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Approaches to quantum gravity suggest that the very small scale structure of space-time becomes inherently “fuzzy”. This suggestion leads to a new perspective on geometry known as noncommutative geometry. The principle idea is to replace the sheaf of functions on a manifold with some noncommutative algebra and treat this as if it were the functions on some “space”. An Introduction to Noncommutative Differential Geometry and its Physical Applications by J. Madore presents an overview of noncommutative differential geometry assessable to physicists and geometers alike. |
There are two main approaches to noncommutative differential geometry. The “mathematics” approach is based on Connes’ ideas of spectral triples. The “physics” approach is deformation and quantised coordinate rings. This books focuses on the “physics” approach and Connes’ spectral triples are mentioned in passing.
The readership is graduate students and researchers in mathematical or theoretical physics interested in noncommutative geometry and modifications to space-time. The prerequisites are some familiarisation with quantum mechanics and differential geometry. Knowing quantum field theory and maybe some string theory would help motivate reading the book, though this is not essential. Noncommutative geometry is a mathematical subject in its own right.
Chapter 1 is an introduction. This chapter gives the basic idea of noncommutative geometry as the reformation of the theory of manifolds in terms of the algebra of functions and then generalise this to more general algebras.
The next 5 chapters lay down the mathematics of noncommutative geometry.
Chapter 2 gives a review of differential manifolds in the language of the structure sheaf. Topics covered include: differential forms, vector fields, connections, metrics and de Rham cohomology. The reader is expected to be familiar with these topics, but not necessarily formulated algebraically in terms of the coordinate ring.
Matrix geometry is the subject of Chapter 3. Matrix geometry can be thought of as a finite dimensional noncommutative geometry and thus all calculations reduce to algebra. One of the main issues in noncommutative differential geometry is how to define differential forms and vector fields. The subtleties are introcuded in matrix geometry ready for more general algebras. Topics include: vector fields as derivations, differential calculi dual to the derivations, differential algebras & universal calculus, metrics and connections.
More general noncommutative geometries are the subject of Chapter 4. Topics include: general algebras, quantised coordinate rings, Poisson structures as “classical limits”, topological algebra (operator algebras) and Hopf algabras.
Chapter 5 discusses vector bundles and K-theory. Topics here include: classical vector bundles as projective modules, matrix analogues thereof and Fredholm modules.
Chapter 6 moves on to cyclic homology. Topics here include: universal calculus, Morita equivalence and the Loday-Quillen theorem.
The next chapters take on a more physics flavour.
Modifications of space-time are the subjects of Chapter 7. Topics here include: noncommutative space-time and “fuzzy” physics.
Chapter 8 discusses some extensions of space-time. Topics here include: the spinning particle, noncommutative electodynamics and Kaluza-Klein theory.
The book is full of examples often well motivated by physics. Each chapter contains short notes that point to the original literature and suggest further reading. The book contains a very health set of references that include arXiv numbers where possible.
Paperback: 380 pages
Publisher: Cambridge University Press; 2 edition (August 13, 1999)
Language: English
ISBN-10: 0521659914
ISBN-13: 978-0521659918