论文标题
3+1维网络中拓扑纺纱器的狄拉克量规理论
Dirac gauge theory for topological spinors in 3+1 dimensional networks
论文作者
论文摘要
图表和网络上的衡量理论不仅吸引了量子重力的方法，而且吸引了量子计算的模型。在这里，我们为$ 3+1 $尺寸网络提供了与任意度量相关的dirac量规理论。拓扑纺纱器是网络上定义的$ 0 $ - 船的直接和$ 1 $ - 船，并描述网络节点和链接上定义的物质字段。最近在参考文献中。 \ cite {bianconi2021Topologicy}已经表明，拓扑纺纱器遵守离散狄拉克操作员驱动的拓扑狄拉克方程。在这项工作中，我们通过在加权和指示$ 3+1 $维度网络上制定狄拉克方程来扩展这些结果，从而可以处理本地理论。狄拉克运算符的换向因子和反交易者是不消失的，它们分别定义了我们理论的曲率张量和磁场。所提出的dirac方程的非相关主义极限证实了这种解释。在所提出的dirac方程的非相关限制中，链路上定义的旋转器的扇区遵循schrödinger方程，具有正确的giromagnetic矩，而在节点上定义的旋转器的扇区遵循klein-gordon方程，并且不可忽略。与拟议的现场理论相关的作用包括狄拉克动作和指标作用。我们描述了在阿贝尔和非阿贝尔转化下的作用的规格不变性，并提出了迪拉克和度量场的田间理论运动方程。该理论可以被解释为在几乎平坦空间的限制下对任何任意网络有效的更通用仪表理论的限制案例。
Gauge theories on graphs and networks are attracting increasing attention not only as approaches to quantum gravity but also as models for performing quantum computation. Here we propose a Dirac gauge theory for topological spinors in $3+1$ dimensional networks associated to an arbitrary metric. Topological spinors are the direct sum of $0$-cochains and $1$-cochains defined on a network and describe a matter field defined on both nodes and links of a network. Recently in Ref. \cite{bianconi2021topological} it has been shown that topological spinors obey the topological Dirac equation driven by the discrete Dirac operator. In this work we extend these results by formulating the Dirac equation on weighted and directed $3+1$ dimensional networks which allow for the treatment of a local theory. The commutators and anti-commutators of the Dirac operators are non vanishing an they define the curvature tensor and magnetic field of our theory respectively. This interpretation is confirmed by the non-relativistic limit of the proposed Dirac equation. In the non-relativistic limit of the proposed Dirac equation the sector of the spinor defined on links follows the Schrödinger equation with the correct giromagnetic moment, while the sector of the spinor defined on nodes follows the Klein-Gordon equation and is not negligible. The action associated to the proposed field theory comprises of a Dirac action and a metric action. We describe the gauge invariance of the action under both Abelian and non-Abelian transformations and we propose the equation of motion of the field theory of both Dirac and metric fields. This theory can be interpreted as a limiting case of a more general gauge theory valid on any arbitrary network in the limit of almost flat spaces.
