Edwin Beggs - noncommutative geometry

There are many ways to extend the ideas of classical differential geometry to a noncommutative world. Which one is correct? There is of course no answer to that question, and these many ways each have a rich pure mathematical theory. However, if we seek to justify what should be named noncommutative differential geometry, four criteria may be used to evaluate possible approaches.
1) There must be a large and diverse collection of examples.
2) The noncommutative theory must reduce to the classical theory, though some parts of the theory may only have a non-trivial meaning in the noncommutative cases.
3) Most constructions in classical differential geometry should have noncommutative analogues.
4) Last but not least, as geometry originated as a practical subject, there should, in principle, be applications, which so far means applications in physics.

The main proposed application of noncommutative geometry is in quantum gravity. The problem with theories of quantum gravity is how they connect to physical measurements. A possible response to this is that the measurable consequences of quantum gravity are at scales far beyond our ability to measure. I no longer consider this an adequate response.

There is an idea that quantum gravity encompasses only effects at incredibly small lengths or incredibly high energies, and is therefore outside foreseeable experimental consequence. However quantum gravity is not only quantum gravity - any measurable gravitational interaction of a quantum system quantises gravity. In particular current work on macroscopic quantum systems (e.g. Bose-Einstein condensates [4]) and technical progress associated to experimental quantum computation raises the possibility that gravitational consequences of quantum effects could be measured -- especially as those quantum effects may themselves provide much more accurate measurements of gravity than we currently have. (A gravity detector using lasers to suspend atoms has already been demonstrated at Porton Down [2,1].)

And, by Einstein, by gravity we mean geometry. The idea of observing a quantum space time has progressed from being inconceivable to being a possibility within the next few decades -- anyone doubting the ability of experimentalists to do this should remember that the accuracy of 10^(-18) m demonstrated by the LIGO gravitational wave observation [3] could easily have been dismissed as impossible for numerous reasons some decades earlier.

Whatever complicated object a fully nonlinear field theory version of quantum gravity might be, there is some consensus that a first approximation as a perturbation to classical theory would be a version of noncommutative geometry -- and differential geometry as our description of physics is via differential equations. If these quantum observations were to happen, the economic consequences would begin with more accurate devices for measuring gravity (and therefore better accelerometers), and go who knows where.

As mathematical analysis of gravity (both noncommutative geometry and string theory) seems to throw up the idea of nonassociativity, it may be asked whether some previously rejected models of gravity may only have been rejected because of the associativity assumption.

[1] BBC Horizon: Project Greenglow, 23 March 2016
[2] B. Farmer, MoD gravity sensor breakthrough to 'see underground or through walls', The Telegraph, 23 March 2016
[3] https://www.ligo.caltech.edu/page/ligos-ifo
[4] Quantum superposition at the half-metre scale, T. Kovachy et al., Nature 528, 530-533 (2015)