Cambridge University Press, 2007. — 323 p. — (Cambridge Studies in Advanced Mathematics 127). — ISBN 0521115779.In this book we attempt to present some of the main ideas of Quantum Field Theory (QFT) for a mathematical audience. As mathematicians, we feel deeply impressed – and at times quite overwhelmed – by the enormous breadth and scope of this beautiful and most successful of physical theories. Throughout centuries, Mathematics has always provided Physics with a variety of tools, oftentimes on demand, for the solution of fundamental physical problems. But the past century has witnessed a new trend in the opposite direction: the strong impact of physical ideas not only in the formulation, but in the very solution to mathematical problems. Some of the most wellknown examples of such impact are (1) the use of renormalization ideas by Feigenbaum, Coullet and Tresser in the study of universality phenomena in one-dimensional dynamics; (2) the use of classical Yang-Mills theory by Donaldson to devise invariants for 4-dimensional manifolds; (3) the use of quantum Yang-Mills by Seiberg and Witten in the construction of new invariants for 4-manifolds; (4) the use of quantum theory in three dimensions leading to the Jones-Witten and Vassiliev invariants. There are several other examples.Classical Mechanics Quantum mechanics Relativity, the Lorentz group and Dirac’s equation Fiber Bundles, Connections and Representations Classical Field Theory Quantization of Classical Fields Perturbative Quantum Field Theory Renormalization The Standard Model Appendixes Hilbert spaces and Operators C∗ Algebras and Spectral Theory
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