龐加萊─史奈德相對論架構下的古典與量子力學

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姓名

李弘義(Hung-Yi Lee) 查詢紙本館藏

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物理學系

論文名稱

龐加萊─史奈德相對論架構下的古典與量子力學
(Classical and Quantum Mechanics in Poincaré-Snyder Relativity)

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摘要(中)

根據最近提出的量子相對論，即一個關於「量子時空」的相對論，我們提出龐加萊-史奈德相對論作為介於一般熟悉的愛因斯坦、伽利略相對論與量子相對論之間的相對論。由於其代數結構的優越性，類伽利略代數的數學性質在建構龐加萊-史奈德的動力學時皆可被使用。我們寫下古典力學和量子力學的正則形式並討論一些它突出的特色和物理意涵。我們特別研究自由粒子的正則實現和一個經過簡化的群量子化過程。在龐加萊-史奈德架構下的正則動力學最重要的結果是勞侖茲變換可以被看成一種正則變換。群量子化的圖像指出不同於愛因斯坦代數，龐加萊-史奈德代數允許非平凡的U(1) 中心荷，正則對易關係從中自然地浮現出來。本篇論文是以龐加萊-史奈德相對論的架構研究量子相對論中關於σ座標的物理的第一步嘗試。未來的研究可能帶出實驗上可檢驗的結果。

摘要(英)

Based on the recently proposed Quantum Relativity theory, a relativity of the `quantum spacetime’’, we introduce the Poincaré-Snyder relativity as an intermediate relativity between the familiar Einstein/Galilean relativity and the quantum one. With the merit of its algebraic structure, mathematical properties of the Galilean-type algebra can be used while formulating dynamics in the Poincaré-Snyder relativity. We write down the canonical formulation of classical and quantum mechanics, discussing some of its salient features and plausible physics implications. In particular, we study the canonical realization of the free particle case and a simplified version of group quantization. The most significant result of canonical dynamics in Poincaré-Snyder relativity is that Lorentz transformation can be treated as a canonical transformation. The group quantization picture indicates that unlike the Einstein one, Poincaré-Snyder algebra admits a non-trivial U(1) central charge from which the canonical commutation relation naturally emerges. The present thesis is a first attempt to study the physic of sigma coordinate of the Quantum Relativity theory in the more practical Poincaré-Snyder setting. Future investigations may lead to experimental verifiable results.

關鍵字(中)

★ 量子相對論
★ 量子時空
★ 龐加萊─史奈德相對論

關鍵字(英)

★ Quantum Relativity
★ Quantum Spacetime
★ Poincaré-Snyder Relativity

論文目次

1 Introduction 1
2 Background 4
2.1 Planck Scale . . . . . . 4
2.2 Contraction and Deformation of Lie Algebra . . . . . . 8
2.2.1 Contraction . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Deformation . . . . . . . . . . . . . . . . . . . . 12
2.3 The Quantum Relativity Framework . . . . . . . . . . 16
3 Classical Mechanics in Poincaré-Snyder Relativity 22
3.1 The Poincaré-Snyder Relativity . . . . . . . . . . . 22
3.2 Canonical Formalism with Poincaré-Snyder relativity 26
3.2.1 sigma-Lagrangian and sigma-Hamiltonian .. . . . . . 26
3.2.2 Canonical Transformations and Lorentz Transformation28
3.2.3 Free Particle sigma-mechanics as an Einstein Limit..33
3.2.4 Two Particles with Interactions . . . . . . . . . . 35
4 Quantum Mechanics in Poincaré-Snyder Relativity 36
4.1 Hilbert Space Formalism of Quantum Mechanics . . . . 37
4.2 Canonical Quantization . . . . . . . . . . . . . . . 38
4.3 Quantization via Group Theoretical Argument . . . . . 40
4.3.1 Central Extension of Poincaré-Snyder Algebra. . . 40
4.3.2 Representation of the Extended Algebra . . . . . . 44
4.4 Comparison and Discussion . . . . . . . . . . . . . . 47
5 Conclusion and Outlook 49
Bibliography 52

參考文獻

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指導教授

江祖永(Otto C.W. Kong)

審核日期

2010-7-26

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