Test of Quantum Effects of Spatial Noncommutativity using Modified Electron Momentum Spectroscopy

The possibility of testing spatial noncommutativity by current experiments on normal quantum scales is investigated. For the case of both position-position and momentum-momentum noncommuting spectra of ions in crossed electric and magnetic fields are studied in the formalism of noncommutative quantum mechanics. In a limit of the kinetic energy approaching its lowest eigenvalue this system possesses non-trivial dynamics. Signals of spatial noncommutativity in the angular momentum are revealed. They are within limits of the measurable accuracy of current experiments. An experimental test of the predictions using a modified electron momentum spectrum is suggested. The related experimental sensitivity and subtle points are discussed. The results are the first step on a realizable way towards a conclusive test of spatial noncommutativity. ∗ Corresponding author Quantum theory in noncommutative space [1–9] presents an attractive possibility as a candidate in the present round in hinting at new physics. In the low energy aspect, quantum mechanics in noncommutative space (NCQM) [10–17], [18] have been studied in detail. But testing their predictions require experiments near the Planck scale which are un-realizable, and /or their modifications to normal quantum theory depending on vanishingly small noncommutative parameters which are outside limits of measurable accuracy of current experiments. It seems that noncommutative quantum theory escapes measurement on normal quantum scales. The possibility of testing spatial noncommutativity by current experiments on normal quantum scales was investigated, and two proposals [18] using Rydberg atoms and ChernSimons processes are suggested. The two proposals revealed that in a limit of diminishing the magnetic field to zero the vanishingly small noncommutative parameters usually present in predictions derived from spatial noncommutativity actually cancel out in the the angular momentum, so that the lowest angular momentum turns out to be h̄/4. This provides a conclusive test of spatial noncommutativity, i. e., a positive experimental result shows an evidence of spatial noncommutativity and a negative one draws a conclusive preclusion of spatial noncommutativity. In practice the magnetic field, however, can only reach some finite value limited by the level of shielding the background magnetic fields. In order to meet the condition of a cancellation between the vanishingly small noncommutative parameters present in the angular momentum derived from spatial noncommutativity, the magnetic field must be decreased to a level of some orders less than the effective intrinsic magnetic field Bη originated from spatial noncommutativity (see below). The field control at that level seems close to impossible in the foreseeable future. According to the present level of shielding the background magnetic fields, this paper explores a realizable way for testing noncommutative quantum effects on normal quantum scale. For the case of both position-position and momentum-momentum noncommuting spectra of ions trapped in crossed electric and magnetic fields are investigated. In a limit of the kinetic energy approaching its lowest eigenvalue this system possesses non-trivial dynamics. The corresponding constraints are analyzed. Signals of spatial noncommutativity in the angular momentum are revealed. They are within limits of the measurable