Modern Canonical Quantum General Relativity pdf epub mobi txt 电子书 下载 2026
出版者:Cambridge University Press
作者:Thomas Thiemann
出品人:
页数:846
译者:
出版时间:2008-12-1
价格:USD 82.99
装帧:Paperback
isbn号码:9780521741873
丛书系列:
图书标签:
- Quantum
- Loop
- Gravity
- 物理
- 量子引力
- 广义相对论
- 规范理论
- 量子场论
- 时空结构
- 黑洞
- 宇宙学
- 量子几何
- 引力量子化
- 现代物理学
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具体描述
Modern physics rests on two fundamental building blocks: general relativity and
quantum theory. General relativity is a geometric interpretation of gravity, while
quantum theory governs the microscopic behaviour of matter. According to Einstein’s
equations, geometry is curved when and where matter is localized. Therefore,
in general relativity, geometry is a dynamical quantity that cannot be prescribed
a priori but is in interaction with matter. The equations of nature are
background independent in this sense; there is no space-time geometry on which
matter propagates without backreaction of matter on geometry. Since matter is
described by quantum theory, which in turn couples to geometry, we need a quantum
theory of gravity. The absence of a viable quantum gravity theory to date is
due to the fact that quantum (field) theory as currently formulated assumes that
a background geometry is available, thus being inconsistent with the principles of general relativity. In order to construct quantum gravity, one must reformulate quantum theory in a background-independent way. Modern Canonical Quantum General Relativity is about one such candidate for a background-independent quantum gravity theory: loop quantum gravity.
This book provides a complete treatise of the canonical quantization of general
relativity. The focus is on detailing the conceptual and mathematical framework,
describing the physical applications, and summarizing the status of this
programme in its most popular incarnation: loop quantum gravity. Mathematical
concepts and their relevance to physics are provided within this book, so
it is suitable for graduate students and researchers with a basic knowledge of
quantum field theory and general relativity.
《现代规范量子引力学》 一场跨越宏观与微观的宇宙探索之旅 本书《现代规范量子引力学》是一部旨在深入探索物理学最核心、最迷人问题的著作。它并非一部简单的理论概览，而是一场严谨而富有洞见的宇宙探索之旅，致力于揭示引力在量子层面的奥秘，并在此基础上构建一个能够统一广义相对论与量子力学的全新框架。本书将引导读者穿越广袤的数学语言，直抵物理学的最前沿，理解我们宇宙结构的最基本构成和运作原理。 为何需要规范量子引力学？ 我们当前对宇宙的理解，主要建立在两个看似截然不同但都极其成功的理论之上：爱因斯坦的广义相对论描述了宏观宇宙的引力现象，从行星运行到黑洞形成，无不精准；而量子力学则支配着微观世界的行为，解释了原子、粒子以及它们之间的相互作用。然而，当我们将目光投向极端条件，例如宇宙大爆炸的奇点，或是黑洞内部，这两个理论便产生了不可调和的矛盾。引力不再仅仅是时空的弯曲，它本身也需要遵循量子规律。发展一个能够囊括这两个理论精髓的“规范量子引力学”理论，是现代物理学面临的最为艰巨的挑战之一。它不仅仅是理论物理学家们的“圣杯”，更是理解我们宇宙起源、演化乃至最终命运的关键。 本书的核心议题与内容深度 《现代规范量子引力学》将从基础概念出发，逐步深入到最前沿的研究领域。本书不回避任何一个关键的数学工具和物理直觉，力求为读者提供一个全面而深刻的认识。 第一部分：理论基石的重塑 广义相对论的现代视角： 本部分将重新审视广义相对论的核心概念，但会以更加现代和数学化的语言呈现。我们将探讨协变导数、黎曼曲率张量、里奇张量和斯卡拉曲率等概念，不仅仅是定义，更侧重于其几何意义以及在描述时空动力学中的作用。本书将详细阐述爱因斯坦-希尔伯特作用量，并深入分析其变分原理如何导出爱因斯坦场方程。此外，还会讨论一些关键性的概念，如类时曲线、类光曲线、类空曲线，以及它们在定义因果结构中的重要性。我们还将回顾广义相对论在强引力场下的预言，如引力红移、引力波，并简要介绍其观测证据。 量子场论的规范性： 紧接着，本书将回顾量子场论的基石——规范对称性。我们将从最简单的阿贝尔规范场（如电磁场）出发，解释规范不变性的概念及其在物理定律中的核心地位。随后，本书将深入到非阿贝尔规范场，如量子色动力学（QCD）中的SU(3)规范群，以及量子电动力学（QED）中的U(1)规范群。我们将详细介绍规范场论的拉格朗日量形式，狄拉克方程的规范协变形式，以及杨-米尔斯方程的推导。本书将重点阐述规范场论的重整化技术，包括重整化群方程、重整化方案（如MSbar方案）以及重整化群的跑动耦合常数，理解其在解决紫外发散问题上的关键作用。 引力为何需要规范化？ 这是本书最为关键的切入点之一。我们将详细解释为何直接将量子场论的框架应用于引力会导致根本性的困难，例如紫外不可重整化。本书将深入探讨引力场的量子化所面临的挑战，特别是其拉格朗日量在能量动量守恒和时空度规上的非平凡耦合。我们将讨论标准量子场论在引力领域的局限性，并引出发展一种新的规范性框架的必要性。 第二部分：前沿规范量子引力学理论的探索 本书将重点介绍当前最具前景的几种规范量子引力学理论，并深入分析它们的数学结构、物理内涵和潜在的实验检验。 