Quantum knots and mosaics
Quantum knots and mosaics
复制标题
量子结和马赛克
DOI：
10.1007/s11128-008-0076-7
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发表时间：
2008
影响因子：
2.5
通讯作者：
L. Kauffman
中科院分区：
文献类型：
--
作者：
S. Lomonaco;L. Kauffman
本文给出了一个精确而可行的量子纽结系统的定义，其状态称为量子纽结。这个定义可以被看作是构建实际物理量子系统的蓝图。此外，量子结系统的这个定义旨在表示闭合打结的物理绳子的“量子体现”。量子结，作为这个系统的一种状态，代表这样一个打结的闭合绳子的状态，即，绳子上打结的特殊空间结构。与量子结系统相关的是一组幺正变换，称为环境群，它代表了所有可能的移动绳子的方式（不切断绳子，也不让绳子穿过自己）。当然，与经典的闭合绳子不同，量子结可以表现出非经典的行为，例如量子叠加和量子纠缠。这就提出了一些关于拓扑和量子纠缠之间关系的有趣而令人困惑的问题。量子结的结型就是量子结在环境群作用下的轨道。我们研究量子可观测量是量子结型的不变量。我们还研究了与周围群的生成元相关的哈密顿量，并简要地研究了量子隧穿的过渡到下渡。本文的一个基本构建块是一个马赛克系统，这是一个正式的（重写）系统的符号串。我们猜想，这个正式的系统完全捕获在一个公理化的方式驯服纽结理论的所有属性。
In this paper, we give a precise and workable definition of a quantum knot system, the states of which are called quantum knots. This definition can be viewed as a blueprint for the construction of an actual physical quantum system. Moreover, this definition of a quantum knot system is intended to represent the “quantum embodiment” of a closed knotted physical piece of rope. A quantum knot, as a state of this system, represents the state of such a knotted closed piece of rope, i.e., the particular spatial configuration of the knot tied in the rope. Associated with a quantum knot system is a group of unitary transformations, called the ambient group, which represents all possible ways of moving the rope around (without cutting the rope, and without letting the rope pass through itself.) Of course, unlike a classical closed piece of rope, a quantum knot can exhibit non-classical behavior, such as quantum superposition and quantum entanglement. This raises some interesting and puzzling questions about the relation between topological and quantum entanglement. The knot type of a quantum knot is simply the orbit of the quantum knot under the action of the ambient group. We investigate quantum observables which are invariants of quantum knot type. We also study the Hamiltonians associated with the generators of the ambient group, and briefly look at the quantum tunneling of overcrossings into undercrossings. A basic building block in this paper is a mosaic system which is a formal (rewriting) system of symbol strings. We conjecture that this formal system fully captures in an axiomatic way all of the properties of tame knot theory.