Back to physics (and math). Last time I stated that geometry requires a generalization, so what does this all mean? There are many ways one can approach this, but let's do it in historical fashion and start with the duality:
Geometry - Algebra
It is informative to remember how ancient Greeks did geometry. For them everything (including the proofs) were a geometric construction with straightedge and compass and they had no concept of coordinates.
It was not until 1637 that geometry and algebra were married by Descartes with what we now call a Cartesian coordinate system. Subsequently mathematicians started realizing that geometry and algebra are nothing but distinct languages describing the very same thing. The first geometry theorem unknown to ancient Greeks was discovered in 1899 and the proof was done by purely algebraic arguments.
The power of algebra is higher than that of geometry because it is easier to formalize abstractions in algebra. In algebra one easily encounters noncommutativity and one example is operator non-commutativity in quantum mechanics. But if algebra is dual to geometry, what kind of geometric spaces would correspond to a non-commutative algebra? What does it mean that "the algebra of coordinates is non-commutative"?
The simplest example is that of a torus. Recall the old fashion arcade games where your character exits through the right side of the screen and re-enters though the left side? Similarly if you move past the top edge you re-emerge at the bottom. Topologically this is a torus. Now suppose you move in straight line in such a way that the ratio of your horizontal and vertical speeds is an irrational number. Slice the torus with trajectory lines respecting this ratio. What you get is a pathological foliation because all "measurable functions" are almost everywhere constant and there are no non-constant continuous functions.
Other pathological examples are: the space of Penrose tilings, deformations of Poisson manifolds, quantum groups, moduli spaces, etc.
From the mathematical side, one can explain away all those pathological cases one by one, but this is missing the forest because of the trees. The duality from above now becomes:
Quotient spaces - Non Commutative Algebra
where we replace the commutative algebra of constant functions along the classes of an equivalence relation by the noncommutative convolution algebra of the equivalence relation.
Basically it all boils down to a generalization of measure theory. It is well known that the proper way to generalize measure theory is by von-Neumann algebras, and this is how quantum mechanics enters the picture (although historically non-commutative geometry arose from quantum mechanics and the work to classify von-Neumann algebras).
Next time we are going to dive deeper into non-commutative geometry and we will encounter the Dirac operator.