Feynman Integrals: A Comprehensive Treatment for Students and Researchers

This textbook provides a comprehensive introduction to Feynman integrals, covering topics such as representations, integration by parts, differential equations, intersection theory, multiple polylogarithms, Gelfand-Kapranov-Zelevinsky systems, actions and symbols, cluster algebras, elliptic Feynman integrals, and motives associated with Feynman integrals. It is aimed at students at the master's level in physics or mathematics, physicists who want to learn how to calculate Feynman integrals, and mathematicians who are interested in the mathematical aspects underlying Feynman integrals.

Format: Paperback / softback
Length: 857 pages
Publication date: 12 June 2023
Publisher: Springer Nature Switzerland AG

This comprehensive textbook on Feynman integrals provides a solid foundation for those seeking to delve into the realm of quantum field theory. Starting from the basics, it assumes only a knowledge of special relativity and undergraduate mathematics, making it accessible to a wide range of readers. Feynman integrals play a crucial role in precision calculations within this field, and their mathematical beauty adds an extra layer of fascination. As the text progresses, it introduces topics from quantum field theory and advanced mathematics as necessary, ensuring a smooth and coherent understanding of the subject matter.

The book encompasses modern developments in the field of Feynman integrals, covering a wide range of topics. These include representations of Feynman integrals, integration-by-parts, differential equations, intersection theory, multiple polylogarithms, Gelfand-Kapranov-Zelevinsky systems, actions and symbols, cluster algebras, elliptic Feynman integrals, and motives associated with Feynman integrals. Each chapter is carefully crafted to provide a clear and concise explanation, accompanied by illustrative examples and exercises to reinforce the concepts.

This textbook is designed to cater to a diverse audience. It is an invaluable resource for students at the master's level in physics or mathematics, offering them a comprehensive introduction to Feynman integrals and their applications. Physicists who wish to learn how to calculate Feynman integrals will find this book particularly useful, as it provides state-of-the-art techniques and computations to facilitate their understanding. Additionally, mathematicians with an interest in the mathematical aspects underlying Feynman integrals will find this text intriguing, as it delves into the intricate connections between physics and mathematics.

What truly makes Feynman integrals captivating is their interwoven nature, where the physical and mathematical aspects merge seamlessly. This fusion of ideas opens up new avenues for exploration and understanding, pushing the boundaries of knowledge and paving the way
The book covers modern developments in the field of Feynman integrals. Topics included are:

Representations of Feynman Integrals: This chapter explores the various representations of Feynman integrals, including the path integral representation, the Feynman parameterization, and the integrand representation. It also discusses the concept of Feynman diagrams and their role in calculating Feynman integrals.

Integration-by-Parts: This chapter introduces the integration-by-parts method, which is a powerful tool for calculating Feynman integrals. It covers the basic principles of integration-by-parts, including the substitution method, the residue theorem, and the integration by parts formula.

Differential Equations: This chapter discusses the use of differential equations in Feynman integrals. It covers the methods of solving differential equations, including the method of characteristics, the Euler-Maclaurin summation formula, and the method of steepest descent. It also discusses the concept of Green's functions and their application in solving differential equations.

Intersection Theory: This chapter introduces the concept of intersection theory in Feynman integrals. It covers the theory of intersection numbers, the theory of Poincaré-Lefschetz numbers, and the theory of intersection operators. It also discusses the concept of scattering amplitudes and their application in calculating Feynman integrals.

Multiple Polylogarithms: This chapter discusses the use of multiple polylogarithms in Feynman integrals. It covers the theory of multiple polylogarithms, including the theory of symmetric functions, the theory of hypergeometric functions, and the theory of rational functions. It also discusses the concept of generating functions and their application in calculating Feynman integrals.

Gelfand-Kapranov-Zelevinsky Systems: This chapter introduces the Gelfand-Kapranov-Zelevinsky (GKZ) systems, which are a class of differential equations that arise in the study of Feynman integrals. It covers the theory of GKZ systems, including the concept of GKZ equations, the method of reduction, and the method of averaging. It also discusses the concept of GKZ equations and their application in calculating Feynman integrals.

Actions and Symbols: This chapter introduces the concept of actions and symbols in Feynman integrals. It covers the theory of actions and symbols, including the concept of gauge transformations, the concept of gauge fixing, and the concept of gauge symmetry. It also discusses the concept of actions and symbols and their application in calculating Feynman integrals.

Cluster Algebras: This chapter introduces the concept of cluster algebras in Feynman integrals. It covers the theory of cluster algebras, including the concept of cluster algebras, the concept of cluster variables, and the concept of cluster operators. It also discusses the concept of cluster algebras and their application in calculating Feynman integrals.

Elliptic Feynman Integrals: This chapter introduces the concept of elliptic Feynman integrals in Feynman integrals. It covers the theory of elliptic Feynman integrals, including the theory of elliptic functions, the theory of elliptic integrals, and the theory of elliptic operators. It also discusses the concept of elliptic Feynman integrals and their application in calculating Feynman integrals.

Motives Associated with Feynman Integrals: This chapter introduces the concept of motives associated with Feynman integrals. It covers the theory of motives, including the concept of motives, the concept of motivic integrals, and the concept of motivic polylogarithms. It also discusses the concept of motives and their application in calculating 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.

Weight: 1312g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030995607
Edition number: 1st ed. 2022