By H. M. Nussenzveig
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Extra info for Causality and Dispersion Relations
6), and is therefore holomorphic in I , . 20 1. Causality and Dispersion Relations Notice, however, that this reasoning cannot be inverted to conclude from that n’(w) = (&‘(o))% is also regular, because branch the analyticity of &’(a) points would not be excluded. 1 1). 5. Physical Origin of Dispersion Relations Can the real refractive index n ( o ) and the extinction coefficient B(o)= ( 2 o / c ) ~ ( oof) a dielectric medium be arbitrary functions of the frequency? 4, n’ = n + i ~is the boundary value of an analytic so that one might expect the existence function that is holomorphic in I , (o), of some relation between its real and imaginary parts.
It is necessary, in addition, that the analytic function be square integrable on any straight line parallel to the real axis, as stated in part (ii) of Titchmarsh’s theorem. 4) is not satisfied. 29) is not a causal transform. As may readily be verified, its inverse Fourier transform vanishes only for t < -a, rather than for t < 0. 7. 1) of square integrability. 7. 1) This usually corresponds to the requirement that the total output energy is at most equal to the total input energy (it may be smaller because of absorption).
Causality and Dispersion Relations The dispersion relation is then of the form + [ (0 - W0)n n! G(w’) - C(w0) - + ( w - uniO ) n + yjm- m X- - (0‘ - w0)n (0‘ -W J + dw‘ (0’ -w ) a * * G(“’(oo) n! 1 8) ’ where we have assumed that G(o) is differentiable up to the order n + 1 at 0 = 00. Additional subtractions can also be made even when dispersion relations with fewer subtractions are valid. 18) is valid for a given n, it is valid a fortiori for n‘ > n, provided that the corresponding higher-order derivatives exist at 0 = a,,.
Causality and Dispersion Relations by H. M. Nussenzveig