複數勞倫茲對稱

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姓名

李承軒(Cheng-Xuan Li) 查詢紙本館藏

畢業系所

物理學系

論文名稱

複數勞倫茲對稱
(Complexified Lorentz symmetry)

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摘要(中)

已經有許多理論預測了超光速粒子以及超光速運動的相對論。我們在這篇文章中提出了複數勞倫茲對稱 (Complexified Lorentz symmetry)，或稱作複數勞倫茲群 (Complex Lorentz group)，我們將勞倫茲群以及它的群表示 (Representation) 複數化 (Complexify) 後，從三加一維的實數時空變成三加一維的複數向量空間，然後證明它是不可約化(irreducible)。然後我們我們使用 SL(4,C) 的餘空間( Coset space)去找到複數的閔可夫斯基時空 (Complex Minkowski space) 以及複數勞倫茲群如何作用在上面。最後我們定義了複數閔可夫斯基時空的均速運動並將之視為一條複數線並且分類。

摘要(英)

There are many theory about faster than light particles, and the new transformation that apply for relative velocities greater than the speed of light. We propose here a complexified Lorentz symmetry, or complex Lorentz group, we complexify Lorentz group and representation from real Lorentz group and 3+1 real space time, and find it is irreducible in 3+1 complex vector space. And we use the coset space of SL(4,C) to find complex Minkowski space and how complex Lorentz group acting on it. Finally we define the uniform motion in complex Minkowski space as complex line and classify it.

關鍵字(中)

★ 勞倫茲群
★ 勞倫茲對稱
★ 複數化
★ 複數閔可夫斯基時空

關鍵字(英)

★ Lorentz symmetry
★ Lorentz group
★ Complexification
★ Complex Minkowski space

論文目次

1 Introduction 1
2 Background 4
2.1 Complex vector space . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.2 Metric tensor . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Lie group and Lie algebra . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Lie algebra and representation . . . . . . . . . . . . . . 7
2.2.2 Lie group . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.3 Real form and real Lie algebra . . . . . . . . . . . . . . 8
2.2.4 Representation of so(1; 3) . . . . . . . . . . . . . . . . 8
3 Complexication of Lie algebra and representation 10
3.1 Real representation of real Lie algebra . . . . . . . . . . . . . 10
3.2 Complexication and realication of Lie algebra . . . . . . . . 11
3.3 Irreducible real representation . . . . . . . . . . . . . . . . . . 13
3.4 Real representation of su(2) . . . . . . . . . . . . . . . . . . . 15
4 Complex Lorentz group through complexication 20
4.1 Complexication of so(1; 3) . . . . . . . . . . . . . . . . . . . 20
4.1.1 Irreducible representation . . . . . . . . . . . . . . . . 20
4.1.2 so(1; 3)C = sl(2;C) sl(2;C) . . . . . . . . . . . . . . . 22
4.2 Spinor representation . . . . . . . . . . . . . . . . . . . . . . . 24
4.3 Complex Lorentz group . . . . . . . . . . . . . . . . . . . . . . 25
5 Complex Minkowski space as a coset space 26
5.1 Coset space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.1.1 Denition and properties . . . . . . . . . . . . . . . . . 26
5.2 Minkowski space . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.2.1 Complex Minkowski space . . . . . . . . . . . . . . . . 27
5.2.2 Action on complex Minkowski sapce . . . . . . . . . . 28
5.2.3 Innitesimal approach . . . . . . . . . . . . . . . . . . 28
5.2.4 Complexied Lorentz symmetry through coset space . 30
5.3 U(1) transformation . . . . . . . . . . . . . . . . . . . . . . . 30
6 Complex Minkowski space 33
6.1 Timelike, spacelike, and lightlike . . . . . . . . . . . . . . . . . 33
6.2 Complex boost and complex rotation . . . . . . . . . . . . . . 34
6.3 Complex four velocity and proper time . . . . . . . . . . . . . 35
6.3.1 Proper time . . . . . . . . . . . . . . . . . . . . . . . . 35
6.3.2 Complex line . . . . . . . . . . . . . . . . . . . . . . . 35
6.3.3 Analytic . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6.3.4 Complex four velocity . . . . . . . . . . . . . . . . . . 37
6.4 Case in real subspace . . . . . . . . . . . . . . . . . . . . . . . 38
6.5 Subliminal and superluminal motion in complex space time . . 44
7 Conclusion and outlook 46
A 47
Bibliography 49

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53

指導教授

江祖永(Otto C.W. Kong)

審核日期

2016-1-12

推文