波的相奇异性的拓扑性质和Friedmann-Robertson-Walker宇宙热力学的量子效应
摘要
本论文在拓扑场论和广义相对论两个领域做了深入的研究。在拓扑场论方面,利用段氏拓扑流理论研究了三种波的相奇异性；在广义相对论方向,主要讨论了Friedmann-Robertson-Walker (FRW)宇宙表观视界处的热力学的量子效应。
首先,我们利用段氏拓扑流理论研究了三种波的相奇异性,分别为电磁场中的Riemann-Silberstein (RS)涡旋、相干光学中的相干涡旋、和卷波的中心轴vortex filaments.对于RS涡旋,我们讨论了它的扭结结构并推导了描述其拓扑性质的Hopf不变量,发现其恰好可以表示成扭结RS涡旋的所有自连接数和连接数之和；对于相干涡旋,我们详细推导其拓扑相干流,并利用该拓扑相干流讨论了相干涡旋的拓扑结构、相干通量量子化、以及相干涡旋的扭结；在卷波系统中,我们同样研究了扭结状vortex filaments的Hopf不变量,并提出可以将Hopf不变量作为系统的拓扑约束。
其次,本文研究了广义不确定关系和扩展不确定关系对FRW宇宙热力学的影响。在广义不确定关系和扩展不确定关系下,传统的熵与面积成正比的熵面积关系不再成立,而是增加了一些修正项。不仅如此,根据表观视界处的热力学第一定律,描述宇宙动力学的Friedmann方程也需要修正。这说明广义不确定关系和扩展不确定关系能够影响FRW宇宙的动力学演化.
最后,我们分别讨论了FRW宇宙表观视界处类Hawking辐射的半经典和超半经典量子隧穿机制。在半经典情况下,通过讨论正则不变的隧穿振幅并考虑作用量的时间部分的贡献,得到了正确的类Hawking温度；通过考虑隧穿粒子的自引力效应,我们还得到了隧穿几率与表观视界熵的改变的关系。在超半经典量子隧穿机制中,我们考虑了表观视界处标量粒子和费米子的量子隧穿,并利用得到的类Hawking温度的量子修正式,我们推导了Einstein引力,Gauss-Bonnet引力、Lovelock引力、f(R)引力、和scalar-tensor引力中FRW宇宙表观视界处的熵的量子修正表达式,我们指出,这个表达式可能不依赖于具体的引力理论.
This thesis involves two main fields of research:topological field theory and General Relativity. On topological field theory, using Duan's topological current theory, we address some topological aspects of phase singularities of wave field; on general relativity, we mainly investigate the quantum effects in thermodynamics of FRW universe.
Firstly, based on Duan's topological current theory, we discuss RS vortices in electro-magnetic field, coherence vortices in coherent optics, and vortex filaments in scroll wave. For RS vortices, we discuss their knot structure and derive the Hopf invariant for knotted RS vortices, we find that the Hopf invariant is just the total sum of all the self-linking numbers and linking numbers of knotted RS vortices. For coherence vortices, we obtain the exact expression of their topological coherence current, and based on this expression, we discuss the topological structures of coherence vortices, quantization of coherence flux, and knotted coherence vortices. In scroll wave system, we also investigate the Hopf in-variant of knotted vortex filaments, and point out that the Hopf invariant can be treated as the topological constraints of the scroll wave system.
Secondly, this thesis investigates the impact of generalized uncertainty principle and extended uncertainty principle on thermodynamics of FRW universe. After consider the effects of generalized uncertainty principle and extended uncertainty principle, the con-ventional entropy area relation on apparent horizon does not hold anymore, but acquires corrections. Moreover, according to first law of thermodynamics on apparent horizon, the Friedmann equations of FRW universe also should corrected. This implies that the generalized uncertainty principle and extended uncertainty principle can influence the dynamics of FRW universe.
Finally, we discuss the semi-classical and beyond semi-classical aspect of Hawking-like radiation as tunneling on apparent horizon of FRW universe. For semi-classical case, using canonically invariant tunneling probability and considering the imaginary parts of both the spatial and temporal contributions of the tunneling amplitude we recover the correct temperature T=1/2πrA; when we consider the self-gravity effect of tunneling particle, we obtain the relationship between the tunneling probability and the change in entropy of apparent horizon. On the quantum tunneling beyond semi-classical approxi- mation, we consider the tunneling of both scalar particles and fermions and obtain the corrected Hawking-like temperature of apparent horizon, with this corrected Hawking-like temperature, the explicit expressions of the corrected entropy of apparent horizon for var-ious gravity theories including Einstein gravity, Gauss-Bonnet gravity, Lovelock gravity, f(R) gravity and scalar-tensor gravity, are computed, we point out that this expression might be independent of gravity theories.
