  We explore the life time of excitations in a dispersive Luttinger liquid. We
perform a bosonization supplemented by a sequence of unitary transformations
that allows us to treat the problem in terms of weakly interacting
quasiparticles. The relaxation described by the resulting Hamiltonian is
analyzed by bosonic and (after a refermionization) by fermionic perturbation
theory. We show that the the fermionic and bosonic formulations of the problem
exhibit a remarkable strong-weak-coupling duality. Specifically, the fermionic
theory is characterized by a dimensionless coupling constant $\lambda= m^*l^2T$
and the bosonic theory by $\lambda^{-1}$, where $1/m^*$ and $l$ characterize
the curvature of the fermionic and bosonic spectra, respectively, and $T$ is
the temperature.
