In this approach, the physical vacuum is viewed as a quantum superfluid which is essentially non-relativistic, whereas Lorentz symmetry is not an exact symmetry of nature but rather the approximate description valid only for the small fluctuations of the superfluid background. Within the framework of the approach, a theory was proposed in which the physical vacuum is conjectured to be a quantum Bose liquid whose ground-state wavefunction is described by the logarithmic Schrödinger equation. It was shown that the relativistic gravitational interaction arises as the small-amplitude collective excitation mode whereas relativistic elementary particles can be described by the particle-like modes in the limit of low momenta. The important fact is that at very high velocities the behavior of the particle-like modes becomes distinct from the relativistic one – they can reach the speed of light limit at finite energy; also, faster-than-light propagation is possible without requiring moving objects to have imaginary mass.