  Using $1/S$ expansion, we discuss the magnon spectrum of Heisenberg
antiferromagnet (AF) on a simple cubic lattice with small dipolar interaction
at small temperature $T\ll T_N$, where $T_N$ is the Neel temperature. Similar
to 3D and 2D ferromagnets, quantum and thermal fluctuations renormalize greatly
the bare gapless spectrum leading to a gap $\Delta\sim \omega_0$, where
$\omega_0$ is the characteristic dipolar energy. This gap is accompanied by
anisotropic corrections to the free energy which make the cube edges easy
directions for the staggered magnetization (dipolar anisotropy). In accordance
with previous results, we find that dipolar forces split the magnon spectrum
into two branches. This splitting makes possible two types of processes which
lead to a considerable enhance of the damping compared to the Heisenberg AF: a
magnon decay into two other magnons and a confluence of two magnons. It is
found that magnons are well defined quasiparticles in quantum AF. We
demonstrate however that a small fraction of long-wavelength magnons can be
overdamped in AFs with $S\gg1$ and in quantum AFs with a single-ion anisotropy
competing with the dipolar anisotropy. Particular materials are pointed out
which can be suitable for experimental observation of this long-wavelength
magnons breakdown that contradicts expectation of the quasiparticle concept.
