  We study a chaotic quantum transport in the presence of a weak spin-orbit
interaction. Our theory covers the whole symmetry crossover regime between
time-reversal invariant systems with and without a spin-orbit interaction. This
situation is experimentally realizable when the spin-orbit interaction is
controlled in a conductor by applying an electric field. We utilize a
semiclassical approach which has recently been developed. In this approach, the
non-Abelian nature of the spin diffusion along a classical trajectory plays a
crucial role. New analytical expressions with one crossover parameter are
semiclassically derived for the average conductance, conductance variance and
shot noise. Moreover numerical results on a random matrix model describing the
crossover from the GOE (Gaussian Orthogonal Ensemble) to the GSE (Gaussian
Symplectic Ensemble) are compared with the semiclassical expressions.
