非对易时空上引力模型的构建和黑洞热力学
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摘要
爱因斯坦的引力理论–广义相对论–在大尺度结构上取得了很大的成功，成功的预言了一系列重要的物理现象。广义相对论认为引力即是时空几何的思想，极大地改变了我们关于时空本质的认识。但是，广义相对论和量子场论之间存在不可调合的矛盾。将广义相对论纳入量子场论框架内的努力遭遇了量子引力不可重整化的失败。寻求一个自洽的量子引力理论，从而实现四种相互作用的统一是绝大多数物理学家的梦想，也是一个极具挑战性的问题。目前，存在多种量子引力的方案，其中包括最流行也是最有希望的弦论，以及圈量子引力理论。非对易作为其中一个尝试，认为时空在极小尺度上将呈现出非对易性。这种思想是对相空间非对易的自然推广。由于时空的非对易性，点和坐标的概念变得模糊，建立在光滑流形上的广义相对论不再适用。在非对易时空上建立或研究引力理论具有很重要的物理意义，它将能在一定程度上将量子效应和引力融合在一起，从而反映出极小尺度或高能标下引力的行为，为迈向最终的量子引力理论提供某些启示。
本博士论文涉及非对易时空上的引力及黑洞热力学问题，我们的工作的主要内容如下：
爱因斯坦的引力理论可以表示成SL(2,C)规范理论形式。我们利用协变坐标技术，将这种表述形式推广到具有辛结构的非对易时空上，在星乘框架下构建出规范不变的作用量。然后，利用Seiberg-Witten映射，在非对易参数的一阶水平上，将非对易的物理自由度映射到对应的对易物理自由度上，从而给出作用量的一阶修正形式。结果显示非对易对SL(2,C)引力作用量的一阶修正在一般情况下并不为零。这一结果是对已有文献结果的推广。其后，我们将上述分析和讨论推广到引力的其它表述形式上，在具有辛结构的非对易时空上构建了U(2,2)引力模型。结果同样表明作用量的一阶修正不为零。我们将在第四章详细阐述这部分内容。这部分内容基于我们以下的工作：Y.-G. Miao and S.-J. Zhang, SL(2,C) gravity on noncommutative space with Pois-son structure, Phys. Rev. D82(2010)084017[arXiv:1004.2118[hep-th]].Y.-G. Miao, Z. Xue and S.-J. Zhang, U(2,2) gravity on noncommuta-tive space with symplectic structure, Phys. Rev. D83(2011)024023[arXiv:1006.4074[hep-th]].
基于坐标相干态方法，已有的文献表明时空非对易导致“点质量”弥散在一个有限范围内。“点质量”的分布不再由狄拉克delta函数描述，而是由高斯函数描述。若同时假设爱因斯坦引力场方程不受影响，可以得到一系列包含非对易效应的黑洞解，通称为“非对易诱导黑洞”。我们利用Parikh-Wilczek隧穿方法对非对易诱导施瓦希（Schwarzschild）黑洞的热力学进行了分析。与已有文献不同的是，我们将Parikh-Wilczek方法推广到有质量粒子隧穿情形。结果表明，隧穿概率被隧穿粒子的质量压低，而有效的辐射温度受到非对易的修正。隧穿概率等于熵变的指数这一众所周知的结论在有质量情形下并不成立。同时，我们讨论了辐射粒子间的关联。结果表明辐射粒子之间存在关联，从而辐射可能包含信息。另外，我们利用Banerjee等人提出的改进的哈密顿-雅可比（Hamilton-Jacobi）方法讨论了非对易诱导克尔（Kerr）黑洞的热力学性质。与已有文献不同的是，我们将WKB方案计算到次一阶，从而包含隧穿粒子的角动量效应。结果表明辐射温度受到非对易修正，隧穿粒子的角动量对辐射温度没有影响。同时，我们分析了熵谱，结果表明熵的量子化既不受时空非对易的影响，也不受波函数量子修正的影响。我们将在第五章详细阐述这部分内容。这部分内容基于我们以下的工作：Y.-G. Miao, Z. Xue and S.-J. Zhang, Massive charged particle’s tunnel-ing from spherical charged black hole, Europhys. Lett.96(2011)10008[arXiv:1012.0390[hep-th]]; Tunneling of massive particles from noncommuta-tive inspired Schwarzschild black hole, Gen. Rel. Gra.44(2012)555[arXiv:1012.2426[hep-th]]; Quantum tunneling and spectroscopy of noncom-mutative inspired Kerr black hole, Int. J. Mod. Phys. D21(2012)1250018[arXiv:1102.0074[hep-th]].
