Supersymmetric Klein–Gordon and Dirac oscillators

This is the authors' abstract. We don't add key points for this paper.

We have recently shown that the space of initial data (covariant phase space) of the relativistic oscillator in Minkowski space R3,1 R 3 , 1 is a homogeneous Kähler–Einstein manifold Z₆=AdS₇/U(1) =U(3,1)/U(3)× U(1) Z 6 = AdS 7 / U ( 1 ) = U ( 3 , 1 ) / U ( 3 ) × U ( 1 ) . It was also shown that the energy eigenstates of the quantum relativistic oscillator form a direct sum of two weighted Bergman spaces of holomorphic (particles) and antiholomorphic (antiparticles) square-integrable functions on the covariant phase space Z₆ Z 6 of the classical oscillator. Here we show that the covariant phase space of the supersymmetric version of the relativistic oscillator (oscillating spinning particle) is the odd tangent bundle of the space Z₆ Z 6 . Quantizing this model yields a Dirac oscillator equation on the phase space whose solution space is a direct sum of two spinor spaces parametrized by holomorphic and antiholomorphic functions on the odd tangent bundle of Z₆ Z 6 . After expanding the general solution in Grassmann variables, we obtain components of the spinor field that are holomorphic and antiholomorphic functions from Bergman spaces on Z₆ Z 6 with different weight functions. Thus, the supersymmetric model under consideration is exactly solvable, Lorentz covariant and unitary.