0 50 20 03 v 2 1 3 Ju n 20 05 Noncommutative Configuration Space . Classical and Quantum Mechanical Aspects ∗

In this work we examine noncommutativity of position coordinates in classical symplectic mechanics and its quantisation. In coordinates {q i , p k } the canonical symplectic two-form is ω 0 = dq i ∧ dp i. It is well known in symplectic mechanics [5, 6, 9] that the interaction of a charged particle with a magnetic field can be described in a Hamil-tonian formalism without a choice of a potential. This is done by means of a modified symplectic two-form ω = ω 0 − eF, where e is the charge and the (time-independent) magnetic field F is closed: dF = 0. With this symplectic structure, the canonical momentum variables acquire non-vanishing Poisson brackets: {p k , p l } = e F kl (q). Similarly a closed two-form in p-space G may be introduced. Such a dual magnetic field G interacts with the particle's dual charge r. A new modified symplectic two-form ω = ω 0 − eF + rG is then defined. Now, both p-and q-variables will cease to Poisson commute and upon quantisation they become noncommuting operators. In the particular case of a linear phase space R 2N , it makes sense to consider constant F and G fields. It is then possible to define, by a linear transformation , global Darboux coordinates: {ξ i , π k } = δ i k. These can then be quantised in the usual way [ ξ i , π k ] = i¯ h δ i k. The case of a quadratic potential is examined with some detail when N equals 2 and 3.