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 $\{q^i,p_k\}$ the canonical symplectic two-form is ${\omega }_0=d{q}^i\wedge d{p}_i$. It is well known in symplectic mechanics {\bf\cite{Souriau,Abraham,Guillemin}} 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 $ω=ω_0-e\F$, where $e$ is the charge and the (time-independent) magnetic field $\F$ is closed: $\dif\F=0$. 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 {\it dual magnetic field} $\G$ interacts with the particle's {\it dual charge} $r$. A new modified symplectic two-form $ω=ω_0-e\F+r\G$ 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 ${\mathbf{R}}^{2N}$, it makes sense to consider constant $\F$ and $\G$ fields. It is then possible to define, by a linear transformation, global Darboux coordinates: $\{{\xi }^i,{\pi }_k\}={{\delta }^i}_k$. These can then be quantised in the usual way $[{\hat{\xi }}^i,{\hat{\pi }}_k]=i\hslash {{\delta }^i}_k$. The case of a quadratic potential is examined with some detail when $N$ equals 2 and 3.

Besides correcting typos, we added a) important references, overlooked in V1, b) a more detailed discussion on the degenerate case, c) a didactical appendix on symplectic reduction