Noncommutative configuration space. Classical and quantum mechanical aspects

Abstract

In this work we examine noncommutativity of position coordinates in classical symplectic mechanics and its quantisation. In coordinates {q i,pk} the canonical symplectic two-form is ω0 = dqi pdi. 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 Hamiltonian formalism without a choice of a potential. This is done by means of a modified symplectic two-form ω = ω0 -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,Pl} = e Fkl)(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 = ω0 -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 R2N, it makes sense to consider constant F and G fields. It is then possible to define, by a linear transformation, global Darboux coordinates: {ξiπ k} = δik. These can then be quantised in the usual way [ξi = πk] = ihδik. The case of a quadratic potential is examined with some detail when N equals 2 and 3.