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研究生：	周靜宜 Chou, Ching-Yi
論文名稱：	廣義相對論的仿射群建構 Affine Group formulation of General Relativity
指導教授：	許祖斌 Soo, Cho-Pin
學位類別：	博士 Doctor
系所名稱：	理學院 - 物理學系 Department of Physics
論文出版年：	2014
畢業學年度：	102
語文別：	英文
論文頁數：	92
中文關鍵詞：	仿射群、廣義相對論
外文關鍵詞：	affine algebra, General Relativity
相關次數：	點閱：68 下載：10
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查詢本校圖書館目錄查詢臺灣博碩士論文知識加值系統勘誤回報

　　在宇宙常數不為零中，四維量子重力哈密頓約束可以改寫為仿射代數關係式。意即：仿射量子機制可以應用到不只是純重力場之下的哈密頓約束，也可延伸到包含費米子場、希格斯場以及其他規範場的粒子物理標準模型。精確量子態可透過仿射群論及仿射基準態來建構。因為哈密頓約束改寫為仿射代數關係式時只需陳–西蒙斯泛函（Chern-Simons functional）的虛部，而非整個泛函。因此可確保仿射代數的生成元（generators）均為厄米特。
　　本論文也介紹Ashtekar變數以及探討維持實相空間的條件。含粒子物理標準模型的Ashtekar變數必須是複數。但是，CPT定理確立了所有符合勞倫茲對稱及自旋–統計規則的場論會將作用量及Ashtekar變數映照至他們的厄米特共軛。即使有費米子所引起的複雜化，我們也可將維持實相空間的條件改寫為CPT–偽厄米關係式。

The Hamiltonian constraint of four-dimensional gravity with Lorentzian signature is reformulated as an affine algebra relation. It is remarkable that the affine quantization program of Klauder is applicable to not just pure gravity but also when fermions, Yang-Mills and Higgs fields of the whole Standard Model are taken into account, provided the cosmological constant is non-vanishing. Techniques of affine group representations are used to construct exact solutions from a fiducial state, and algebraic affine group methodology and results are employed to study them. The crucial observation that just the imaginary part, rather than the full Chern-Simons functional, is needed in the reformulation of the Hamiltonian constraint of pure gravity theory as an affine algebra ensures that the generators of the algebra are all Hermitian. The extension to full Standard Model incorporating Weyl fermions (with Hermitian action) with Higgs and Yang-Mills fields is also carried out. Within the context of this work, all physical states of quantum gravity must come from representations of the affine algebra. It is intriguing the formulation of the Hamiltonian constraint as an affine algebra is predicated upon a non-vanishing cosmological constant.

Ashtekar variables are introduced and the associated reality conditions are studied. With Standard Model fermions, the necessity of using complex (anti-)self-dual Ashtekar connection is pointed out. A key observation is that CPT maps the Ashtekar variable to its Hermitian adjoint. Moreover, with Lorentz invariance and spin-statistics rule, the CPT theorem ensures that the action is mapped to its Hermitian adjoint under CPT. Thus despite the complications which arise from fermions, the generic reality condition for the Ashtekar connection can be formulated as a pseudo-Hermiticity condition with respect to CPT even in the presence of fermions and other Standard Model fields.

Contents
1 Introduction and overview---4
2 Preliminaries, Ashtekar variables in General Relativity
, and pseudo-Hermiticity---8
2.1 Ashtekar connection---10
2.1.1 Gravity with Standard Model Weyl fermions should 
couple to anti-self-dual (complex) Ashtekar connection---12
2.1.2 Samuel-Jacobson-Smolin action, and Weyl coupling 
of fermion to Ashtekar connection---13
2.2 Pseudo-Hermiticity formulation and the 
Ashtekar-Wheeler-DeWitt operator---15
2.2.1 Pure gravity and constraints---15
2.2.2 Pseudo-Hermiticity relations of basic variables 
and the Ashtekar-Wheeler-DeWitt operator---16
2.2.3 Factor-ordering of super-Hamiltonian operator and 
pseudo-Hermiticity relations---17
2.2.4 Constraint algebra of general factor-ordering of 
super-Hamiltonian constraints---18
2.3 Pseudo-Hermiticity, and Ashtekar-Wheeler-DeWitt 
operator with inclusion of spin-1/2 fermions---20
2.3.1 Fermion contribution to super-Hamiltonian 
constraint---21
2.3.2 Pseudo-Hermiticity relations of super-Hamiltonian with inclusion of spin-1/2 fermions---22
2.3.3 Closure of super-Hamiltonian constraint algebra with 
fermionic fields---24
2.4 Hermitian super-Hamiltonian operator with inclusion of 
fermionic matter and Higgs fields---25
2.4.1 Spin-0 Higgs field and Spin-1 Yang-Mills field---27
2.4.2 Hermitian spin-1/2 theory and its contribution to the 
super-Hamiltonian constraint---27
2.5 Parity transformation of holonomy and flux operators in 
Loop Quantum Gravity---29
2.6 Summary of reality conditions and pseudo-Hermicity 
conditions for pure gravity, and for Standard Model 
coupled to gravity---30
3 Affine group representation formalism for four-dimensional
, Lorentzian, quantum gravity---32
3.1 Introduction---32
3.2 Dirac quantization procedure---34
3.3 Reformulation of the Hamiltonian constraint---35
3.4 Affine group and quantum gravity with Ashtekar variables---38
3.5 Algebraic quantization---40
3.5.1 Construction of the Hilbert space H_Phys---42
3.5.2 General quantum affine group element---44
3.5.3 Normalizability and inner product---45
3.6 Standard Model coupled to pure General Relativity---45
3.6.1 Results from the metric-ADM formalism---46
3.6.2 Inclusion of fermions, and Ashtekar-ADM triad variables---47
3.6.3 Fermionic contribution to Hamiltonian constraints---50
3.6.4 The kinematic constraints for Fermionic field in 
Standard Model---51
3.6.5 Affine algebra representation of Hamiltonian constraint in Standard Model coupled to Gravity---53
3.7 Summary of affine algebraic formulation in four-dimensional Lorentzian quantum gravity---55
4 Supplementary computations, verifications and proofs---57
4.1 P or CPT-pseudo-Hermiticity relation for Ashtekar connection---57
4.2 The proof of PAP = A† and P(~E)P = −(~E)---58
4.2.1 The proof of PAP = A†---58
4.2.2 The proof P(~E)P = −(~E)---60
4.3 Solution of factor ordering of super-Hamiltonian constraint from consideration of P/CPT-pseudo-Hermiticity---61
4.4 Computation of constraint algebra for general ordering of super-Hamiltonian constraint---61
4.4.1 Explicit calculations---64
4.5 Detailed derivation of the torsion part of Ashtekar connection---70
4.5.1 Generating function for torsion part of Ashtekar connection---72
4.6 CPT transformation of fermionic contributions to 
super-Hamiltonian constraint for Weyl and Hermitian Weyl actions---72
4.7 pseudo-Hermiticity and commutation relation of fundamental loop variables---74
5 Conclusions and summary of major results---79
A Measure, inner product and coherent states associated with the affine group---81
B Proof that the real part of the Chern-Simons functional Poisson-commutes with the volume element---85
C CPT and transformation properties of the affine states---88
Bibliography---90
                                    

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校內：2019-07-24公開
校外：2020-01-01公開

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