A many-body theory is presented here for the Van Hove pair (or density) correlation function, in a quantum fluid obeying Boltzmann statistics, of which the Fourier transform $S(q,\ensuremath{\omega})$ determines the differential cross section for inelastic coherent scattering of slow neutrons. The theoretical formalism of two-particle thermodynamic Green's functions is based on Baym and Kadanoff's conserving
T-matrix approximation, developed for strong short-range (intermolecular) potentials. The only other approximations made are: (i) the smallness of the activity or fugacity $z\ensuremath{\ll}1$, which implies Boltzmann statistics for the single-particle states; and (ii) the
T matrices are replaced by elastic scattering amplitudes. Our results for $S(q,\ensuremath{\omega})$ developed to order
z2 are found to consist of Boltzmann averages of Lorentzian distributions in the energy transfer $\ensuremath{\hbar}\ensuremath{\omega}$ (except for a three-particle scattering term which is developed in the $\ensuremath{\delta}$-function limit of the Lorentzians). Our results are general enough to satisfy detailed balancing at all stages and are found to generalize Van Leeuwen and Yip's results for the classical case. The only unknown in our results is the intermolecular scattering amplitude or phase shifts. Our results seem, however, to break down in the hydrodynamic limit $\ensuremath{\omega}$ and $q\ensuremath{\rightarrow}0$ since each term in our expansion diverges in this limit.
The static scaling laws may be generalized to dynamic phenomena by making assumptions on the behavior of time-dependent correlation functions near the magnetic critical point. At any temperature different from Tc, the correlation functions are assumed to reflect the ``hydrodynamic'' behavior of the system, for sufficiently long wavelengths and low frequencies. As the critical temperature is approached, however, the range ξ of spatial correlations in the system diverges, and the domain of applicability of hydrodynamics is reduced to very small wavenumbers, satisfying the inequality kξ≪1. The dynamic scaling assumptions relate the behavior of the time-dependent correlation functions in the ``hydrodynamic'' (kξ≪1) and ``critical'' (kξ≫1) regions. It is thus possible to predict certain features of the dynamic behavior at Tc, from a knowledge of the behavior away from Tc. Application is made to the spin-correlation function of the isotropic Heisenberg antiferromagnet, which is directly measurable by inelastic neutron scattering. Detailed predictions are presented on the form of the cross section as a function of energy and momentum transfer, and of temperature near Tc.