By H. M. Nussenzveig
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Additional resources for Causality and Dispersion Relations
19) which contains an additional power of 0’in the denominator for o’-+Go. 8. Dispersion Relations and Distributions 33 In the dispersion relations employed in high-energy physics, subtraction constants are usually related with the coupling constants that characterize the interactions. 8. 3, the assumption that the input, the output, and Green's function are ordinary functions is too restrictive for many physical applications, in which we have to deal with distributions. We shall now extend the preceding discussion of the relation between causality and analyticity to this more general case.
That is O ( w - ' - ' ) or O(o-'[Inlol]-'-") ii. , b' c gL' (cf. Section A6). 7), with all F,(o) having compact support. 9) If F ( o ) is a summable function, so is F(o)cp (o), whenever cp (0)is bounded. Similarly, let cp(o)be a C" function that is bounded, as well as all its derivatives. Then it can be shown that, i f T, E gnL' and cp(o) E a, (Tw,cp) exists, although neither T, nor cp need have compact support. In particular, cp(o)= 1 E a, and we have (Tu, 1) = ST, do. 10) To characterize the distributions G, that are convolutionable with P ( l / o ) , we might require that w-'C, be summable.
We assume that this distance is much larger than the wavelength, so that k z $ 1. 10. The Optical Theorem 49 point P' of the medium (Fig. 4) where r = PIP and a is the unit vector along the direction P'P. Let ( p , cp) be the polar coordinates of P' in the (x,y)-plane, and 8 the angle between P ' P and Oz. 2. 5) The contribution to the scattered field at P from a volume element p dp d q . 6 around P' is dE, = Ne,p dp dcp 6, where N is the number of dipoles per unit volume and 6 is the thickness of the layer.