在已有的大量文献中，采用坐标相干态方法处理非对易的同时采用了非标准的量子化过程。这种非标准的量子化过程推演出一个重要结论：点质量分布不是由狄拉克delta函数描述，而是由高斯函数描述。在非对易闵氏平面上，我们沿着标准的量子化过程重新考察了坐标相干态方法，推演出截然不同的结论：点质量仍然由狄拉克delta函数描述。此结论表明若采用坐标相干态方法处理非对易，则点粒子的概念仍然成立。为了探讨两种不同的量子化过程带来的后果，我们分别采用它们分析了安鲁（Unruh）效应和霍金（Hawking）辐射。结果表明：在非标准量子化过程下，安鲁温度和安鲁谱不受非对易影响，霍金温度受非对易修正而霍金辐射谱不受影响；在标准量子化过程下，安鲁温度和霍金温度都不受影响，但是两种谱都修正了一个有效灰度因子，此灰度因子源自时空非对易。我们将在第六章详细阐述这部分内容。这部分内容基于我们以下的工作：Y.-G. Miao and S.-J. Zhang, The coordinate coherent states approach revisited,arXiv:1105.4025[hep-th].
Einstein’s gravity theory, general relativity, has achieved great success in predict-ing and explaining physical phenomenons at large scale. In general relativity, the grav-ity is considered to be the geometry of spacetime, and this idea has greatly refreshedour cognition of spacetime. However, there exist irreconcilable contradictions betweengeneral relativity and quantum field theory. Any attempts on bringing the general rela-tivity into the frame of quantum field theory all fail in the renormalizability of quantumgravity. It is the dream of most physicists to look for a consistent theory of quan-tum gravity so that the four fundamental interactions can be unified, and also it is agreat challenge. At present, there exist some ansatzes of quantum gravity includingthe most promising one, string theory, and loop quantum gravity. Noncommutativityof spacetime is one of the various attempts whose essential idea is that the spacetimewill appear noncommutativity at very small scale. It is a natural generalization of theidea of noncommutativity of phase space in quantum mechanics. Because of the non-commutavity of spacetime, the concepts of point and coordinate become ambiguity,and general relativity basing on smooth manifolds is not valid any more. It is greatimportant to establish a corresponding gravity theory on noncommutative spacetimesand investigate its properties. This gravity theory could in some sense combine theeffects of gravity and quantum theory, and then reflects the behavior of gravity at verysmall scale (or at high energy), and so highlights the road to the final theory of quantumgravity.
This thesis deals with the problems of gravity and thermodynamics of black holeson noncommutative spacetimes. The main contents of our work are organized as fol-lows.
Einstein’s gravity theory can be reformulated as a gauge theory with gauge groupSL(2, C). With the help of the technique of covariant coordinates, we extend thisformulation to noncommutative spacetimes with symplectic structure and builda gauge invariant action in the frame of star-product. Using the Seiberg-Wittenmap, the physical degrees of freedom are expressed in terms of their commutativecounterparts up to the first order in noncommutative parameter and then the firstorder correction to the commutative action is obtained. The result shows that ingeneral the first order correction does not vanish. This is the extension of the ex-isted results in literatures. We then extend the above analysis to the construction of an U (2,2) gravity model on the same class of noncommutative spacetimes.The result also shows that the first order correction does not vanish. This part ofcontents is based on our following workY.-G. Miao and S.-J. Zhang, SL(2,C) gravity on noncommutative space with Pois-son structure, Phys. Rev. D82(2010)084017[arXiv:1004.2118[hep-th]].Y.-G. Miao, Z. Xue and S.-J. Zhang, U(2,2) gravity on noncommutative space withsymplectic structure, Phys. Rev. D83(2011)024023[arXiv:1006.4074[hep-th]].and will be given in detail in chapter4;
