  We study the energy-momentum relations of the Nambu-Goldstone modes in
quantum antiferromagnetic Heisenberg models on a hypercubic lattice. This work
is a sequel to the previous series about the models. We prove that the
Nambu-Goldstone modes show a linear dispersion relation for the small momenta,
i.e., the low energies of the Nambu-Goldstone excitations are proportional to
the momentum of the spin wave above the infinite-volume ground states with
symmetry breaking. Our method relies on the upper bounds for the
susceptibilities which are derived from the reflection positivity of the
quantum Heisenberg antiferromagnets. The method is also applicable to other
systems when the corresponding upper bounds for the susceptibilities are
justified.
