Klein-Gordon oscillators, RQM and quantum time

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.
Comment: 24 pages; v2: minor corrections