刚体量子化的协变方法
A covariant approach to the quantisation of a rigid body
摘要：
本文讨论了在具有绝对时间的弯曲时空上，在“协变量子力学”框架下的刚体量子化问题。其基本思想是将多组态空间，即$n$粒子的组态空间，视为一个粒子的组态空间的$n$ -倍积。然后，我们施加一个刚性约束的多配置空间。由此产生的空间，然后处理作为一个单一的抽象的“粒子”的配置空间。同样的想法适用于所有的几何和动力结构。我们证明了上述组态空间符合“协变量子力学”的一般框架。因此，该理论的方法可以应用于刚体。因此，我们发现刚体的量子结构有两种不等价的选择。然后，我们评估的量子能量和动量运营商和'旋转部分'的光谱。我们用一个新的非平凡量子丛的单值截面给出了SO（3）上二值波函数的一个新的数学解释。这些结果与自旋有明显的相似之处。
英文摘要：
This paper concerns the quantisation of a rigid body in the framework of ``covariant quantum mechanics'' on a curved spacetime with absolute time. The basic idea is to consider the multi-configuration space, i.e. the configuration space for $n$ particles, as the $n$-fold product of the configuration space for one particle. Then we impose a rigid constraint on the multi-configuration space. The resulting space is then dealt with as a configuration space of a single abstract `particle'. The same idea is applied to all geometric and dynamical structures. We show that the above configuration space fits into the general framework of ``covariant quantum mechanics''. Hence, the methods of this theory can be applied to the rigid body. Accordingly, we find exactly two inequivalent choices of quantum structures for the rigid body. Then, we evaluate the quantum energy and momentum operators and the `rotational part' of their spectra. We provide a new mathematical interpretation of two-valued wavefunctions on SO(3) in terms of single-valued sections of a new non-trivial quantum bundle. These results have clear analogies with spin.
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