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arXiv 2401.07307gr-qc
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圈量子引力导论：Holst作用量与协变形式

Introduction to Loop Quantum Gravity. The Holst's action and the covariant formalism

L. Fatibene, A. Orizzonte, A. Albano, S. Coriasco, M. Ferraris, S. Garruto, N. Morandi

备注：
19 pages
发表信息：
International Journal of Geometric Methods in Modern Physics (2024)
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AI总结：

本文回顾圈量子引力中的Holst形式，证明其与标准广义相对论动力学等价，并通过约化分裂引入Barbero-Immirzi联络构造协变模型，指出最终只有Immirzi参数$β$保留在边界场方程中。

摘要：

我们回顾Holst形式，并讨论其与标准广义相对论（在四维时空中）的动力学等价性。Holst形式是针对时空$M$上的自旋余标架场$e^I_μ$和$Spin(3,1)$-联络$ω^{IJ}_μ$写出的，它依赖于Holst参数$γ\in \mathbb{R}-\{0\}$。我们证明该模型与标准广义相对论动力学等价，其含义是：在作用于标架指标上的逐点$Spin(3,1)$-规范变换范围内，两个模型的解之间存在一一对应关系。因此，这两个模型在经典意义上是等价的。我们还可以通过将自旋联络分解为一个$Spin(3)$-联络$A^i_μ$和一个$Spin(3)$-值1-形式$k^i_μ$来引入新变量。这些新变量的构造依赖于一种特定的代数结构，称为约化分裂。约化分裂是一种比要求规范群分裂为两个子群的乘积更弱的结构，正如在欧几里得签名下由Ashtekar最初在此语境中引入的自对偶表述中所发生的情形，它仍然允许在不借助复化的情况下处理洛伦兹签名。$SL(2, \mathbb{C})$的约化分裂不是唯一的，它由一个实参数$β$参数化，该参数称为Immirzi参数。这里的分裂是在时空上进行的，而不是在空间上，以获得一个$Spin(3)$-联络$A^i_μ$，它被称为时空上的Barbero-Immirzi联络。由此得到一个依赖于场$(e^I_μ, A^i_μ, k^i_μ)$的协变模型，该模型再次与标准广义相对论（以及Holst作用量）动力学等价。通常，在文献中为了简单起见，人们设定$β=γ$。在这里，我们将Holst参数和Immirzi参数保持为不同的量，以表明最终只有$β$会保留在边界场方程中。

英文摘要：

We review Holst formalism and we discuss dynamical equivalence with standard GR (in dimension 4). Holst formalism is written for a spin coframe field $e^I_μ$ and a $Spin(3,1)$-connection $ω^{IJ}_μ$ on spacetime $M$ and it depends on the Holst parameter $γ\in \mathbb{R}-\{0\}$. We show the model is dynamically equivalent to standard GR, in the sense that up to a pointwise $Spin(3,1)$-gauge transformation acting on frame indices, solutions of the two models are in one-to-one correspondence. Hence the two models are classically equivalent. One can also introduce new variables by splitting the spin connection into a pair of a $Spin(3)$-connection $A^i_μ$ and a $Spin(3)$-valued 1-form $k^i_μ$. The construction of these new variables relies on a particular algebraic structure, called a reductive splitting. A reductive splitting is a weaker structure than requiring that the gauge group splits as the products of two sub-groups, as it happens in Euclidean signature in the selfdual formulation originally introduced in this context by Ashtekar, and it still allows to deal with the Lorentzian signature without resorting to complexifications. The reductive splitting of $SL(2, \mathbb{C})$ is not unique and it is parameterized by a real parameter $β$, called the Immirzi parameter. The splitting is here done on spacetime, not on space, to obtain a $Spin(3)$-connection $A^i_μ$, which is called the Barbero-Immirzi connection on spacetime. One obtains a covariant model depending on the fields $(e^I_μ, A^i_μ, k^i_μ)$ which is again dynamically equivalent to standard GR (as well as the Holst action). Usually, in the literature one sets $β=γ$ for the sake of simplicity. Here we keep the Holst and Immirzi parameters distinct to show that eventually, only $β$ will survive in boundary field equations.

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。

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