Jan Cnops

An Introduction to Dirac Operators on Manifolds

• Broschiertes Buch

Produktbeschreibung

Dirac operators play an important role in several domains of mathematics and physics, for example: index theory, elliptic pseudodifferential operators, electromagnetism, particle physics, and the representation theory of Lie groups. In this essentially self-contained work, the basic ideas underlying the concept of Dirac operators are explored. Starting with Clifford algebras and the fundamentals of differential geometry, the text focuses on two main properties, namely, conformal invariance, which determines the local behavior of the operator, and the unique continuation property dominating its global behavior. Spin groups and spinor bundles are covered, as well as the relations with their classical counterparts, orthogonal groups and Clifford bundles. The chapters on Clifford algebras and the fundamentals of differential geometry can be used as an introduction to the above topics, and are suitable for senior undergraduate and graduate students. The other chapters are also accessible at this level so that this text requires very little previous knowledge of the domains covered. The reader will benefit, however, from some knowledge of complex analysis, which gives the simplest example of a Dirac operator. More advanced readers---mathematical physicists, physicists and mathematicians from diverse areas---will appreciate the fresh approach to the theory as well as the new results on boundary value theory.

Produktdetails

• Progress in Mathematical Physics 24
• Verlag: Birkhäuser / Birkhäuser Boston / Springer, Basel
• Artikelnr. des Verlages: 978-1-4612-6596-2
• Softcover reprint of the original 1st ed. 2002
• Seitenzahl: 228
• Erscheinungstermin: 1. November 2012
• Englisch
• Abmessung: 235mm x 155mm x 13mm
• Gewicht: 353g
• ISBN-13: 9781461265962
• ISBN-10: 1461265967
• Artikelnr.: 39560621

Autorenporträt

Dirac operators play an important role in several domains of
mathematics and mathematical physics. In this self-contained text,
the basic theories underlying the concept of Dirac operators are
explored. Starting with preliminary material, the book covers Clifford
algebras, manifolds, conformal maps, unique continuation and the
Cauchy kernel, and boundary values. Only real analysis is required,
although complex analysis is helpful. A good textbook for senior
undergrad and graduate students, it will also be a useful resource for
math physicists and theoretical physicists.

Inhaltsangabe

1 Clifford Algebras.- 1 Definition and basic properties.- 2 Dot and wedge products.- 3 Examples of Clifford algebras.- 4 Modules over Clifford algebras.- 5 Subgroups.- 2 Manifolds.- 1 Manifolds.- 2 Derivatives and differentials.- 3 The Spin group as a Lie group.- 4 Exterior derivatives and curvature.- 5 Spinors.- 6 Spinor fields.- 3 Dirac Operators.- 1 The vector derivative.- 2 The spinor Dirac operator.- 3 The Hodge-Dirac operator.- 4 Gradient, divergence and Laplace operators.- 4 Conformal Maps.- 1 Möbius transformations.- 2 Liouville's Theorem.- 3 Conformal embeddings.- 4 Maps between manifolds.- 5 Unique Continuation and the Cauchy Kernel.- 1 The unique continuation property.- 2 Sobolev spaces.- 3 The Cauchy kernel.- 4 The case of Euclidean space.- 6 Boundary Values.- 1 The Cauchy transform.- 2 Boundary values and boundary spinors.- 3 Boundary spinors and integral operators.- Appendix. General manifolds.- 1 Vector bundles.- 2 Connections.- 3 Connections on SO(M).- 4 Spinor bundles.- List of Symbols.

Rezensionen

"The text should be accessible for senior undergraduate and graduate students. It requires very little previous knowledge of the domains covered. More advanced readers could perhaps appreciate the new approach to the theory as well as some new results on boundary value theory."

-Mathematical Reviews "This book gives an introduction to Dirac operators on manifolds for readers with little knowledge in differential geometry and analysis.... Compared to other books treating similar subjects...the present book is considerably more elementary and is mostly restricted to results that can easily be obtained out of the definitions."

-Zentralblatt Math "The extraordinary importance of Dirac operators in variuos domains of mathematics and physics is well known. So, although there are some remakrable monographs on Dirac operators, the high number of recent papers covering several subjects needs periodical surveys...

The book is excellent for beginners offering several ideas of research and a global picture of a fascinating theory!" ---Memoriile Sectiilor Stiintifice