We point out that the symplectic structure, written in terms of the Sen-Ashtekar-Immirzi-Barbero variables, of a spacetime admitting an isolated horizon as the inner boundary, involves a positive constant parameter, say $\sigma$, if $\gamma\neq\pm i$, where $\gamma$ is the Barbero-Immirzi parameter. The parameter $\sigma$ represents the rescaling freedom that characterizes the equivalence class of null generators of the isolated horizon. We reiterate the fact that the laws of mechanics associated with the isolated horizon does not depend on the choice of $\sigma$ and, in particular, while one uses the value of standard surface gravity as input, that does not fix $\sigma$ to a particular value. This fact contradicts the claims made in certain parts of the concerned literature that we duly refer to. We do the calculations by taking Schwarzschild metric as an example so that the contradiction with the referred literature, where similar approaches were adopted, becomes apparent. The contribution to the symplectic structure that comes from the isolated horizon, diverges for $\sigma^2=(1+\gamma^2)^{-1}$, implying that the rescaling symmetry of the isolated horizon is violated for any real $\gamma$. Since the quantum theory of $SU(2)$ isolated horizon in the LQG framework exists only for real values of $\gamma$, it is founded on this flawed classical setup. Nevertheless, if the flaw is ignored, then two different viewpoints persist in the literature for entropy calculation. We highlight the main features of those approaches and point out why one is logically viable and the other is not.

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关于SU (2) 孤立地平线的说明

我们指出，以 Sen-Ashtekar-Immirzi-Barbero 变量的形式写成的时空以孤立视界为内边界的辛结构涉及一个正常数参数，比如 $\sigma$，如果 $\gamma \neq\pm i$，其中 $\gamma$ 是 Barbero-Immirzi 参数。参数 $\sigma$ 表示重新标度的自由度，它表征了孤立视界的零生成器的等价类。我们重申这样一个事实，即与孤立视界相关的力学定律不依赖于 $\sigma$ 的选择，特别是当人们使用标准表面重力值作为输入时，这并不能解决 $\sigma$到一个特定的值。这一事实与我们适当参考的相关文献的某些部分中提出的主张相矛盾。我们以 Schwarzschild 度量为例进行计算，以便与采用类似方法的参考文献的矛盾变得明显。来自孤立视界对辛结构的贡献，发散为 $\sigma^2=(1+\gamma^2)^{-1}$，这意味着对于任何实数都违反了孤立视界的重新缩放对称性$\伽马$。由于 LQG 框架中 $SU(2)$ 孤立视界的量子理论仅存在于 $\gamma$ 的真实值，它建立在这个有缺陷的经典设置上。然而，如果忽略该缺陷，那么在熵计算的文献中仍然存在两种不同的观点。我们强调了这些方法的主要特点，并指出为什么一种在逻辑上可行而另一种不可行。