As Kholodenko points out, Dirac was looking at a multiplication rule for Fourier amplitudes and observed that they did not necessarily commute. In fact, one can derive the quantization relation $[x,p] = i \hbar$ using this observation.
In more recent times Kapranov has studied a non-commutative Fourier transform. He begins by considering cubical paths representing monomials in letters x, y and z. 
Arcadian Functor 
Matti Pitkanen said,
April 18, 2007 @ 4:05 am
Dear Kea,
thank you for an interesting link. I read the introduction of Khodolenko’s paper although I deeply know that this practical side is not for me at this age!
The question was whether it might be possible to construct Veneziano like amplitudes by taking products of already constructed Veneziano amplitudes by using the conservation rule defined by Veneziano alpha(s)+alpha(t)+alpha(u)=-1 or its generalization. The answer was affirmative and practical rules were deduced.
If I understood correctly, you suggest some non-commutative generalization of Veneziano amplitudes? This brings into my mind TGD inspired quantum measurement theory.
In the hyper-finite II_1 vision a non-commutative generalization of Veneziano type amplitudes could emerge when complex rays in the state space are replaced by N-rays, where N is the included algebra in N subset M. N defines measurement resolution. Amplitudes would become N-valued and state space would be (usually) finite-dimensional M/N with N-valued coefficients.
One might imagine of having commutative fundamental amplitudes associated with M and of deriving infinite variety of non- commutative amplitudes by using various inclusions N subset M. This would produce huge variety of non-commutative amplitudes with a clear physical interpretation. The product for non-commutative amplitudes would be induced from that for the commutative amplitudes.
In the paper also more general situation in which momenta are quasimomenta conserved only modulo lattice momentum as in conformal theories is considered so that Veneziano condition alpha(s)+alpha(t)+alpha(u)=-1 is generalized. In TGD the “momenta” associated with the representations of vertex operators using target space: target space is now purely formal auxiliary tool rather than the 10- or 11-D fundamental space-time as in string theory context. These momenta are not real four-momenta which saves from tachyon problems, the pains of compactification, and landscape and could be conserved only modulo lattice momenta.
Cheers, Matti
Anonymous said,
April 18, 2007 @ 7:53 pm
Dirac was not the first who noticed this commutation rule (e.g. read references in Kholodenko’s paper). Furthermore, careful reading of Heisenberg’s paper (and that part of Kholodenko’s which discusses Heisenberg’s contributions) reveals that, actually, “new” quantum mechanics is very much the same thing as the old one since, rigorously speaking, there are as many x-p commutators as there are Bohr-Sommerfeld adiabatic invariants (this point is subtle and requires careful reading of both Heisenberg’s and Kholodenko’s papers). This means that already Helium atom is quantized incorrectly. The way out of this difficulty is discussed in Kholodenko’s paper.
Kea said,
April 18, 2007 @ 9:45 pm
Thank you both for interesting comments. Yes, Matti this is more or less where we are heading. I tried to track down Heisenberg’s paper in English, but failed, and I’m afraid my German is not that great. anonymous, if you have links to English versions that would be greatly appreciated.
Anonymous said,
April 18, 2007 @ 11:35 pm
Gladly.But first things first. By noticing that Heisenberg’s paper is NOT readily awailable in English you, like everybody else, silently acknowledge that there is some kind of mistery around this work. I am sure, that Schrodinger’s work is much more available in English. The reasons why Heisenberg’s is not are manyfold. I just mention a few. First, most likely, it was Kramers who invented QM. N.Bohr had put much older Kramers in charge of 22 yers old Heisenberg and Kramers for free gave away his ideas which is acknowleged in Heisenberg’s paper (albeit indirectly). Second, paper does contain some typos…Third,… OK, the only known to me Engish translation is in old and issued only once book “Sources of Quantum Mechanics” by B.L.Van Der Waerden, North-Holland, 1967. It is the very same guy who edited the complete works by Heisenberg. His very thoughtful remarks in these complete works are worth reading by all means! Notice, that practically none of the works which are in collection are translated into English! Only Editors comments guide you through these papers…
Kea said,
April 18, 2007 @ 11:45 pm
Gee, thanks! That book is in fact in my library, so I will run down and get it. I am reluctant to buy the line that Heisenberg did not have major input into the original idea, given his very insightful later writings. Anyway, cheers.
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