F eb 2 00 5 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, 7] that the interaction of a charged particle with a magnetic field can be described without a choice of a potential in a Lagrangian formalism. This is done introducing 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 we introduce a dual magnetic field G, which is a closed two-form in p-space interacting 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 quanti-sation 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 global Darboux coordinates through a linear transformation. These can then be quantised in the usual way. Quadratic Hamiltonians are examined with some detail in the two-and three-dimensional cases.