AI 中文总结
该研究利用时空代数建立了非厄米两能级量子系统的布洛赫球几何推广，将态空间扩展为未来光锥，分类了哈密顿量动力学类型并研究了PT对称量子力学的异常点拓扑特征。
AI 中文摘要
我们利用时空代数（STA）形式体系，为具有非厄米哈密顿量的两能级量子系统建立了布洛赫球的几何推广。通过将态密度算子从偶子代数提升至完整STA，我们证明态空间从单位2-球面扩展为未来光锥。由一般非厄米哈密顿量产生的非幺正时间演化对应于零矢量上的固有正时洛伦兹变换。我们将哈密顿量动力学分为四类不同的几何类别：空间旋转（对应PT对称系统）、纯推进（反PT对称系统）、零旋转（异常点）及一般混合类型。我们还利用该形式体系研究了PT对称量子力学的若干结果，包括异常点的拓扑特征。
英文摘要
We establish a geometric generalization of the Bloch sphere for two-level quantum systems with non-Hermitian Hamiltonians using the Spacetime Algebra (STA) formulation. By lifting the state density operator from the even subalgebra to the full STA, we show that the state space expands from the unit 2-sphere to a future light cone. The non-unitary time evolution generated by a general non-Hermitian Hamiltonian corresponds to proper orthochronous Lorentz transformations on the null vectors. We classify the Hamiltonian dynamics into four distinct geometric classes: spatial rotations (corresponding to $\mathcal{PT}$-symmetric systems), pure boosts (anti-$\mathcal{PT}$-symmetric systems), null rotations (exceptional points), and general mixtures. We also use this formulation to study several results from PT-symmetric quantum mechanics, including the topological features of the exceptional points.
