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ix | |
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xiii | |
Preface |
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xv | |
Units, Conventions and Common Abbreviations |
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xviii | |
Acknowledgments |
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xxi | |
Introduction |
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xxiii | |
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1 | (6) |
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1.1 Global Scale Transformations |
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1 | (2) |
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1.2 Local Scale Transformations |
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3 | (4) |
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7 | (12) |
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2.1 The Coastline Paradox |
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7 | (1) |
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8 | (7) |
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2.2.1 The Topological Dimension |
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8 | (1) |
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2.2.2 The Hausdorff Dimension |
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9 | (1) |
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2.2.3 The Spectral Dimension |
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10 | (3) |
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13 | (1) |
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2.2.5 Myrheim-Meyer Dimension |
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13 | (1) |
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2.2.6 Correlation Dimension |
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14 | (1) |
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2.3 Fractals Above Us and Below Us |
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15 | (4) |
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2.3.1 Fractals in Cosmology |
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15 | (1) |
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2.3.2 Fractals in Quantum Mechanics |
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16 | (3) |
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19 | (20) |
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19 | (3) |
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3.2 Evidence for a Minimal Length |
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22 | (12) |
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3.2.1 A Lower Bound on Distance Measurements |
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22 | (1) |
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3.2.2 Black Hole Limitations |
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23 | (2) |
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3.2.3 Heisenberg's Microscope |
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25 | (3) |
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3.2.4 High-Energy Convergence |
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28 | (1) |
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3.2.5 Fluctuations of the Conformal Factor |
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29 | (3) |
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3.2.6 Modified Feynman Propagator |
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32 | (1) |
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3.2.7 Lattice Quantum Gravity |
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33 | (1) |
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3.3 Special Relativity and a Minimal Length |
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34 | (2) |
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3.4 Phenomenological Quantum Gravity |
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36 | (3) |
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4 The Renormalisation Group |
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39 | (6) |
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39 | (1) |
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4.2 Kadanoff's Block-Spin Model |
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40 | (1) |
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41 | (2) |
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4.4 Renormalisation Group Operators |
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43 | (2) |
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5 The Asymptotic Safety Scenario |
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45 | (8) |
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5.1 Weinberg's Great Idea |
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45 | (4) |
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49 | (4) |
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6 Quantum Gravity on the Lattice |
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53 | (16) |
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6.1 Lattice Regularisation |
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53 | (1) |
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6.2 Geometric Observables |
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54 | (5) |
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6.3 Euclidean Dynamical Triangulations |
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59 | (6) |
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6.3.1 Conformal Instability |
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63 | (2) |
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6.4 Causal Dynamical Triangulations |
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65 | (4) |
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7 Is the Dimension of Spacetime Scale Dependent? |
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69 | (32) |
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69 | (2) |
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7.2 The Evidence for Dimensional Reduction |
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71 | (13) |
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71 | (1) |
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7.2.2 Causal Dynamical Triangulations |
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71 | (4) |
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7.2.3 Euclidean Dynamical Triangulations |
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75 | (3) |
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7.2.4 Horava-Lifshitz Gravity |
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78 | (1) |
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79 | (2) |
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7.2.6 Loop Quantum Gravity |
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81 | (1) |
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7.2.7 The Wheeler-DeWitt Equation |
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82 | (1) |
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83 | (1) |
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7.2.9 Non-Commutative Geometry |
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83 | (1) |
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7.3 A Possible Solution to an Old Problem |
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84 | (1) |
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7.4 Dimensional Reduction in the Sky |
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85 | (3) |
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88 | (3) |
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88 | (2) |
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90 | (1) |
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91 | (1) |
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7.6 What is Dimensional Reduction Really Telling Us? |
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91 | (10) |
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91 | (1) |
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7.6.2 Scale Dependent Length |
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92 | (5) |
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7.6.3 A Dual Description? |
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97 | (4) |
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8 Scale Dependent Spacetime |
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101 | (16) |
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101 | (5) |
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8.2 Renormalising Spacetime |
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106 | (11) |
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106 | (3) |
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109 | (1) |
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8.2.3 Immediate Implications |
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110 | (1) |
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8.2.4 Implications for Quantum Gravity |
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111 | (6) |
Final Thoughts |
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117 | (2) |
References |
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119 | (24) |
About the Author |
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143 | (2) |
Index |
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145 | (4) |
Related Nova Publications |
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149 | |