the evolution of randomness as a mode of artistic expression. She analyzes the relation between graphical illustration and scientific text, and offers her own watercolor-decorated mathematical notebooks. Throughout, she balances discussions of science with explorations of art, using one to inform the other. (She employs some formal notation, which can easily be skipped by general readers.) Marcolli is not simply explaining art to scientists and science to artists; she charts unexpected interdependencies that illuminate the universe.