# Noncommutative Geometry and quantum space-time

This generalization of ordinary Riemannian geometry arises naturally when one considers physics at very small distances and takes both gravity and quantum theory seriously: Quantum theory implies that position and momentum cannot be determined simultaneously with arbitrary precision. Special relativity then links momentum to energy and mass and finally Einstein's gravity tells us about gravity and black holes.

So if you try to measure distances too precisely, it could happen that the uncertainty in mass becomes so big that a black hole forms and the object under consideration vanishes behind the horizon. So there seems to be a natural size limit under which objects can no longer be "looked at". It is likely that this fundamental limit is a limit of space itself and that it makes no sense to distinguish points closer than this limiting distance.

The geometry of space as a dense collection of points has to be replaced by a more general concept. One possibility is to promote coordinates to noncommuting operators. Then, as position and momentum in quantum mechanics, they cannot be measured with great precision at the same time. This has an implication for functions on space: While normally they are multiplied point wise as f(x)g(x) and this equals g(x)f(x), in noncommutative geometry, this is no longer true. In fact, it turns out, all of the geometry of a space is encoded in this algebra of functions.

There are several approaches to noncommutative geometry:
• One can deform an ordinary (commutative) space by introducing a small parameter (like Planck's constant) and treat the noncommutativity in a perturbative series.
• In open string theory, points on the boundary of a string world-sheet cannot be moved around each other. This results in non-commutativity of the low energy theory of open strings.
• It is also possible to define the noncommutative algebra abstractly and then study the resulting geometry. This is the approach pioneered by Alain Connes.
In the end, one would expect all three approaches to be equivalent but technically they pose different challenges.