Elastic field causing noncommutativity

We study how a uniform torsion background —modeling a continuous density of screw dislocations— induces effective spatial noncommutativity and reshapes the energy spectrum of a free quantum particle. Within the geometric theory of defects, the metric yields a first-order (magnetic-like) coupling in the transverse dynamics, equivalent to an effective magnetic field $B_{\mathrm {eff}}\propto p_{z}\,\Omega $ , where Ω encodes the torsion strength. In the strong-coupling (Landau) regime, the planar coordinates obey $[x,y]\neq0$ and the spectrum organizes into Landau-like levels with a slight electric-field–driven tilt and a uniform shift. Thus, increasing Ω drives the system continuously toward the familiar Landau problem in flat space, with torsion setting the noncommutativity scale and controlling the approach to the Landau limit.