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NC plane waves, Casimir effect and flux tube potential with Lüscher terms

We analyze plane waves in a model of quantum mechanics in a three dimensional noncommutative (NC) space $R^3_λ$. Signature features of NC models are impossibility of probing distances smaller than a certain length scale λ and a presence of natural energetic cut-off at energy scale of order $1/λ^2$ (in convenient units). We analyze consequences of such restrictions on a 1 dimensional Casimir effect. The result shows resemblance to flux tube potential for quark-antiquark pairs and to effective bosonic string theories with Lüscher terms. Such behavior might effect the radius of possible compact (fuzzy) dimensions.