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Springer eBooks

On a theorem for complex symmetric matrices and its relevance in the study of decay phenomena

Charles E. Reid, Erkki Brändas

Published Apr 6, 2008• 39 citations
Publisher
DOI
10.1007/3-540-50994-1_54

Related papers

Holonomy in quantum physics

Alexander Robert Rutherford

Holonomy in nonrelativistic quantum mechanics is examined in the context of the adiabatic theorem. This theorem is proven for sufficiently regular unbounded hamiltoni-ans. Then, simplifying to matrix hamiltonians, it is proven that the adiabatic theorem defines a connection on vector bundles constructed out of eigenspaces of the hamiltonian. Similar degeneracy regions, the natural base spaces for these bundles, are defined in terms of stratifications for the spaces of complex, hermitian matrices and real, symmetric matrices. The algebraic topology of similar degeneracy regions is studied in detail, and the results are used to classify and calculate all possible adiabatic phases for time-reversal invariant matrix hamiltonians in terms of the relevant topological data. It is shown how vector bundles may be used to impose transversality on the helicity vector of a photon. This is used to give a calculation, which is consistent with transversality, of quantum adiabatic phase for photons in a coiled optical fibre. As an additional application, the importance of quantum adiabatic in the dynamical Jahn-Teller effect is briefly explained. An introduction is given to some important aspects of algebraic topology, which are used herein. Moreover, a number of mathematical results for flag manifolds are obtained. These results are applied to quantum adiabatic holonomy.

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On spin and matrix models in the complex plane

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arXiv:hep-lat/9307016•Nuclear Physics B•Dec 1, 1993•22 citations

Complex-Hilbert Rigidity under Solèr, Local Tomography, and Standard Composition A Conditional MTT Reconstruction Theorem

Peter Nero

Solèr's theorem classifies an orthomodular generalized Hilbert space containing an infinite orthonormal sequence as a real, complex, or quaternionic Hilbert space. This classification does not by itself select the complex field. We add a separate finite-composition requirement: a system of dimension d over the selected field has the full self-adjoint matrix space as its unnormalized state/effect space; a composite of dimensions a and b has dimension ab over the same field; and independent product effects are tomographically complete. The resulting dimension identity holds for complex Hermitian matrices and fails, by explicit nonzero defects, for real symmetric and quaternionic Hermitian matrices whenever a and b are both greater than one. Complex Hilbert space is therefore rigid within this declared reconstruction class. The conclusion is conditional and representation-sensitive: complex quantum theory can be realified by adjoining an orthogonal complex-structure operator, and alternative composite rules fall outside the theorem. Applied to MTT, the result identifies exact missing premises rather than deriving them. The current q79 finite quantum model and finite complex polarization are compatible with the complex branch, but MTT has not yet sourced the required infinite orthogonality and standard locally tomographic composition from one physical upper construction.

•Jan 21, 2026•0 citations

Remarks on pseudo-Hermitian matrices and their exceptional points

G. Scolarici, Luigi Solombrino

Czechoslovak Journal of Physics•Sep 1, 2005•4 citations

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International Journal of Theoretical Physics•Jul 1, 1974•0 citations

Complex classical paths in quantum reflections and tunneling

Job Feldbrugge, Dylan L. Jow, Ue‐Li Pen

The real-time propagator of the symmetric Rosen-Morse, also known as the symmetric modified Pöschl-Teller, barrier is expressed in the Picard-Lefschetz path integral formalism using real and complex classical paths. We explain how the interference pattern in the real-time propagator and energy propagator is organized by caustics and Stokes phenomena and list the relevant real and complex classical paths as a function of the initial and final position. We discover the occurrence of singularity crossings, where the analytic continuation of the complex classical path no longer satisfies the boundary value problem and needs to be analytically continued. Moreover, we demonstrate how these singularity crossings play a central role in the real-time description of quantum tunneling.

arXiv:2309.12420•Physical review. D/Physical review. D.•Apr 23, 2025•4 citations
Holonomy in nonrelativistic quantum mechanics is examined in the context of the adiabatic theorem. This theorem is proven for sufficiently regular unbounded hamiltoni-ans. Then, simplifying to matrix hamiltonians, it is proven that the adiabatic theorem defines a connection on vector bundles constructed out of eigenspaces of the hamiltonian. Similar degeneracy regions, the natural base spaces for these bundles, are defined in terms of stratifications for the spaces of complex, hermitian matrices and real, symmetric matrices. The algebraic topology of similar degeneracy regions is studied in detail, and the results are used to classify and calculate all possible adiabatic phases for time-reversal invariant matrix hamiltonians in terms of the relevant topological data. It is shown how vector bundles may be used to impose transversality on the helicity vector of a photon. This is used to give a calculation, which is consistent with transversality, of quantum adiabatic phase for photons in a coiled optical fibre. As an additional application, the importance of quantum adiabatic in the dynamical Jahn-Teller effect is briefly explained. An introduction is given to some important aspects of algebraic topology, which are used herein. Moreover, a number of mathematical results for flag manifolds are obtained. These results are applied to quantum adiabatic holonomy.
Solèr's theorem classifies an orthomodular generalized Hilbert space containing an infinite orthonormal sequence as a real, complex, or quaternionic Hilbert space. This classification does not by itself select the complex field. We add a separate finite-composition requirement: a system of dimension d over the selected field has the full self-adjoint matrix space as its unnormalized state/effect space; a composite of dimensions a and b has dimension ab over the same field; and independent product effects are tomographically complete. The resulting dimension identity holds for complex Hermitian matrices and fails, by explicit nonzero defects, for real symmetric and quaternionic Hermitian matrices whenever a and b are both greater than one. Complex Hilbert space is therefore rigid within this declared reconstruction class. The conclusion is conditional and representation-sensitive: complex quantum theory can be realified by adjoining an orthogonal complex-structure operator, and alternative composite rules fall outside the theorem. Applied to MTT, the result identifies exact missing premises rather than deriving them. The current q79 finite quantum model and finite complex polarization are compatible with the complex branch, but MTT has not yet sourced the required infinite orthogonality and standard locally tomographic composition from one physical upper construction.
The real-time propagator of the symmetric Rosen-Morse, also known as the symmetric modified Pöschl-Teller, barrier is expressed in the Picard-Lefschetz path integral formalism using real and complex classical paths. We explain how the interference pattern in the real-time propagator and energy propagator is organized by caustics and Stokes phenomena and list the relevant real and complex classical paths as a function of the initial and final position. We discover the occurrence of singularity crossings, where the analytic continuation of the complex classical path no longer satisfies the boundary value problem and needs to be analytically continued. Moreover, we demonstrate how these singularity crossings play a central role in the real-time description of quantum tunneling.