What Non-commutative Geometry Is and Can Do

[arivero]

The diagram you have there is, I believe from Newton's Propostion 1. I just learned http://www.arxiv.org/PS_cache/physics/pdf/0504/0504093.pdf that Hooke, independently, reached the same construction (see the diagram in the paper). Have you read Stephenson's Baroque Trilogy? The first volume is much about the rivalry of Hooke and Newton, two thinkers equal in power but very different in application. Almost as it were, Witten and Connes!

Hi selfAdjoint, thanks for the pointer. I had missed that one. Another very important one from Nauenberg is math.HO/0112048, in fact it was the article that suggested me to include the figure in the webpage. It is very important to be aware (I think Nauenberg tell it explicitly, if not other researchers) that the goal of this postulate is to geometrise time while avoiding circular logic. It is an amusing consequence that time in classical mechanics becomes not a line but an area.


As for Stephenson's Baroque Trilogy, I haven't keep reading the next two volumes because I felt disappointed about the documentation effort of Stephenson, far from Criptonomicon.

 

[arivero]

Some more notes:

Supersymmetric quantum theory, non-commutative geometry, and gravitation Lecture Notes, Les Houches 1995
J. Froehlich, O. Grandjean, A. Recknagel
http://arxiv.org/abs/hep-th/9706132

I was there, I think we stole some chocolate from either Recknagel or Grandjean. Beware that Froehlich is selling there an extension of Connes non-commutative geometry. There was various such extensions in the gamefield, with different objectives. In fact Froehlich's was close to the Connesian standards. Coquereaux have some more different models, now deprecated, but amusingly one of them could contain a Higgs at the expected mass value. Connes himself has two different models, either using two algebras or using a single algebra and an additional "Reality" postulate. And then there are also the possibility of starting from an action principle instead of from differential forms.

 

[arivero]

Some interesting action in NCG is being taken by V. Gayral, who has started to analise the Moyal Plane from the point of view of spectral triple. This can be a cornerstone for two unifications: NCG and NCFieldTheory, on one side, and NCG and Manin Plane, on another.

http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=find+a+gayral&FORMAT=WWW&SEQUENCE=


About the Connes-Lott (and Coquereaux et al.) views of the standard model, the newest article is hep-ph/0503147, from Macesanu and Wali, not very usual authors in the field, but involved enough.

http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+A+MACESANU+or+a+wali

 

[Spin_Network]

Some interesting action in NCG is being taken by V. Gayral, who has started to analise the Moyal Plane from the point of view of spectral triple. This can be a cornerstone for two unifications: NCG and NCFieldTheory, on one side, and NCG and Manin Plane, on another.

http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=find+a+gayral&FORMAT=WWW&SEQUENCE=


About the Connes-Lott (and Coquereaux et al.) views of the standard model, the newest article is hep-ph/0503147, from Macesanu and Wali, not very usual authors in the field, but involved enough.

http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+A+MACESANU+or+a+wali

You may want to read the Feynman lost lecture:http://www.mathsnet.net/cabri/feynman05.html

if you back up the pages in the link:http://www.mathsnet.net/cabri/feynman.html
it gives a little overview, the book itself is an amazing and well written expose, and I think you will enjoy the read, there was a link in PF webpage from some time ago:http://kitap.tubitak.gov.tr/k177.html

I have not listened to it (as I have the book).

 

[selfAdjoint]

About the Connes-Lott (and Coquereaux et al.) views of the standard model, the newest article is hep-ph/0503147, from Macesanu and Wali, not very usual authors in the field, but involved enough.

http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+A+MACESANU+or+a+wali

Hummm, two-sheeted spacetime. Consider complexified relativity. The Riemann Surface of
\gamma(w) = \frac {1}{\sqrt{1 - \frac{w^2}{c^2}}}
is two-sheeted, where w is a complex veloctiy.

 

[s3nn0c]

There are many interesting papers written by Polish scientists, especially by Michael Heller (a priest, btw).

http://arxiv.org/find/gr-qc/1/au:+Heller_M/0/1/0/all/0/1

Personally I'm waiting for their next paper, where they will discuss physical interpretations of their NCG unification theory.

 

[s3nn0c]

Here is another paper about quantum mechanics in noncommutative regime (without spacetime).

http://arxiv.org/abs/gr-qc/0510042

Quantum mechanics without spacetime: a case for noncommutative geometry

Quantum mechanics in its presently known formulation requires an external classical time for its description. A classical spacetime manifold and a classical spacetime metric are produced by classical matter fields. In the absence of such classical matter fields, quantum mechanics should be formulated without reference to a classical time. If such a new formulation exists, it follows as a consequence that standard linear quantum mechanics is a limiting case of an underlying non-linear quantum theory. A possible approach to the new formulation is through the use of noncommuting spacetime coordinates in noncommutative differential geometry. Here, the non-linear theory is described by a non-linear Schrodinger equation which belongs to the Doebner-Goldin class of equations, discovered some years ago. This mass-dependent non-linearity is significant when particle masses are comparable to Planck mass, and negligible otherwise. Such a non-linearity is in principle detectable through experimental tests of quantum mechanics for mesoscopic systems, and is a valuable empirical probe of theories of quantum gravity. We also briefly remark on the possible connection our approach could have with loop quantum gravity and string theory.