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arXiv 2406.01466hep-thgr-qchep-ph
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关于量子引力与黑洞量子力学的矩阵模型提案

A Matrix Model Proposal for Quantum Gravity and the Quantum Mechanics of Black Holes

Chong-Sun Chu

AI总结:

提出基于大$N$非交换几何的量子引力矩阵模型,发现半满费米海的模糊球可描述量子黑洞并导出黑洞熵与质量-半径关系,且双球系统在大$N$极限下可还原牛顿引力。

摘要:

我们提出了一种由大 $N$ 非交换几何描述的量子空间的量子力学理论,作为量子引力的模型。该模型允许模糊球作为静态解。在模糊几何上,费米子的量子力学由等频率的振子之和给出。恰好半数费米海被填满的能态包含最大程度的简并。如果将模糊球等同于黑洞视界,则模糊球的这种状态遵循 Schwarzschild 黑洞的质量-半径关系。此外,费米海中的状态集精确地给出了 Bekenstein-Hawking 熵。因此,我们提出在我们的模型中,量子黑洞由具有半满费米海的模糊球来描述。我们还考虑了两个模糊球的系统,将它们作为块嵌入到矩阵量子力学中。当两个模糊球之间的距离 $r$ 很小时,可以使用微扰理论计算系统的总能量。我们证明,在大 $N$ 极限的主阶下,相互作用能精确地以 $- G M_1 M_2$ 的方式依赖于距离,与牛顿引力完全相同。为了在长程范围内重现正确的 $r$ 依赖性,我们预计需要包含大 $N$ 修正和量子效应。

英文摘要:

We propose a quantum mechanical theory of quantum spaces described by large $N$ noncommutative geometry as a model for quantum gravity. The model admits fuzzy sphere as static solution. Over the fuzzy geometry, the quantum mechanics of the fermions is given by a sum of oscillators with equal frequency. The energy state where exactly half of the Fermi sea is filled contains the maximal amount of degeneracy. This state of the fuzzy sphere obeys the mass-radius relation of a Schwarzschild black hole if the fuzzy sphere is identified with the black hole horizon. Moreover the set of states in the Fermi sea gives precisely the Bekenstein-Hawking entropy. We thus propose that quantum black holes are described by fuzzy spheres with a half-filled Fermi sea in our model. We also consider a system of two fuzzy spheres by embedding them as blocks in the matrix quantum mechanics. When the distance $r$ between the two fuzzy spheres is small, the total energy of the system can be computed using perturbation theory. We show that in the leading order of large $N$ limit, the interaction energy depends on $- G M_1 M_2$ exactly the manner as in Newton gravity. To reproduce the correct $r$ dependence in the long range, we expect the inclusion of large $N$ corrections and quantum effects will be needed.

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