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…

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