量子结和晶格，或者是做绳索把戏的量子系统的蓝图
Quantum Knots and Lattices, or a Blueprint for Quantum Systems that Do Rope Tricks
摘要：
使用欧几里得三维空间的立方蜂窝（立方镶嵌），我们定义了一个量子系统，其状态称为量子结，表示一个封闭的打结绳子，即，表示在三维空间中系在绳子上的结的特定空间配置。这个量子系统，称为量子结系统，在物理上是可实现的，就像Shor的量子因子分解算法是可实现的一样。为了定义一个量子结系统，我们将标准的三个里德迈斯特结移动替换为一组等效的三个移动，分别称为摆动，摇摆和拖船，这样命名是因为它们模仿了狗可能会摇尾巴的方式。我们认为，这些举动实际上是更“物理友好”，因为，不像Reidemeister的举动，他们尊重微分几何的3-空间，而且他们可以转化为无穷小的举动。这三种移动产生了一个幺正群，称为晶格环境群，它作用于量子系统的状态空间。晶格环境光组表示在三维空间中移动绳子而不切断绳子以及不让绳子穿过自身的所有可能方式。然后，我们研究了量子纽结系统的那些量子可观测量，它们是纽结不变量。我们还研究了与格周围群的生成元相关的哈密顿算子。最后，我们提出了一系列开放性问题。
英文摘要：
Using the cubic honeycomb (cubic tessellation) of Euclidean 3-space, we define a quantum system whose states, called quantum knots, represent a closed knotted piece of rope, i.e., represent the particular spatial configuration of a knot tied in a rope in 3-space. This quantum system, called a quantum knot system, is physically implementable in the same sense as Shor's quantum factoring algorithm is implementable. To define a quantum knot system, we replace the standard three Reidemeister knot moves with an equivalent set of three moves, called respectively wiggle, wag, and tug, so named because they mimic how a dog might wag its tail. We argue that these moves are in fact more "physics friendly" because, unlike the Reidemeister moves, they respect the differential geometry of 3-space, and moreover they can be transformed into infinitesimal moves. These three moves wiggle, wag, and tug generate a unitary group, called the lattice ambient group, which acts on the state space of the quantum system. The lattice ambient group represents all possible ways of moving a rope around in 3-space without cutting the rope, and without letting the rope pass through itself. We then investigate those quantum observables of the quantum knot system which are knot invariants. We also study Hamiltonians associated with the generators of the lattice ambient group. We conclude with a list of open questions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
