Connection to spin 1/2 electron system?

Question

In another Physics stack exchange thread here, Spin matrix for various spacetime fields I obtained the generator of rotations of the SO(2) rotation group for an infinitesimal rotation of 2D vectors. I then tried to relate this to the spin-1/2 electron system, but it appears vectors representing states for that system transform under the Pauli matrices instead. I believe this is because those sets of matrices, when scaled properly, yield the correct eigenvalues for the operators.

However, I also noticed that $$D(\omega) = \text{Id} + \omega \begin{pmatrix} 0&1\\-1&0 \end{pmatrix} = \text{Id} + i \omega \begin{pmatrix} 0&-i \\i&0 \end{pmatrix},$$ so I seemed to have made contact with one of the Pauli matrices, (i.e the scaled form of which the spin states do transform). It looks like the above is the infinitesimal version of $\exp(i\omega \hat n \cdot \sigma)$ with $$\hat n \cdot \sigma = \sigma_2 = \begin{pmatrix} 0&-i \\ i&0 \end{pmatrix}$$ which seems to mean $\hat n = (0,1,0)$. In particular, c.f Sakurai P.159, eqn (3.2.3), he says that the physical spin 1/2 electron system does transform under the operator $D(\phi) = \exp(-iS_z\phi/\hbar)$ which is analogous to what I have above with the replacements $\phi \rightarrow \omega$ and $ S_z/\hbar \rightarrow \hat n \cdot \sigma$.

So is there really a connection to the matrix I derived and the spin 1/2 electron system? I do actually think not, since the matrix I derived in that other thread came from analyzing the rotation of vectors in space-time but the spin states live in another space. On the other hand, the analysis above makes me think otherwise at the same time.

Many thanks for clarification!

Answer 1

It is not surprising that you found one of the Pauli matrices as the generator of your rotation. Let us see how it can be seen algebraically that it is to be expected:

Observe that 2D rotations embed naturally into 2D unitary matrices, as $\mathrm{SO}(2) \subset \mathrm{SU}(2)$, corresponding to the subgroup of real matrices. Now, as we know, the $\mathrm{SU}(2)$ is generated by the Pauli matrices, and Lie subgroups must be generated by Lie subalgebras. The question is - which Lie subalgebra is generating the $\mathrm{SO}(2)$?

Well, the one generating the real matrices of course! If you examine¹ the Pauli matrices, you will see that only the one you found by directly linearizing a 2D rotation is giving real instead of complex matrices when exponentiated. Now, how can we be sure that's it? Because 2D rotations are isomorphic to the $\mathrm{U}(1)$, the one-dimensional Lie group, and so we expect only one generator.

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¹ Observe that $\mathrm{e}^{\mathrm{i}\phi (\hat n \cdot \vec \sigma)} = \boldsymbol{1}\mathrm{cos}(\phi) + \mathrm{i}(\hat n \cdot \vec \sigma)\mathrm{sin}(\phi) $. Plugging in the explicit form of the Pauli matrices, we find that the summand from $n_y\sigma_y$ is real (since $\mathrm{i}\sigma_y$ is real), while the others are complex.