重力場中準局域角動量的旋子表述｜國立中央大學博碩士論文系統

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研究生：	李曉菁 Siao-Jing Li
論文名稱：	重力場中準局域角動量的旋子表述 Spinor Formulations for the Quasilocal Angular Momentum of Gravitational Fields
指導教授：	聶斯特 James M. Nester
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
畢業學年度：	94
語文別：	英文
論文頁數：	80
中文關鍵詞：	準局域量、Witten 的正能量證明、旋子、QSL 、角動量、質心距、Clifford 代數、Clifform 、differential form 、spinor-valued form 、Witten''s Hamiltonian 、重力場
外文關鍵詞：	Witten''s positive energy proof, Witten''s Hamiltonian, quasilocal quantities, Clifform, spinor formulation, quasilocal angular momentum, differential form, clifford algebra, spinor-valued form, quadratic spinor Lagrangian, QSL, center-of-mass moment, spinor parameterization, gravitational fields, covariant derivative, spinor-curvature identity
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Witten 的Hamiltonian 和QSL方法提供了兩個漸進平坦時空重力系統能量動量的局域化和重立場正能量的證明, 但關於角動量和質心距則一直尚未被詳細地探討. 本論文先研究是否這兩種 spinor 表示式也能提供準局域角動量和質心距的表示式.
首先由於角動量是一個 pseudovector, 我們嘗試將Witten Hamiltonian 中位移向量的spinor 向量參數化改成spinor pesudovector 的參數化. 但這樣將導致spinor的發散, 因此我們考慮一個新的由四項二次covaraint 微分構成的 Hamiltonian. 經過仔細檢查後, 我們發現它不能提供一個角動量的表示式. 我們還個別研究由其中兩個不同的兩項構成的 Hamiltonian,但都失敗了. 我們希望能夠找到某些spinor gauge 條件或加上某些項使得 Witten 的方法能成功.
至於 QSL 方法, 我們發現選擇一個轉動的位移參數化及可得到準局域角動量的表示式. 這是在本論文我們所有測試的表示式中唯一可以成功用來描述重立場角動量的方法.
一但我們有角動量的表示式, 根據過去的研究, 我們認為只要將此用來描述角動量的 Hamiltonian 加上一個必須的項即可得到質心距表示式, 但我們仍需要作深入的研究.
Spinor 能量動量表示式提供了重立場正能量的證明. 我們期待一個成功的spinor 角動量表示式能提供某種未被發現的重力場中能量和動量的關係. 如果這樣一個關係真的被找到且證明存在,那將會是重力理論領域的一個重要的里程碑.

Witten''s spinor formulation and Tung''s QSL approach have provided localizations
for the energy-momentum of asymptotically flat gravitational systems plus positive
energy proofs. The spinor formulation for angular momenta, however, had not been
sufficiently investigated in the past. In this thesis we discuss Witten''s approach and
the QSL approach adapted for angular momenta. Noting that replacing the vectorial
spinor parameterization of the displacement in Witten''s Hamiltonain by a pseudovec-
torial spinor parameterization leads to unsatisfying behavior of the spinor field, we
consider a new quadratic spinor Hamiltonian containing four quadratic spinor terms,
whose parameterization of the displacement is a satisfying antisymmetric tensor. This
Hamiltonian is, however, found to be unsuccessful after testing. From the process of
testing this Hamiltonian, we guess it may be successful if we just pick two promising
terms to compose the spinor Hamiltonian for the angular momentum. After a detailed
survey, we find it still has a problem| its boundary term has a redundant term in
addition to the expected form asymptotically. Nor is the Hamiltonian composed of
the other two terms a good choice for it has other problems in addition to that of
the former 2-term Hamiltonian. Then we speculate that Witten''s approach might be
successful for angular momenta by assuming some spinor gauge conditions, which is
being sought. We also explore the QSL approach, finding that the QSL Hamiltonian
is successful for the quasilocal angular momentum as long as we choose certain spinor
gauge conditions. Since the QSL approach is successful for angular momenta, we are
ready to the exploration of the quasilocal center-of-mass moment, for which a necessary
term should be added to the QSL Hamiltonian. From some clues, we expect a suc-
cessful spinor angular momentum expression will lead us to some connection between
the energy and angular momentum of a asymptotically flat gravitational system. If
this connection can be figured out and proved to exist, this will be a grand milestone in the area of gravitational research, for the connection will provide a further norm
for a good expression for the angular momentum of a gravitational system.

Table of Contents iv
Abstract vi
1 Introduction 1
2 Clifford Algebras, Clifforms and Spinor-valued Forms 6
2.1 Clifford Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Clifforms and Spinor-valued Forms . . . . . . . . . . . . . . . . . . . 18
3 The Hamiltonian Formulation and the Failure of the Witten Spinor
Hamiltonian for Angular Momenta 23
4 Spinor Hamiltonian Formulation for Angular Momenta 29
4.1 Full 4 Terms of the Spinor Hamiltonian . . . . . . . . . . . . . . . . . 29
4.2 2 Terms among the Spinor Hamiltonian . . . . . . . . . . . . . . . . . 41
4.3 The Other 2 Terms among the Spinor Hamiltonian . . . . . . . . . . 48
5 The Quadratic Spinor Lagrangian 51
5.1 Angular Momenta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2 Center-Of-Mass Moments . . . . . . . . . . . . . . . . . . . . . . . . 63
6 Conclusion 65
Bibliography 68
A Some Basic Geometric Objects 71
A.1 Geometric Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
A.2 Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
A.3 Covector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
A.4 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
A.5 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
A.6 Lie Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
A.7 Covariant Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
A.8 Einstein''s Field Equation . . . . . . . . . . . . . . . . . . . . . . . . . 80

                                

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