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Usually it is pointed out that the relation $[x,p]=i\hbar$ comes from the promoting the poisson bracket to commutator but as I know this process of quantization is called deformation quantization which got it's mathematical foundation in 1960's. I'm not quite sure but I read it in some book that originally Heisenberg came up with the CCR due to the nature radiation. Can anyone enlighten me the real thought process which went in Heisenberg's mind during the conception of CCR?

Also Connes in his noncommutative geometry book shows how one arrives at the CCR since wavelengths of hydrogen spectrum forms a groupoid? Was this the idea behind Heisenberg CCR, did physicists started using group theory in early 20th century?

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Replacing the Poisson brackets with the commutator is usually called canonical quantisation and attributed to Dirac in his PhD thesis, 1926.

Heisenberg's original 1925 paper was recognised as incomplete, and was developed by Dirac and by Born & Jordan (leading to the three man paper). The CCR was first expressed in the clarifications of Heisenberg's original 1925 paper, by Born & Jordan and by Dirac.

You can find more on Heisenberg's original paper at Heisenberg's entryway to matrix mechanics, and in Understanding Heisenberg’s ‘magical’ paper of July 1925: a new look at the calculational details. Whether it will cast any light on Heisenberg's thought processes, I am not sure. The author quotes Weinberg

‘If the reader is mystified at what Heisenberg was doing, he or she is not alone. I have tried several times to read the paper that Heisenberg wrote on returning from Heligoland, and, although I think I understand quantum mechanics, I have never understood Heisenberg’s motivations for the mathematical steps in his paper. Theoretical physicists in their most successful work tend to play one of two roles: they are either sages or magicians....It is usually not difficult to understand the papers of sage-physicists, but the papers of magician-physicists are often incomprehensible. In that sense, Heisenberg’s 1925 paper was pure magic'.

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  • $\begingroup$ Thanks! I never knew about that wiki post and I have also read that Weinberg quote but I let it slide thinking it was meant to please common folks. $\endgroup$ – aitfel Apr 1 at 8:01

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