基于段氏拓扑流理论的极早期宇宙演化理论和反对称背景场存在时的开弦理论
摘要
本论文利用段一士教授提出的规范势分解理论和φ-映射拓扑流理论，详细研究了宇宙学中宇宙弦的拓扑性质，宇宙弦的自旋与额外维空间挠率的关系，宇宙弦、暴涨在宇宙极早期演化中的作用，宇宙极早期的演化图景，和平面物理学中anyon的拓扑特性等问题。同时还研究了弦理论中Bose开弦、超对称开弦边界条件和时空坐标非对易的关系问题。
首先，通过对于宇宙弦形成机制和拓扑性质的分析，我们给出，宇宙弦是在宇宙演化的极早期Higgs复标量场对称性自发破缺相变之后形成的，它的拓扑结构、拓扑荷等可以由退化到真空期望值的Higgs场的分布来描述。结合额外维空间存在时的4-维时空有效理论，我们指出额外维空间的挠率可以用一个U(1)规范场来描述，这个规范场在宇宙弦形成时与Higgs场发生有效的耦合而禁闭到宇宙弦的核中，并进而作为一种“源”使宇宙弦产生自旋。我们同时分析指出，相变之后产生宇宙弦的Higgs复标量场在相变之前具有整体的对称性，它可以作为暴涨子场推动宇宙空间在极早期的暴涨。进而结合对宇宙弦、空间挠率和暴涨的所有讨论，我们构建了一个宇宙在极早期的演化模型。
其次，作为拓扑流理论的一个简单应用，我们研究了平面物理学中anyon的一些拓扑性质，并指出anyon的基本荷是有结构的，它可以由系统的两个特征耦合常数完全确定，进而不同anyon之间Aharonov-Bohm型相互作用的相因子与理论的Chem-Simons耦合常数成正比。
我们还研究了弦理论中开弦与时空结构的一些关系问题，指出当时空中存在一个与度规场地位相平等的2-阶反对称背景场时，开弦原始的Neumann和Dirichlet边界条件将发生混合；混合的结果，对于Bose开弦，其端点的时空坐标，无论在与D-膜平行还是垂直的维度方向上，都变得非对易；而对于超对称开弦，其端点处的整个相空间(坐标和动量)都变得非对易。
In this thesis, using the gauge potential decomposition theory and theφ-mapping topological current theory which were originally proposed by Prof.Duan, we have studied in detail the topological properties of the cosmic strings,the relationship between the spin of the cosmic strings and the torsion of theextra dimensional spaces, the roles of the cosmic strings and the inflation inthe very early evolution of the universe, the evolution picture of the very earlyuniverse in cosmology, and the topological properties of the anyons in planarphysics. Also we have studied the relationship between the boundary conditionsof the bosonic open string and the open superstring and the noncommutativityof the spacetime coordinates in string theory.
Firstly, through the analysis of the formation mechanism and topologicalproperties of the cosmic strings, we have showed that, the cosmic stringsare formed after the automatically symmetric broken phase transition of thecomplex scalar Higgs field in the very early history of the universe, their topologicalstructures, topological charges etc. can all be described in terms of theHiggs field which had reached its expectation values. Combining with the 4-dimensional spacetime effective theory when there exist extra dimensions, wehave studied that the torsion of the extra dimensional space can be described bya U(1) gauge potential, and this gauge potential would get coupled to the Higgsfield and get trapped into the cosmic strings when the cosmic strings form, andthen it would act as some kind of source of the spins of the cosmic strings.Meanwhile, we have discussed that before the phase transition the symmetryof the Higgs field is global, and the field could act as the inflaton to derive theuniverse space to inflate. Then combining the whole discussion of the cosmicstrings, the torsion and the inflation, we have constructed a brief evolution pictureof the very early universe.
Secondly, as a simple application of the topological current theory, wehave studied some topological properties of the anyons in planar physics. I haveshowed that, the basic charge of the anyons actually has structures, namely it istotally decided by the two coupling constants of the system; also the phase factorof the Aharonov-Bohm type interactions among the anyons is proportionalto the Chern-Simons coupling constant.
