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nLab > Latest Changes: special unitary group

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeAug 5th 2013

    added some very basic facts on SU(2)SU(2) here to special unitary group. Just so as to be able to link to them.

    • CommentRowNumber2.
    • CommentAuthorjim_stasheff
    • CommentTimeAug 5th 2013
    Is there a corresponding link for basic facts about SO(n) for small n?
    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeAug 5th 2013
    • (edited Aug 5th 2013)

    not yet, no. The entry special orthogonal group is still mostly a stub

    • CommentRowNumber4.
    • CommentAuthorjim_stasheff
    • CommentTimeAug 5th 2013
    I prepared a lecture on that topic here at Penn.
    Should I e-mail to you for approval or guest post or???
    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeAug 5th 2013

    Sure, if you have some pdf or the like, we can upload it to the nLab and announce it suitably. Just send me the file by email.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeAug 6th 2013

    Okay, Jim has sent me his slides and I have uploaded them to special orthogonal group. They are here.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMay 5th 2020

    fixed factor of 2 in the Pauli matrix Lie algebra

    diff, v17, current

    • CommentRowNumber8.
    • CommentAuthorzskoda
    • CommentTimeJun 10th 2020
    • (edited Jun 10th 2020)

    I wanted to write few things about relation of facts about rotations and Euler angles to (the exponentials of) Pauli matrices, but our entry special unitary group 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’), Landau-Lifschitz (1989) III (55,7), Ryder (1985) (2.50), which all have

    σ 1=σ x=(0 1 1 0),σ 2=σ y=(0 i −i 0),σ 3=σ z=(1 0 0 −1) \sigma_1 = \sigma_x = \left(\array{0 & 1\\ 1& 0}\right),\,\,\,\,\sigma_2= \sigma_y = \left(\array{0 & i\\ -i & 0}\right),\,\,\,\,\,\sigma_3 = \sigma_z = \left(\array{1 & 0 \\ 0 & -1}\right) [σ a,σ b]=2iε abcσ c,{σ a,σ b}=2δ ab [\sigma_a,\sigma_b] = 2i\epsilon_{abc}\sigma_c,\,\,\,\,\,\,\,\,\,\{\sigma_a,\sigma_b\} = 2\delta_{a b}

    One problem with the choice in the entry is that the square root normalization is put into the definition and it messes the commutation relations stated at special unitary group (you have three square roots in the equation so it can not give a rational ratio), 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. Then, as it is usual in physics treatments of similar situations (antihermitian generators for real Lie groups), there is an imaginary coefficient in the commutation relation. 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.

    • CommentRowNumber9.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 10th 2020
    • (edited Jun 10th 2020)

    @Zoran off-topic for this thread, but I have a question for you here about an old reference you added that is unclear. Scroll back to comment #40 for context.

    • CommentRowNumber10.
    • CommentAuthorzskoda
    • CommentTimeJun 10th 2020

    For your convenience, the rescalings (which appear in applications but are usually not renamed as σ\sigma matrices themselves) have commutation relations

    [σ a2,σ b2]=iε abcσ c2,[σ a2,σ b2]=iε abcσ c=−2ε abcσ c2. \left[\frac{\sigma_a}{2},\frac{\sigma_b}{2}\right] = i\epsilon_{abc}\frac{\sigma_c}{2},\,\,\,\,\,\,\,\,\,\,\,\,\,\left[\frac{\sigma_a}{\sqrt{2}},\frac{\sigma_b}{\sqrt{2}}\right] = i\epsilon_{abc}\sigma_c = \sqrt{-2}\,\epsilon_{abc}\frac{\sigma_c}{\sqrt{2}}.
    • CommentRowNumber11.
    • CommentAuthorzskoda
    • CommentTimeJun 10th 2020
    • (edited Jun 10th 2020)

    Never mind the remark to David, understood the thread question.

    • CommentRowNumber12.
    • CommentAuthorzskoda
    • CommentTimeMar 2nd 2023

    Removed 12\frac{1}{\sqrt{2}} from the (wrong) definition of Pauli matrices. There is a factor of 1/2 for spin, which rescales the commutation relations by factor of 2, but 12\frac{1}{\sqrt{2}} is of no use. Wikipedia https://en.wikipedia.org/wiki/Pauli_matrices has no coefficients for the matrices, but rather just once one talks on spin, then one rescales without square root. This entry refers to page [Pauli matrices] which oddly returns to this same page special unitary group.

    Please check if my edit was sensitive, as far as I see the current state is correct and conservative.

    diff, v20, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeMar 2nd 2023

    Thanks for catching. This was a silly glitch that I had introduced — as an Anonymous had already noticed here.

    Eventually there should be a stand-alone entry Pauli matrix. As long as that does not exist, it redirects to the next best entry. Better than not pointing anywhere.

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