  We consider the many-body spectra of interacting bosonic quantum fields on a
lattice in the semiclassical limit of large particle number $N$. We show that
the many-body density of states can be expressed as a coherent sum over
oscillating long-wavelength contributions given by periodic, non-perturbative
solutions of the, typically non-linear, wave equation of the classical
(mean-field) limit. To this end we construct the semiclassical approximation
for both the smooth and oscillatory part of the many-body density of states in
terms of a trace formula starting from the exact path integral form of the
propagator between many-body quadrature states. We therefore avoid the use of a
complexified classical limit characteristic of the coherent state
representation. While quantum effects like vacuum fluctuations and gauge
invariance are exactly accounted for, our semiclassical approach captures
quantum interference and therefore is valid well beyond the Ehrenfest time
where naive quantum-classical correspondence breaks down. Remarkably, due to a
special feature of harmonic systems with incommesurable frequencies, our
formulas are generically valid also in the free-field case of non-interacting
bosons.
