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Over the last century quantum field theory has made a significant impact on the formulation and solution of mathematical problems and inspired powerful advances in pure mathematics. However, most accounts are written by physicists, and mathematicians struggle to find clear definitions and statements of the concepts involved. This graduate-level introduction presents the basic ideas and tools from quantum field theory to a mathematical audience. Topics include classical and quantum mechanics, classical field theory, quantization of classical fields, perturbative quantum field theory, renormalization, and the standard model. The material is also accessible to physicists seeking a better understanding of the mathematical background, providing the necessary tools from differential geometry on such topics as connections and gauge fields, vector and spinor bundles, symmetries and group representations.

Quantum field theory --- Field theory (Physics) --- Classical field theory --- Continuum physics --- Physics --- Continuum mechanics --- Relativistic quantum field theory --- Quantum theory --- Relativity (Physics) --- Mathematics.

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Electromagnetic fields. --- Field theory (Physics) --- Classical field theory --- Continuum physics --- Physics --- Continuum mechanics --- Fields, Electromagnetic --- Magnetic fields --- Electric fields --- Quantum mechanics. Quantumfield theory --- Classical mechanics. Field theory --- Electromagnetic fields --- Champs électromagnétiques --- Champs, Théorie des (Physique)

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Knowledge of the renormalization group and field theory is a key part of physics, and is essential in condensed matter and particle physics. Written for advanced undergraduate and beginning graduate students, this textbook provides a concise introduction to this subject. The textbook deals directly with the loop expansion of the free energy, also known as the background field method. This is a powerful method, especially when dealing with symmetries, and statistical mechanics. In focussing on free energy, the author avoids long developments on field theory techniques. The necessity of renormalization then follows.

Phase transformations (Statistical physics) --- Field theory (Physics) --- Statistical mechanics. --- Mechanics --- Mechanics, Analytic --- Quantum statistics --- Statistical physics --- Thermodynamics --- Classical field theory --- Continuum physics --- Physics --- Continuum mechanics --- Phase changes (Statistical physics) --- Phase transitions (Statistical physics) --- Phase rule and equilibrium

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This is a masterly exposition and an encyclopedic presentation of the theory of hyperbolic conservation laws. It illustrates the essential role of continuum thermodynamics in providing motivation and direction for the development of the mathematical theory while also serving as the principal source of applications. The reader is expected to have a certain mathematical sophistication and to be familiar with (at least) the rudiments of analysis and the qualitative theory of partial differential equations, whereas prior exposure to continuum physics is not required. The target group of readers would consist of (a) experts in the mathematical theory of hyperbolic systems of conservation laws who wish to learn about the connection with classical physics; (b) specialists in continuum mechanics who may need analytical tools; (c) experts in numerical analysis who wish to learn the underlying mathematical theory; and (d) analysts and graduate students who seek introduction to the theory of hyperbolic systems of conservation laws. New to the 3rd edition is an account of the early history of the subject, spanning the period between 1800 to 1957. Also new is a chapter recounting the recent solution of open problems of long standing in classical aerodynamics. Furthermore, the presentation of a number of topics in the previous edition has been revised and brought up to date, and the collection of applications has been substantially enriched. The bibliography, also expanded and updated, now comprises over fifteen hundred titles. From the reviews of the 2nd edition: "The author is known as one of the leading experts in the field. His masterly written book is, surely, the most complete exposition in the subject." Evgeniy Panov, Zentralblatt MATH "This book is sure to convince every reader that working in this area is challenging, enlightening, and joyful." Katarina Jegdic, SIAM Review.

Conservation laws (Physics). --- Differential equations, Hyperbolic. --- Field theory (Physics). --- Conservation laws (Physics) --- Differential equations, Hyperbolic --- Field theory (Physics) --- Mathematics --- Physics --- Nuclear Physics --- Calculus --- Physical Sciences & Mathematics --- Classical field theory --- Continuum physics --- Hyperbolic differential equations --- Mathematics. --- Partial differential equations. --- Mechanics. --- Thermodynamics. --- Continuum mechanics. --- Structural mechanics. --- Partial Differential Equations. --- Continuum Mechanics and Mechanics of Materials. --- Structural Mechanics. --- Continuum mechanics --- Differential equations, Partial --- Physical laws

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On November 3, 2005, Alexander Vasil’evich Kazhikhov left this world, untimely and unexpectedly. He was one of the most in?uential mathematicians in the mechanics of ?uids, and will be remembered for his outstanding results that had, and still have, a c- siderablysigni?cantin?uenceinthe?eld.Amonghis manyachievements,werecall that he was the founder of the modern mathematical theory of the Navier-Stokes equations describing one- and two-dimensional motions of a viscous, compressible and heat-conducting gas. A brief account of Professor Kazhikhov’s contributions to science is provided in the following article “Scienti?c portrait of Alexander Vasil’evich Kazhikhov”. This volume is meant to be an expression of high regard to his memory, from most of his friends and his colleagues. In particular, it collects a selection of papers that represent the latest progress in a number of new important directions of Mathematical Physics, mainly of Mathematical Fluid Mechanics. These papers are written by world renowned specialists. Most of them were friends, students or colleagues of Professor Kazhikhov, who either worked with him directly, or met him many times in o?cial scienti?c meetings, where they had the opportunity of discussing problems of common interest.

Fluid mechanics -- Mathematics. --- Fluid mechanics. --- Kazhikhov, Alexander V. --- Fluid mechanics --- Engineering & Applied Sciences --- Physics --- Physical Sciences & Mathematics --- Applied Mathematics --- Physics - General --- Mathematics --- Continuum mechanics. --- Mathematics. --- Mechanics of continua --- Hydromechanics --- Physics. --- Continuum physics. --- Mechanics. --- Fluids. --- Classical Continuum Physics. --- Fluid- and Aerodynamics. --- Elasticity --- Mechanics, Analytic --- Field theory (Physics) --- Continuum mechanics --- Classical and Continuum Physics. --- Classical Mechanics. --- Classical mechanics --- Newtonian mechanics --- Dynamics --- Quantum theory --- Classical field theory --- Continuum physics --- Hydraulics --- Mechanics --- Hydrostatics --- Permeability