An Introduction to Dirac Operators on Manifolds

Softcover Nachdruck of the original 1. Auflage 2002
ISBN: 978-1-4612-6596-2
Verlag: Birkhäuser Boston

Buch, Englisch, 211 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 353 g

Reihe: Progress in Mathematical Physics

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.

Zielgruppe

Research

Autoren/Hrsg.

Cnops, Jan

Fachgebiete

• Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik
• Mathematik | Informatik Mathematik Geometrie Differentialgeometrie
• Mathematik | Informatik Mathematik Algebra Algebraische Strukturen, Gruppentheorie
• Mathematik | Informatik Mathematik Mathematische Analysis Funktionentheorie, Komplexe Analysis
• Mathematik | Informatik Mathematik Mathematische Analysis Funktionalanalysis

Inhaltsverzeichnis

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.