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marcus

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https://www.physicsforums.com/showpost.php?p=1363886&postcount=609

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marcus

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I think that it is interesting to be aware that nowadays there are two main aproachs to noncunmutative goemtery. On one side it is the one taht the papers you linked are about, i.e. the Alain Connes "spectral" noncounmutative geometry. I´s later incarnation is explained in his new book which can be downloaded from his web page: http://www.alainconnes.org/downloads.html

There you can see also a previous book. You can read a blog entry about that vertient of noncunmutative goemtry here: http://sbseminar.wordpress.com/2007/06/24/review-of-14th-of-connes-marcollis-new-book/

An extract of that web clarifies a litle the line of attack:

As all the Connes work it is very heavy mathemathically and it requires agood knowledge of things like dirac operators, k-theory, C* álgebras and maybe to have some familiarity with the sheaves (abstract) aproach to algebraic geometry.

On the "observational" point to say that these is the theory whcih makes the famous prediction of the higss boson beiwn around 170 TeV.

There is a difrent, and easier, aproach to noncumutative geometry. I call it "field theoretic" NCG. Roughtly speaking the idea is that you translate the noncunmutativity of the space time into a nonconumtative product of the common fields of the standar model. There a few candidates for the way to make that product but the main one is the Moyal product. You can read an introduction, with a great discusion on the observational posibilities, here:

arXiv:hep-ph/0205040 v3

A most extensive introduction can be found here:

http://arxiv.org/abs/hep-th/0106048v4

These introduction takes account of an important advance in NCG, the Seiberg-Witten map. In a famous paper (cited in the link i give) Seiberg and Witten (who is using ideas of NCG at leass since his fist string field theorie) analized a model in which the antisimetric B field of the NS spectrum comon to all the superstring theories could have a non-vanishing vacuum value. AS a resoult of it the space tourned non-conmutative. Based on that aproach Witten and Seiberg obtained an improved version of the Moyal product which resolved some of the previous probles of NCG, in particular it allowed to do SU(N) NCG gauge theories while previously it ony was allowed to do U(N) gauge theorie.

The string theorie comunitie seems to look these "field theoretic" NCG as some kind of effective theorie which reflects some aspectos of the most fundamental stringy aproach.

On the other side Connes seems to think that his aproach is in the end a fundamental one not depending in anything else and that it is, in some weaks sense at least, an unifed theorie of the standard model + quantum gravity. Even thought in its actual realization it is still some kind of effective action of an still unknown theorie.

Anyway, both aproachs seem to be interesting because they seem to make predictions observable in a nearly reasonable time.

There you can see also a previous book. You can read a blog entry about that vertient of noncunmutative goemtry here: http://sbseminar.wordpress.com/2007/06/24/review-of-14th-of-connes-marcollis-new-book/

An extract of that web clarifies a litle the line of attack:

I would add that an important point of the new aproach to NCG of Connes respecto to his previous one is that it solves a problem of fermion duplication.They write down a very simple, essentially gravitational Lagrangian on a spacetime which is a direct product of 4d Minkowski and a “very small” non-commutative space F. Then they ask what this Lagrangian looks like in 4d. The answer is, of course, determined by F, and it turns out that there’s a very simple choice of F that gives exactly the Standard Model.

As all the Connes work it is very heavy mathemathically and it requires agood knowledge of things like dirac operators, k-theory, C* álgebras and maybe to have some familiarity with the sheaves (abstract) aproach to algebraic geometry.

On the "observational" point to say that these is the theory whcih makes the famous prediction of the higss boson beiwn around 170 TeV.

There is a difrent, and easier, aproach to noncumutative geometry. I call it "field theoretic" NCG. Roughtly speaking the idea is that you translate the noncunmutativity of the space time into a nonconumtative product of the common fields of the standar model. There a few candidates for the way to make that product but the main one is the Moyal product. You can read an introduction, with a great discusion on the observational posibilities, here:

arXiv:hep-ph/0205040 v3

A most extensive introduction can be found here:

http://arxiv.org/abs/hep-th/0106048v4

These introduction takes account of an important advance in NCG, the Seiberg-Witten map. In a famous paper (cited in the link i give) Seiberg and Witten (who is using ideas of NCG at leass since his fist string field theorie) analized a model in which the antisimetric B field of the NS spectrum comon to all the superstring theories could have a non-vanishing vacuum value. AS a resoult of it the space tourned non-conmutative. Based on that aproach Witten and Seiberg obtained an improved version of the Moyal product which resolved some of the previous probles of NCG, in particular it allowed to do SU(N) NCG gauge theories while previously it ony was allowed to do U(N) gauge theorie.

The string theorie comunitie seems to look these "field theoretic" NCG as some kind of effective theorie which reflects some aspectos of the most fundamental stringy aproach.

On the other side Connes seems to think that his aproach is in the end a fundamental one not depending in anything else and that it is, in some weaks sense at least, an unifed theorie of the standard model + quantum gravity. Even thought in its actual realization it is still some kind of effective action of an still unknown theorie.

Anyway, both aproachs seem to be interesting because they seem to make predictions observable in a nearly reasonable time.

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- #3

garrett

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"u"'s and "n"'s shouldn't commute. ;)

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