Klein-Gordon oscillators and Bergman spaces

• URL: http://arxiv.org/abs/2405.14349v3
• Date: Wed, 23 Oct 2024 09:52:55 GMT
• Title: Klein-Gordon oscillators and Bergman spaces
• Authors: Alexander D. Popov,
• Abstract summary: We consider classical and quantum dynamics of relativistic oscillator in Minkowski space $mathbbR3,1$. The general solution of this model is given by functions from the weighted Bergman space of square-integrable holomorphic (for particles) and antiholomorphic functions on the K"ahler-Einstein manifold $Z_6$.
• Score: 55.2480439325792
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• 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.

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