Quantum Field Theory I: Basics in Mathematics and Physics

Prof. Dr. Dr. h.c. Eberhard Zeidler works at the Max Planck Institute for Mathematics in the Sciences in Leipzig (Germany). In 1996 he was one of the founding directors of this institute. He is a member of the Academy of Natural Scientists Leopoldina. In 2006 he was awarded the "Alfried Krupp Wissenschaftspreis" of the Alfried Krupp von Bohlen und Halbach-Stiftung. The author wrote the following books. (a) E. Zeidler, Nonlinear Functional Analysis and its Applications, Vols. I-IV, Springer Verlag New York, 1984-1988 (third edition 1998). (b) E. Zeidler, Applied Functional Analysis, Vol. 1: Applications to Mathematical Physics, 2nd edition, 1997, Springer Verlag, New York. (c) E. Zeidler, Applied Functional Analysis, Vol. 2: Main Principles and Their Applications, Springer-Verlag, New York, 1995. (d) E. Zeidler, Oxford Users' Guide to Mathematics, Oxford University Press, 2004 (translated from German).

• Prologue.- Historical Introduction.- Phenomenology of the Standard Model for Elementary Particles.- The Challenge of Different Scales in Nature.- Basic Techniques in Mathematics.- Analyticity.- A Glance at Topology.- Many-Particle Systems in Mathematics and Physics.- Rigorous Finite-Dimensional Magic Formulas of Quantum Field Theory.- Rigorous Finite-Dimensional Perturbation Theory.- Fermions and the Calculus for Grassmann Variables.- Infinite-Dimensional Hilbert Spaces.- Distributions and Green’s Functions.- Distributions and Physics.- Heuristic Magic Formulas of Quantum Field Theory.- Basic Strategies in Quantum Field Theory.- The Response Approach.- The Operator Approach.- Peculiarities of Gauge Theories.- A Panorama of the Literature.

From the reviews: "Quantum field theory combines relativity, quantum mechanics, and many-particle physics to provide a theoretical basis for the most fundamental understanding of our universe. ... it is a fun book for practicing quantum field theorists to browse, and it may be similarly enjoyed by mathematical colleagues. Its ultimate value may lie in encouraging students to enter this challenging interdisciplinary area of mathematics and physics. Summing Up: Recommended. Upper-division undergraduates through faculty." (M. C. Ogilvie, CHOICE, Vol. 44 (9), May, 2007) "This is the first volume of a six volume book on quantum field theory. ... The main object is to explain mathematics for students or researchers in physics how to use them, and to show students and researchers in mathematics how to use them, and to show students and researchers in mathematics how to use mathematics in physics. Translations of different languages used by mathematicians and physicists for the same mathematical objects are also presented." (Akira Asada, Zentralblatt MATH, Vol. 1124 (1), 2008) "This book on quantum field theory, consisting of over a thousand pages, is the first volume of a projected six-volume series. It alone has seventeen chapters and an appendix ... . The appendix covers notation, units, and dimensional analysis. ... Quantum field theory is one of the great intellectual edifices in the history of human thought. ... This volume differs from other books on quantum field theory in its greater emphasis on the interaction of physics with mathematics. ... an impressive work of scholarship." (William G. Faris, SIAM Review, Vol. 50 (2), 2008) "Volume I is divided in three parts. Part 1 has an introductory nature. ... Part II accompanies the reader through a forest of basic techniques in mathematics. ... In Part III, Zeidler presents the essential available information on heuristic magic formulas of quantum field theory. ... The presentation of the material is clear. ... For a scholar aiming to work in quantum field theory ... Zeidler can be a reference text. ... Mathematicians can find a list of topics who are still waiting rigorous treatments." (Paolo Maria Mariano, Meccanica, Vol. 46, 2011)