What Non-commutative Geometry Is and Can Do

[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.