We investigate full dynamical behaviors of the spin-Peierls transition in
S=1/2 Heisenberg antiferromagnetic chains by means of two-time Green's functions. The effect of temperature-dependent lattice displacements is incorporated self-consistently. Depending on temperature T and transfer wave number k, the imaginary part of the dynamical susceptibility ${\ensuremath{\chi}}_{\mathrm{zz}}^{\ensuremath{''}}(k,\ensuremath{\omega})$ shows rich structures: At
T=0, it is nonvanishing in the frequency region ${\ensuremath{\Omega}}_{M}(k)<\ensuremath{\Omega} (\ensuremath{\equiv}\ensuremath{\Elzxh}\ensuremath{\omega}/J)<{\ensuremath{\Omega}}_{H}(k).$ ${\ensuremath{\Omega}}_{M}(k)$ denotes a lower edge of a triplet magnetic excitation continuum, and is finite at
k=0 due to lattice distortion. The upper edge ${\ensuremath{\Omega}}_{H}(k)$ increases as
k=0 is approached, which is completely different from the feature without lattice distortion. An asymmetric line shape of ${\ensuremath{\chi}}_{\mathrm{zz}}^{\ensuremath{''}}(k,\ensuremath{\omega})$ is found with a peak at the frequency slightly higher than ${\ensuremath{\Omega}}_{M}(k).$ At
0<T<TSP (spin-Peierls transition point) there appears another structure due to thermal fluctuations in the lower-frequency region, $0<\ensuremath{\Omega}<{\ensuremath{\Omega}}_{L}(k).$ Since ${\ensuremath{\Omega}}_{L}(k)<{\ensuremath{\Omega}}_{M}(k),$ the forbidden band always exists at
T<TSP. Above
TSP, ${\ensuremath{\chi}}_{\mathrm{zz}}^{\ensuremath{''}}(k,\ensuremath{\omega})$ reduces to a smooth curve characterized by a single broad maximum. These features in dynamics are systematically explained in terms of different quasiparticle scattering processes proper to the spin-Peierls phase. The results are compared with recent experimental data of neutron scattering for the inorganic compound
CuGeO3.We discuss a competition between a spin-Peterls state and the antiferromagnetic state for a Heisenberg antiferromagnetic linear chain system including a phonon Hamiltonian. A Hartree-Fock approximation is applied to a Hamiltonian redescribed in terms of spin-less fermions. We get Dyson's equation of phonons which is consistent with this Hartree-Fock approximation. We point out that Pytte's work is not still complete even in the framework of the Hartree-Fock theory since a term was left out there which gave a quantitative difference between the Heisenberg model and the X Y model. We find the softening of phonons in the antiferromagnetic state which has overcome the spin-Peierls state.