圈量子引力（Loop Quantum Gravity, LQG）： 作为一种几何化的量子引力理论，LQG将时空离散化，引入了“圈”和“自旋网络”作为基本量子单元。本书将详细阐述 LQG 的数学基础，包括阿希特-霍金（Ashtekar-Hovart）变量，以及它们如何将广义相对论的动力学转化为一种场论的样式。我们将深入讲解自旋网络（Spin Networks）的结构，它们如何编码时空的量子几何，以及面积和体积算符的谱。本书还将讨论 LQG 中“圈”和“节点”的动力学，并介绍“圈量子宇宙学”（Loop Quantum Cosmology）如何尝试解决宇宙大爆炸的奇点问题。 弦理论（String Theory）与M理论（M-Theory）： 尽管弦理论最初并非直接面向量子引力，但它自然地包含了引力子，并提供了一个统一所有基本力的框架。本书将介绍弦理论的基本概念，包括弦的振动模式、狄拉克-拉蒙（D-brane）、背景独立性等。我们将探讨超弦理论的五种不同类型，以及它们如何在低能量下统一为M理论。本书将重点关注弦理论如何解决量子引力的问题，例如其潜在的紫外行为，以及如何通过对偶性（Duality）连接不同的弦理论。此外，还将讨论弦理论的景观（Landscape）概念及其对宇宙学常数问题的启示。 因果动力学三角剖分（Causal Dynamical Triangulations, CDT）： CDT 是一种基于离散化和微扰展开的量子引力模型，它在数学上更加严谨，并成功地避开了某些传统方法中的病态。本书将详细介绍 CDT 的核心思想，包括如何通过三角剖分将时空离散化，以及如何引入因果关系。我们将分析 CDT 的路径积分方法，以及其在不同维度的表现。本书将重点讨论 CDT 如何从微观的离散单元中涌现出宏观的、光滑的时空，以及它在解决黑洞信息佯谬和奇点问题上的潜力。 渐进安全性（Asymptotic Safety）： 这种方法试图寻找一个量子引力的有效场论，其在高能量下行为良好，不受紫外发散的困扰。本书将阐述渐进安全性的核心思想，即引力耦合常数在高能下可能存在一个非平凡的紫外固定点。我们将介绍函数重整化群（Functional Renormalization Group）在分析渐进安全性中的应用，以及如何利用它来研究引力场的量子行为。本书还将讨论渐进安全性与弦理论和圈量子引力等其他方法的潜在联系。 第三部分：未解之谜与未来展望 黑洞信息佯谬的规范量子引力学视角： 黑洞是检验量子引力理论的理想场所。本书将深入探讨黑洞信息佯谬，并从不同规范量子引力学理论的角度提出可能的解决方案，例如霍金辐射的量子纠缠、纠缠熵的计算，以及量子信息理论在其中的作用。 宇宙学常数问题与量子引力： 宇宙学常数问题是理论物理学中最令人费解的难题之一。本书将分析量子引力理论如何尝试解释观测到的极小的宇宙学常数，并探讨其与暗能量的联系。 观测的挑战与实验信号： 尽管规范量子引力学理论在数学上非常优美，但其直接的实验检验仍是巨大的挑战。本书将讨论可能存在的观测信号，例如早期宇宙的引力波、黑洞附近的时空结构、甚至可能存在的微观黑洞。我们将评估不同理论预测的可检验性，并展望未来的观测技术。 连接物理学不同分支的桥梁： 本书将强调规范量子引力学理论对于连接粒子物理学、凝聚态物理学乃至生物学等领域的重要性，探索跨学科的潜在联系和新的研究方向。 本书的独特价值 《现代规范量子引力学》并非一本面向初学者的入门读物，它旨在为那些已经具备一定物理学和数学基础的读者提供一次深度探索的机会。本书最大的特点在于： 理论的深度与广度兼备： 它既深入到最前沿的数学推导，又力求保持物理直觉的清晰。 跨理论的视角： 它不局限于某一种量子引力理论，而是提供了对多种主流方法的全面介绍和比较。 严谨的数学框架： 本书的论证过程将严格遵循数学逻辑，确保结论的可靠性。 面向未来的思考： 它不仅回顾了已有的成就，更积极地展望了未来的研究方向和可能突破。 本书的读者群体将包括物理学专业的研究生、博士后，以及对量子引力学有浓厚兴趣的资深研究人员。通过阅读本书，读者将能够： 构建对量子引力学研究的宏观认知： 理解该领域的核心问题、主要挑战和不同理论的优劣。 掌握关键的数学工具： 熟悉理解量子引力学理论所必需的数学语言和技术。 深入理解前沿理论的细节： 能够独立地阅读和理解相关的学术论文。 激发新的研究思路： 为自身的科研工作提供理论支持和灵感。 《现代规范量子引力学》是一部献给所有渴望理解宇宙最深层奥秘者的重要著作。它将带领我们踏上一段充满挑战但也极具回报的智慧之旅，去揭示隐藏在万物之下的终极规律。
作者简介
Thomas Thiemann is Staff Scientist at the Max Planck Institut fur Gravitationsphysik (Albert Einstein Institut), Potsdam, Germany. He is also
a long-term researcher at the Perimeter Institute for Theoretical Physics and
Associate Professor at the University of Waterloo, Canada. Thomas Thiemann
obtained his Ph.D. in theoretical physics from the Rheinisch-Westf¨alisch Technische Hochschule, Aachen, Germany. He held two-year postdoctoral positions at The Pennsylvania State University and Harvard University. As of 2005 he holds a guest professor position at Beijing Normal University, China.
目录信息
Foreword, by Chris Isham page xvii
Preface xix
Notation and conventions xxiii
Introduction: Defining quantum gravity 1
Why quantum gravity in the twenty-first century? 1
The role of background independence 8
Approaches to quantum gravity 11
Motivation for canonical quantum general relativity 23
Outline of the book 25
I CLASSICAL FOUNDATIONS, INTERPRETATION AND THE
CANONICAL QUANTISATION PROGRAMME
1 Classical Hamiltonian formulation of General Relativity 39
1.1 The ADM action 39
1.2 Legendre transform and Dirac analysis of constraints 46
1.3 Geometrical interpretation of the gauge transformations 50
1.4 Relation between the four-dimensional diffeomorphism group and
the transformations generated by the constraints 56
1.5 Boundary conditions, gauge transformations and symmetries 60
1.5.1 Boundary conditions 60
1.5.2 Symmetries and gauge transformations 65
2 The problem of time, locality and the interpretation of
quantum mechanics 74
