We review Holst formalism and dynamical equivalence with standard GR (in dimension $4$). Holst formalism is written for a spin coframe field $e_{\mu }^I$ and a Spin(3,1)-connection ${\omega }_{\mu }^{IJ}$ on spacetime $M$ and it depends on the Holst parameter$\gamma \in ℝ-\{0\}$. 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 (uppercase Latin) 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 $A_{\mu }^i$ and a Spin(3)-valued 1-form $k_{\mu }^i$. The construction of these new variables relies on a particular algebraic structure, called a reductive splitting. 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 $\text{SL}(2,ℂ)$ is not unique and it is parameterized by a real parameter $\beta$ which is called the Immirzi parameter. The splitting is here done on spacetime, not on space as it is usually done in the literature, to obtain a Spin(3)-connection $A_{\mu }^i$, which is called the Barbero–Immirzi connection on spacetime. One obtains a covariant model depending on the fields $(e_{\mu }^I,A_{\mu }^i,k_{\mu }^i)$ which is again dynamically equivalent to standard GR. Usually in the literature one sets $\beta =\gamma$ for the sake of simplicity. Here, we keep the Holst and Immirzi parameters distinct to show that eventually only $\beta$ will survive in boundary field equations.

我们回顾霍尔斯特形式主义和标准 GR 的动力学等价性（维度$4$）。霍尔斯特形式主义是为自旋共框域编写的$e_{\mu }^{我}$和 Spin(3,1) 连接${\omega }_{\mu }^{我J}$在时空上$\mathrm{中号}$这取决于Holst 参数$\gamma \epsilon ℝ-\{0\}$。我们证明该模型在动态上等效于标准 GR，从某种意义上说，直到作用于（大写拉丁文）框架索引的点式 Spin(3,1) 规范变换，两个模型的解是一一对应的。因此，这两个模型在经典上是等效的。还可以通过将自旋连接拆分为一对 Spin(3) 连接来引入新变量$A_{\mu }^{我}$和 Spin(3) 值 1 形式$k_{\mu }^{我}$。这些新变量的构造依赖于一种特殊的代数结构，称为还原分裂。这是一个比要求规范群分裂为两个子群的乘积更弱的结构，正如 Ashtekar 最初在这种情况下引入的自对偶公式中的欧几里得签名中所发生的那样，并且它仍然允许处理洛伦兹签名而无需诉诸复杂化。的还原分裂$\text{SL}（2,ℂ）$不唯一，并且由实数参数化$\beta$这称为Immirzi 参数。这里的分割是在 spacetime 上完成的，而不是像文献中通常在 space 上完成的那样，以获得 Spin(3) 连接$A_{\mu }^{我}$，这被称为时空上的Barbero-Immirzi 连接。根据字段获得协变模型$（e_{\mu }^{我},A_{\mu }^{我},k_{\mu }^{我}）$这又动态地等同于标准 GR。文献中通常有一套$\beta =\gamma$为了简单起见。在这里，我们保持 Holst 和 Immirzi 参数不同，以表明最终只有$\beta$将存在于边界场方程中。