弦論的對稱及其高能實現｜國立清華大學博碩士論文庫

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研究生：	葉啟賢 Yeh, Chi-Hsien
論文名稱：	弦論的對稱及其高能實現 Stringy Symmetries and Their High-Energy Realizations
指導教授：	張敬民 Kingman Cheung 詹傳宗 Chuan-Tsung Chan
口試委員:
學位類別：	碩士 Master
系所名稱：	理學院 - 物理學系 Department of Physics
論文出版年：	2006
畢業學年度：	94
語文別：	英文
論文頁數：	98
中文關鍵詞：	零模態、規範對稱、五點散射振福、Ward恆等式、高能極限
外文關鍵詞：	zero-norm states, gauge symmetry, 5-point scattering amplitudes, Ward identities, High-energy limit
相關次數：	點閱：3 下載：0
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摘要
本論文主要研究弦論的規範變換與零模態，以及弦的散射振幅在高能極限下所具有的對稱性質。零模態具有豐富的性質，甚至能夠決定弦的散射振幅在高能極限下所具有的對稱性。本文主要由四個部分組成。
首先，本文解釋了如何計算在玻色開弦理論的物理態的自由度。並且本文也完整計算了所有質量平方等於8的玻色開弦零模態，以及質量平方等於8的玻色開弦的物理態的自由度，這些計算提供了一個明確的例子檢驗Polyakov書中[1]計算自由度的公式。而且本文給了一個具體的答案顯示出質量平方等於8的玻色開弦零模態確實具有簡併。
第二部分，主要顯現質量平方等於6的玻色開弦物理震盪態（正模態）的規範變化。所有物理上有等效結果但具有不同表現形式的正模態之間可藉由零模態來做聯結，這就是弦的規範變化。
第三部分，計算了1個具有質量的玻色開弦與4個迅子的散射振幅，並檢驗了此散射振幅滿足Ward恆等式。本文也發現Ward恆等式與超幾何函數的遞迴關係有關連。
最後，在特別的運動學變數選取下（高能極限），本文驗證了部分的五點散射振幅具有與四點散射振幅相同的比例關係，這些比例關係顯現了弦在高能極限下所具有的對稱性質，這結果顯現了弦在高能極限下所具有的對稱性質也許是獨立於散射過程的。

Abstract
This paper consists of four parts. Firstly, we explain how to calculate the degrees of
freedom in bosonic open string theory. We then calculate the zero-norm states at mass
level m^2 = 8. This calculation provides a concrete example for the formula of count-
ing between degrees of freedom in the book [1]. Moreover, we give an explicit answer to show the existence of zero-norm state degeneracy at m^2 = 8.
Secondly, we show the gauge transformations on the positive-norm states at mass level m^2 = 6. Thus, explaining all the
equivalent representations of positive-norm states are indeed differ by zero-norm states.
Thirdly, we check the stringy Ward identities for one massive tensor scattering with 4 tachyon states, and we find an interesting connection between 5-point Ward identities
and recursion relations of hypergeometric functions. Finally, by taking special kinemat-
ical limits, we verify the linear relations among these 5-point amplitudes. These linear
relations do not depend upon the scattering processes.

Table of Contents
Introduction . . . . . . . . . . . . . . . . . . . . . 1
1 Gauge symmetry and zero-norm states . . . . . . . . . . . . . . . . . . . . 3
1.1 gauge symmetry of electromagnetism . . . . . . . . . . . . . . . . . 3
1.2 general properties of zero-norm states . . . . . . . . . . . . . . . . . 7
1.3 gauge symmetry of string theory . . . . . . . . . . . . . . . . . . . 8
2 Scattering amplitudes and symmetry . . . . . . . . . . . . . . . . . . . . . 14
2.1 An example from hadronic physics . . . . . . . . . . . . . . . . . . 15
2.2 Ward identities and high-energy limits of scattering amplitudes in
bosonic open string theory . . . . . . . . . . . . . . . . . . . . . . . 17
Degeneracy among two types of zero-norm states 21
1 Degrees of freedom for physical states in bosonic open string theory . . . . 22
2 Example of degenerate zero-norm states . . . . . . . . . . . . . . . . . . . 24
2.1 Degrees of freedom of physical states at m^2=8 . . . . . . . . . 24
2.2 Degeneracy of m^2 = 8 zero-norm states . . . . . . . . . . . . . . . . 27
Gauge transformations on positive-norm states 32
1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2 Examples of m^2 = 6 positive-norm states . . . . . . . . . . . . . . . . . . . 37
2.1 General parametrization of physical states at
m^2 = 6 . . . . . . . . 37
2.2 Irreducible decomposition of a totally symmetric spin-four tensor . 38
2.3 Equivalent representations of singlet positive-norm states . . . . . . 39
2.4 Equivalent representations of spin-two positive-norm states . . . . . 45
Stringy Ward identities for 5-point amplitudes 48
1 Five-tachyon scattering amplitude . . . . . . . . . . . . . . . . . . . . . . . 49
2 Five-point scattering amplitudes and Ward Identities at m^2 = 0 . . . . . . 50
3 Five-point scattering amplitudes and Ward Identities at m^2 = 2 . . . . . . 51
4 Five-point scattering amplitudes and Ward Identities at m^2 = 4 . . . . . . 55
5 Ward identities and Recursion relations . . . . . . . . . . . . . . . . . . . . 61
High-energy limits and linear relations for 5-point amplitudes 63
1 Saddle-point approximation of 5-point stringy amplitudes . . . . . . . . . . 65
2 The ratios of 5-point scattering amplitudes in the high-energy limit . . . . 68
Summary and Conclusions 74
Appendix 75
A Notations and Conventions 76
B On the equivalence between Nambu-Goto and Polyakov actions of bosonic
string theory 77
C Spectrum of m^2 = 8 zero-norm states 80
D Spectrum of physical states at m^2 = 6 91
D.1 Positive-norm states of m^2 = 6 . . . . . . .  . 91
D.2 Zero-norm state of m^2 =6 . . . . . . . . . . . . 92
References 94

                                

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