Non-Hermitian Generalization of Bloch Sphere in Spacetime Algebra

Abstract

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.