The dynamical properties of nuclei, carried by the concept of phonon
quasiparticles (QP), are central to the field of condensed matter. While the
harmonic approximation can reproduce a number of properties observed in real
crystals, the inclusion of anharmonicity in lattice dynamics is essential to
accurately predict properties such as heat transport or thermal expansion. For
highly anharmonic systems, non perturbative approaches are needed, which result
in renormalized theories of lattice dynamics. In this article, we apply the
Mori-Zwanzig projector formalism to derive an exact generalized Langevin
equation describing the quantum dynamics of nuclei in a crystal. By projecting
this equation on quasiparticles in reciprocal space, and with results from
linear response theory, we obtain a formulation of vibrational spectra that
fully accounts for the anharmonicity. Using a mode-coupling approach, we
construct a systematic perturbative expansion in which each new order is built
to minimize the following ones. With a truncation to the lowest order, we show
how to obtain a set of self-consistent equations that can describe the
lineshapes of quasiparticles. The only inputs needed for the resulting set of
equations are the static Kubo correlation functions, which can be computed
using (fully quantum) path-integral molecular dynamics or approximated with
(classical or ab initio) molecular dynamics. We illustrate the theory with an
application on fcc 4He, an archetypal quantum crystal with very strong
anharmonicity