Supersymmetric Klein-Gordon and Dirac oscillators

TL;DR

The work develops a geometric, phase-space formulation of the supersymmetric Klein–Gordon and Dirac oscillators in Minkowski space. The Klein–Gordon oscillator is associated with the covariant phase space $Z_6=\mathrm{AdS}_7/\mathrm{U}(1)$, with energy eigenstates occupying weighted Bergman spaces of holomorphic/antiholomorphic functions. Its supersymmetric extension places the dynamics on the odd tangent bundle $\Pi TZ_6$, where quantization yields a Dirac oscillator whose spinor components live in Bergman spaces on the odd tangent directions with weights $\mu_{N+k}$ ($k=0\ldots3$). The model is shown to be exactly solvable, Lorentz covariant and unitary, providing a geometric realization of the Dirac equation as an odd covariant Laplacian on the covariant phase space.

Abstract

We have recently shown that the space of initial data (covariant phase space) of the relativistic oscillator in Minkowski space $\mathbb{R}^{3,1}$ is a homogeneous Kähler-Einstein manifold $Z_6$=AdS$_7$/U(1)=U(3,1)/U(3)$\times$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 $Z_6$ 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 $Z_6$. 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 $Z_6$. 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 $Z_6$ with different weight functions. Thus, the supersymmetric model under consideration is exactly solvable, Lorentz covariant and unitary.