In Loop Quantum Gravity (LQG) the quantisation of General Relativity leads to precise predictions for the eigenvalues of geometrical observables like volume and area, up to the value of the only free parameter in the theory, the Barbero-Immirzi (BI) parameter. With the help of the eigenvalues equation for the area operator, LQG successfully derives the Bekenstein-Hawking entropy of large black holes with isolated horizons, fixing in this limit the BI parameter as $\gamma \approx 0.274$. In the present paper we show that a black hole with angular momentum $\hbar$ and Planck mass is eigenstate of the area operator provided that $\gamma = \sqrt{3}/6 \approx 1.05 \times 0.274$. As the black hole is extremal, there is no Hawking radiation and the horizon is isolated. We also show that such a black hole can be formed in the head-on scattering of two parallel Standard Model neutrinos in the mass state $m_2$ (assuming $m_1 = 0$). Furthermore, we use the obtained BI parameter to numerically compute the entropy of isolated horizons with areas ranging up to $250\,l_P^2$, by counting the number of micro-states associated to a given area. The resulting entropy has a leading term ${\cal S} \approx 0.25\, {\cal A}$, in agreement to the Bekenstein-Hawking entropy. As the identification of the above eigenstate rests on the matching between classical areas and quantum area eigenvalues, we also present, on the basis of an effective quantum model for the Schwarzschild black hole recently proposed by Ashtekar, Olmedo and Singh, an expression for the quantum corrected area of isolated horizons, valid for any black hole mass. Quantum corrections are shown to be negligible for a Planck mass black hole, of order $10^{-3}$ relative to the classical area.

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关于 Immirzi 参数和视界熵的值

在循环量子引力 (LQG) 中，广义相对论的量化导致对体积和面积等几何可观测量的特征值进行精确预测，直至理论中唯一的自由参数 Barbero-Immirzi (BI) 参数的值。借助面积算子的特征值方程，LQG 成功推导出具有孤立视界的大黑洞的 Bekenstein-Hawking 熵，将 BI 参数固定为 $\gamma\approx 0.274$。在本文中，我们证明了角动量 $\hbar$ 和普朗克质量的黑洞是面积算子的本征态，条件是 $\gamma = \sqrt{3}/6 \approx 1.05 \times 0.274$。由于黑洞是极值，没有霍金辐射，视界是孤立的。我们还表明，这样的黑洞可以在质量状态 $m_2$（假设 $m_1 = 0$）的两个平行标准模型中微子的正面散射中形成。此外，我们使用获得的 BI 参数通过计算与给定区域相关联的微观状态的数量，以数值方式计算区域范围高达 $250\,l_P^2$ 的孤立层的熵。由此产生的熵有一个前导项 ${\cal S} \approx 0.25\, {\cal A}$，与 Bekenstein-Hawking 熵一致。由于上述本征态的识别依赖于经典面积和量子面积本征值的匹配，我们还根据Ashtekar、Olmedo和Singh最近提出的Schwarzschild黑洞的有效量子模型，给出了量子场的一个表达式孤立视界的校正区域，适用于任何黑洞质量。