  The traditional Kubo formula is generalized to describe the linear response
with respect to non-Abelian fields. To fulfil the demand for studying spin
transport, the SU(2) Kubo formulae are derived by two conventional approaches
with different gauge fixings. Those two approaches are shown to be equivalent
where the non-conservation of the SU(2) current plays an essential role in
guaranteeing the consistency. Some concrete examples relating Spin Hall Effect
are considered. The dc spin conductivity vanishes in the system with parabolic
unperturbed dispersion relation. By applying a time-dependent Rashba field, the
spin conductivity can be measured directly. Our formula is also applied to the
high-dimensional representation for the interests of some important models,
such as Luttinger model and bilayer spin Hall system.
