Category Archives: quantization

Dirac and integrality


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In the first paper on “second quantization”, namely the paper of Dirac called “The quantum theory of the emission and absorption of radiation” the process of second quantization is introduced and is related again to “integrality”. This time it is not the Fredholm index that is behind the integrality but the following simple fact : if an operator a satisfies [a,a*]=1, then the spectrum of a*a is contained in N, the set of positive integers (as follows from the equality of the spectra of a a* and a* a except possibly for 0)…. Second quantization is obtained simply by replacing the ordinary complex numbers a_j which label the Fourier expansion of the electromagnetic field by non-commutative variables fulfilling [a_j,a_j*]=1….(more precisely the 1 is replaced by hbar nu where nu is the frequency of the Fourier mode). This example shows of course that integrality and non-commutativity are deeply related… While the Fredholm index is a good model of relative integers (positive or negative), the a a* for [a,a*]=1
is a good model for positive integers…

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.