2.1 The classical problem of time: Dirac observables 75
2.2 Partial and complete observables for general constrained systems 81
2.2.1 Partial and weak complete observables 82
2.2.2 Poisson algebra of Dirac observables 85
2.2.3 Evolving constants 89
2.2.4 Reduced phase space quantisation of the algebra of Dirac
observables and unitary implementation of the
multi-fingered time evolution 90
2.3 Recovery of locality in General Relativity 93
2.4 Quantum problem of time: physical inner product and
interpretation of quantum mechanics 95
2.4.1 Physical inner product 95
2.4.2 Interpretation of quantum mechanics 98
3 The programme of canonical quantisation 107
3.1 The programme 108
4 The new canonical variables of Ashtekar for
General Relativity 118
4.1 Historical overview 118
4.2 Derivation of Ashtekar’s variables 123
4.2.1 Extension of the ADM phase space 123
4.2.2 Canonical transformation on the extended phase space 126
II FOUNDATIONS OF MODERN CANONICAL QUANTUM
GENERAL RELATIVITY
5 Introduction 141
5.1 Outline and historical overview 141
6 Step I: the holonomy–flux algebra P 157
6.1 Motivation for the choice of P 157
6.2 Definition of P: (1) Paths, connections, holonomies and
cylindrical functions 162
6.2.1 Semianalytic paths and holonomies 162
6.2.2 A natural topology on the space of generalised connections 168
6.2.3 Gauge invariance: distributional gauge transformations 175
6.2.4 The C∗ algebraic viewpoint and cylindrical functions 183
6.3 Definition of P: (2) surfaces, electric fields, fluxes and vector fields 191
6.4 Definition of P: (3) regularisation of the holonomy–flux
Poisson algebra 194
6.5 Definition of P: (4) Lie algebra of cylindrical functions and
flux vector fields 202
7 Step II: quantum ∗-algebra A 206
7.1 Definition of A 206
7.2 (Generalised) bundle automorphisms of A 209
8 Step III: representation theory of A 212
8.1 General considerations 212
8.2 Uniqueness proof: (1) existence 219
8.2.1 Regular Borel measures on the projective limit:
the uniform measure 220
8.2.2 Functional calculus on a projective limit 226
8.2.3 + Density and support properties of A,A/G with respect
to A,A/G 233
8.2.4 Spin-network functions and loop representation 237
8.2.5 Gauge and diffeomorphism invariance of μ0 242
8.2.6 + Ergodicity of μ0 with respect to spatial diffeomorphisms 245
8.2.7 Essential self-adjointness of electric flux momentum
operators 246
8.3 Uniqueness proof: (2) uniqueness 247
8.4 Uniqueness proof: (3) irreducibility 252
9 Step IV: (1) implementation and solution of the
kinematical constraints 264
9.1 Implementation of the Gauß constraint 264
9.1.1 Derivation of the Gauß constraint operator 264
9.1.2 Complete solution of the Gauß constraint 266
9.2 Implementation of the spatial diffeomorphism constraint 269
9.2.1 Derivation of the spatial diffeomorphism constraint
operator 269
9.2.2 General solution of the spatial diffeomorphism constraint 271
10 Step IV: (2) implementation and solution of the
Hamiltonian constraint 279
10.1 Outline of the construction 279
10.2 Heuristic explanation for UV finiteness due to background
independence 282
10.3 Derivation of the Hamiltonian constraint operator 286
10.4 Mathematical definition of the Hamiltonian constraint operator 291
10.4.1 Concrete implementation 291
10.4.2 Operator limits 296
10.4.3 Commutator algebra 300
10.4.4 The quantum Dirac algebra 309
10.5 The kernel of the Wheeler–DeWitt constraint operator 311
10.6 The Master Constraint Programme 317
