The scattering of a classical Maxwell field by a fixed spherically symmetric scatterer of finite range is considered. This problem is easily related to the scattering of a scalar field which is considered first. The condition of strict causality is imposed on the forward scattering of a sequence of primary scalar wave packets { ψ p ( n) }, n = 0, 1, 2, …, in which the nth primary wave contains only partial waves for l ≥ n. This implies that the dispersion relation is satisfied by each member of an infinite set of causal amplitudes which are expressed as combinations of the elements of the S-matrix { Sl( ω)}. Inverting the relation for the causal amplitudes in terms of the S-matrix, one obtains a set of dispersion formulas which give J mS m(ω), for each m, in terms of combinations of the R eS l(ω)'s plus the low energy limits of [ S l(ω) ω 2l+1 ] for all l ≥ m. Hence the real part of the scattering amplitude, for all angles and frequencies, is determined by the knowledge of its imaginary part plus the low energy limits of [ S l(ω) ω 2l+1 ] . Corresponding formulas exist for R eS m(ω) and the imaginary part of the scattering amplitude. If a perturbation expansion method is valid, then the restriction is much stronger, namely, low energy limits determine all phase shifts, or, equivalently, the s-wave phase shift determines the phase shift for all higher angular momenta. The scattering of the Maxwell field is described by three matrices, corresponding to the two transverse modes and one longitudinal mode; each of these matrices is shown to satisfy the same set of dispersion relations as that for scalar waves, except for the omission of the s-wave equation in the case of transverse modes. The equivalent formulation of the derivation of the dispersion relations for the S-matrix in terms of the vanishing of commutators is given in an Appendix for the case of the scattering of scalar wave by a static potential.