# Rotation operator as discussed in S. Weinberg

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.

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.
• 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. – Ptheguy Oct 3 '19 at 21:37
• @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! – ACuriousMind Oct 3 '19 at 21:45
• 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. – Ptheguy Oct 3 '19 at 21:51
• @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 – ACuriousMind Oct 3 '19 at 22:13