  We show that turning on the interaction of relativistic spinless particles
with the vacuum of relativistic quantum mechanics (RQM) leads to the
replacement of the Klein-Gordon equation with the Klein-Gordon oscillator
equation. In this case, coordinate time becomes an operator and free
relativistic particles go into a virtual state. We discuss geometry associated
with classical and quantum Klein-Gordon oscillators, and its relation to the
geometry underlying the description of free particles.
