  We study electrical and optical conductivities of hole gas in $p$-doped bulk
III-V semiconductors described by the Luttinger Hamiltonian. We provide exact
analytical expressions of the Drude conductivity, inverse relaxation time for
various impurity potentials, Drude weight and optical conductivity in terms of
the Luttinger parameters $\gamma_1 $ and $\gamma_2$. The back scattering is
completely suppressed as a result of the helicity conservation of the heavy and
light hole states. We find that the inverse relaxation time of heavy holes is
much less than that of the light holes for Coulomb-type and Gaussian-type
impurity potentials and vice-versa for short-range impurity potentials. The
Drude conductivity increases non-linearly with the increase of the hole
density. The exponent of the density dependence of the conductivity is obtained
in Thomas-Fermi limit. The Drude weight varies linearly with the density even
in presence of the spin-orbit coupling. The finite-frequency optical
conductivity goes as $\sqrt{\omega}$ and its amplitude strongly depends on the
Luttinger parameters. The Luttinger parameters can be extracted from the
optical conductivity measurement.
