  Localization marks the breakdown of thermalization in subregions of quantum
many-body systems in the presence of sufficiently large disorder. In this
paper, we use numerical techniques to study thermalization and localization in
a many-body system of coupled quantum harmonic oscillators obtained by
discretizing a scalar quantum field theory in Minkowski spacetime. We consider
a Gaussian initial state, constructed through a global mass quench, with a
quadratic Hamiltonian, and solve for the system's exact dynamics without and
with disorder in one and two spatial dimensions. We find that finite-size
systems localize for sufficiently large disorder in both cases, such that the
entanglement entropy of subregions retains its initial area-law behavior, and
the system no longer develops long-range correlations. To probe the
thermalization-to-localization transition further, we define a frequency gap
ratio that measures adjacent gaps in the phase space eigenvalues of the
Hamiltonian and study how it varies with disorder strength and system size. We
find signatures of a chaotic regime at intermediate disorder in two spatial
dimensions and argue that it is a finite-size effect, such that the system
would localize for arbitrarily small disorder in the continuum in both one and
two spatial dimensions, consistent with Anderson localization. Lastly, we use
the frequency gap ratio to argue that in three spatial dimensions, on the other
hand, the system would only localize for disorder strengths above a critical
value in the continuum, again consistent with Anderson localization.
