Here is a picture of the Moon I took on the 3rd February 2012. The picture was taken using my 7MP Advent digital camera (“point and click”) directly through the eyepiece of my Bresser Skylux NG 70-700 retractor. I used a Moon filter and 20mm eyepiece.
The results are ok. I will post more as I take them.
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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
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Quantum field theory is the theory that describes all the known forces apart from gravity. However, the standard approach to quantum field theory via path-integrals is fraught with mathematical difficulties. One mathematical approach is to remove the fields as the primary objects and consider the algebras of observables as the fundamental objects of interest. Local Quantum Physics by Rudolf Haag introduces the reader to the ideas of constructive field theory and algebraic field theory. |
Haag is a major player in algebraic field theory and this book gives his perspectives on the subject. The main mathematical tools employed in algebraic field theory are nets of c*-algebras. The book does not assume the reader is an expert in operator algebra, though some familiarity with quantum mechanics, quantum field theory and special relativity is a prerequisite. The book is not aimed at experts in constructive or algebraic field theory and so serves as a great introduction to the subject. The reader will be exposed to the main concepts and main theorems of algebraic field theory.
The readership is graduate students and researchers in mathematical physics interested in rigorous approaches to quantum field theory. The book may also be of interest to mathematicians working in operator algebra who would like to understand how to make contact with advanced physics.
In the following I will be referring to the second revised and enlarged edition of the book.
The book is divided into 8 chapters each containing between 3 and 5 sections.
Chapter I lays down the background of quantum field theory. Topics here include: basic concepts of quantum mechanics, locality in relativistic theories, Poincare invariant quantum field theory, the action principle and basic quantum field theory like canonical quantisation, free fields and gauge invariance.
General quantum field theory is the topic of Chapter II. Here one encounters the ideas of constructive field theory via the Wightman axioms. Topics discussed here include: the Wightman axioms, generating functionals, time ordered functions, covariant perturbation theory via Feynman diagrams, asymptotic configurations and particles, the S-matrix and the LSZ-formulation. The CPT theorem, spin-statistics theorem and analytical properties of the S-matrix are also discussed.
Chapter III moves on to algebraic field theory and the algebra of local observables. Topics here include: operator algebras (von Neumann, c* and w*), factors, positive linear forms and states, the GNS construction, nets of algebras of local observables and vacuum states. This chapter discusses the “guts” of algebraic field theory.
The next two chapters discuss some of more advanced aspects of algebraic field theory. That is superselction rules and the KMS-states.
Superselection sectors and symmetry is the topic of Chapter IV. Topic discussed include: charge superselction sectors, the DHR-anaysis, the Buchholz-Fredenhagen analysis, low dimensional space-time and braid statistics.
Chapter V moves on to thermal states and modular automorphisms. Topics here include Gibbs ensembles, the KMS condition, the Tomita-Takesaki theorem, equilibrium states, modular automorphisms of local algebras and nuclearity.
Chapter VI discusses the particle picture of quantum field theory. Topics here include: asymptotic particle cofigurations, particles & infraparticles and the physical state space of QED.
The interpretations and mathematical formalism of quantum physics is the topic of Chapter VII. Topics here include: the Copenhagen interpretation, the classical approximation, “quantum logic” and the EPR-effect.
The final part of the book, Chapter VIII is a concluding chapter which presents a retrospective look at algebraic field theory and presents some future challenges. Topics discussed here include: a comparison with Euclidean quantum field theory, supersymmetry and general relativity. The book presents the challenges faced by gravity, for instance QFT on curved space-times, Hawking radiation and the possibility of quantum gravity.
The book contains a healthy bibliography as well as an author index with references. This allows the reader to chase up the original literature, if desired.
Paperback: 390 pages
Publisher: Springer; 2nd. rev. and enlarged ed. edition (5 Aug 1996)
Language English
ISBN-10: 3540610499
ISBN-13: 978-3540610496