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 $\theta =\infty$.

arXiv comment31 pp, 3 figures

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