熵界和全息原理
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摘要
自从Hawking于1975年发现霍金辐射以来,黑洞热力学及其背后的量子引力描述一直得到广泛的关注和研究。尤其是九十年代以后受黑洞面积熵定律的启发,由't Hooft和Susskind先后提出并发展了全息原理的概念,即一个物理系统的信息容量不超过其Planck单位下的边界面积。随后又出现了ADS／CFT对偶对全息原理的实现,这使得全息原理在量子引力的研究中逐渐成为一个中心问题。
在本学位论文中,我们介绍了熵界和全息原理的基本知识,并以熵界研究作为出发点,考察了全息原理相关的几个问题。
首先,我们研究了引力约束下定域场论的熵界,第一次用数微观态的方法严格证明了玻色场和费米场都遵从同样的熵界A3／4lp-3／2,其中A是体系的边界面积,lp=1.616×10-35m是Planck尺度。这支持了'tHooft借助热态所做的定性分析的结论,也纠正了以往认为定域场论可以饱和全息熵界Alp-2的错误的认识。
随后,我们研究了由无穷大统计所描述的量子玻尔兹曼场的热力学熵界和统计熵界,发现在引力约束下,它刚好遵从全息熵界Alp-2。在一般的情况下,这个熵界正是Bekenstein熵界El／(hc),其中E和l分别是体系的能量和尺度。我们的结果增进了对从定域场论的熵界A3／4lp-3/2到全息熵界Alp-2之间的熵隙的认识,而这个熵隙对应的自由度的来源在之前一直是不清楚的。这个结果也暗示无穷大统计和量子引力之间存在深刻的联系。
最后,我们借助量子计算的观点,结合量子和引力的效应,导出了一个新的时空不确定性关系。这个关系式可以应用于包括宇宙的计算容量和效率,宇宙热力学等问题的分析上。这个关系式看以看作是全息原理的一个拓展。全息原理一般来说只对系统的信息容量给出限制,而我们的关系式可以推出包括系统的信息处理能力在内的更多性质。
Since Hawking radiation was discovered in 1975, the thermodynamics of black holes and the quantum gravitational mechanism behind it have gained a lot of interest. Especially, illuminated by the area law of black hole entropy,'t Hooft and Susskind subsequently de-veloped the concept of holographic principle, which states that the information capacity of a system will not exceed its boundary area in Planck units. Later, the holographic principle was realized by the famous ADS/CFT dual, which soon turns holographic principle to be a central topic in the research of quantum gravity.
In this thesis, we introduce the fundamental lore in the area of entropy bounds and holographic principle. Starting from the study of entropy bounds, we have investigated several questions related to holographic principle.
First, we investigate the entropy bound for local quantum field theory under gravita-tional constraint. We strictly proved that bosonic and fermionic fields conform to the same entropy bound A3/4lp-3/2, where A is the boundary area of the system, lp=1.616×10-35m is Planck length. The results have supported the qualitative analysis given by't Hooft some years ago, and rectified the incorrect viewpoint that local quantum field theory could satu-rate the holographic entropy.
Second, we investigate the thermodynamical and statistical entropy bound for quan-tum Boltzmann fields which obey infinite statistics. We find infinite statistics complies with the holographic entropy bound Alp-2 very well. In general cases, the entropy bound comes back to the Bekenstein bound El/(hc), where E and l are respectively the energy and size of the system. Our results have improved the understanding of the gap between the entropy bound of local quantum field theory and the holographic entropy, that is, from A3/4lp-3/2 to Alp-2, the corresponding degrees of freedom of which are very obscure before our work. Our results also suggest a close relationship between infinite statistics and quantum gravity.
Finally, starting from a quantum computational perspective, combining quantum and gravitational principles, we derived a new space-time uncertainty relation, which can be used to facilitate the discussion of several profound questions, such as computational ca-pacity and thermodynamic properties of the universe. This uncertainty relation can be viewed as an extension of holographic principle. Holographic principle only limits the information of a system, while from our relation one can obtain some other important properties of the system such as its capacity of information processing.
在本学位论文中,我们介绍了熵界和全息原理的基本知识,并以熵界研究作为出发点,考察了全息原理相关的几个问题。
首先,我们研究了引力约束下定域场论的熵界,第一次用数微观态的方法严格证明了玻色场和费米场都遵从同样的熵界A3／4lp-3／2,其中A是体系的边界面积,lp=1.616×10-35m是Planck尺度。这支持了'tHooft借助热态所做的定性分析的结论,也纠正了以往认为定域场论可以饱和全息熵界Alp-2的错误的认识。
随后,我们研究了由无穷大统计所描述的量子玻尔兹曼场的热力学熵界和统计熵界,发现在引力约束下,它刚好遵从全息熵界Alp-2。在一般的情况下,这个熵界正是Bekenstein熵界El／(hc),其中E和l分别是体系的能量和尺度。我们的结果增进了对从定域场论的熵界A3／4lp-3/2到全息熵界Alp-2之间的熵隙的认识,而这个熵隙对应的自由度的来源在之前一直是不清楚的。这个结果也暗示无穷大统计和量子引力之间存在深刻的联系。
最后,我们借助量子计算的观点,结合量子和引力的效应,导出了一个新的时空不确定性关系。这个关系式可以应用于包括宇宙的计算容量和效率,宇宙热力学等问题的分析上。这个关系式看以看作是全息原理的一个拓展。全息原理一般来说只对系统的信息容量给出限制,而我们的关系式可以推出包括系统的信息处理能力在内的更多性质。
Since Hawking radiation was discovered in 1975, the thermodynamics of black holes and the quantum gravitational mechanism behind it have gained a lot of interest. Especially, illuminated by the area law of black hole entropy,'t Hooft and Susskind subsequently de-veloped the concept of holographic principle, which states that the information capacity of a system will not exceed its boundary area in Planck units. Later, the holographic principle was realized by the famous ADS/CFT dual, which soon turns holographic principle to be a central topic in the research of quantum gravity.
In this thesis, we introduce the fundamental lore in the area of entropy bounds and holographic principle. Starting from the study of entropy bounds, we have investigated several questions related to holographic principle.
First, we investigate the entropy bound for local quantum field theory under gravita-tional constraint. We strictly proved that bosonic and fermionic fields conform to the same entropy bound A3/4lp-3/2, where A is the boundary area of the system, lp=1.616×10-35m is Planck length. The results have supported the qualitative analysis given by't Hooft some years ago, and rectified the incorrect viewpoint that local quantum field theory could satu-rate the holographic entropy.
Second, we investigate the thermodynamical and statistical entropy bound for quan-tum Boltzmann fields which obey infinite statistics. We find infinite statistics complies with the holographic entropy bound Alp-2 very well. In general cases, the entropy bound comes back to the Bekenstein bound El/(hc), where E and l are respectively the energy and size of the system. Our results have improved the understanding of the gap between the entropy bound of local quantum field theory and the holographic entropy, that is, from A3/4lp-3/2 to Alp-2, the corresponding degrees of freedom of which are very obscure before our work. Our results also suggest a close relationship between infinite statistics and quantum gravity.
Finally, starting from a quantum computational perspective, combining quantum and gravitational principles, we derived a new space-time uncertainty relation, which can be used to facilitate the discussion of several profound questions, such as computational ca-pacity and thermodynamic properties of the universe. This uncertainty relation can be viewed as an extension of holographic principle. Holographic principle only limits the information of a system, while from our relation one can obtain some other important properties of the system such as its capacity of information processing.
引文
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