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- W2123951602 abstract "One-dimensional motion of a quantum point particle is usually described by its wave function $ensuremath{psi}(x)$, where the argument $xensuremath{in}mathbb{R}$ 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 ${ensuremath{Theta}}_{(x,{x}^{ensuremath{'}})}ensuremath{ne}ensuremath{delta}(xensuremath{-}{x}^{ensuremath{'}})$ 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 $ensuremath{theta}>0$, 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 $|xensuremath{-}{x}^{ensuremath{'}}|ensuremath{le}ensuremath{theta}$. 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." @default.
- W2123951602 created "2016-06-24" @default.
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- W2123951602 date "2009-08-21" @default.
- W2123951602 modified "2023-10-18" @default.
- W2123951602 title "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" @default.
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- W2123951602 doi "https://doi.org/10.1103/physrevd.80.045022" @default.
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