  We study the correlation functions of quantum spin $1/2$ ladders at finite
temperature, under a magnetic field, in the gapless phase at various relevant
temperatures $T\neq 0$, momentum $q$ and frequencies $\omega$. We compute those
quantities using the time dependent density matrix renormalization group
(T-DMRG) in some optimal numerical scheme. We compare these correlations with
the ones of dimerized quantum spin chains and simple spin chains, that we
compute by a similar technique. We analyze the intermediate energy modes and
show that the effect of temperature lead to the formation of an essentially
dispersive mode corresponding to the propagation of a triplet mode in an
incoherent background, with a dispersion quite different from the one occurring
at very low temperatures. We compare the low energy part of the spectrum with
the predictions of the Tomonaga-Luttinger liquid field theory at finite
temperature. We shows that the field theory describes in a remarkably robust
way the low energy correlations for frequencies or temperatures up to the
natural cutoff (the effective dispersion) of the system. We discuss how our
results could be tested in e.g. neutron scattering experiments.
