A Hermitian bypass to the non-Hermitian quantum theory
arXiv:2310.10263 · doi:10.1088/1751-8121/adbac6
Abstract
Describing systems with non-Hermitian (NH) operators remains a challenge in quantum theory due to instabilities (e.g., exceptional points and decoherence) arising from interactions with the environment. We propose a framework to express the energy states of NH Hamiltonians using a well-defined basis (dub computational basis) derived from a related Hermitian operator. This suitably shifts the singularities from the basis states to the expansion coefficients, allowing for their easier mathematical treatment and parametric control. Furthermore, we introduce a `space-time' transformation on the computational basis that defines a generic dual space map for the energy states. Interestingly, this transformation leads to a symmetry for real/imaginary energy values, revealing the existence of weaker condition than hermiticity or the symmetry. This leads to clearer understanding and novel interpretations of key features like exceptional points, dual space, and weaker symmetry-enforced real eigenvalues. The applicability of our framework extends to various branches of physics where NH operators manifest as ladder operators, order parameters, self-energies, projectors, and other entities.
20 pages including Supplementary Material, two column, 6 figures. The second version of this manuscript includes new examples and more detailed discussions at appropriate places
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