Based on the coordinate coherent states approach, the results in the existed lit-eratures show that a”point mass” should be smeared in a finite size of rangeand should be described by a gaussian function rather than the usual Dirac deltafunction. Assuming that the Einstein equation is untack, a series of solutions ofnoncommutative inspired black holes can be obtained. We first apply the Parikh-Wilczek’s tunneling method to analyze the thermodynamics of the noncomm-tative inspired Schwarzschild black hole. Different from the existed literatures,we extend the Parikh-Wilczek method to the tunneling of massive particles. Theresults show that the tunneling rate is suppressed by the mass of the tunneling par-ticles and the effective radiation temperature is modified by noncommutativity.The well-known result that the tunneling rate equals the exponent of the differ-ence of entropy does not hold in the massive case. Moreover, we also discuss thecorrelations between tunneling particles. The result shows that there exist corre-lations which imply that the radiation may contain informations. We then analyzethe thermodynamics of the noncommutative inspired Kerr black hole using thereformulated Hamilton-Jacobi method proposed by Banerjee et al. We calculatethe WKB ansatz to the first order in order to include the effect of the angular mo-mentum of tunneling particles. The results show that the radiation temperatureis modified by noncommutativity but not by the angular momentum of tunnelingparticles. Moreover, We analyze the spectra of entropy and show that the entropyquantum is affected neither by noncommutativity or by the quantum correctionof wave function. This part of contents is based on our following workY.-G. Miao, Z. Xue and S.-J. Zhang, Massive charged particle’s tunnel-ing from spherical charged black hole, Europhys. Lett.96(2011)10008 [arXiv:1012.0390[hep-th]]; Tunneling of massive particles from noncommu-tative inspired Schwarzschild black hole, Gen. Rel. Gra.44(2012)555[arXiv:1012.2426[hep-th]]; Quantum tunneling and spectroscopy of noncom-mutative inspired Kerr black hole, Int. J. Mod. Phys. D21(2012)1250018[arXiv:1102.0074[hep-th]].and will be given in detail in chapter5;
In the existing literatures, the coordinate coherent states approach is associatedwith a non-standard quantization procedure. The essential outcome of this non-standard quantization procedure is that a”point mass” now should be describedby a gaussian function rather than the usual Dirac delta function. We revisit thecoordinate coherent states approach with the standard quantization procedure onnoncommutative Minkowski plane and deduce a significantly different result: a”point mass” is still described by the Dirac delta function. This implies that theconcept of point particle is still valid if we deal the noncommutativity with thecoordinate coherent states approach associating with the standard quantizationprocedure. To investigate the physical consequences on the quantization proce-dures, we apply the two quantization procedures in discussing the Unruh effectand Hawking radiation. The results are: Under the non-standard quantizationprocedure, Unruh temperature and Unruh spectrum are not modified by noncom-mutativity, but the Hawking temperature is modified while the Hawking radiationspectrum is untack; while under the standard quantization procedure, Unruh tem-perature and Hawking temperature are untack but the both spectra are modifiedby an effective greybody factor. This grebody factor originates from noncommu-tativity. This part of contents is based on our following workY.-G. Miao and S.-J. Zhang, The coordinate coherent states approach revisited,arXiv:1105.4025[hep-th].and will be given in detail in chapter6.
本博士论文涉及非对易时空上的引力及黑洞热力学问题，我们的工作的主要内容如下：
爱因斯坦的引力理论可以表示成SL(2,C)规范理论形式。我们利用协变坐标技术，将这种表述形式推广到具有辛结构的非对易时空上，在星乘框架下构建出规范不变的作用量。然后，利用Seiberg-Witten映射，在非对易参数的一阶水平上，将非对易的物理自由度映射到对应的对易物理自由度上，从而给出作用量的一阶修正形式。结果显示非对易对SL(2,C)引力作用量的一阶修正在一般情况下并不为零。这一结果是对已有文献结果的推广。其后，我们将上述分析和讨论推广到引力的其它表述形式上，在具有辛结构的非对易时空上构建了U(2,2)引力模型。结果同样表明作用量的一阶修正不为零。我们将在第四章详细阐述这部分内容。这部分内容基于我们以下的工作：Y.-G. Miao and S.-J. Zhang, SL(2,C) gravity on noncommutative space with Pois-son structure, Phys. Rev. D82(2010)084017[arXiv:1004.2118[hep-th]].Y.-G. Miao, Z. Xue and S.-J. Zhang, U(2,2) gravity on noncommuta-tive space with symplectic structure, Phys. Rev. D83(2011)024023[arXiv:1006.4074[hep-th]].
