2025/09/11 by Popov, Alexander D.
#510 #Bergman spaces #Quantum relativistic oscillator #Segal-Bargmann representations
paper · doi:10.15488/19574
We consider classical and quantum dynamics of relativistic oscillator in Minkowski space R3,1 . 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 Z6 = AdS7/U(1) = U(3, 1)/U(3) × U(1). In the limit ω → 0, this manifold is deformed into the covariant phase space T ∗ H3 of a free relativistic particle, where H3 = H3 + ∪ H3 – is a two-sheeted hyperboloid in momentum space. Quantization of this model with ω = 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 Z6 . This relativistic model is Lorentz covariant, unitary and does not contain non-physical states.