## Friday, February 16, 2007

### Alain's comment

This has been posted as a comment to Masoud's post. Since it is almost invisible there, I am posting a copy of it here:

This topic of "quantization and NCG" is very relevant. The word quantum', from the beginning, is not so much related to non-commutativity' but rather to integrality'. In the word quantum' there is really this discovery by Planck, of the formula for blackbody radiation, from which he understood that energy had to be quantized in quanta of $\hbar \nu$. There is a confusion, created by people doing deformation theory who let one believe that quantizing an algebra just means deforming it to a non-commutative one. They take a commutative space and since they deform the product into a non-commutative algebra, they believe they are quantizing. But this is wrong: you succeed in quantizing a space only if you give a deformation into a very specific algebra : the algebra of compact operators. And then, there is an integrality, the integrality of the Fredholm index. The use of the wrong vocabulary, creates confusion and does not help at all to understand. That's why I am so reluctant to use the word quantum' - instead of "non-commutative" and am against talking about "quantum spaces" or "quantum geometry".... this looks more flashy, perhaps, but the truth is that you are doing something quantum only in very particular cases, otherwise you are doing something non-commutative, that's all. Then this may be less fashionable at the linguistic level, but never mind: it is much closer to reality.

Anonymous said...

What a wonderful blog.

Yes, the heart of quantum mechanics is that things come in integral numbers.

Classical mechanics, for exmaple, E&M, allows things to show up at any level, 1.05 times a solution of an E&M problem is a new solution, which corresponds to a slightly stronger physical situation.

But quantum states cannot be continuously deformed to increase the number of particles from 1 to 1.05. That the theory is usually written in a linear form is confusing; it can seem just like classical mechanics.

I prefer to write quantum states in the density operator language where everything is inherently bilinear instead of linear. Translated back into QM, this means that I don't separate the bras from the kets.

Anonymous said...

I completely agree with Alain on the fact that the word quantum' is quite misleading and misused and that one should rather use non-commutative (or noncommutative ?).

This is expecially so after a series of recent papers showing that quantum groups' and quantum spaces' associated with them are examples of (modified -- to some `infinitesimal' extent) noncommutative spaces.