Abstract

Black holes violate the third law of thermodynamics, and this gives rise to difficulties with the microscopic description of their entropy. Recently, it has been shown that the microscopic description of the Schwarzschild black hole thermodynamics in \(D = 4\) space–time dimensions is provided by the analytic continuation of the entropy of Bose gas with a nonrelativistic one-particle energy to \(d =-4\) negative spatial dimensions. In this paper, we show that the \(D=5\) and \(D=6\) Schwarzschild black holes thermodynamics can be modeled by the \(d\)-dimensional Bose gas, \(d=1,2,3,\dots\,\), with the one-particle energy \(\varepsilon(k)=k^\alpha\) under the respective conditions \(\alpha=-d/3\) and \(\alpha=-d/4\). In these cases, the free energy of the Bose gas has divergences, and we introduce a cut-off and perform the minimal renormalizations. We also perform renormalizations using analytic regularization and prove that the minimal cut-off renormalization gives the same answer as the analytic regularization by the Riemann zeta-function.

中文翻译：

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史瓦西黑洞热力学的玻色气体模型

摘要

黑洞违反了热力学第三定律，这给微观描述黑洞的熵带来了困难。最近，研究表明，在\(D = 4\)时空维度上史瓦西黑洞热力学的微观描述是由非相对论性单粒子能量的玻色气体熵的解析延拓提供的\( d =-4\)负空间维度。在本文中，我们证明了\(D=5\)和\(D=6\)史瓦西黑洞热力学可以通过\(d\)维玻色气体来建模，\(d=1,2, 3,\dots\,\)，在相应条件\(\alpha=-d/3\)和\ (\alpha= -d/4\)。在这些情况下，玻色气体的自由能存在发散，我们引入截止并执行最小重正化。我们还使用解析正则化执行重整化，并证明最小截止重整化给出与黎曼 zeta 函数的解析正则化相同的答案。