  Spatial correlation between atoms can generate a depletion in the energy
dispersion of acoustic phonons. Two well known examples are rotons in
superfluid helium and the Kohn anomaly in metals. Here we report on the
observation of a large softening of the transverse acoustic mode in quantum
paraelectric SrTiO$_3$ by means of inelastic neutron scattering. In contrast to
other known cases, this softening occurs at a tiny wave vector implying spatial
correlation extending over a distance as long as 40 lattice parameters. We
attribute this to the formation of mesoscopic fluctuating domains due to the
coupling between local strain and quantum ferroelectric fluctuations. Thus, a
hallmark of the ground state of insulating SrTiO$_3$ is the emergence of
hybridized optical-acoustic phonons. Mesoscopic fluctuating domains play a role
in quantum tunneling, which impedes the emergence of a finite macroscopic
polarisation.
