Anisotropy and asymptotic degeneracy of the physical-Hilbert-space inner-product metrics in an exactly solvable crypto-unitary quantum model

DOI10.3390/SYM16030353arXiv1212.0734MaRDI QIDQ6502946

Abstract: An an exactly solvable quantum model is found to sample the evolution towards the sudden and complete loss of observability. For the purpose we choose an N-level system. While the time runs from 0 to 1, the process (leading to the collapse) is controlled by a toy-model Hamiltonian H and by a unitary-evolution guaranteeing minimally anisotropic (i.e., unique) Hilbert-space metric. The process of the degeneracy of the real N-plet of energy levels is studied without the usual assumption of adiabaticity. The initial Hamiltonian is diagonal, the initial metric is chosen as identity. Once the system reaches the observability horizon, the metric becomes singular (of rank one) while the end-point Hamiltonian acquires the canonical Jordan-block form (i.e., it loses its diagonalizability). An optimal measure of the distance from the final catastrophe is finally found in a universal, exact formula for the spectrum of the metric.

Mathematics Subject Classification ID

Symmetry breaking in quantum theory (81R40) Selfadjoint operator theory in quantum theory, including spectral analysis (81Q10) General quantum mechanics and problems of quantization (81S99) Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) (81P10) Nonselfadjoint operator theory in quantum theory including creation and destruction operators (81Q12) Special quantum systems, such as solvable systems (81Q80)

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