Living Reviews in Relativity

"Discrete Approaches to Quantum Gravity
in Four Dimensions"
by
Renate Loll  

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Abstract
1 Introduction
2 Gauge-Theoretic Discretizations of Gravity
2.1 Lagrangian treatment. Introduction
2.2 Smolin’s lattice model
2.3 Numerical implementation
2.4 Other gauge formulations
2.5 Proving reflection positivity
2.6 The measure
2.7 Assorted topics
2.8 Summary
2.9 Hamiltonian treatment. Introduction
2.10 Hamiltonian lattice gravity
2.11 The measure
2.12 The constraint algebra
2.13 Solutions to the Wheeler–DeWitt equation
2.14 The role of diffeomorphisms
2.15 The volume operator
2.16 The real dynamics
2.17 Summary
3 Quantum Regge Calculus
3.1 Path integral for Regge calculus
3.2 Higher-derivative terms
3.3 First simulations
3.4 The phase structure
3.5 Influence of the measure
3.6 Evidence for a second-order transition?
3.7 Avoiding collapse
3.8 Two-point functions
3.9 Non-hypercubic lattices
3.10 Coupling to SU(2)-gauge fields
3.11 Coupling to scalar matter
3.12 Recovering the Newtonian potential
3.13 Gauge invariance in Regge calculus?
3.14 Assorted topics
3.15 Summary
4 Dynamical triangulations
4.1 Introduction
4.2 Path integral for dynamical triangulations
4.3 Existence of an exponential bound?
4.4 Performing the state sum
4.5 The phase structure
4.6 Evidence for a second-order transition?
4.7 Influence of the measure
4.8 Higher-derivative terms
4.9 Coupling to matter fields
4.10 Non-spherical lattices
4.11 Singular configurations
4.12 Renormalization group
4.13 Exploring geometric properties
4.14 Two-point functions
4.15 Summary
5 Conclusions and Outlook
Acknowledgements
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