We study the analytic structure for the eigenvalues of the one-dimensional Dirac oscillator, by analytically continuing its frequency on the complex plane. A twofold Riemann surface is found, connecting the two states of a pair of particle and antiparticle. One can, at least in principle, accomplish the transition from a positive energy state to its antiparticle state by moving the frequency continuously on the complex plane, without changing the Hamiltonian after transition. This result provides a visual explanation for the absence of a negative energy state with the quantum number n=0.

中文翻译：

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1+1狄拉克振荡器的解析特征值结构

我们通过在复平面上分析延续其频率来研究一维狄拉克振荡器的特征值的解析结构。发现了双重黎曼曲面，连接了一对粒子和反粒子的两种状态。至少在原则上，可以通过在复平面上连续移动频率来完成从正能态到其反粒子态的跃迁，而无需在跃迁后改变哈密顿量。该结果为不存在量子数 n=0 的负能态提供了直观的解释。