标题： 非交换构型空间。 经典和量子力学方面 显示英文标题

标题： Noncommutative configuration space. Classical and quantum mechanical aspects

摘要： 在这项工作中，我们研究了经典辛力学中位置坐标之间的非交换性及其量子化。 在坐标$\{q^i,p_k\}$中，典范辛二形式为$\omega_0=dq^i\wedge dp_i$。 在辛力学中众所周知{\bf \cite{Souriau,Abraham,Guillemin}}，带电粒子与磁场的相互作用可以在没有选择势的情况下用哈密顿形式描述。 这是通过修正的辛二形式$\omega=\omega_0-e\F$实现的，其中$e$是电荷，(时间无关的)磁场$\F$是闭合的：$\dif\F=0$。 具有这一辛结构后，正则动量变量获得了非零的泊松括号：$\{p_k,p_l\} = e F_{kl}(q)$。类似地，可以在$p$-空间$\G$中引入一个闭的二形式。这样的{\it 双磁场} $\G$ 与粒子的{\it 双重电荷} $r$相互作用。 定义一个新的修正的辛二形式$\omega=\omega_0-e\F+r\G$。 现在，$p$- 和$q$- 变量将不再泊松对易，并且在量子化后它们成为非对易算符。 在相空间${\bf R}^{2N}$为线性的特殊情况下，考虑常数$\F$和$\G$场是有意义的。 然后可以通过一个线性变换来定义全局达布坐标：$\{\xi^i,\pi_k\}= {\delta^i}_k$。 这些随后可以以通常的方式进行量化，见$[\hat{\xi}^i,\hat{\pi}_k]=i\hbar {\delta^i}_k$。 当$N$等于 2 和 3 时，详细研究了二次势的情况。 显示英文摘要

摘要： 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.