10.6.1 Motivation for the Master Constraint Programme in
General Relativity 317
10.6.2 Definition of the Master Constraint 320
10.6.3 Physical inner product and Dirac observables 326
10.6.4 Extended Master Constraint 329
10.6.5 Algebraic Quantum Gravity (AQG) 331
10.7 + Further related results 334
10.7.1 The Wick transform 334
10.7.2 Testing the new regularisation technique by models of
quantum gravity 340
10.7.3 Quantum Poincar´e algebra 341
10.7.4 Vasiliev invariants and discrete quantum gravity 344
11 Step V: semiclassical analysis 345
11.1 + Weaves 349
11.2 Coherent states 353
11.2.1 Semiclassical states and coherent states 354
11.2.2 Construction principle: the complexifier method 356
11.2.3 Complexifier coherent states for diffeomorphism-invariant
theories of connections 362
11.2.4 Concrete example of complexifier 367
11.2.5 Semiclassical limit of loop quantum gravity: graph-changing
operators, shadows and diffeomorphism-invariant
coherent states 376
11.2.6 + The infinite tensor product extension 385
11.3 Graviton and photon Fock states from L2(A, dμ0) 390
III PHYSICAL APPLICATIONS
12 Extension to standard matter 399
12.1 The classical standard model coupled to gravity 400
12.1.1 Fermionic and Einstein contribution 401
12.1.2 Yang–Mills and Higgs contribution 405
12.2 Kinematical Hilbert spaces for diffeomorphism-invariant theories
of fermion and Higgs fields 406
12.2.1 Fermionic sector 406
12.2.2 Higgs sector 411
12.2.3 Gauge and diffeomorphism-invariant subspace 417
12.3 Quantisation of matter Hamiltonian constraints 418
12.3.1 Quantisation of Einstein–Yang–Mills theory 419
12.3.2 Fermionic sector 422
12.3.3 Higgs sector 425
12.3.4 A general quantisation scheme 429
13 Kinematical geometrical operators 431
13.1 Derivation of the area operator 432
13.2 Properties of the area operator 434
13.3 Derivation of the volume operator 438
13.4 Properties of the volume operator 447
13.4.1 Cylindrical consistency 447
13.4.2 Symmetry, positivity and self-adjointness 448
13.4.3 Discreteness and anomaly-freeness 448
13.4.4 Matrix elements 449
13.5 Uniqueness of the volume operator, consistency with the flux
operator and pseudo-two-forms 453
13.6 Spatially diffeomorphism-invariant volume operator 455
14 Spin foam models 458
14.1 Heuristic motivation from the canonical framework 458
14.2 Spin foam models from BF theory 462
14.3 The Barrett–Crane model 466
14.3.1 Plebanski action and simplicity constraints 466
14.3.2 Discretisation theory 472
14.3.3 Discretisation and quantisation of BF theory 476
14.3.4 Imposing the simplicity constraints 482
14.3.5 Summary of the status of the Barrett–Crane model 494
14.4 Triangulation dependence and group field theory 495
14.5 Discussion 502
15 Quantum black hole physics 511
15.1 Classical preparations 514
15.1.1 Null geodesic congruences 514
15.1.2 Event horizons, trapped surfaces and apparent horizons 517
15.1.3 Trapping, dynamical, non-expanding and (weakly) isolated
horizons 519
15.1.4 Spherically symmetric isolated horizons 526
15.1.5 Boundary symplectic structure for SSIHs 535
15.2 Quantisation of the surface degrees of freedom 540
15.2.1 Quantum U(1) Chern–Simons theory with punctures 541
15.3 Implementing the quantum boundary condition 546
15.4 Implementation of the quantum constraints 548
15.4.1 Remaining U(1) gauge transformations 549
15.4.2 Remaining surface diffeomorphism transformations 550
15.4.3 Final physical Hilbert space 550
15.5 Entropy counting 550
