Hermitian versus non-Hermitian representations for minimal length uncertainty relations

Dey, S., Fring, A. & Khantoul, B. (2013). Hermitian versus non-Hermitian representations for minimal length uncertainty relations. Journal of Physics A: Mathematical and Theoretical, 46(33), doi: 10.1088/1751-8113/46/33/335304

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Abstract

We investigate four different types of representations of deformed canonical variables leading to generalized versions of Heisenberg's uncertainty relations resulting from noncommutative spacetime structures. We demonstrate explicitly how the representations are related to each other and study three characteristically different solvable models on these spaces, the harmonic oscillator, the manifestly non-Hermitian Swanson model and an intrinsically noncommutative model with Pöschl-Teller type potential. We provide an analytical expression for the metric in terms of quantities specific to the generic solution procedure and show that when it is appropriately implemented expectation values are independent of the particular representation. A recently proposed inequivalent representation resulting from Jordan twists is shown to lead to unphysical models. We suggest an anti PT-symmetric modification to overcome this shortcoming.