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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeDec 27th 2018

    for the moment just to satisfy links

    v1, current

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeJun 7th 2020

    Wrote few defining formulas for SU(2)SU(2) case.

    diff, v5, current

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeJun 7th 2020

    I wanted to write few things about relation to Pauli matrices, but our entry takes unusual conventions disagreeing with wikipedia and wolframworld, as well as textbooks I have at hand, Sudbery (4.39), Ramond (4.18), Blohincev 1983 (59.9, 59.9’). One is that the square root normalization is put into the definition which then messes the commutation relations stated at special unitary group, another is that the choices themselves are made antihermitian (with role of xx and yy interchanged in a way). The usual choice is that iσ x,iσ y,iσ zi\sigma_x,i\sigma_y,i\sigma_z are the antihermitian generators of the real Lie group su(2)su(2) and σ x 2=σ y 2=σ z 2=1\sigma_x^2 = \sigma_y^2 = \sigma_z^2 =1. There is much variation in the literature on which representation is taken for γ\gamma-matrices but I think that for what we call Pauli matrices, the choice is standard, I think. But please let me know how to resolve this.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeJun 7th 2020

    Sketch of the geometric construction for SO(3)SO(3).

    diff, v6, current

    • CommentRowNumber5.
    • CommentAuthorNikolajK
    • CommentTimeJun 8th 2020
    • (edited Jun 8th 2020)

    Surprised the word tori doesn’t fall on that page.

    Just two days ago, I incidentally wrote a script to generate a bunch of rotation matrices via a sweep over Euler angles, and I used this parametrization since it’s probably the one with the least trigonomentric functions. The “SO(3) article” on Wikipedia is for better or worse actually spread over about 8 articles - whenever you want some matrix representation you gotta check them all. I list a few of them here at the start. I have a few hands-on clips about the start of this but also axis-angle stuff.

    As another recommendation, if you’re looking for some more formulas, including the Jacobian matrices as they are computed on your phone, I like this cheatsheet of an article. Btw. I came across this text on quaternion algebras (but that’s not just the Hamiltonian quaternions).

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeJun 9th 2020

    It seems the furthest general analogue in the setup of compact Lie groups so far is the scheme recently discovered in

    • S. L. Cacciatori, F. Dalla Piazza, A. Scotti, Compact Lie groups: Euler constructions and generalized Dyson conjecture, Trans. Amer. Math. Soc. 369 (2017), 4709-4724 doi
    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeJun 9th 2020

    5, NikolajK, thank you.

    Quaternionic presentation of rotations to which you allude deserve and will eventually have a page separate from Euler angles. Although Euler invented both the Euler rotation formula and the Euler-Rodrigues parameters, which are quaternions in a disguise. The latter have a hyperbolic analogue

    • D. Brezov, C. Mladenova, I. Mladenov, Vector parameters in classical hyperbolic geometry, Proc. 15th Int. Conf. on Geometry, Integrability and Quantization, Ivaïlo M. Mladenov, A. Ludu, A. Yoshioka, eds. (Sofia: Avangard Prima, 2014) 79-105 euclid

    Euler rotation formula is often mistakenly attributed to also to Rodrigues, see

    • Hui Cheng, K. C. Gupta, An historical note on finite rotations, J. Appl. Mech. Mar 1989, 56(1): 139-145 (7 pages) doi
    • CommentRowNumber8.
    • CommentAuthorzskoda
    • CommentTimeJun 9th 2020

    5 NIkolajK, I wrote few lines on the role of quaternions under rotation, following Berger’s book, chapter 8.

  1. There was a missing minus sign in the parameterization of uu. It was not equal to the decomposition below. Due to this error, the determinant of uu was not 11. I.e., the upper right and bottom left entries were not complex conjugates of each other.

    Anonymous

    diff, v10, current

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