Supersymmetric Klein–Gordon and Dirac oscillators
Year: 2025
Type: article
Abstract: We have recently shown that the space of initial data (covariant phase space) of the relativistic oscillator in Minkowski space $$\mathbb {R}^{3,1}$$ R 3 , 1 is a homogeneous Kähler–Einstein manifold $$Z_6=\textrm{AdS}_7/\textrm{U}(1) =\textrm{U}(3,1)/\textrm{U}(3)\times \textrm{U}(1)$$ 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 $$Z_6$$ 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 $$Z_6$$ 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 $$Z_6$$ 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 $$Z_6$$ Z 6 with different weight functions. Thus, the supersymmetric model under consideration is exactly solvable, Lorentz covariant and unitary.
Source: Letters in Mathematical Physics
Author: Alexander D. Popov
Institution: Leibniz University Hannover
Language: English
Subfield: Nuclear and High Energy Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,990
OpenAlex ID: W4408741894