  By considering correlations between classical orbits we derive semiclassical
expressions for the decay of the quantum fidelity amplitude for classically
chaotic quantum systems, as well as for its squared modulus, the fidelity or
Loschmidt echo. Our semiclassical results for the fidelity amplitude agree with
random matrix theory (RMT) and supersymmetry predictions in the universal Fermi
golden rule regime. The calculated quantum corrections can be viewed as arising
from a static random perturbation acting on nearly self-retracing interfering
paths, and hence will be suppressed for time-varying perturbations. Moreover,
using trajectory-based methods we show a relation, recently obtained in RMT,
between the fidelity amplitude and the cross-form factor for parametric level
correlations. Beyond RMT, we compute Ehrenfest-time effects on the fidelity
amplitude. Furthermore our semiclassical approach allows for a unified
treatment of the fidelity, both in the Fermi golden rule and Lyapunov regimes,
demonstrating that quantum corrections are suppressed in the latter.
