Klein-Gordon oscillators and Bergman spaces

Abstract

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 K\"ahler-Einstein manifold $Z_6=\mathrm{Ad}S_7/\mathrm{U}(1)=\mathrm{U}(3,1)/\mathrm{U}(3)\times \mathrm{U}(1)$. 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 K\"ahler-Einstein manifold $Z_6$. This relativistic model is Lorentz covariant, unitary and does not contain non-physical states.