膜世界中的物质场及其KK模式
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摘要
膜世界理论的提出,为解决层次问题、宇宙学常数等难题提供了新的思路,因而倍受关注.本文将着重研究膜世界中的物质场及其KK模式,研究目的是通过分析膜世界中物质场的KK模式,提供一种理论上验证膜世界理论的方法。
膜世界理论,先后经历了薄膜和厚膜的发展阶段.初期阶段的膜世界理论是薄膜理论,如ADD理论、RS理论.这两种理论,着重解决了层次问题,至此也揭开了膜世界理论发展的序幕.不过薄膜理论忽略了膜本身的厚度,是数学上的一种理想化,而现实的膜是应该有厚度的,因而厚膜理论得到了发展。在厚膜理论中,通过引入背景标量场,使得膜的能量密度沿额外维有了一定的展宽.不过因为没有限定标量势的具体形式,所以厚膜世界解具有多样性.
膜世界理论认为,标准模型中的粒子都被束缚在膜上,引力可以在膜之外的时空传播。因此,需要一种可以使得这些物质场自然地束缚在膜上的机制.同时,这样才能保证膜世界理论不与现实的世界相矛盾,这也是现实的膜世界模型首先要保证的。本文重点探讨一种使得物质场能够被束缚在膜上的机制,即局域化机制。
另外,在膜世界理论中,时空的维度是大于四维的,即有额外的空间维度存在。那么如何验证额外维存在的真实性也成了一个重要的科研问题。事实上,额外维理论的发展是早于膜世界理论的,不过膜世界理论的提出为验证额外维的真实存在性提供了一个重要的方法,就是上文提到的膜世界中物质场的局域化。
通过物质场的局域化方法,我们可以得到这些物质场的KK模式。所谓的KK模式,是指高维时空中的物质场在低维膜上的表现形式。因此,这种KK模式必然隐含了高维时空(额外维空间)的性质,并通过质量表现出来。于是,通过分析物质场的KK模式的质量谱,就可以了解到高维时空的性质。
在研究的过程中,我们还发现一些特殊的模式(零模)。利用一定的局域化条件,可以使得它们被束缚在膜上.被束缚在膜上的零模式,与额外维无关,不受额外维的影响,因此可以认为这些模式代表着膜上已知的物质场。这样就实现了自然地局域化膜上已知物质场的目的,由此,可以看出研究膜世界中的物质场及其KK模式的重要性.而KK模式的质量谱是依赖于不同的膜世界模型的。
本文重点讨论了三种有趣的膜世界模型中的物质场及其KK模式。首先,讨论了五维时空中的GRS-inspired膜世界中的情况,这种膜的特点是薄膜和厚膜同时存在.标量场和费米场的零模解都可以被束缚在该膜上,并且存在有质量的束缚态和共振态KK模式。不过矢量场却不能被束缚在这种膜世界中。于是,我们针对矢量场的局域化问题,提出额外维有限的厚膜世界模型(包括单膜和劈裂膜).在这中膜世界模型中,我们借助额外维的有限性使得矢量场也能够被局域化在膜上。最后,我们将膜世界理论从五维推广到任意维膜世界,即p—膜世界,求解出几种具有代表性的p—膜世界解。同时研究了这些模型中任意q—形式场的局域化问题,发现了膜的空间维数p对于不同的q—形式场局域化的影响,所得到的关于标量场(q=0)、矢量场(q=1)的局域化结果不仅与五维时空中的情况相吻合,还给出了一些新的预言。
但是,本文所采用的局域化机制只能从理论上验证膜世界理论的合理性(使得膜上已知的物质场自然地束缚在膜上)与真实性(额外维的存在性).而通过进一步地计算引力的KK模式对于牛顿势的修正,或规范场的KK模式对于库伦势的修正,就可以寻求实验上验证的可能性,这也是我们的下一步工作。
The brane world theory has received more and more attention, as it opened up new avenues to explain some questions, such as the hierarchy problem, and the cosmological constant problem. In this thesis, we focus on the matter fields and their KK modes in brane world, and the aim is to find a way to theoretically prove the brane world theory, by analyzing the KK modes of matter fields.
In the development of the brane world theory, there are two types of brane models proposed, i.e., the thin brane and the thick brane models. The famous thin brane models are the ADD and the RS. These two theories mainly tried to solve the hierarchy problem, and they open the prelude of the brane theory. However, the thickness of the brane was ignored in these theories, and thin brane was merely a mathematical idea. So the thick brane theory was built up. In the thick brane theory, the energy of the system can spread along the extra dimension. And the thick brane is generated by a back-ground scalar field. Due to the freedom in choosing scalar potential, there are various solutions of thick branes.
