Fundamental length in quantum theories with<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>P</mml:mi><mml:mi>T</mml:mi></mml:math>-symmetric Hamiltonians

One-dimensional motion of a quantum point particle is usually described by its wave function $\ensuremath{\psi}(x)$, where the argument $x\ensuremath{\in}\mathbb{R}$ represents a (measurable) coordinate and where the integrated probability density is normalized to one, $\ensuremath{\int}{\ensuremath{\psi}}^{*}(x)\ensuremath{\psi}(x)=1$. The direct observability of $x$ may be lost in $\mathcal{P}\mathcal{T}$-symmetric quantum mechanics where a ``smeared'' metric kernel ${\ensuremath{\Theta}}_{(x,{x}^{\ensuremath{'}})}\ensuremath{\ne}\ensuremath{\delta}(x\ensuremath{-}{x}^{\ensuremath{'}})$ may enter the double-integral normalization $∬{\ensuremath{\psi}}^{*}(x){\ensuremath{\Theta}}_{(x,{x}^{\ensuremath{'}})}\ensuremath{\psi}({x}^{\ensuremath{'}})=1$. We argue that such a formalism proves particularly suitable for the introduction of a nonvanishing fundamental length $\ensuremath{\theta}&gt;0$, which would characterize the ``smearing width'' of the kernel ${\ensuremath{\Theta}}_{(x,{x}^{\ensuremath{'}})}$. The technical feasibility of such a project is illustrated via a toy family of Hamiltonians ${H}^{(N)}(\ensuremath{\lambda})$ taken from Ref. [11]. For each element of this family the complete set of all the eligible metric kernels ${\ensuremath{\Theta}}_{(x,{x}^{\ensuremath{'}})}^{(N)}(\ensuremath{\lambda})$ is constructed in closed form. We show that at any preselected non-negative fundamental length these metrics can be made to vanish unless $|x\ensuremath{-}{x}^{\ensuremath{'}}|\ensuremath{\le}\ensuremath{\theta}$. The strictly local inner product of Ref. [11] recurs at $\ensuremath{\theta}=0$, while the popular $\mathcal{C}\mathcal{P}\mathcal{T}$-symmetric option requires $\ensuremath{\theta}=\ensuremath{\infty}$ in this language.

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