Topological Aspects Of Phase Singularities In Wave Field And Quantum Effects In Thermodynamics Of Friedmann-Robertson-Walker Universe

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=1/2Ï€rA; 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.