The optical theorem
σtot=4πkIm(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†=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†=T−T†=2Im(T)
TT† is nothing but the crosssection (I suppressed some integral signs here for brevity) the optical theorem is right there. Hence one finds σ∼Im(T)
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