臺灣博碩士論文加值系統

荷蘭弦論學家 Erik Verlinde 在2010年提出的熵引力理論(entropic gravity theory)中重現了牛頓的重力定律。在此架構下，幾個包含量子力學意義的假設竟推導出純古典的牛頓重力定律，我們不禁疑問：熵引力理論中的量子效應究竟隱藏在哪裡？
為了回答這個問題，我們首先回顧前人將重力描述成演生現象(emergent phenomena)的動機與背景，接著重新檢視Verlinde熵引力理論中的幾個概念：全像屏的意義，量子糾纏熵(entanglement entropy)和熵的變化式等，以期獲得自洽的解決方法。我們進一步主張當由於在熵引力理論中已經隱含一最小尺度的存在，而在最小尺度的存在底下我們必考慮廣義測不準原理(generalized uncertainty principle, GUP)。在廣義測不準原理的架構之下，Bekenstein所提出的的黑洞熵(black hole entropy)之簡單形式無論在強烈或微弱的重力作用底下都需要加以修正。而在微弱重力的極限下，廣義測不準原理給予黑洞熵一個對數的首項修正。我們將廣義測不準原理對黑洞熵的修正應用在重力表現為一種熵力的詮釋上，得到一個GUP修正的熵引力形式，此形式也確實包含了古典牛頓重力定律以外的量子修正項。我們所得到的量子修正項，在將來或許能進一步作為熵引力理論的驗證依據。

The entropic gravity scenario recently proposed by Erik Verlinde reproduced the Newton''s law of purely classical gravity yet the key assumptions of this approach all have quantum mechanical origins. So one naturally wonders: where is $hbar$ hiding in entropic gravity?
To address this question, we first reformulate the entropic derivation of Newton''s gravitation force law to address a self-consistent approach to the problem. Next we argue that as the concept of minimal length has been invoked in the Bekenstein entropic derivation, the generalized uncertainty principle (GUP), which is a direct consequence of the minimal length, should be taken into consideration in the entropic interpretation of gravity. Indeed based on GUP it has been demonstrated that the black hole Bekenstein entropy area law must be modified not only in the strong but also in the weak gravity regime where in the weak gravity limit the GUP modified entropy exhibits a logarithmic correction.
In the weak gravity limit, such a GUP modified entropy exhibits a logarithmic correction term. When applying it to the entropic interpretation, we demonstrate that the resulting gravity force law does include sub-leading order correction terms that depend on $hbar$. Such deviation from the classical Newton''s law may serve as a probe to the validity of the entropic gravity postulate.

誌謝 i
中文摘要 ii
Abstract i ii
Contents i v
List of Figures vi
1 Introduction 1
2 Emergent theory of gravity 4
2.1 The need for a quantum gravity theory 4
2.2 Unsolved puzzles in gravity 5
2.3 Gravity as an emergent phenomenon 6
3 Holographic principle and holographic entanglement entropy 9
3.1 Holographic principle 9
3.2 Entropy of entanglement 10
3.3 Holographic entanglement entropy 11
4 Verlinde’s entropic gravity scenario 13
4.1 Meaning of holographic screen 13
4.1.1 Entropy variation law 14
4.1.2 Verlinde’s approach 14
4.1.3 Fursaev’s approach 16
4.1.4 Our approach 18
vi4.2 Temperature 21
5 Black hole entropy and GUP 24
6 Quantum Effects in Entropic Gravity 27
6.1 Quantum corrections to entropic gravity: other approaches 27
6.2 Entropic gravity under GUP 28
7 Conclusion and discussion 30
Reference 32

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