Klein-Gordon oscillators and Bergman spaces

We consider classical and quantum dynamics of relativistic oscillator in Minkowski space $R^{3,1}$. It is shown that for a non-zero frequency parameter ω the covariant phase space of the classical Klein-Gordon oscillator is a homogeneous Kähler-Einstein manifold $Z_6=\mathrm{Ad}{S}_7/U\left(1\right)=U(3,1)/U\left(3\right)\times U\left(1\right)$. In the limit $\omega \to 0$, this manifold is deformed into the covariant phase space $T^{⁎}{H}^3$ of a free relativistic particle, where $H^3=H_{+}^3\cup H_{-}^3$ is a two-sheeted hyperboloid in momentum space. Quantization of this model with $\omega \ne 0$ leads to the Klein-Gordon oscillator equation which we consider in the Segal-Bargmann representation. It is shown that the general solution of this model is given by functions from the weighted Bergman space of square-integrable holomorphic (for particles) and antiholomorphic (for antiparticles) functions on the Kähler-Einstein manifold $Z_6$. This relativistic model is Lorentz covariant, unitary and does not contain non-physical states.