In the brane world theory, the particles in the Standard Model are supposed to be confined on the brane, while gravity can propagate freely outside of the brane. Thus, we need a mechanism to localize the matter fields naturally, which can also make the brane world be consistent with the actual world. And this is to be sure for a realized brane world model. In this thesis, we study one of the mechanisms that can confine the matter fields on the brane, namely, the localization.
On the other hand, in the brane world theory, the dimension of the space-time is larger than four, i.e., there are extra dimensions. Then how to prove the existence of these extra dimensions becomes an important problem. In fact, the development of the extra dimension theory is earlier than the brane world theory, but the latter theory provides a useful method to prove the former, and this method is the localization mechanism mentioned above.
By the mechanism of localization, we can obtain the KK modes of matters. What are the KK modes? They are the matter fields in higher dimensional space-time, which are seen in our four-dimensional world. So they must reflect the property of higher di-mensional space-time (extra dimensional space-time) through their masses. By analyzing the mass spectrum of the KK modes, we can know the properties of higher dimensional space-time.
In the investigation, we also find some special KK modes (zero modes). They can be localized on the brane with some conditions. And the localized zero modes are in-dependent and free from the extra dimensions. Thus the localized zero modes might be regarded as the matter fields on the brane. And the aim of localizing the known matter field on the brane is realized. So it can be seen the importance of studying the localization of matter fields and their KK modes on the brane. The mass spectrum of the KK modes is determined by the type of brane models.
We mainly investigate the matter fields and their KK modes in three interesting types of brane models in this thesis. Firstly, we consider the GRS-inspired brane world. In this model, there are both thin and thick branes at the same time. The zero modes of the scalar and fermion fields can be localized on the brane, and some massive bound or resonant KK modes appear. However, the vector field cannot be localized on the brane. In order to solve this problem, we propose the thick brane model with finite extra dimension (including the single brane model and the split brane one). Because of the finiteness of the extra dimension, the vector can be trapped on the brane. At last, we generalize the brane world in five-dimensional space-time to any dimension, i.e., the p-brane. We get the solutions of some representative types of p-brane, and investigate the localization of q-form fields in this brane. The effect of the spatial dimensions p in the localization of any form field is found, and this result not only covers the cases in five dimensional space-time for the scalar field (q=0) and the vector field (q=1), but also gives some new predictions for the higher dimensional space-time.
Although the localization mechanism in this thesis is ideal and preliminary, it also theoretically offers a way to prove the rationality (to localize the known matter on the brane) and the truth (the existence of the extra dimensions) of the brane theory. The existence of extra dimension(s) might be proved in future experiments on the correction to Newton potential from KK gravitons, and the correction to Coulomb potential from KK gauge bosons. And this is our further work.
膜世界理论,先后经历了薄膜和厚膜的发展阶段.初期阶段的膜世界理论是薄膜理论,如ADD理论、RS理论.这两种理论,着重解决了层次问题,至此也揭开了膜世界理论发展的序幕.不过薄膜理论忽略了膜本身的厚度,是数学上的一种理想化,而现实的膜是应该有厚度的,因而厚膜理论得到了发展。在厚膜理论中,通过引入背景标量场,使得膜的能量密度沿额外维有了一定的展宽.不过因为没有限定标量势的具体形式,所以厚膜世界解具有多样性.
膜世界理论认为,标准模型中的粒子都被束缚在膜上,引力可以在膜之外的时空传播。因此,需要一种可以使得这些物质场自然地束缚在膜上的机制.同时,这样才能保证膜世界理论不与现实的世界相矛盾,这也是现实的膜世界模型首先要保证的。本文重点探讨一种使得物质场能够被束缚在膜上的机制,即局域化机制。
另外,在膜世界理论中,时空的维度是大于四维的,即有额外的空间维度存在。那么如何验证额外维存在的真实性也成了一个重要的科研问题。事实上,额外维理论的发展是早于膜世界理论的,不过膜世界理论的提出为验证额外维的真实存在性提供了一个重要的方法,就是上文提到的膜世界中物质场的局域化。
通过物质场的局域化方法,我们可以得到这些物质场的KK模式。所谓的KK模式,是指高维时空中的物质场在低维膜上的表现形式。因此,这种KK模式必然隐含了高维时空(额外维空间)的性质,并通过质量表现出来。于是,通过分析物质场的KK模式的质量谱,就可以了解到高维时空的性质。
在研究的过程中,我们还发现一些特殊的模式(零模)。利用一定的局域化条件,可以使得它们被束缚在膜上.被束缚在膜上的零模式,与额外维无关,不受额外维的影响,因此可以认为这些模式代表着膜上已知的物质场。这样就实现了自然地局域化膜上已知物质场的目的,由此,可以看出研究膜世界中的物质场及其KK模式的重要性.而KK模式的质量谱是依赖于不同的膜世界模型的。
本文重点讨论了三种有趣的膜世界模型中的物质场及其KK模式。首先,讨论了五维时空中的GRS-inspired膜世界中的情况,这种膜的特点是薄膜和厚膜同时存在.标量场和费米场的零模解都可以被束缚在该膜上,并且存在有质量的束缚态和共振态KK模式。不过矢量场却不能被束缚在这种膜世界中。于是,我们针对矢量场的局域化问题,提出额外维有限的厚膜世界模型(包括单膜和劈裂膜).在这中膜世界模型中,我们借助额外维的有限性使得矢量场也能够被局域化在膜上。最后,我们将膜世界理论从五维推广到任意维膜世界,即p—膜世界,求解出几种具有代表性的p—膜世界解。同时研究了这些模型中任意q—形式场的局域化问题,发现了膜的空间维数p对于不同的q—形式场局域化的影响,所得到的关于标量场(q=0)、矢量场(q=1)的局域化结果不仅与五维时空中的情况相吻合,还给出了一些新的预言。