基于坐标相干态方法，已有的文献表明时空非对易导致“点质量”弥散在一个有限范围内。“点质量”的分布不再由狄拉克delta函数描述，而是由高斯函数描述。若同时假设爱因斯坦引力场方程不受影响，可以得到一系列包含非对易效应的黑洞解，通称为“非对易诱导黑洞”。我们利用Parikh-Wilczek隧穿方法对非对易诱导施瓦希（Schwarzschild）黑洞的热力学进行了分析。与已有文献不同的是，我们将Parikh-Wilczek方法推广到有质量粒子隧穿情形。结果表明，隧穿概率被隧穿粒子的质量压低，而有效的辐射温度受到非对易的修正。隧穿概率等于熵变的指数这一众所周知的结论在有质量情形下并不成立。同时，我们讨论了辐射粒子间的关联。结果表明辐射粒子之间存在关联，从而辐射可能包含信息。另外，我们利用Banerjee等人提出的改进的哈密顿-雅可比（Hamilton-Jacobi）方法讨论了非对易诱导克尔（Kerr）黑洞的热力学性质。与已有文献不同的是，我们将WKB方案计算到次一阶，从而包含隧穿粒子的角动量效应。结果表明辐射温度受到非对易修正，隧穿粒子的角动量对辐射温度没有影响。同时，我们分析了熵谱，结果表明熵的量子化既不受时空非对易的影响，也不受波函数量子修正的影响。我们将在第五章详细阐述这部分内容。这部分内容基于我们以下的工作：Y.-G. Miao, Z. Xue and S.-J. Zhang, Massive charged particle’s tunnel-ing from spherical charged black hole, Europhys. Lett.96(2011)10008[arXiv:1012.0390[hep-th]]; Tunneling of massive particles from noncommuta-tive inspired Schwarzschild black hole, Gen. Rel. Gra.44(2012)555[arXiv:1012.2426[hep-th]]; Quantum tunneling and spectroscopy of noncom-mutative inspired Kerr black hole, Int. J. Mod. Phys. D21(2012)1250018[arXiv:1102.0074[hep-th]].
在已有的大量文献中，采用坐标相干态方法处理非对易的同时采用了非标准的量子化过程。这种非标准的量子化过程推演出一个重要结论：点质量分布不是由狄拉克delta函数描述，而是由高斯函数描述。在非对易闵氏平面上，我们沿着标准的量子化过程重新考察了坐标相干态方法，推演出截然不同的结论：点质量仍然由狄拉克delta函数描述。此结论表明若采用坐标相干态方法处理非对易，则点粒子的概念仍然成立。为了探讨两种不同的量子化过程带来的后果，我们分别采用它们分析了安鲁（Unruh）效应和霍金（Hawking）辐射。结果表明：在非标准量子化过程下，安鲁温度和安鲁谱不受非对易影响，霍金温度受非对易修正而霍金辐射谱不受影响；在标准量子化过程下，安鲁温度和霍金温度都不受影响，但是两种谱都修正了一个有效灰度因子，此灰度因子源自时空非对易。我们将在第六章详细阐述这部分内容。这部分内容基于我们以下的工作：Y.-G. Miao and S.-J. Zhang, The coordinate coherent states approach revisited,arXiv:1105.4025[hep-th].
Einstein’s gravity theory, general relativity, has achieved great success in predict-ing and explaining physical phenomenons at large scale. In general relativity, the grav-ity is considered to be the geometry of spacetime, and this idea has greatly refreshedour cognition of spacetime. However, there exist irreconcilable contradictions betweengeneral relativity and quantum field theory. Any attempts on bringing the general rela-tivity into the frame of quantum field theory all fail in the renormalizability of quantumgravity. It is the dream of most physicists to look for a consistent theory of quan-tum gravity so that the four fundamental interactions can be unified, and also it is agreat challenge. At present, there exist some ansatzes of quantum gravity includingthe most promising one, string theory, and loop quantum gravity. Noncommutativityof spacetime is one of the various attempts whose essential idea is that the spacetimewill appear noncommutativity at very small scale. It is a natural generalization of theidea of noncommutativity of phase space in quantum mechanics. Because of the non-commutavity of spacetime, the concepts of point and coordinate become ambiguity,and general relativity basing on smooth manifolds is not valid any more. It is greatimportant to establish a corresponding gravity theory on noncommutative spacetimesand investigate its properties. This gravity theory could in some sense combine theeffects of gravity and quantum theory, and then reflects the behavior of gravity at verysmall scale (or at high energy), and so highlights the road to the final theory of quantumgravity.
This thesis deals with the problems of gravity and thermodynamics of black holeson noncommutative spacetimes. The main contents of our work are organized as fol-lows.
