Talk:Pauli matrices

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Somebody should redirect Pauli Gate to go to this page, I have no experience doing this, and it wasn't as easy as #REDIRECT Pauli Gate so I didn't do it.

I removed this:

, and the Pauli matrices generate the corresponding Lie group SU(2)

I don't see in what sense four matrices can "generate" an uncountable group, especially if they aren't even elements of that group. AxelBoldt 00:34 Apr 29, 2003 (UTC)

By exponentiation. -- CYD

Looking at the replacement

so the Pauli matrices are a representation of the generators of the corresponding Lie group SU(2).

I think I see the source of my confusion. We are not talking about generators in the sense of group theory, but rather "infinitesimal generators" of a Lie group, i.e. the elements of its Lie algebra. This should be clarified somewhere. So what we are really saying is that σ1,σ2 and σ3 form an R-basis of the Lie algebra su(2) of all Hermitian 2x2 matrices with trace 0, is that correct?

Also, the above link to group representation is misleading, since we are really representing a Lie algebra, not a group. I'll try to weave that into the article. AxelBoldt 20:02 Apr 29, 2003 (UTC)

That sounds right to me. -- CYD

[edit] Sign of second Pauli matrix

I think the sign of σ2 has been inadvertently flipped. Indeed, I don't know what books y'all are looking at, but it at least some textbooks this guy does appear with the other sign.

Why is the other sign preferable? Self-consistency, but more for aesthetics than anything else. The problem is that with the present sign, multiplying by t > 0 and exponentiating gives clockwise rotation, whereas σ1,σ3 give counterclockwise rotation. That's a bit awkward and is a minor annoyance in related articles like Lorentz group. OK, this might be my most pedantic quibble yet, but if anyone agrees the sign needs fixing, please do it (don't forget to check the commutators, which you'll probably also need to modify).---CH (talk) 16:42, 13 July 2005 (UTC)

No, the sign given in the article is correct:
\sigma_2 = \begin{bmatrix}0 & -i \\ i & 0\end{bmatrix}
Please don't change it. This sign gives rise to a counterclockwise rotation, just as do σ1,σ3. Check your math. (The Lorentz group article needs to be changed as well). -- Fropuff 17:48, 13 July 2005 (UTC)
Actually, let me qualify my previous response. The spinor map SL(2,C) → SO+(3,1) depends on the isomorphism chosen between Minkowski space and the space of 2×2 Hermitian matrices. As long as this isomorphism is chosen to be
\{t,x,y,z\} \leftrightarrow t\sigma_0 + x\sigma_1 + y\sigma_2 + z\sigma_3
it doesn't matter which sign is chosen for σ2 (choosing a different sign means choosing a different spinor map). With this choice of isomorphism the image under the spinor map of the exponential of a Pauli matrix always represents a counterclockwise rotation about the corresponding axis.
However, I again request that no one change the sign of σ2 as this sign is conventional in physics papers and textbooks worldwide. -- Fropuff 05:02, 16 July 2005 (UTC)

[edit] Here you are:

\sigma_x \times \sigma_y = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} \times \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix} = i \sigma_z = i\begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}
\sigma_x \times \sigma_z = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} \times \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} = \begin{pmatrix} 0&-1\\ 1&0 \end{pmatrix} =\frac{\sigma_y}{i}
\sigma_y \times \sigma_x = \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix} \times \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} = \begin{pmatrix} -i&0\\ 0&i \end{pmatrix}
\sigma_y \times \sigma_z = \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix} \times \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} = \begin{pmatrix} 0&i\\ i&0 \end{pmatrix} =i\sigma_x
\sigma_z \times \sigma_x = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} \times \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} = \begin{pmatrix} 0&1\\ -1&0 \end{pmatrix}
\sigma_z \times \sigma_y = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} \times \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix} = \begin{pmatrix} 0&-i\\ -i&0 \end{pmatrix} =-i\sigma_x

At this point, I feel, it may be useful to emphasize the anticommutator relations of the σ matrices as their defining equations. This clarifies their relationship to the invariant metric tensor defining SO(3) and their role in the corresponding Clifford algebra. This algebraic definition allows for a manifold of alternative σ representations. Please, have a look at the γ matrices for an analogy.

You may, of course, insist on [σx,σy] = 2iσz etc. to keep conventions of chirality in 3-dimensional space.

[edit] Real algebra

Direct calculation shows that the Lie algebra su(2) is the 3 dimensional real algebra spanned by the set {i σj}.

What is meant by a real algebra here? Surely the elements of the set {i σj} are complex.Wiki me (talk) 22:30, 27 February 2008 (UTC)

The coefficients are only allowed to be real. Compare with the statement "the complex numbers are the real algebra spanned by the set {1, i}." -- Fropuff (talk) 06:38, 28 February 2008 (UTC)