# Quantizing the orbital angular momentum of a free Electromagnetic field

It can be shown that the total angular momentum of a free electromagnetic field is given by (for example, in the book A Modern Introduction to quantum field theory by Maggiore, page 98, Eq. (4.82)) $$J^{jk}=L^{jk}+S^{jk}\tag{1}$$ where $$L^{jk}=\int d^3x[\partial_0A^i(x^j\partial^k-x^k\partial^j)A^i]\tag{2}$$ represents the components of the orbital angular momentum and $$S^{jk}=\int d^3x[A^j\partial_0A^k-A^k\partial_0A^j]\tag{3}$$ represents the components of the spin angular momentum.

The expression (2), with normal ordering, has been expressed in terms of creation and annihilation operators (in Eq. 4.84, of the same reference) as $$S^{ij}=i\int \frac{d^3q}{(2\pi)^3}\sum\limits_{\lambda,\lambda^\prime}\Big[\epsilon^i(\textbf{q},\lambda^\prime)\epsilon^{j*}(\textbf{q},\lambda)-\epsilon^{i*}(\textbf{q},\lambda)\epsilon^{j}(\textbf{q},\lambda^\prime)\Big]a^\dagger_{\textbf{q},\lambda}a_{\textbf{q},\lambda^\prime}\tag{4}$$ which is then used to act on one-particle photon states to determine the helicity or spin projection along the direction of propagation.

However, nothing is said about $L^{ij}$ i.e., whether it can be expressed in terms of creation and annihilation operators. This is important since a quantized expression for $L^{ij}$ can be similarly operated on one-particle photon states, to determine whether it has any non-vanishing orbital angular momentum unambiguously.

Question : Can someone give a reference which expresses $L^{jk}$ in terms of the creation and annihilation operators? The presence of $x^j$ and $\partial^k$ in it makes the process of quantizing $L^{jk}$ non-trivial compared to that of $S^{jk}$. Therefore, any hint on how to obtain $L^{jk}$, especially what to substitute for $x^j$ and $\partial^k$, will also be helpful. The ultimate motive is to act it on one-photon states and draw some conclusion.