Quantum Field Theory I: Basics in Mathematics and Physics

This is the first volume of a modern introduction to quantum field theory which addresses both mathematicians and physicists ranging from advanced undergraduate students to professional scientists. The book tries to bridge the existing gap between the different languages used by mathematicians and physicists. For students of mathematics it is shown that detailed knowledge of the physical background helps to motivate the mathematical subjects and to discover interesting interrelationships between quite different mathematical topics. For students of physics, fairly advanced mathematics is presented, which is beyond the usual curriculum in physics. It is the author's goal to present the state of the art of realizing Einstein's dream of a unified theory for the four fundamental forces in the universe (gravitational, electromagnetic, strong, and weak interaction).

From the reviews:

"... 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)

"... 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)

Auteur

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).

Résumé
1 Daß ich erkenne, was die Welt im Innersten zusammenh¨ alt. Faust Concepts without intuition are empty, intuition without concepts is blind. Immanuel Kant (17241804) The greatest mathematicians like Archimedes, Newton, and Gauss have always been able to combine theory and applications into one. Felix Klein (18491925) The present comprehensive introduction to the mathematical and physical aspects of quantum ?eld theory consists of the following six volumes: Volume I: Basics in Mathematics and Physics Volume II: Quantum Electrodynamics Volume III: Gauge Theory Volume IV: Quantum Mathematics Volume V: The Physics of the Standard Model Volume VI: Quantum Gravity and String Theory. Since ancient times, both physicists and mathematicians have tried to und- stand the forces acting in nature. Nowadays we know that there exist four fundamental forces in nature: • Newton's gravitational force, • Maxwell's electromagnetic force, • the strong force between elementary particles, and • the weak force between elementary particles (e.g., the force responsible for the radioactive decay of atoms). In the 20th century, physicists established two basic models, namely, • the Standard Model in cosmology based on Einstein's theory of general relativity, and • the Standard Model in elementary particle physics based on gauge theory. 1 So that I may perceive whatever holds the world together in its inmost folds.

Contenu
Preface.- Part I. Introduction.- Prologue.- 1. Historical Introduction.- 2. Phenomenology of the Standard Model for Elementary Particles.- 3. The Challenge of Different Scales in Nature.- Part II. Basic Techniques in Mathematics.- 4. Analyticity.- 5. A Glance at Topology.- 6. Many-Particle Systems in Mathematics and Physics.- 7. Rigorous Finite-Dimensional Magic Formulas of Quantum Field Theory.- 8. Rigorous Finite-Dimensional Perturbation Theory.- 9. Fermions and the Calculus for Grassmann Variables.- 10. Infinite-Dimensional Hilbert Spaces.- 11. Distributions and Green's Functions.- 12. Distributions and Physics.- Part III. Heuristic Magic Formulas of Quantum Field Theory.- 13. Basic Strategies in Quantum Field Theory.- 14. The Response Approach.- 15. The Operator Approach.- 16. Peculiarities of Gauge Theories.- 17. A Panorama of the Literature.- Appendix.- Epilogue.- References.- List of Symbols.- Index.