I am reading Subleading soft dressings of asymptotic states in QED and perturbative quantum gravity by Choi and Akhoury. The authors, at some point, write the following expressions for the total angular momentum operators $$ \langle\textbf{p}|J^{\mu\nu}= -i \left( p^{\mu}\frac{\partial}{\partial p_{\nu}}- p^{\nu}\frac{\partial}{\partial p_{\mu}} \right) \langle\textbf{p}| \tag{4.10} \label{Eq. (4.10)}$$ and $$ J^{\mu\nu}|\textbf{p}\rangle= i \left( p^{\mu}\frac{\partial}{\partial p_{\nu}}- p^{\nu}\frac{\partial}{\partial p_{\mu}} \right) |\textbf{p}\rangle \tag{4.11} \label{Eq. (4.11)}$$
What do these expressions mean? How does one act with a derivative on bras and kets? Do the authors mean something entirely different, like that the derivatives are meant to act on the scattering amplitude?
Thanks in advance.