Einstein’s gravity theory can be reformulated as a gauge theory with gauge groupSL(2, C). With the help of the technique of covariant coordinates, we extend thisformulation to noncommutative spacetimes with symplectic structure and builda gauge invariant action in the frame of star-product. Using the Seiberg-Wittenmap, the physical degrees of freedom are expressed in terms of their commutativecounterparts up to the first order in noncommutative parameter and then the firstorder correction to the commutative action is obtained. The result shows that ingeneral the first order correction does not vanish. This is the extension of the ex-isted results in literatures. We then extend the above analysis to the construction of an U (2,2) gravity model on the same class of noncommutative spacetimes.The result also shows that the first order correction does not vanish. This part ofcontents is based on our following workY.-G. Miao and S.-J. Zhang, SL(2,C) gravity on noncommutative space with Pois-son structure, Phys. Rev. D82(2010)084017[arXiv:1004.2118[hep-th]].Y.-G. Miao, Z. Xue and S.-J. Zhang, U(2,2) gravity on noncommutative space withsymplectic structure, Phys. Rev. D83(2011)024023[arXiv:1006.4074[hep-th]].and will be given in detail in chapter4;
Based on the coordinate coherent states approach, the results in the existed lit-eratures show that a”point mass” should be smeared in a finite size of rangeand should be described by a gaussian function rather than the usual Dirac deltafunction. Assuming that the Einstein equation is untack, a series of solutions ofnoncommutative inspired black holes can be obtained. We first apply the Parikh-Wilczek’s tunneling method to analyze the thermodynamics of the noncomm-tative inspired Schwarzschild black hole. Different from the existed literatures,we extend the Parikh-Wilczek method to the tunneling of massive particles. Theresults show that the tunneling rate is suppressed by the mass of the tunneling par-ticles and the effective radiation temperature is modified by noncommutativity.The well-known result that the tunneling rate equals the exponent of the differ-ence of entropy does not hold in the massive case. Moreover, we also discuss thecorrelations between tunneling particles. The result shows that there exist corre-lations which imply that the radiation may contain informations. We then analyzethe thermodynamics of the noncommutative inspired Kerr black hole using thereformulated Hamilton-Jacobi method proposed by Banerjee et al. We calculatethe WKB ansatz to the first order in order to include the effect of the angular mo-mentum of tunneling particles. The results show that the radiation temperatureis modified by noncommutativity but not by the angular momentum of tunnelingparticles. Moreover, We analyze the spectra of entropy and show that the entropyquantum is affected neither by noncommutativity or by the quantum correctionof wave function. This part of contents is based on our following workY.-G. Miao, Z. Xue and S.-J. Zhang, Massive charged particle’s tunnel-ing from spherical charged black hole, Europhys. Lett.96(2011)10008 [arXiv:1012.0390[hep-th]]; Tunneling of massive particles from noncommu-tative inspired Schwarzschild black hole, Gen. Rel. Gra.44(2012)555[arXiv:1012.2426[hep-th]]; Quantum tunneling and spectroscopy of noncom-mutative inspired Kerr black hole, Int. J. Mod. Phys. D21(2012)1250018[arXiv:1102.0074[hep-th]].and will be given in detail in chapter5;
In the existing literatures, the coordinate coherent states approach is associatedwith a non-standard quantization procedure. The essential outcome of this non-standard quantization procedure is that a”point mass” now should be describedby a gaussian function rather than the usual Dirac delta function. We revisit thecoordinate coherent states approach with the standard quantization procedure onnoncommutative Minkowski plane and deduce a significantly different result: a”point mass” is still described by the Dirac delta function. This implies that theconcept of point particle is still valid if we deal the noncommutativity with thecoordinate coherent states approach associating with the standard quantizationprocedure. To investigate the physical consequences on the quantization proce-dures, we apply the two quantization procedures in discussing the Unruh effectand Hawking radiation. The results are: Under the non-standard quantizationprocedure, Unruh temperature and Unruh spectrum are not modified by noncom-mutativity, but the Hawking temperature is modified while the Hawking radiationspectrum is untack; while under the standard quantization procedure, Unruh tem-perature and Hawking temperature are untack but the both spectra are modifiedby an effective greybody factor. This grebody factor originates from noncommu-tativity. This part of contents is based on our following workY.-G. Miao and S.-J. Zhang, The coordinate coherent states approach revisited,arXiv:1105.4025[hep-th].and will be given in detail in chapter6.
引文
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