Noncommutative configuration space. Classical and quantum mechanical aspects

TL;DR

This paper explores how noncommutativity of position coordinates arises in classical and quantum mechanics through modified symplectic structures influenced by magnetic-like fields, leading to noncommuting operators upon quantization.

Contribution

It introduces a dual magnetic field concept in phase space, extending noncommutative geometry in classical and quantum mechanics with explicit models for linear phase spaces.

Findings

Noncommutative Poisson brackets for position and momentum variables.

Quantization yields noncommuting operators in phase space.

Explicit analysis for quadratic potentials in 2D and 3D cases.

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 $\omega_0=dq^i\wedge dp_i$. 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 $\omega=\omega_0-e\F$, 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 $\omega=\omega_0-e\F+r\G$ 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 ${\bf R}^{2N}$, 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 $[\hat{\xi}^i,\hat{\pi}_k]=i\hbar {\delta^i}_k$. The case of a quadratic potential is examined with some detail when $N$ equals 2 and 3.