标题： 超对称克莱因-戈登和狄拉克振子 显示英文标题

标题： Supersymmetric Klein-Gordon and Dirac oscillators

摘要： 我们最近证明了闵可夫斯基空间中相对论振子的初始数据空间（协变相空间）$\mathbb{R}^{3,1}$是一个齐次Kähler-Einstein流形$Z_6$=AdS$_7$/U(1)=U(3,1)/U(3)$\times$U(1)。还证明了量子相对论振子的能量本征态是经典振子的协变相空间$Z_6$上全纯（粒子）和反全纯（反粒子）平方可积函数的两个加权Bergman空间的直和。在这里，我们证明了相对论振子的超对称版本（振荡自旋粒子）的协变相空间是空间$Z_6$的奇切丛。对该模型进行量子化会在相空间上得到一个狄拉克振子方程，其解空间是由$Z_6$的奇切丛上的全纯和反全纯函数参数化的两个旋量空间的直和。 在展开Grassmann变量的通解后，我们得到自旋场的分量，这些分量是从Bergman空间在$Z_6$上的全纯和反全纯函数，具有不同的权函数。 因此，所考虑的超对称模型是精确可解的，洛伦兹协变的且幺正的。 显示英文摘要

摘要： 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 K\"ahler-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.