Also we have studied the relationship of the open string and the spacetimestructure in string theory, we have discussed that, when there exists an antisym-metric2-order background field besides the metric field, the original Neumannand Dirichlet boundary conditions of the open string would mix; and as a result,for bosonic open string, the spacetime coordinates of its endpoints become noncommutative,in the directions both parallel and perpendicular to the D-branes,and for open superstring, the whole phase space, namely the space of coordinatesand momentum, of the string endpoints becomes noncommutative.
首先，通过对于宇宙弦形成机制和拓扑性质的分析，我们给出，宇宙弦是在宇宙演化的极早期Higgs复标量场对称性自发破缺相变之后形成的，它的拓扑结构、拓扑荷等可以由退化到真空期望值的Higgs场的分布来描述。结合额外维空间存在时的4-维时空有效理论，我们指出额外维空间的挠率可以用一个U(1)规范场来描述，这个规范场在宇宙弦形成时与Higgs场发生有效的耦合而禁闭到宇宙弦的核中，并进而作为一种“源”使宇宙弦产生自旋。我们同时分析指出，相变之后产生宇宙弦的Higgs复标量场在相变之前具有整体的对称性，它可以作为暴涨子场推动宇宙空间在极早期的暴涨。进而结合对宇宙弦、空间挠率和暴涨的所有讨论，我们构建了一个宇宙在极早期的演化模型。
其次，作为拓扑流理论的一个简单应用，我们研究了平面物理学中anyon的一些拓扑性质，并指出anyon的基本荷是有结构的，它可以由系统的两个特征耦合常数完全确定，进而不同anyon之间Aharonov-Bohm型相互作用的相因子与理论的Chem-Simons耦合常数成正比。
我们还研究了弦理论中开弦与时空结构的一些关系问题，指出当时空中存在一个与度规场地位相平等的2-阶反对称背景场时，开弦原始的Neumann和Dirichlet边界条件将发生混合；混合的结果，对于Bose开弦，其端点的时空坐标，无论在与D-膜平行还是垂直的维度方向上，都变得非对易；而对于超对称开弦，其端点处的整个相空间(坐标和动量)都变得非对易。
In this thesis, using the gauge potential decomposition theory and theφ-mapping topological current theory which were originally proposed by Prof.Duan, we have studied in detail the topological properties of the cosmic strings,the relationship between the spin of the cosmic strings and the torsion of theextra dimensional spaces, the roles of the cosmic strings and the inflation inthe very early evolution of the universe, the evolution picture of the very earlyuniverse in cosmology, and the topological properties of the anyons in planarphysics. Also we have studied the relationship between the boundary conditionsof the bosonic open string and the open superstring and the noncommutativityof the spacetime coordinates in string theory.
Firstly, through the analysis of the formation mechanism and topologicalproperties of the cosmic strings, we have showed that, the cosmic stringsare formed after the automatically symmetric broken phase transition of thecomplex scalar Higgs field in the very early history of the universe, their topologicalstructures, topological charges etc. can all be described in terms of theHiggs field which had reached its expectation values. Combining with the 4-dimensional spacetime effective theory when there exist extra dimensions, wehave studied that the torsion of the extra dimensional space can be described bya U(1) gauge potential, and this gauge potential would get coupled to the Higgsfield and get trapped into the cosmic strings when the cosmic strings form, andthen it would act as some kind of source of the spins of the cosmic strings.Meanwhile, we have discussed that before the phase transition the symmetryof the Higgs field is global, and the field could act as the inflaton to derive theuniverse space to inflate. Then combining the whole discussion of the cosmicstrings, the torsion and the inflation, we have constructed a brief evolution pictureof the very early universe.
Secondly, as a simple application of the topological current theory, wehave studied some topological properties of the anyons in planar physics. I haveshowed that, the basic charge of the anyons actually has structures, namely it istotally decided by the two coupling constants of the system; also the phase factorof the Aharonov-Bohm type interactions among the anyons is proportionalto the Chern-Simons coupling constant.
Also we have studied the relationship of the open string and the spacetimestructure in string theory, we have discussed that, when there exists an antisym-metric2-order background field besides the metric field, the original Neumannand Dirichlet boundary conditions of the open string would mix; and as a result,for bosonic open string, the spacetime coordinates of its endpoints become noncommutative,in the directions both parallel and perpendicular to the D-branes,and for open superstring, the whole phase space, namely the space of coordinatesand momentum, of the string endpoints becomes noncommutative.
引文
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