  A microscopic theory of the electrical conductivity $\sigma(\omega)$ within
the t-J model is developed. An exact representation for $\sigma(\omega)$ is
obtained using the memory-function technique for the relaxation function in
terms of the Hubbard operators, and the generalized Drude law is derived. The
relaxation rate due to the decay of charge excitations into particle-hole pairs
assisted by antiferromagnetic spin fluctuations is calculated in the
mode-coupling approximation. Using results for the spectral function of spin
excitations calculated previously, the relaxation rate and the optical and dc
conductivities are calculated in a broad region of doping and temperatures. The
reasonable agreement of the theory with experimental data for cuprates proves
the important role of spin-fluctuation scattering in the charge dynamics.
