  An ab initio quantum Monte Carlo method is introduced for calculating total
rates of muon weak capture in light nuclei with mass number $A \leq 12$. As a
first application of the method, we perform a calculation of the rate in $^4$He
in a dynamical framework based on realistic two- and three-nucleon interactions
and realistic nuclear charge-changing weak currents. The currents include one-
and two-body terms induced by $\pi$- and $\rho$-meson exchange, and
$N$-to-$\Delta$ excitation, and are constrained to reproduce the empirical
value of the Gamow-Teller matrix element in tritium. We investigate the
sensitivity of theoretical predictions to current parametrizations of the
nucleon axial and induced pseudoscalar form factors as well as to two-body
contributions in the weak currents. The large uncertainties in the measured
values obtained from bubble-chamber experiments (carried out over 50 years ago)
prevent us from drawing any definite conclusions.
