标题： 三维非对易量子力学与旋转对称性 显示英文标题

标题： Non-commutative Quantum Mechanics in Three Dimensions and Rotational Symmetry

摘要： 我们将非对易量子力学的公式推广到三维非对易空间。 特别关注于确定物理状态应被表示的量子Hilbert空间，构建该空间上的旋转群表示，伴随该表示的Leibnitz法则的变形以及为了恢复旋转对称性自动同态而必须进行的余乘法的变形。 这也意味着在薛定谔作用量和方程以及哈密顿量层面，即使对于旋转不变势能，旋转不变性也会被破坏。 对于旋转不变势能，对称性的破坏纯粹来自于变形，因为哈密顿量和角动量的对易子与变形成正比。 显示英文摘要

摘要： We generalize the formulation of non-commutative quantum mechanics to three dimensional non-commutative space. Particular attention is paid to the identification of the quantum Hilbert space in which the physical states of the system are to be represented, the construction of the representation of the rotation group on this space, the deformation of the Leibnitz rule accompanying this representation and the implied necessity of deforming the co-product to restore the rotation symmetry automorphism. This also implies the breaking of rotational invariance on the level of the Schroedinger action and equation as well as the Hamiltonian, even for rotational invariant potentials. For rotational invariant potentials the symmetry breaking results purely from the deformation in the sense that the commutator of the Hamiltonian and angular momentum is proportional to the deformation.