Fundamental length in quantum theories with PT-symmetric Hamiltonians

TL;DR

This paper explores the concept of a fundamental length scale in PT-symmetric quantum theories, demonstrating how different hermitizing metrics relate to the non-locality measure and local limits within toy models.

Contribution

It introduces a constructive method to assign a complete set of hermitizing metrics parameterized by a fundamental length, clarifying the transition from non-local to local metrics in PT-symmetric models.

Findings

The local metric corresponds to zero length scale.

The most common CPT-symmetric metric is infinitely non-local.

A continuum of metrics with varying non-locality is explicitly constructed.

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 $\theta=\infty$.