15.6 Discussion 557
16 Applications to particle physics and quantum cosmology 562
16.1 Quantum gauge fixing 562
16.2 Loop Quantum Cosmology 563
17 Loop Quantum Gravity phenomenology 572
IV MATHEMATICAL TOOLS AND THEIR CONNECTION
TO PHYSICS
18 Tools from general topology 577
18.1 Generalities 577
18.2 Specific results 581
19 Differential, Riemannian, symplectic and complex
geometry 585
19.1 Differential geometry 585
19.1.1 Manifolds 585
19.1.2 Passive and active diffeomorphisms 587
19.1.3 Differential calculus 590
19.2 Riemannian geometry 606
19.3 Symplectic manifolds 614
19.3.1 Symplectic geometry 614
19.3.2 Symplectic reduction 616
19.3.3 Symplectic group actions 621
19.4 Complex, Hermitian and K¨ahler manifolds 623
20 Semianalytic category 627
20.1 Semianalytic structures on Rn 627
20.2 Semianalytic manifolds and submanifolds 631
21 Elements of fibre bundle theory 634
21.1 General fibre bundles and principal fibre bundles 634
21.2 Connections on principal fibre bundles 636
22 Holonomies on non-trivial fibre bundles 644
22.1 The groupoid of equivariant maps 644
22.2 Holonomies and transition functions 647
23 Geometric quantisation 652
23.1 Prequantisation 652
23.2 Polarisation 662
23.3 Quantisation 668
24 The Dirac algorithm for field theories with constraints 671
24.1 The Dirac algorithm 671
24.2 First- and second-class constraints and the Dirac bracket 674
25 Tools from measure theory 680
25.1 Generalities and the Riesz–Markov theorem 680
25.2 Measure theory and ergodicity 687
26 Key results from functional analysis 689
26.1 Metric spaces and normed spaces 689
26.2 Hilbert spaces 691
26.3 Banach spaces 693
26.4 Topological spaces 694
26.5 Locally convex spaces 694
26.6 Bounded operators 695
26.7 Unbounded operators 697
26.8 Quadratic forms 699
27 Elementary introduction to Gel’fand theory for
Abelian C∗-algebras 701
27.1 Banach algebras and their spectra 701
27.2 The Gel’fand transform and the Gel’fand isomorphism 709
28 Bohr compactification of the real line 713
28.1 Definition and properties 713
28.2 Analogy with loop quantum gravity 715
29 Operator ∗-algebras and spectral theorem 719
29.1 Operator ∗-algebras, representations and GNS construction 719
29.2 Spectral theorem, spectral measures, projection valued measures,
functional calculus 723
30 Refined algebraic quantisation (RAQ) and direct integral
decomposition (DID) 729
30.1 RAQ 729
30.2 Master Constraint Programme (MCP) and DID 735
31 Basics of harmonic analysis on compact Lie groups 746
31.1 Representations and Haar measures 746
31.2 The Peter and Weyl theorem 752
32 Spin-network functions for SU(2) 755
32.1 Basics of the representation theory of SU(2) 755
32.2 Spin-network functions and recoupling theory 757
32.3 Action of holonomy operators on spin-network functions 762
32.4 Examples of coherent state calculations 765
33 + Functional analytic description of classical connection
dynamics 770
33.1 Infinite-dimensional (symplectic) manifolds 770
References 775
Index 809
· · · · · · (收起)
Preface xix
Notation and conventions xxiii
Introduction: Defining quantum gravity 1
Why quantum gravity in the twenty-first century? 1
The role of background independence 8
Approaches to quantum gravity 11
Motivation for canonical quantum general relativity 23
Outline of the book 25
I CLASSICAL FOUNDATIONS, INTERPRETATION AND THE
CANONICAL QUANTISATION PROGRAMME
1 Classical Hamiltonian formulation of General Relativity 39
1.1 The ADM action 39
