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Following the discussion on rotation operators in Sakurai made clear sense, however, due to coursework, I need to also understand the discussion provided in Quantum Mechanics by S. Weinberg.

In Chapter 4, page 100, he writes the infinitesimal rotation matrix as

$$\tag{1} \mathbb{R}=\mathbb{1}+\mathbb{W}$$

where $\mathbb{W}$ is a matrix. The orthogonality condition readily gives $\mathbb{W}^T=-\mathbb{W}$. So far so good.

Then, Weinberg goes on to write

$$\tag{2} U(1+\mathbb{W})\rightarrow 1+\frac{i}{2\hbar}\sum_{ij}\mathbb{W}_{ij}\mathbb{J}_{ij}+\mathcal{O}(\mathbb{W}^2)$$

where $\mathbb{J}$ is a matrix of operators. He doesn't really explain much how we can write this down.

  1. We can write an infinitesimal unitary operator as $U(\epsilon)=1+i\epsilon\ \hat{T}$ for some Hermitian operator $\hat{T}$. Applying this to Eq (2), why don't we write $$U(1+\mathbb{W})\rightarrow 1+\frac{i}{2\hbar}\sum_{ij}(\delta_{ij}+\mathbb{W}_{ij})\mathbb{J}_{ij}+\mathcal{O}(\mathbb{W}^2)\ ?$$
  2. I have seen vector of operators, but I have never encountered a matrix of operators until now. What allows us to guess that we actually need to introduce a matrix of operators in Eq (2)?
  3. On page 101, he says that in three-dimensions, we can define a vector of operators $\mathbf{J}$ with components $$J_1=\mathbb{J}_{23} \quad J_2=\mathbb{J}_{31} \quad J_3=\mathbb{J}_{12}.$$ How? Why these entries of the matrix of operators $\mathbb{J}$?

Understanding rotations in QM made a lot of sense when reading Sakurai or other references! Weinberg's treatment of the problem seems too convoluted and there's a great lack of contextual justification in steps taken. Any help in understanding the concept behind a matrix of operators that shows up in Eq (2) and picking the $J_x,J_y,J_z$ components would be appreciated.

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  1. What you mean by $U(\epsilon)$ Weinberg here seems to write as $U(1+\epsilon)$ - it's a difference in notation, but you both mean the same thing.

  2. Weinberg is writing $\mathbb{W} = \sum_{i,j} \mathbb{W}_{ij}\mathbb{J}_{ij}$, where the $\mathbb{J}_{ij}$ are the basic antisymmetric matrices with a $1$ at the $ij$-th position, a $-1$ at the $ji$-th position and zeroes elsewhere. Since $\mathbb{W}$ is an antisymmetric matrix and these form a basis for the vector space of antisymmetric matrices, $\mathbb{W}$ can be expressed as this linear combination of them.

  3. Due to $\mathbb{W}^T = -\mathbb{W}$, these are the independent non-vanishing components of $\mathbb{W}$ in three dimensions.

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  • $\begingroup$ 2. Could you please expand on some details here? Is $E$ the identity matrix? Moreover, $$\sum_{ij}W_{ij}J_{ij}=\sum_{ij}-W_{ji}^TJ_{ij}=\sum_j(-W^TJ)_j=-\mathrm{Tr}(W^TJ),$$ which is a number, whereas we need to end up with a Hermitian operator. $$$$ 3. But this doesn't explain how we choose which one is x, y, or z. $\endgroup$ – Ptheguy Oct 3 '19 at 21:37
  • $\begingroup$ @Ptheguy 2. There is no "$E$" here. The $E_{ij}$ are not components of a matrix, each of them is a matrix on its own (e.g. $E_{12}$ is the matrix with a 1 in the 1st row and 2nd column, and zeros everywhere else). 3. It doesn't matter. Which label you attach to which axis is an arbitrary choice! $\endgroup$ – ACuriousMind Oct 3 '19 at 21:45
  • $\begingroup$ 2. Ah I see. Ok so then are we saying that $\mathbb{J_{ij}}$ are these $E_{ij}$ matrices? I though they were the angular momenta operators. $\endgroup$ – Ptheguy Oct 3 '19 at 21:51
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    $\begingroup$ @Ptheguy They are just the elementary ones. As a consistency check in 3d, e.g. when you set $\mathbb{W}_{12} = 1$ and $\mathbb{W}_{13} = \mathbb{W}_{23} = 0$, you get $\mathbb{W} = \mathbb{J}_{12} - \mathbb{J}_{21}$, which is the infinitesimal rotation matrix Wikipedia calls $L_z$ here $\endgroup$ – ACuriousMind Oct 3 '19 at 22:13
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    $\begingroup$ @Ptheguy I've corrected the answer. $\endgroup$ – ACuriousMind Oct 7 '19 at 16:42

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