A Simple Signal of Noncommutative Space

We propose a simple low-energy classical experiment in which the effects of noncommutativity can be clearly separated from commutative physics. The ensuing bound on the noncommutative scale is remarkable, especially in view of its elementary derivation. Noncommutative mechanics has recently been under intensive investigation [1, 2]. It uses the following basic commutators for the coordinates qi and momenta pi, [qi, qj] = iθij , [qi, pj] = iδij , [pi, pj] = iFij , (1) which are a generalization of the Heisenberg commutation relations. The time evolution of an operator follows from its commutator with the prescribed Hamiltonian. In the classical case the commutators in (1) are replaced by Poisson brackets, {qi, qj} = θij , {qi, pj} = δij , {pi, pj} = Fij. (2) Above, δij is a unit matrix, whereas Fij and θij are functions of both the coordinates and the momenta, to be restricted only by the Jacobi identities. This note will consider the case in which θ is a nonvanishing constant diagonal matrix, whereas a generic F (qi, pj) is allowed. A dynamical effect due to the gradient of F will be reported. For simplicity in presentation we will work in two dimensions, i, j = 1, 2 in (2), although our considerations extend ∗On leave from: National Institute of Nuclear Physics and Engineering P.O. Box MG-6, 76900 Bucharest, Romania; e-mail: [email protected].