1.2 Legendre transform and Dirac analysis of constraints 46
1.3 Geometrical interpretation of the gauge transformations 50
1.4 Relation between the four-dimensional diffeomorphism group and
the transformations generated by the constraints 56
1.5 Boundary conditions, gauge transformations and symmetries 60
1.5.1 Boundary conditions 60
1.5.2 Symmetries and gauge transformations 65
2 The problem of time, locality and the interpretation of
quantum mechanics 74
2.1 The classical problem of time: Dirac observables 75
2.2 Partial and complete observables for general constrained systems 81
2.2.1 Partial and weak complete observables 82
2.2.2 Poisson algebra of Dirac observables 85
2.2.3 Evolving constants 89
2.2.4 Reduced phase space quantisation of the algebra of Dirac
observables and unitary implementation of the
multi-fingered time evolution 90
2.3 Recovery of locality in General Relativity 93
2.4 Quantum problem of time: physical inner product and
interpretation of quantum mechanics 95
2.4.1 Physical inner product 95
2.4.2 Interpretation of quantum mechanics 98
3 The programme of canonical quantisation 107
3.1 The programme 108
4 The new canonical variables of Ashtekar for
General Relativity 118
4.1 Historical overview 118
4.2 Derivation of Ashtekar’s variables 123
4.2.1 Extension of the ADM phase space 123
4.2.2 Canonical transformation on the extended phase space 126
II FOUNDATIONS OF MODERN CANONICAL QUANTUM
GENERAL RELATIVITY
5 Introduction 141
5.1 Outline and historical overview 141
6 Step I: the holonomy–flux algebra P 157
6.1 Motivation for the choice of P 157
6.2 Definition of P: (1) Paths, connections, holonomies and
cylindrical functions 162
6.2.1 Semianalytic paths and holonomies 162
6.2.2 A natural topology on the space of generalised connections 168
6.2.3 Gauge invariance: distributional gauge transformations 175
6.2.4 The C∗ algebraic viewpoint and cylindrical functions 183
6.3 Definition of P: (2) surfaces, electric fields, fluxes and vector fields 191
6.4 Definition of P: (3) regularisation of the holonomy–flux
Poisson algebra 194
6.5 Definition of P: (4) Lie algebra of cylindrical functions and
flux vector fields 202
7 Step II: quantum ∗-algebra A 206
7.1 Definition of A 206
7.2 (Generalised) bundle automorphisms of A 209
8 Step III: representation theory of A 212
8.1 General considerations 212
8.2 Uniqueness proof: (1) existence 219
8.2.1 Regular Borel measures on the projective limit:
the uniform measure 220
8.2.2 Functional calculus on a projective limit 226
8.2.3 + Density and support properties of A,A/G with respect
to A,A/G 233
8.2.4 Spin-network functions and loop representation 237
8.2.5 Gauge and diffeomorphism invariance of μ0 242
8.2.6 + Ergodicity of μ0 with respect to spatial diffeomorphisms 245
8.2.7 Essential self-adjointness of electric flux momentum
operators 246
8.3 Uniqueness proof: (2) uniqueness 247
8.4 Uniqueness proof: (3) irreducibility 252
9 Step IV: (1) implementation and solution of the
kinematical constraints 264
9.1 Implementation of the Gauß constraint 264
9.1.1 Derivation of the Gauß constraint operator 264
9.1.2 Complete solution of the Gauß constraint 266
9.2 Implementation of the spatial diffeomorphism constraint 269
9.2.1 Derivation of the spatial diffeomorphism constraint
operator 269
9.2.2 General solution of the spatial diffeomorphism constraint 271
10 Step IV: (2) implementation and solution of the
Hamiltonian constraint 279
10.1 Outline of the construction 279
