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In theoretical [[particle physics]], the '''non-commutative Standard Model''', mainly due to the French mathematician [[Alain Connes]], uses his [[noncommutative geometry]] to devise an extension of the [[Standard Model]] to include a modified form of [[general relativity]]. This unification implies a few constraints on the parameters of the Standard Model. Under an additional assumption, known as the "big desert" hypothesis, one of these constraints determines the mass of the [[Higgs boson]] to be around 170 [[GeV]], comfortably within the range of the [[Large Hadron Collider]]. Recent [[Tevatron]] experiments exclude a Higgs mass of 158 to 175 GeV at the 95% confidence level and recent experiments at [[CERN]] suggest a Higgs mass of between 125 GeV and 127 GeV.<ref name="CERN March 2013">{{cite web|last=Pralavorio|first=Corinne|title=New results indicate that new particle is a Higgs boson|url=http://home.web.cern.ch/about/updates/2013/03/new-results-indicate-new-particle-higgs-boson|accessdate=14 March 2013|date=2013-03-14|publisher=CERN}}</ref><ref name=nbc14032013>{{cite news |last=Bryner |first=Jeanna |title=Particle confirmed as Higgs boson |url=http://science.nbcnews.com/_news/2013/03/14/17311477-particle-confirmed-as-higgs-boson |date=14 March 2013 |work=[[NBC News]] |accessdate=14 March 2013}}</ref><ref name="Huffington 14 March 2013">{{cite news|url=http://www.huffingtonpost.com/2013/03/14/higgs-boson-discovery-confirmed-cern-large-hadron-collider_n_2874975.html?icid=maing-grid7%7Cmain5%7Cdl1%7Csec1_lnk2%26pLid%3D283596|title= Higgs Boson Discovery Confirmed After Physicists Review Large Hadron Collider Data at CERN|publisher= Huffington Post|accessdate=14 March 2013|date=14 March 2013}}</ref> However, the previously computed Higgs mass was found to have an error, and more recent calculations are in line with the measured Higgs mass.<ref>[https://arxiv.org/abs/1208.1030 Resilience of the Spectral Standard Model]</ref><ref>Asymptotic safety, hypergeometric functions, and the Higgs mass in spectral action models [https://arxiv.org/abs/1208.5023]</ref>
==Background==
Current physical theory features four [[fundamental interactions|elementary forces]]: the [[gravitation|gravitational force]], the [[electromagnetism|electromagnetic force]], the [[weak force]], and the [[strong force]]. Gravity has an elegant and experimentally precise theory: [[Einstein]]'s [[general relativity]]. It is based on [[Riemannian geometry]] and interprets the gravitational force
as curvature of [[space-time]]. Its [[Lagrangian (field theory)|Lagrangian]] formulation requires only two empirical parameters, the [[gravitational constant]] and the [[cosmological constant]].
The other three forces also have a Lagrangian theory, called the [[Standard Model]]. Its underlying idea is that they are mediated by the exchange of [[spin (physics)|spin]]-1 particles, the so-called [[gauge bosons]]. The one responsible for electromagnetism is the [[photon]]. The weak force is mediated by the [[W and Z bosons]]; the strong force, by [[gluon]]s. The gauge Lagrangian is much more complicated than the gravitational one: at present, it involves some 30 real parameters, a number that could increase. What is more, the gauge Lagrangian must also contain a [[spin (physics)|spin]] 0 particle, the [[Higgs boson]], to give mass to the spin 1/2 and spin 1 particles.
[[Alain Connes]] has generalized [[Bernhard Riemann]]'s geometry to [[noncommutative geometry]]. It
describes spaces with curvature and uncertainty. Historically, the first example of such a geometry is [[quantum mechanics]], which introduced [[Werner Heisenberg|Heisenberg]]'s [[uncertainty relation]] by turning the classical observables of position and momentum into noncommuting operators. Noncommutative geometry is still sufficiently similar to [[Riemannian geometry]] that Connes was able to rederive [[general relativity]]. In doing so, he obtained the gauge [[Lagrangian (field theory)|Lagrangian]] as a companion of the gravitational one, a truly geometric unification of all four [[fundamental interaction]]s. Connes has thus devised a fully geometric formulation of the [[Standard Model]], where all the parameters are geometric invariants of a noncommutative space. A result is that parameters like the [[electron mass]] are now analogous to purely mathematical constants like [[pi]]. In 1929 Weyl wrote Einstein that any unified theory would need to include the metric tensor, a gauge field, and a matter field. Einstein considered the Einstein-Maxwell-Dirac system by 1930. He probably didn't develop it because he was unable to geometricize it. It can now be geometricized as a non-commutative geometry.
It is worth stressing that, however, a fundamental physical drawback plagues this interesting and very remarkable attempt. Barring some relevant partial results, all the noncommutative geometry structure is a generalization of Riemannian geometry, that is a geometry where the [[metric tensor]] is positively defined. Conversely physics deals with the geometric structure known as [[pseudo-Riemannian manifold]] that allows one to give a mathematically rigorous description of [[causality (physics)]].
In particular cases (in presence of a [[timelike]] [[Killing vector]] field in the Lorentzian picture) one passes from the Riemannian picture to the Lorentzian (pseudo-Riemannian) one by means of the so-called [[Wick rotation]], without loss of information. Up to now no generalization of the Wick rotation exists in the noncommutative case.
==See also==
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