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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeAug 5th 2013

added some very basic facts on $SU(2)$ 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

• CommentRowNumber8.
• CommentAuthorzskoda
• CommentTimeJun 9th 2020
• (edited Jun 9th 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

$\sigma_1 = \sigma_x = \left(\array{0 & 1\\ 1& 0}\right),\,\,\,\,\sigma_2= \sigma_y = \left(\array{0 & i\\ -i & 0}\right),\,\,\,\,\,\sigma_3 = \sigma_z = \left(\array{1 & 0 \\ 0 & -1}\right)$ $[\sigma_a,\sigma_b] = 2i\epsilon_{abc}\sigma_c,\,\,\,\,\,\,\,\,\,\{\sigma_a,\sigma_b\} = 2\delta_{a b}$

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.

• CommentRowNumber9.
• CommentAuthorDavidRoberts
• CommentTimeJun 9th 2020
• (edited Jun 9th 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 9th 2020

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

$\left[\frac{\sigma_a}{2},\frac{\sigma_b}{2}\right] = i\epsilon_{abc}\frac{\sigma_c}{2},\,\,\,\,\,\,\,\,\,\,\,\,\,\left[\frac{\sigma_a}{\sqrt{2}},\frac{\sigma_b}{\sqrt{2}}\right] = i\epsilon_{abc}\sigma_c = \sqrt{-2}\,\epsilon_{abc}\frac{\sigma_c}{\sqrt{2}}.$
• CommentRowNumber11.
• CommentAuthorzskoda
• CommentTimeJun 9th 2020
• (edited Jun 9th 2020)

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