10.2 Heuristic explanation for UV finiteness due to background
independence 282
10.3 Derivation of the Hamiltonian constraint operator 286
10.4 Mathematical definition of the Hamiltonian constraint operator 291
10.4.1 Concrete implementation 291
10.4.2 Operator limits 296
10.4.3 Commutator algebra 300
10.4.4 The quantum Dirac algebra 309
10.5 The kernel of the Wheeler–DeWitt constraint operator 311
10.6 The Master Constraint Programme 317
10.6.1 Motivation for the Master Constraint Programme in
General Relativity 317
10.6.2 Definition of the Master Constraint 320
10.6.3 Physical inner product and Dirac observables 326
10.6.4 Extended Master Constraint 329
10.6.5 Algebraic Quantum Gravity (AQG) 331
10.7 + Further related results 334
10.7.1 The Wick transform 334
10.7.2 Testing the new regularisation technique by models of
quantum gravity 340
10.7.3 Quantum Poincar´e algebra 341
10.7.4 Vasiliev invariants and discrete quantum gravity 344
11 Step V: semiclassical analysis 345
11.1 + Weaves 349
11.2 Coherent states 353
11.2.1 Semiclassical states and coherent states 354
11.2.2 Construction principle: the complexifier method 356
11.2.3 Complexifier coherent states for diffeomorphism-invariant
theories of connections 362
11.2.4 Concrete example of complexifier 367
11.2.5 Semiclassical limit of loop quantum gravity: graph-changing
operators, shadows and diffeomorphism-invariant
coherent states 376
11.2.6 + The infinite tensor product extension 385
11.3 Graviton and photon Fock states from L2(A, dμ0) 390
III PHYSICAL APPLICATIONS
12 Extension to standard matter 399
12.1 The classical standard model coupled to gravity 400
12.1.1 Fermionic and Einstein contribution 401
12.1.2 Yang–Mills and Higgs contribution 405
12.2 Kinematical Hilbert spaces for diffeomorphism-invariant theories
of fermion and Higgs fields 406
12.2.1 Fermionic sector 406
12.2.2 Higgs sector 411
12.2.3 Gauge and diffeomorphism-invariant subspace 417
12.3 Quantisation of matter Hamiltonian constraints 418
12.3.1 Quantisation of Einstein–Yang–Mills theory 419
12.3.2 Fermionic sector 422
12.3.3 Higgs sector 425
12.3.4 A general quantisation scheme 429
13 Kinematical geometrical operators 431
13.1 Derivation of the area operator 432
13.2 Properties of the area operator 434
13.3 Derivation of the volume operator 438
13.4 Properties of the volume operator 447
13.4.1 Cylindrical consistency 447
13.4.2 Symmetry, positivity and self-adjointness 448
13.4.3 Discreteness and anomaly-freeness 448
13.4.4 Matrix elements 449
13.5 Uniqueness of the volume operator, consistency with the flux
operator and pseudo-two-forms 453
13.6 Spatially diffeomorphism-invariant volume operator 455
14 Spin foam models 458
14.1 Heuristic motivation from the canonical framework 458
14.2 Spin foam models from BF theory 462
14.3 The Barrett–Crane model 466
14.3.1 Plebanski action and simplicity constraints 466
14.3.2 Discretisation theory 472
14.3.3 Discretisation and quantisation of BF theory 476
14.3.4 Imposing the simplicity constraints 482
14.3.5 Summary of the status of the Barrett–Crane model 494
14.4 Triangulation dependence and group field theory 495
14.5 Discussion 502
15 Quantum black hole physics 511
15.1 Classical preparations 514
15.1.1 Null geodesic congruences 514
15.1.2 Event horizons, trapped surfaces and apparent horizons 517
15.1.3 Trapping, dynamical, non-expanding and (weakly) isolated
horizons 519
15.1.4 Spherically symmetric isolated horizons 526
