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added some very basic facts on $SU(2)$ here to special unitary group. Just so as to be able to link to them.
not yet, no. The entry special orthogonal group is still mostly a stub
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.
Okay, Jim has sent me his slides and I have uploaded them to special orthogonal group. They are here.
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
$\sigma_1 = \sigma_x = \left(\array{0 & 1\\ 1& 0}\right),\,\,\,\,\sigma_2= \sigma_y = \left(\array{0 & i\\ -i & 0}\right),\,\,\,\,\,\sigma_3 = \sigma_z = \left(\array{1 & 0 \\ 0 & -1}\right)$ $[\sigma_a,\sigma_b] = 2i\epsilon_{abc}\sigma_c,\,\,\,\,\,\,\,\,\,\{\sigma_a,\sigma_b\} = 2\delta_{a b}$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 $x$ and $y$ interchanged in a way). The usual choice is that $i\sigma_x,i\sigma_y,i\sigma_z$ are the antihermitian generators of the real Lie group $su(2)$ and $\sigma_x^2 = \sigma_y^2 = \sigma_z^2 =1$. 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 $\gamma$-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.
@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.
For your convenience, the rescalings (which appear in applications but are usually not renamed as $\sigma$ matrices themselves) have commutation relations
$\left[\frac{\sigma_a}{2},\frac{\sigma_b}{2}\right] = i\epsilon_{abc}\frac{\sigma_c}{2},\,\,\,\,\,\,\,\,\,\,\,\,\,\left[\frac{\sigma_a}{\sqrt{2}},\frac{\sigma_b}{\sqrt{2}}\right] = i\epsilon_{abc}\sigma_c = \sqrt{-2}\,\epsilon_{abc}\frac{\sigma_c}{\sqrt{2}}.$Never mind the remark to David, understood the thread question.
Removed $\frac{1}{\sqrt{2}}$ 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 $\frac{1}{\sqrt{2}}$ 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.
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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