A Hermitian bypass to the non-Hermitian quantum theory

• URL: http://arxiv.org/abs/2310.10263v1
• Date: Mon, 16 Oct 2023 10:39:25 GMT
• Title: A Hermitian bypass to the non-Hermitian quantum theory
• Authors: Priyanshi Bhasin, Tanmoy Das
• Abstract summary: Non-Hermitian (NH) operators are gaining growing significance in all branches of physics and beyond. Here, we propose a quantum theory that resolves challenges with singularities, instabilities, and violations of standard linear algebra and differential geometry. Our formalism elucidates the origin and interpretation of several features associated with NH operators, including exceptional points, normal operators, dual-space mapping, dynamical metric manifold, and emergent symmetry-enforced real eigenvalues.
• Score: 0.2538209532048867
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Non-Hermitian (NH) operators are gaining growing significance in all branches of physics and beyond. However, NH quantum theory faces challenges with singularities, instabilities, and violations of standard linear algebra and differential geometry. Here, we propose a quantum theory that resolves these instabilities by reassigning them to the expansion parameters of a suitably defined basis state of a Hermitian operator. We discover a computational basis, defined by the eigenspace of $H^\dagger H$, in which the exceptional points of $H$ are positioned as vacua on the two boundaries. The theory also introduces a generic dual space map that functions as a dynamical `space-time' transformation within the computational space. When this transformation assumes a static symmetry, it ensures real energies, unraveling a hidden symmetry beyond hermiticity or parity-time reversal symmetries. Our formalism elucidates the origin and interpretation of several features associated with NH operators, including exceptional points, normal operators, dual-space mapping, dynamical metric manifold, and emergent symmetry-enforced real eigenvalues. Our general framework broadens the application of NH theory across numerous branches of physics where NH operators manifest as ladder operators, order parameters, self-energies, projectors, and other entities.

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