Year: 2024
Type: preprint
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 $ω$ the covariant phase space of the classical Klein-Gordon oscillator is a homogeneous Kähler-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 $ω\to 0$, 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 $ω\ne 0$ 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.
Source: arXiv (Cornell University)
Author: Popov, Alexander D.
Institution: Leibniz University Hannover
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4398796869