  We present a study of the dynamic structure factor of the antiferromagnetic
spin-1/2 Heisenberg chain at finite temperatures and finite magnetic fields.
Using Quantum-Monte-Carlo based on the stochastic series expansion and
Maximum-Entropy methods we evaluate the longitudinal and the transverse dynamic
structure factor from vanishing magnetic fields up to and above the threshold
$B_c$ for ferromagnetic saturation, as well as for high and for intermediate
temperatures. We study the field-induced redistribution of spectral weight
contrasting longitudinal versus transverse excitations. At finite fields below
saturation incommensurate low-energy modes are found consistent with zero
temperature Bethe-Ansatz. The crossover between the field induced ferromagnet
above $B_c$ and the Luttinger liquid below $B_c$ is analyzed in terms of the
transverse spin-dynamics. Evaluating sum-rules we assess the quality of the
analytic continuation and demonstrate excellent consistency of the
Maximum-Entropy results.
