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I’m a bit confused about time averages in electromagnetism using complex amplitudes. Specifically about the angular momentum of the fields- the spin part is proportional to $\bf{E} \times \bf{A}$. In the Coulomb gauge in the absence of any sources, $\bf{E} = -c^{-1} \partial_t \bf{A}$, and so the electric field is proportial to the vector potential for a monochromatic plane wave, and the spin part seems to vanish identically. However, using the time average of the angular momentum we have that $<L_{spin} \propto \bf{E} \times \bf{A^*}$, in which case the spin is not necessarily zero. How is the time average of zero a non zero number?

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