但是,本文所采用的局域化机制只能从理论上验证膜世界理论的合理性(使得膜上已知的物质场自然地束缚在膜上)与真实性(额外维的存在性).而通过进一步地计算引力的KK模式对于牛顿势的修正,或规范场的KK模式对于库伦势的修正,就可以寻求实验上验证的可能性,这也是我们的下一步工作。
The brane world theory has received more and more attention, as it opened up new avenues to explain some questions, such as the hierarchy problem, and the cosmological constant problem. In this thesis, we focus on the matter fields and their KK modes in brane world, and the aim is to find a way to theoretically prove the brane world theory, by analyzing the KK modes of matter fields.
In the development of the brane world theory, there are two types of brane models proposed, i.e., the thin brane and the thick brane models. The famous thin brane models are the ADD and the RS. These two theories mainly tried to solve the hierarchy problem, and they open the prelude of the brane theory. However, the thickness of the brane was ignored in these theories, and thin brane was merely a mathematical idea. So the thick brane theory was built up. In the thick brane theory, the energy of the system can spread along the extra dimension. And the thick brane is generated by a back-ground scalar field. Due to the freedom in choosing scalar potential, there are various solutions of thick branes.
In the brane world theory, the particles in the Standard Model are supposed to be confined on the brane, while gravity can propagate freely outside of the brane. Thus, we need a mechanism to localize the matter fields naturally, which can also make the brane world be consistent with the actual world. And this is to be sure for a realized brane world model. In this thesis, we study one of the mechanisms that can confine the matter fields on the brane, namely, the localization.
On the other hand, in the brane world theory, the dimension of the space-time is larger than four, i.e., there are extra dimensions. Then how to prove the existence of these extra dimensions becomes an important problem. In fact, the development of the extra dimension theory is earlier than the brane world theory, but the latter theory provides a useful method to prove the former, and this method is the localization mechanism mentioned above.
By the mechanism of localization, we can obtain the KK modes of matters. What are the KK modes? They are the matter fields in higher dimensional space-time, which are seen in our four-dimensional world. So they must reflect the property of higher di-mensional space-time (extra dimensional space-time) through their masses. By analyzing the mass spectrum of the KK modes, we can know the properties of higher dimensional space-time.
In the investigation, we also find some special KK modes (zero modes). They can be localized on the brane with some conditions. And the localized zero modes are in-dependent and free from the extra dimensions. Thus the localized zero modes might be regarded as the matter fields on the brane. And the aim of localizing the known matter field on the brane is realized. So it can be seen the importance of studying the localization of matter fields and their KK modes on the brane. The mass spectrum of the KK modes is determined by the type of brane models.
We mainly investigate the matter fields and their KK modes in three interesting types of brane models in this thesis. Firstly, we consider the GRS-inspired brane world. In this model, there are both thin and thick branes at the same time. The zero modes of the scalar and fermion fields can be localized on the brane, and some massive bound or resonant KK modes appear. However, the vector field cannot be localized on the brane. In order to solve this problem, we propose the thick brane model with finite extra dimension (including the single brane model and the split brane one). Because of the finiteness of the extra dimension, the vector can be trapped on the brane. At last, we generalize the brane world in five-dimensional space-time to any dimension, i.e., the p-brane. We get the solutions of some representative types of p-brane, and investigate the localization of q-form fields in this brane. The effect of the spatial dimensions p in the localization of any form field is found, and this result not only covers the cases in five dimensional space-time for the scalar field (q=0) and the vector field (q=1), but also gives some new predictions for the higher dimensional space-time.
Although the localization mechanism in this thesis is ideal and preliminary, it also theoretically offers a way to prove the rationality (to localize the known matter on the brane) and the truth (the existence of the extra dimensions) of the brane theory. The existence of extra dimension(s) might be proved in future experiments on the correction to Newton potential from KK gravitons, and the correction to Coulomb potential from KK gauge bosons. And this is our further work.
引文
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