scattering - Optical theorem and conservation of particle current

The optical theorem

$$ \sigma_{tot} = \frac{4\pi}{k} \text{Im}(f(0)) $$

links the total cross section with the imaginary part of the scattering amplitude.

My lecture notes say that this is a consequence of the conservation of the particle current. How do I get to this consequence?

Answer

Conservation of particle current is nothing but the statement that a theory has to be unitary. In other words the scattering matrix $S$ has to obey

$SS^\dagger=1$

Defining $S=1+iT$ i.e. rewriting the scattering matrix as a trivial part plus interactions (encoded in $T$ which corresponds to your $f$) one finds from the unitarity condition:

$iTT^\dagger=T-T^\dagger=2Im(T)$

$TT^\dagger$ is nothing but the crosssection (I suppressed some integral signs here for brevity) the optical theorem is right there. Hence one finds $\sigma\sim Im(T)$