My dear friend the physics student, (who knows how to diagonalize a matrix) enter here and forge your hope (trying to inhabit the noncommutative space)!

Here is a friendly introduction to the concepts of noncommutative geometry written by Fabien Besnard, a young professor and mathematician, educated in physics (and a blogger as well). His text is very pedagogical and not technical. Moreover the following excerpt makes an interesting point addressing the subtle issue of quantization in the specific context of noncommutative geometry. 

Noncommutative geometry is such an impressive field that giving an introductory talk to it is quite daunting. Many threads lead to it and it has ramifications in many branches of mathematics, most of them I have meagre knowledge about. So instead of rushing through so many subject, counting on speed to conceal my incompetence, I will follow only one thread, the one that starts with quantum physics, where it all began, and comes back to it. Moreover, I will keep as much a low-tech profile as possible, so that everything should be easy to follow with an undergrad level in math. In fact, if you have ever diagonalized a matrix, you should be able to keep up. (I’m overselling a little bit, since I will need bounded operators, but if you don’t know what a bounded operator is, you can imagine it to be an infinite matrix and it should be OK. You will have to know a bit of topology too, but very basic)... 

A word of caution before you proceed : even though our trip will start with quantum mechanics and end with the standard model of particle physics, noncommutative geometry is not quantum in the same way quantum mechanics is. The reason is simple : noncommutative geometry is not about mechanics, it’s about space. So its “quantumness” applies inside the configuration space, not between configuration and impulsion variables like in quantum mechanics. Take a look at figures 4 and 5. They illustrate the difference between commutative and noncommutative geometry. The same figures could be used to illustrate the difference between classical and quantum mechanics, but only if we understand that the space in this case is the phase space. Of course we may hope that one day the whole of physics will be reduced to some kind of noncommutative geometry, so that the distinction will disappear. But for the moment it is still a dream...

Fabien Besnard, EPF Friendly introduction to the concepts of noncommutative geometry  Notes for the seminar “Philosophy and Physics”, SPHERE Laboratory, Paris 7

May 24, 2013