15.1.5 Boundary symplectic structure for SSIHs 535
15.2 Quantisation of the surface degrees of freedom 540
15.2.1 Quantum U(1) Chern–Simons theory with punctures 541
15.3 Implementing the quantum boundary condition 546
15.4 Implementation of the quantum constraints 548
15.4.1 Remaining U(1) gauge transformations 549
15.4.2 Remaining surface diffeomorphism transformations 550
15.4.3 Final physical Hilbert space 550
15.5 Entropy counting 550
15.6 Discussion 557
16 Applications to particle physics and quantum cosmology 562
16.1 Quantum gauge fixing 562
16.2 Loop Quantum Cosmology 563
17 Loop Quantum Gravity phenomenology 572
IV MATHEMATICAL TOOLS AND THEIR CONNECTION
TO PHYSICS
18 Tools from general topology 577
18.1 Generalities 577
18.2 Specific results 581
19 Differential, Riemannian, symplectic and complex
geometry 585
19.1 Differential geometry 585
19.1.1 Manifolds 585
19.1.2 Passive and active diffeomorphisms 587
19.1.3 Differential calculus 590
19.2 Riemannian geometry 606
19.3 Symplectic manifolds 614
19.3.1 Symplectic geometry 614
19.3.2 Symplectic reduction 616
19.3.3 Symplectic group actions 621
19.4 Complex, Hermitian and K¨ahler manifolds 623
20 Semianalytic category 627
20.1 Semianalytic structures on Rn 627
20.2 Semianalytic manifolds and submanifolds 631
21 Elements of fibre bundle theory 634
21.1 General fibre bundles and principal fibre bundles 634
21.2 Connections on principal fibre bundles 636
22 Holonomies on non-trivial fibre bundles 644
22.1 The groupoid of equivariant maps 644
22.2 Holonomies and transition functions 647
23 Geometric quantisation 652
23.1 Prequantisation 652
23.2 Polarisation 662
23.3 Quantisation 668
24 The Dirac algorithm for field theories with constraints 671
24.1 The Dirac algorithm 671
24.2 First- and second-class constraints and the Dirac bracket 674
25 Tools from measure theory 680
25.1 Generalities and the Riesz–Markov theorem 680
25.2 Measure theory and ergodicity 687
26 Key results from functional analysis 689
26.1 Metric spaces and normed spaces 689
26.2 Hilbert spaces 691
26.3 Banach spaces 693
26.4 Topological spaces 694
26.5 Locally convex spaces 694
26.6 Bounded operators 695
26.7 Unbounded operators 697
26.8 Quadratic forms 699
27 Elementary introduction to Gel’fand theory for
Abelian C∗-algebras 701
27.1 Banach algebras and their spectra 701
27.2 The Gel’fand transform and the Gel’fand isomorphism 709
28 Bohr compactification of the real line 713
28.1 Definition and properties 713
28.2 Analogy with loop quantum gravity 715
29 Operator ∗-algebras and spectral theorem 719
29.1 Operator ∗-algebras, representations and GNS construction 719
29.2 Spectral theorem, spectral measures, projection valued measures,
functional calculus 723
30 Refined algebraic quantisation (RAQ) and direct integral
decomposition (DID) 729
30.1 RAQ 729
30.2 Master Constraint Programme (MCP) and DID 735
31 Basics of harmonic analysis on compact Lie groups 746
31.1 Representations and Haar measures 746
31.2 The Peter and Weyl theorem 752
32 Spin-network functions for SU(2) 755
32.1 Basics of the representation theory of SU(2) 755
32.2 Spin-network functions and recoupling theory 757
32.3 Action of holonomy operators on spin-network functions 762
32.4 Examples of coherent state calculations 765
33 + Functional analytic description of classical connection
dynamics 770
33.1 Infinite-dimensional (symplectic) manifolds 770
References 775
Index 809
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