Klein-Gordon oscillators and Bergman spaces

We consider classical and quantum dynamics of relativistic oscillator in Minkowski space ${\mathbb{R}}^{3,1}$. It is shown that for a non-zero frequency parameter $\omega$ the covariant phase space of the classical Klein-Gordon oscillator is a homogeneous Kähler-Einstein manifold $Z_6=\mathrm{A}\mathrm{d}{S}_7\mathrm{/}\mathrm{U}(1)=\mathrm{U}(3,1)\mathrm{/}\mathrm{U}(3)\times \mathrm{U}(1)$. In the limit $\omega \to 0$, this manifold is deformed into the covariant phase space $T^{\ast }{H}^3$ 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 $\omega \mathrm{\ne }0$ 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.