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Where it says
Here we can take $\theta = \exp(2\pi i \hbar)$, where $\hbar$ is Planck’s constant.
maybe we should add for clarity something like:
(here $\hbar$ is thought of, and could be replaced by, any irrational number)
?
At the end where the $GL(2,\mathbb{Z})$-action is given, I have put the usual brackets around the array of matrix components.
Let’s see, there are two things $\hbar$ could mean here:
Either it’s thought of as a real number, specifically an irrrational number.
Or it’s thought of as the formal variable in a formal power series.
In the entry I thought you had in mind the former. But now it sounds you need the latter?
It could be both: the former in the first construction mentioned, and the latter in the second construction.
(I’d have to remind myself, but don’t feel I have the leisure right now.)
Maybe best to clarify either way!
Please take this as pure ignorance on my part, and sorry for the bother, but I don’t understand why $\hbar$, as a physical constant, is brought into this mathematical context, but more specifically why it is asserted or assumed that $\hbar$ is irrational – or even what it means to say that $\hbar$ is not rational. Can someone enlighten me, please?
Regarding the specific technical question:
The angular parameter by which the circle is quotiented out needs to be irrational for the result not be equivalent to a circle again. If the angular parameter is rational, hence if its exponential is a root of unity, then the result of the construction is not a fancy non-commutative space, but just the circle. That’s Rieffel’s famous theorem, quoted in the entry.
Regarding the broader conceptual question:
Since the only example of non-commutative geometry securely seen in nature remains the non-commutative phase spaces of quantum theory, people tend to try to think of every non-commutative geometry as the quantization of some pre-quantum geometry, hence as the deformation of some commutative geometry. In nature, the relevant deformation parameter is Planck’s constant, and so people tend to refer to any non-commutative deformation parameter as “Planck’s constant”. Just read it as shorthand for: “the pertinent noncommutativitity deformation parameter which vanishes in the commutative case”.
Regarding the entry:
I have now tweaked the text slightly to address #2 - #4, to some minimum at least. Don’t have the leisrure to do more.
added more original references, and added pointers to page numbers here:
Thank you, Urs – that was very helpful.
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