首先,我们利用段氏拓扑流理论研究了三种波的相奇异性,分别为电磁场中的Riemann-Silberstein (RS)涡旋、相干光学中的相干涡旋、和卷波的中心轴vortex filaments.对于RS涡旋,我们讨论了它的扭结结构并推导了描述其拓扑性质的Hopf不变量,发现其恰好可以表示成扭结RS涡旋的所有自连接数和连接数之和；对于相干涡旋,我们详细推导其拓扑相干流,并利用该拓扑相干流讨论了相干涡旋的拓扑结构、相干通量量子化、以及相干涡旋的扭结；在卷波系统中,我们同样研究了扭结状vortex filaments的Hopf不变量,并提出可以将Hopf不变量作为系统的拓扑约束。
其次,本文研究了广义不确定关系和扩展不确定关系对FRW宇宙热力学的影响。在广义不确定关系和扩展不确定关系下,传统的熵与面积成正比的熵面积关系不再成立,而是增加了一些修正项。不仅如此,根据表观视界处的热力学第一定律,描述宇宙动力学的Friedmann方程也需要修正。这说明广义不确定关系和扩展不确定关系能够影响FRW宇宙的动力学演化.
最后,我们分别讨论了FRW宇宙表观视界处类Hawking辐射的半经典和超半经典量子隧穿机制。在半经典情况下,通过讨论正则不变的隧穿振幅并考虑作用量的时间部分的贡献,得到了正确的类Hawking温度；通过考虑隧穿粒子的自引力效应,我们还得到了隧穿几率与表观视界熵的改变的关系。在超半经典量子隧穿机制中,我们考虑了表观视界处标量粒子和费米子的量子隧穿,并利用得到的类Hawking温度的量子修正式,我们推导了Einstein引力,Gauss-Bonnet引力、Lovelock引力、f(R)引力、和scalar-tensor引力中FRW宇宙表观视界处的熵的量子修正表达式,我们指出,这个表达式可能不依赖于具体的引力理论.
This thesis involves two main fields of research:topological field theory and General Relativity. On topological field theory, using Duan's topological current theory, we address some topological aspects of phase singularities of wave field; on general relativity, we mainly investigate the quantum effects in thermodynamics of FRW universe.
Firstly, based on Duan's topological current theory, we discuss RS vortices in electro-magnetic field, coherence vortices in coherent optics, and vortex filaments in scroll wave. For RS vortices, we discuss their knot structure and derive the Hopf invariant for knotted RS vortices, we find that the Hopf invariant is just the total sum of all the self-linking numbers and linking numbers of knotted RS vortices. For coherence vortices, we obtain the exact expression of their topological coherence current, and based on this expression, we discuss the topological structures of coherence vortices, quantization of coherence flux, and knotted coherence vortices. In scroll wave system, we also investigate the Hopf in-variant of knotted vortex filaments, and point out that the Hopf invariant can be treated as the topological constraints of the scroll wave system.
Secondly, this thesis investigates the impact of generalized uncertainty principle and extended uncertainty principle on thermodynamics of FRW universe. After consider the effects of generalized uncertainty principle and extended uncertainty principle, the con-ventional entropy area relation on apparent horizon does not hold anymore, but acquires corrections. Moreover, according to first law of thermodynamics on apparent horizon, the Friedmann equations of FRW universe also should corrected. This implies that the generalized uncertainty principle and extended uncertainty principle can influence the dynamics of FRW universe.
Finally, we discuss the semi-classical and beyond semi-classical aspect of Hawking-like radiation as tunneling on apparent horizon of FRW universe. For semi-classical case, using canonically invariant tunneling probability and considering the imaginary parts of both the spatial and temporal contributions of the tunneling amplitude we recover the correct temperature T=1/2πrA; when we consider the self-gravity effect of tunneling particle, we obtain the relationship between the tunneling probability and the change in entropy of apparent horizon. On the quantum tunneling beyond semi-classical approxi- mation, we consider the tunneling of both scalar particles and fermions and obtain the corrected Hawking-like temperature of apparent horizon, with this corrected Hawking-like temperature, the explicit expressions of the corrected entropy of apparent horizon for var-ious gravity theories including Einstein gravity, Gauss-Bonnet gravity, Lovelock gravity, f(R) gravity and scalar-tensor gravity, are computed, we point out that this expression might be independent of gravity theories.
引文
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