What Non-commutative Geometry Is and Can Do

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In summary, Alain Connes has written a paper discussing how noncommutative geometry can be used to derive the standard model of particle physics. However, the mathematical support offered is tentative and iffy. If you are interested in QG, I suggest you check out other threads on the topic.
  • #36
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.
 
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  • #37
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.
 
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