  A time-domain formulation of the equilibrium quantum fluctuation-dissipation
theorem (FDT) in the whole range of temperatures is presented. In the classical
limit, the FDT establishes a proportionality relation between the dissipative
part of the linear response function and the derivative of the corresponding
equilibrium correlation function. At zero temperature, the FDT takes the form
of Hilbert transform relations between the dissipative part of the response
function and the corresponding symmetrized equilibrium correlation function,
which allows to establish a connection with analytic signal theory. The
time-domain formulation of the FDT is especially valuable when
out-of-equilibrium dynamics is concerned, as it is for instance the case in the
discussion of aging phenomena.
