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nLab > Latest Changes: noncommutative torus

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  1.  
    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 1st 2021

    Added definitions. Added the classification.

    diff, v4, current

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

    Where it says

    Here we can take θ=exp(2πiℏ)\theta = \exp(2\pi i \hbar), where ℏ\hbar is Planck’s constant.

    maybe we should add for clarity something like:

    (here ℏ\hbar is thought of, and could be replaced by, any irrational number)

    ?

    At the end where the GL(2,ℤ)GL(2,\mathbb{Z})-action is given, I have put the usual brackets around the array of matrix components.

    diff, v6, current

    • 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 ℏ\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 ℏ\hbar, as a physical constant, is brought into this mathematical context, but more specifically why it is asserted or assumed that ℏ\hbar is irrational – or even what it means to say that ℏ\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:

    diff, v7, current

    • 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.

1 to 9 of 9

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