1
$\begingroup$

I'm looking for sources on non-commutative geometry and integration theory. I wonder if this theory might replace the standard theorey in the long run, as it seems to be more general. What are possible applications of this theory to physics? Connes has a lot to say about the Standard Model, but I don't really understand what that theory gains from Conne's point of view. I haven't been able to get an overview of what learning non-Commutative geometry exactly entails and what it is used for.

I tried reading the book by Connes, but got stuck very early, at the part where he starts going on about leaves and foliations. I have some background in Hilbert space theory (Unbounded operators, spectral theorems, Schatten classes) and also very basic manifold theory (Vector bundles, basic statements about de Rham cohomology...). I was planing to learn the foliations stuff anyway because of the connections to PDE, but haven't been able to find sources I like so far.

I tried looking for other sources on non-Commutative measure theory to get into the subject a bit, but found only very specific and technical accounts of certain aspects of the theory. Does anyone know good sources to start leaning this?

$\endgroup$

Before answering, please see our policy on resource recommendation questions. Please write substantial answers that detail the style, content, and prerequisites of the book, paper or other resource. Explain the nature of the resource so that readers can decide which one is best suited for them rather than relying on the opinions of others. Answers containing only a reference to a book or paper will be removed!

  • $\begingroup$ Non-Commutative Geometry, in general, will likely be a significant part of whatever the final quantum gravity theory is. However, Connes style geometry is unlikely to be very useful. $\endgroup$ – user73352 Mar 25 '17 at 22:15
  • $\begingroup$ I edited the question in accordance to the recomendations in the comments. @Jake you seem to differentiate between Non-Commutative Geometry and Connes style geometry. What do you mean by Non-Commutative Geometry? I thought it's the theory that builds on the work by Connes. $\endgroup$ – Adomas Baliuka Mar 25 '17 at 22:18
  • $\begingroup$ Related: physics.stackexchange.com/q/238650/2451 $\endgroup$ – Qmechanic Mar 25 '17 at 22:22
  • $\begingroup$ @AdomasBaliuka There are various things called Non-Commutative Geometry, Connes is just one type. An example in physics, that is not based on Connes, would be the derived geometry of mirror symmetry. $\endgroup$ – user73352 Mar 25 '17 at 22:28

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.