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• CommentRowNumber1.
• CommentAuthorDmitri Pavlov
• CommentTimeJan 1st 2021

Added definitions. Added the classification.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJan 2nd 2021
• (edited Jan 2nd 2021)

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.

• CommentRowNumber3.
• CommentAuthorDmitri Pavlov
• CommentTimeJan 2nd 2021
• (edited Jan 2nd 2021)
Re #2: Yes, we can also say this. I was trying to relate to canonical commutation relations and the Stone-von Neumann theorem here, in the Weyl form: U(s)V(t)=exp(-ist)V(t)U(s),
where U(-) and V(-) denote the one-parameter semigroups generated by U and V.
• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJan 2nd 2021

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!

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeJan 2nd 2021

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?

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJan 2nd 2021

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.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeJan 2nd 2021

added more original references, and added pointers to page numbers here:

• CommentRowNumber8.
• CommentAuthorTodd_Trimble
• CommentTimeJan 2nd 2021

Thank you, Urs – that was very helpful.

• CommentRowNumber9.
• CommentAuthorDmitri Pavlov
• CommentTimeJan 2nd 2021
The changes look good to me.

I certainly did not mean to say that h is a formal parameter.

Recall that deformation quantization can also be done nonformally,
in fact, we have an article about this:
C* algebraic deformation quantization.

In this case, the noncommutative tori for various h assemble into a bundle over a circle (which itself lives inside U(1)),
so h can be thought of as a (nonformal) parameter for deformation quantization.

Maybe we can also mention connections the physical system corresponding to the algebra of observables on a noncommutative torus?
I don't think I have enough knowledge to write about this myself.
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