Topology and Geometry for Physicists |
Geometry and topology are now a well established tools in the theoretical physicists tool kit. Topology and geometry for physicists by C. Nash & S. Sen gives a very accessible introduction to the subject without getting bogged down with mathematical rigour. Examples from condensed matter physics, statistical physics and theoretical high energy physics appear throughout the book. |
However, one obvious topic missing is general relativity. As the authors state, good books on geometry & topology in general relativity existed at the time of writing.
The first 8 chapters present the key ideas of topology and differential geometry.
Chapter 1 discusses basic topology. Topics include homomorphisms, homotopy, the idea of topological invariants, compactness and connectedness. The reader is introduced to “topological thinking”.
Manifolds are the subject of Chapter 2. Topics include: the definition of manifolds, orientablilty, calculus on manifolds and differential structures.
Chapter 3 discusses the fundamental group. Topics include: the definition of the fundamental group, simplexes, triangulation and the fundamental group of a product of spaces.
Chapter 4 moves on to the homology group. Topics include: the definition of homology groups, relative homology, exact sequences, the Kunneth formula and the Poincare-Euler formula.
The higher homotopy groups are the subject of Chapter 5. Topics covered include: the definition of higher homotopy groups, the abelian nature of higher homotopy groups and the exact homotopy sequence.
The de Rham cohomology of a manifold is the subject of Chapter 6. Topics include: Poincare lemma, calculation of de Rham cohomology for simple examples, the cup product and a comparison of homology with cohomology.
Chapter 7 presents the core concepts of differential geometry. Topics here include: fibre bundles, sections, the Lie derivative, connections on bundles, curvature, parallel transport, geodesics, the Yang-Mills connection and characteristic classes.
Chapter 8 outlines Morse theory. Topics include: the Morse inequalities and the Morse lemma. Connection with physics is established via symmetry breaking selection rules in crystals.
The next two chapters look at application in physics of some of the ideas presented earlier in the book.
Defects and homotopy theory is the subject of Chapter 9. Topics include: planar spin in 2d, ordered mediums and the stability of defects theorem.
Chapter 10 discusses instantons and monopoles in Yang-Mills theory. Topics here include: instantons, instanton number & the second Chern class, instantons in terms of quaternions, twistor methods, monopoles and the Aharanov-Bohm effect.
Paperback: 311 pages
Publisher: Academic Press Inc; New edition edition (Jun 1987)
Language English
ISBN-10: 0125140819
ISBN-13: 978-0125140812
The book has also been reprinted by Dover Books in 2011. Paperback: 311 pages |
I have a hard time understanding why “topology and geometry for physicists” is different from topology and geometry. I also don’t quite see how one can shortcut the rigor in these subjects and have much left. The rigor is there for a reason, and the reason is to avoid talking nonsense.
Particularly puzzling is how the geometry and topology of general relativity is different from the toplogy and geometry of anything else. Mathematics should serve to unify, not discriminate unnecessarily. Since general relativity is the stereotypical model for applications of geometry to physics, the exclusion of it from the text only amplifies my puzzlement.
It seems to me that this might be a good text for demonstrating selected applications of the relevant concepts to topics in physics — after one understands the concepts themselves. At under $10 for the Dover book, it is quite a bargain for that purpose.
@DrRocket:
“Since general relativity is the stereotypical model for applications of geometry to physics, the exclusion of it from the text only amplifies my puzzlement.”
According to the authors the initial idea was to include applications in general relativity. They decided not to in order to keep the length of the book down and because good texts already exist. In particular they cite Hawking and Ellis.
“It seems to me that this might be a good text for demonstrating selected applications of the relevant concepts to topics in physics — after one understands the concepts themselves”
I found the book great for exposing oneself to the ideas of geometry & topology with applications in physics in mind. There is quite a lot of physical intuition in the book and less worry about formal definitions.
It is not the book to learn geometry and topology properly. The book is good for giving you the ideas, but will leave you hungry for more mathematics. In this respect the book has been a great success.