tag:blogger.com,1999:blog-6912603287930240451.post8926969657942856254..comments2024-03-29T07:40:52.031+00:00Comments on Noncommutative Geometry: HEART BIT # 1Unknownnoreply@blogger.comBlogger13125tag:blogger.com,1999:blog-6912603287930240451.post-812375383530686022008-04-02T10:59:00.000+00:002008-04-02T10:59:00.000+00:00This might appear quite unrelated to the heart bit...This might appear quite unrelated to the heart bit, but we usually start from the set of rationals as a dense subset of completion of its algebraic closure, i.e. arriving finally at the complex numbers. It might be very interesting to start from a dense subset of transcendental numbers; not just rational multiples of a given transcendental number, t, but choosing from the uncountably many t's, Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-17932379826744160612008-02-15T05:46:00.000+00:002008-02-15T05:46:00.000+00:00This question may be naive, so I apologize in adva...This question may be naive, so I apologize in advance for that. <BR/><BR/>It is mostly based on John Baez's week 175 where he said: <BR/>"... it's easy to construct a type II1 factor. Start with the algebra of 1 x 1 matrices, and stuff it into the algebra of 2 x 2 matrices ... This doubles the trace, so define a new trace on the algebra of 2 x 2 matrices which is half the usual one. Now keep Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-53682753755227369852007-11-13T22:55:00.000+00:002007-11-13T22:55:00.000+00:00Dear Professor Connes,Do you take requests for exp...Dear Professor Connes,<BR/><BR/>Do you take requests for expository pieces with respect to older work?<BR/><BR/>I would love to have a post or series of posts explaining renormalization from the basics, and what is special about your approach. I am deeply confused about what wild fundamental groups are and why Riemann-Hilbert problems should play a role in renormalization.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-68361734040051765802007-11-08T19:07:00.000+00:002007-11-08T19:07:00.000+00:00Hello, Alain Connes. (I think we once had dinner t...Hello, Alain Connes. (I think we once had dinner together along with some other people at the Legal Sea Foods restaurant at MIT after you gave a talk there.)<BR/><BR/>In this paper, the Russian mathematicians Vershik and Yakubovich give a representation of a continuous (hyperfinite) partition lattice in both the lattice of projectors of the hyperfinite factor type II_1 and in the (profinite) Unknownhttps://www.blogger.com/profile/11244827310014342524noreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-58207590560864944712007-11-02T14:42:00.000+00:002007-11-02T14:42:00.000+00:00Dear AnonymousThe space-time which allows to recov...Dear Anonymous<BR/><BR/>The space-time which allows to recover the Sandard Model coupled to gravity is of type I, since it is the product of a manifold M by a finite space F i. e. a space whose algebra of coordinates is finite dimensional. It is not at this level that we expect to get "emergent time" but rather at the level of the algebra of observables in QG. The origin of this idea comes from AChttps://www.blogger.com/profile/10951419541401211230noreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-73491485554213542092007-11-02T12:15:00.000+00:002007-11-02T12:15:00.000+00:00First thank you for your post,very exciting and so...First thank you for your post,very exciting and so I would like to ask you some qualitative questions.<BR/>First you said that commutative space are encoded by a Von Neumann algebra of type I, and so I would like to know if in your noncommutative standard model you use Von Neumann algebra of type II or III. Or Von Neumann algebra must appear only at the quantization level of the noncommutative Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-60228475283861006002007-11-01T17:49:00.000+00:002007-11-01T17:49:00.000+00:00Dear UrsIt is not really nicely spelled out anywhe...Dear Urs<BR/><BR/>It is not really nicely spelled out anywhere, so the best is to understand the basic idea in an example without entering in the technicalities. Consider the spin system on an infinite lattice. The algebra of observables is the inductive limit of the finite dimensional algebras that come from tensor products of matrix algebras over finite subsets of the lattice. By construction AChttps://www.blogger.com/profile/10951419541401211230noreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-13491655848007789472007-11-01T15:50:00.000+00:002007-11-01T15:50:00.000+00:00Alain Connes explained:"inner automorphisms only c...Alain Connes explained:<BR/><BR/>"inner automorphisms only control a total of finitely many such degrees of freedom!"<BR/><BR/>Thanks, that's very helpful to know. <BR/><BR/>Where can I find this spelled out in detail for some simple examples? Is this in your recent book?Unknownhttps://www.blogger.com/profile/01006460812296046328noreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-56148841566956052792007-11-01T00:10:00.000+00:002007-11-01T00:10:00.000+00:00These are likely very dumb questions?How are degre...These are likely very dumb questions?<BR/><BR/>How are degrees of freedom [DOF] related to dimensions [D]?<BR/><BR/>Are there constraints?<BR/><BR/>For example, is a DOF the trajectory from source to target [optimal or not], constrained in and by 3D Euclidean space and time-D, from start to finish?<BR/><BR/>The trajectory from target to source would appear to be noncommutative over the same Doughttps://www.blogger.com/profile/07643919214761722345noreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-88967475353864245712007-10-31T22:16:00.000+00:002007-10-31T22:16:00.000+00:00Dear UrsThe time evolution is as "canonical" as it...Dear Urs<BR/><BR/>The time evolution is as "canonical" as it can be since any noncommutative algebra has inner automorphisms. Moreover one can show that the time evolution belongs to the center of Out(A)!<BR/>If you take very simple examples as the lattice case you will find that an inner automorphism essentially ony affects what happens on finitely many lattice sites. In a simple translation AChttps://www.blogger.com/profile/10951419541401211230noreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-91397003615807767492007-10-31T21:46:00.000+00:002007-10-31T21:46:00.000+00:00Alain Connes wrote:"What you need to understand is...Alain Connes wrote:<BR/><BR/>"What you need to understand is that all the interesting stuff here occurs when the number of degrees of freedom involved is infinite."<BR/><BR/>Thanks for your reply. <BR/><BR/>Yes, I understand that. I mentioned the finite DOF case just to highlight that inner automorphisms matter for physical time evolution.<BR/><BR/>I understand that for infinitely many DOFs thereUnknownhttps://www.blogger.com/profile/01006460812296046328noreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-17475357731945369722007-10-31T21:14:00.000+00:002007-10-31T21:14:00.000+00:00Dear UrsWhat you need to understand is that all th...Dear Urs<BR/><BR/>What you need to understand is that all the interesting stuff here occurs when the number of degrees of freedom involved is infinite. A typical example is quantum statistical mechanics (such as a spin system on a lattice). Systems occuring in quantum field theory, in the examples related to prime numbers are all involiving infinitely many degrees of freedom and are most often ofAChttps://www.blogger.com/profile/10951419541401211230noreply@blogger.comtag:blogger.com,1999:blog-6912603287930240451.post-31037762156658493372007-10-31T20:29:00.000+00:002007-10-31T20:29:00.000+00:00When I last tried to understand the statement that...When I last tried to understand the statement that type III factors come equipped with a canonical time evolution, I tried to see if there are any known physical examples where one sees this canonical time evolution realized as the actual time evolution.<BR/><BR/>My understanding was that the time evolution induced by a <A HREF="http://en.wikipedia.org/wiki/KMS_state" REL="nofollow">KMS state</A>Unknownhttps://www.blogger.com/profile/01006460812296046328noreply@blogger.com