Fundamental length in quantum theories with PT-symmetric Hamiltonians

Abstract

The direct observability of coordinates x is often lost in PT-symmetric quantum theories. A manifestly non-local Hilbert-space metric $\Theta$ enters the double-integral normalization of wave functions $\psi (x)$ there. In the context of scattering, the (necessary) return to the asymptotically fully local metric has been shown feasible, for certain family of PT-symmetric toy Hamiltonians H at least, in paper I (M. Znojil, Phys. Rev. D 78 (2008) 025026). Now we show that in a confined-motion dynamical regime the same toy model proves also suitable for an explicit control of the measure or width $\theta$ of its non-locality. For this purpose each H is assigned here, constructively, the complete menu of its hermitizing metrics $\Theta ={\Theta }_{\theta }$ distinguished by their optional "fundamental lengths" $\theta \in (0,\infty )$. The local metric of paper I recurs at $\theta =0$ while the most popular CPT-symmetric hermitization proves long-ranged, with $\theta =\infty$.

31 pp, 3 figures

References in corpus (19)

• Making Sense of Non-Hermitian Hamiltonians
• Formulation, Interpretation and Application of non-Commutative Quantum Mechanics
• On "full" twisted Poincare' symmetry and QFT on Moyal-Weyl spaces
• Delta-Function Potential with a Complex Coupling
• Interface between Hermitian and non-Hermitian Hamiltonians in a model calculation
• Scattering theory with localized non-Hermiticities
• Construction of a unique metric in quasi-Hermitian quantum mechanics: non-existence of the charge operator in a 2 x 2 matrix model
• Discrete PT-symmetric models of scattering
• A Note on Quantum Field Theories with a Minimal Length Scale
• Calculation of the metric in the Hilbert space of a PT-symmetric model via the spectral theorem
• Fundamental length scale of quantum spacetime foam
• Nonunique C operator in PT Quantum Mechanics
• Matching method and exact solvability of discrete PT-symmetric square wells
• Non-Hermitian quantum mechanics in non-commutative space
• Matrix Hamiltonians with an algebraic guarantee of unbroken PT-symmetry
• Short-fragment Na-DNA dilute aqueous solutions: fundamental length scales and screening
• A fundamental length as a candidate for dark energy: a DSR inspired FRW spacetime
• The construction of a general inner product in non-Hermitian quantum theory and some explanation for the nonuniqueness of the C operator in PT quantum mechanics
• Conditional observability versus self-duality in a schematic model

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• PT-symmetric dynamical confinement: Fermi acceleration, quantum force and Berry phase
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