I'm trying to calculate the angular momentum in the chern simons theory. But equivalently, I was trying a calculation of angular momentum in the Maxwell field theory, which will hopefully be insightful in getting the answer for CS theory.
$\mathcal{L} \propto f^{\mu\nu}f_{\mu\nu} $
The generalised angular momentum is defined as:
$p^i = \frac{\partial \mathcal{L}}{\partial (\partial_0 a_i)} = E^i$
Now, the angular momentum would be defined as:
$L^i = \epsilon^{ijk}a_j p_k$
However, the angular momentum in a Maxwell theory is given by $L^i \propto x \times (E\times B)$, which is nowhere near the answer I get for the angular momentum I've defined. Am I wrong in the definition of the angular momentum? There should be a B field present to get the correct answer, which is nowhere to be found in my answer.
I would also appreciate the calculation for a pure CS theory, if there is any fundamental difference in the steps for either theory. The answer for the CS theory: $\mathcal{L} = \frac{k}{4\pi}ada$
$L = \frac{-k}{4\pi} \int x^i \epsilon^{ij} (a_j f + f a_j)$
Which is completely against any intuition I have. This is given in Carl Turner's notes (https://arxiv.org/abs/1905.12656) in eq 3.67.
Edit: pasting a clarification from the comments, I'm not sure if my definition for angular momentum is correct (I haven't seen any mention of generalised angular momentum in my quick research). I'm literally just basing it out of a naive generalisation of angular momentum in classical mechanics. Hence my question.