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Introductory reading: Non-commutative geometry

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In its most general form, the idea of non-commutative geometry (or, as some say, noncommutative geometry) is a generalisation of geometry so that the functions on the space no longer commute, fg≠gf.

Possibly the first example of this idea was in quantum mechanics, where the coordinates on the phase space of a particle in classical mechanics are replaced by operators on a Hilbert space of wavefunctions. One wants to think of the quantum system in classical terms where this is possible, so, for example, the Hamiltonian operator in quantum mechanics is a non-commutative analogue of the Hamiltonian function of classical mechanics, and has somewhat similar properties.

There are other types of non-commutative geometry, not necessarily related to phase spaces. They all have in common the idea that the coordinates don’t commute, but beyond that, schemes differ. The point is that more structure is needed to get any interesting sort of geometry, and there isn’t a consensus on what that “more” is.

The most well-developed aspects of non-commutative geometry arise from the ideas of Alain Connes, as laid out in the Red Book, published with the title Noncommutative Geometry. This is very much a mathematician’s project but inspired by ideas from physics, with the ultimate application the mathematical structure of the standard model of particle physics. It starts with ideas about the non-commutative generalisation of topology, vector bundles and cohomology, moving on to differential calculus (the “quantized calculus”) and finally, the non-commutative version of Riemannian geometry. This is based around the Dirac operator, the main aspect of metric geometry that survives in this non-commutative approach. Of course, the Dirac operator is also the main ingredient in high-energy physics. The mathematical structure that contains the Dirac operator is called a spectral triple and this is the key gadget in my research work.

Other approaches to non-commutative geometry arise out of various physics projects. [More…]

Presentations of non-commutative geometry differ greatly depending on the level of mathematical sophistication assumed. Some of the finest introductions (such as the Red Book) assume you are already a research-level mathematician, which makes them impossible to read if you are not. Also, it is often difficult to extract the big ideas from a lot of mathematical detail.

So I will start by presenting the mathematically simpler introductions first. Fabien Besnard has written some very accessible notes Friendly introduction to the concepts of noncommutative geometry covering the basic ideas for the non-commutative algebra, the quantized calculus and the Dirac operator, with some simple examples, including a very brief section on the standard model. Another fairly low-tech introduction more biased towards the physics is Jean Petitot: Noncommutative geometry and physics. It contains a lot more material about spectral triples and explains Yang-Mills-Higgs theory from this point of view.

An introduction to the more mathematical aspects is provided by Landi: An Introduction to Noncommutative Spaces and their Geometry, which appears to be similar to his book with the same name. A nice feature is that it has a lot of explicit examples, in particular ones based on matrices and non-commutative versions of a lattice. It follows the same development, through differential calculus to some aspects of the standard model. Differential forms and connections are particularly detailed.

An introduction that assumes a bit more mathematical knowledge is Khalkhali:  Very Basic Noncommutative Geometry, also similar to his book Basic Noncommutative Geometry, which heads for cyclic cohomology and the non-commutative Chern character instead of the physics.

 

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Written by johnwbarrett

11 December 2018 at 15:29

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