标题： 圈量子引力的洛伦兹协变性 显示英文标题

标题： Lorentz covariance of loop quantum gravity

摘要： 圈引力的运动学可以被赋予一个明显的洛伦兹协变形式：传统的SU(2)自旋网络希尔伯特空间可以映射到SL(2,C)函数的空间K，其中洛伦兹协变性是显而易见的。 K可以用Livine、Alexandrov和Dupuis研究过的“投影”自旋网络的某个子集来描述。 它由完全由其在SU(2)上的限制确定的SL(2,C)函数组成。 这些函数在SU(2)数量积下是平方可积的，但在SL(2,C)数量积下不是。 因此，SU(2)自旋网络态可以通过洛伦兹协变的SL(2,C)函数来表示，就像两个分量光子可以在洛伦兹协变的古普塔-布莱勒形式主义中描述一样。 正如Wolfgang Wieland在一篇相关论文中所展示的，这种明显的洛伦兹协变形式也可以直接从正则量子化获得。 我们证明了圈量子引力的自旋泡沫动力学在内部是局部SL(2,C)-不变的，并且在边界上产生正好属于K的状态。 这阐明了SL(2,C)自旋泡沫形式主义如何在边界上产生SU(2)理论。 这些结构定义了一个整洁的洛伦兹协变形式主义用于圈引力。 显示英文摘要

摘要： The kinematics of loop gravity can be given a manifestly Lorentz-covariant formulation: the conventional SU(2)-spin-network Hilbert space can be mapped to a space K of SL(2,C) functions, where Lorentz covariance is manifest. K can be described in terms of a certain subset of the "projected" spin networks studied by Livine, Alexandrov and Dupuis. It is formed by SL(2,C) functions completely determined by their restriction on SU(2). These are square-integrable in the SU(2) scalar product, but not in the SL(2,C) one. Thus, SU(2)-spin-network states can be represented by Lorentz-covariant SL(2,C) functions, as two-component photons can be described in the Lorentz-covariant Gupta-Bleuler formalism. As shown by Wolfgang Wieland in a related paper, this manifestly Lorentz-covariant formulation can also be directly obtained from canonical quantization. We show that the spinfoam dynamics of loop quantum gravity is locally SL(2,C)-invariant in the bulk, and yields states that are preciseley in K on the boundary. This clarifies how the SL(2,C) spinfoam formalism yields an SU(2) theory on the boundary. These structures define a tidy Lorentz-covariant formalism for loop gravity.