Searching for effects of Spatial Noncommutativity via a Penning Trap

The possibility of testing spatial noncommutativity via a Penning trap is explored. The case of both space-space and momentum-momentum noncommuting is considered. Spatial noncommutativity leads to the spectrum of the orbital angular momentum of a Penning trap possessing fractional values, and in the limits of vanishing kinetic energy and subsequent vanishing magnetic field, this system has non-trivial dynamics. The dominant value of the lowest orbital angular momentum is h̄/4, which is a clear signal of spatial noncommutativity. An experimental verification of this prediction by a Stern-Gerlach-type experiment is suggested. ∗ E-mail address: [email protected] Studies of low energy effective theory of superstrings show that space is noncommutative [1]. Spatial non-commutativity is apparent near the Planck scale. Its modifications to ordinary quantum theory are extremely small. We ask whether one can find some low energy detectable relics of physics at the Planck scale by current experiments. Such a possibility is inferred from the incomplete decoupling between effects at high and low energy scales. For the purpose of clarifying phenomenological low energy effects, quantum mechanics in noncommutative space (NCQM) is available. If NCQM is a realistic physics, all the low energy quantum phenomena should be reformulated in it. In the literature, NCQM have been studied in detail [2–7]; many interesting topics, from the Aharonov-Bohm effect to the quantum Hall effect have been considered [8]. Recent investigation of the non-perturbation aspect of the deformed Heisenberg-Weyl algebra (the NCQM algebra) in noncommutative space shows [5] that, when the state vector space of identical bosons is constructed by generalizing one-particle quantum mechanics, in order to maintain Bose-Einstein statistics at the non-perturbation level described by deformed annihilation-creation operators, the consistent ansatz of commutation relations of the phase space canonical variables should include both space-space noncommutativity and momentum-momentum noncommutativity. This explores some new features of effects of spatial noncommutativity [5–7]. In [6] the possibility of testing spatial noncommutativity via Rydberg atoms is explored. But the measurement in experiments of Rydberg atoms depends on a characteristic frequency which is extremely high and may be difficult to reach by current experiments. For Penning traps [9–12] the situation is different. In this paper we show a possibility of testing spatial noncommutativity via a Penning trap. In [13] properties of a Penning trap at the level of NCQM are investigated. Spatial noncommutativity leads to the spectrum of the orbital angular momentum of a Penning trap possessing a fractional zero-point angular momentum, fractional values and fractional intervals. In the limit of vanishing kinetic energy this system has non-trivial dynamics. We find that in both limits of vanishing kinetic energy and the subsequent diminishing magnetic field the dominant value of the lowest orbital angular momentum in a Penning trap is h̄/4. This result is a clear signal of spatial noncommutativity, and can be verified by a Stern-Gerlach-type experiment. The objects trapped in a Penning trap are charged particles or ions. The trapping