Noncommutative standard model

Today we think that there are four elementary forces, the gravitational force describing the motion of apples, planets and maybe even galaxies, the electromagnetic force allowing to understand light, cell phones, X-rays and gamma-decay, the weak force for beta-decay and the strong force binding protons and neutrons to form atomic nuclei and thereby describing fusion, fission and alpha-decay.

Gravity has an extremely 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. It has a Lagrangian formulation with only two real parameters, Newton's constant and a cosmological constant.

The other three forces also have a Lagrangian theory, called the standard model of electro-weak and strong forces. Its underlying idea is that they are mediated by the exchange of spin-one particles, the so-called gauge bosons. The one responsible for electromagnetism is the photon. The weak force has the W- and Z-bosons, the strong force is thought to result from gluon exchange. The gauge Lagrangian is much more complicated than the gravitaional one: at present, it involves some 30 real parameters. This number may still increase. What is more, the gauge Lagrangian must also contain a spin-zero particle, the `Higgs-boson'. So far this particle has not been observed and if it does not show up at the Large Hadron Collider in Geneva, the consistency of the standard model is questionable.

Alain Connes has generalized Riemannian geometry to noncommutative geometry. It describes spaces with curvature and uncertainty. Historically, the first example of such a geometry is quantum mechanics, which introduces Heisenberg's uncertainty relation by turning the classical observables of position and momentum into noncommuting operators. Still, noncommutative geometry is close enough to Riemannian geometry and Connes was able to redo general relativity with it. In doing so he obtained the gauge Lagrangian as a companion of the gravitaional one, a truly geometric unification of all four forces. This unification implies a few constraints on the parameters of the standard model. If their number does not increase further, one of the constraints determines the mass of the Higgs-boson to be around 170 GeV, comfortably within the range of the Large Hadron Collider.

References:

A. Connes, Noncommutative Geometry, Academic Press (1994) A. Connes, Noncommutative geometry and reality, J. Math. Phys. 36 (1995) 6194 A. Connes, Gravity coupled with matter and the foundation of noncommutative geometry, hep-th/9603053, Comm. Math. Phys. 155 (1996) 109 A. Chamseddine & A. Connes, The spectral action principle, hep-th/9606001, Comm. Math. Phys. 182 (1996) 155 A. Chamseddine, A. Connes, M. Marcolli, Gravity and the standard model with neutrino mixing, hep-th/0610241

References available at http://www.alainconnes.org/