Notation of angular momentum operators vs numbers

Question

I'm reading about finding the mass of quarks in mesons. In the lecture notes, it says

We need to find $\langle\boldsymbol{s}_q\cdot\boldsymbol{s}_\bar{q}\rangle$. Since $L=0$, then $$\boldsymbol{J}=\boldsymbol{s}_q+\boldsymbol{s}_\bar{q}$$ so $$J^2={s_q}^2+{s_\bar{q}}^2+2\boldsymbol{s}_q\cdot\boldsymbol{s}_\bar{q}$$ which means $$\langle\boldsymbol{s}_q\cdot\boldsymbol{s}_\bar{q}\rangle=\frac{1}{2}\langle J^2-{s_q}^2-{s_\bar{q}}^2\rangle=\frac{1}{2}\left[J(J+1)-s_q(s_q+1)-s_\bar{q}(s_\bar{q}+1)\right]\hbar^2$$

My question is, why has he used $\boldsymbol{J}$ not $\mathbf{\hat{J}}$, and why has he dropped the bold from $\boldsymbol{J}$ to $J^2$. I find this confusing, considering it basically says $\langle J^2\rangle=J(J+1)\hbar^2$. I would be used to seeing $\langle\mathbf{\hat{J}}^2\rangle=j(j+1)\hbar^2$

Answer 1

You're probably used to the convention where a hat is used to denote that something is an operator. But that convention is not universal. In many cases, when it's clear from the context whether something is an operator or not, we just write it without a hat either way. For this case in particular, $\boldsymbol{J}$ is defined to be an operator. The fact that you take its expectation value is another reminder that it's an operator. So the hat is omitted.

The bold, on the other hand, just denotes that $\boldsymbol{J}$ is a vector. It's a very common convention to use a bold letter to denote a vector and the corresponding non-bold letter to denote the $L_2$ norm (or magnitude) of that vector, which is defined such that $J^2 = \boldsymbol{J}^2$.

Actually, the way I've usually seen it, the letter that denotes a vector is set in a bold upright font, like $\mathbf{J}$, whereas your lecture notes evidently use a bold italic fond, $\boldsymbol{J}$.