  We consider the quantum dynamics of a magnetic domain wall at low
temperatures, where dissipative couplings to magnons and electrons are very
small. The wall motion is then determined by its coupling to phonons and
nuclear spins, and any pinning potentials. In the absence of nuclear spins
there is a dominant superOhmic 1-phonon coupling to the wall velocity, plus a
strongly T-dependent Ohmic coupling to pairs of phonons. There is also a
T-independent Ohmic coupling between single phonons and the wall chirality,
which suppresses ``chirality tunneling''. We calculate the effect of these
couplings on the T-dependent tunneling rate of a wall out of a pinning
potential. Nuclear spins have a very strong and hitherto unsuspected influence
on domain wall dynamics, coming from a hyperfine-mediated coupling to the
domain wall position. For $k_B T \gg \omega_{0}$ this coupling yields a
spatially random potential, fluctuating at a rate governed by the nuclear
$T_2$. When $k_B T \ll \omega_{0}$, the hyperfine potential fluctuates around a
linear binding potential. The wall dynamics is influenced by the fluctuations
of this potential, ie., by the nuclear spin dynamics. Wall tunneling can occur
when fluctuations open an occasional ``tunneling window''. This changes the
crossover to tunneling and also causes a slow "wandering", in time, of the
energy levels associated with domain wall motion inside the pinning potential.
This effect is fairly weak in $Ni$- and $Fe$- based magnets, and we give an
approximate treatment of its effect on the tunneling dynamics, as well as a
discussion of the relationship to recent domain wall tunneling experiments.
