Lorentz covariance of loop quantum gravity
Lorentz covariance of loop quantum gravity
复制标题
圈量子引力的洛伦兹协方差
DOI：
10.1103/physrevd.83.104029
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发表时间：
2010
影响因子：
5
通讯作者：
Simone Speziale
中科院分区：
文献类型：
--
作者：
C. Rovelli;Simone Speziale
回路重力的运动学可以给出明显的Lorentz协变形式：传统的SU(2)-自旋网络Hilbert空间可以映射到SL(2，C)函数的空间K，其中Lorentz协方差是明显的。K可以用Livine、Alexandrov和Dupuis研究的投影自旋网络的某个子集来描述。它是由SL(2，C)函数构成的，完全由它们对SU(2)的限制决定。它们在SU(2)标积中是平方可积的，但在SL(2，C)积中不是平方可积的。因此，SU(2)-自旋网络态可以用Lorentz协变SL(2，C)函数来表示，因为两分量光子可以用Lorentz协变Gupta-Bleuler形式来描述。沃尔夫冈·威兰德在一篇相关论文中指出，这种明显的洛伦兹协变形式也可以直接从正则量子化中得到。我们证明了圈量子引力的自旋泡沫动力学在整体中是局部SL(2，C)不变的，并且在边界上产生精确在K中的态。这阐明了SL(2，C)自旋泡沫形式如何在边界上产生SU(2)理论。这些结构为环路引力定义了一个整齐的洛伦兹协变形式。
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 precisely 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.