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研究生:卓岱寧
研究生(外文):Dai-Ning Cho
論文名稱(外文):Lie Algebra Contraction and Relativity Symmetries
指導教授:江祖永
指導教授(外文):Otto C.W. Kong
學位類別:碩士
校院名稱:國立中央大學
系所名稱:物理研究所
學門:自然科學學門
學類:物理學類
論文種類:學術論文
論文出版年:2011
畢業學年度:100
語文別:英文
論文頁數:95
中文關鍵詞:李代數
外文關鍵詞:Cayley-Klein algebrarelativity symmetrycontractionLie algebra
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  • 被引用被引用:0
  • 點閱點閱:201
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  • 下載下載:3
  • 收藏至我的研究室書目清單書目收藏:0
變形(deformation)作為李代數的數學工具,於近來被應用於推導量子相對論(Quantum Relativity)及其對應之架構;而壓縮(contraction)是做為另外一個李代數之下的數學工具,乃為變形的相反運算。在這篇論文中,我們著重於壓縮的基本概念以及數種主要運算過程,並加以適當的範例作為演示。我們首先將壓縮應用在目前科學界熟悉的五維時空,並得到許多作為可能容納動力學的時空。再來我們將壓縮應用於量子相對論的架構,並得到許多與前述時空相呼應的可能容納動力學的幾何結構。
Deformation as a mathematical tool is used in deducing the recently developed Quantum Relativity and its framework, whereas contraction is the operation opposite to against deformation. In this dissertation we focus on and illustrate the idea of contraction and discuss several general approaches to contraction in detail. We first use contraction as an illustration in deducing the algebras of the so-called possible kinematical groups under the framework of the SO(1,4) and SO(2,3) groups. Then we apply contraction under the framework of Quantum Relativity (SO(2,4)) and examine the algebras we obtained that may be of the kinematical groups in its framework.
1 Introduction 1
2 Background 4
3 Contraction 11
3.1 General Formalism of Contraction 11
3.1.1 Wigner-Ino ̈nu ̈ Contraction 11
3.1.2 Generalized Wigner-Inonu Contraction 14
3.1.3 Graded Contraction 16
3.2 Bacry and Leblond’s Analysis of Possible Kinematics 20
3.2.1 The Method in the Paper 21
3.2.2 Description of SO(5) Contraction 26
3.3 Graded Contraction Focused on the Cayley-Klein Algebra 44
3.3.1 Grading Process of SO(N+1) Lie Algebra by Using Z_2^(⊗N) 44
3.3.2 Graded Contraction by Using the Cayley-Klein Lie Algebra 47
3.3.3 Description of the SO(5) Case 59
3.3.4 Connection of Graded and Generalized Contraction 64
4 Contraction in the Quantum Relativity Framework 70
4.1 Wigner-Inonu Contraction of SO(2,4) 70
4.2 One Step Further 80
4.3 Graded Contraction Focused on the Caley-Klein Algebra of SO(6) 85
5 Conclusion and Outlook 92
Bibliography 93
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[20] M. Planck, “Uber irreversible Strahlungsvorgange. Funfte Mitteilung,” Koniglich Preussiche Akademie der Wissenschaften (Berlin). Sitzungsberichte p. 440 (1899).
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[22] Robert Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications (John Wiley and Sons. Inc., 1941).
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