Quantum Gravity : Mathematical Models and Experimental Bounds

Contents
vii
1.2. Why topology change? Continuum 3rd quantization of gravity 103
1.3. Why going discrete? Matrix models and simplicial quantum gravity 105
1.4. Why groups and representations? Loop quantum gravity/spin foams 107
2. Group field theory: What is it? The basic GFT formalism 109
2.1. A discrete superspace 109
2.2. The field and its symmetries 111
2.3. The space of states or a third quantized simplicial space 112
2.4. Quantum histories or a third quantized simplicial spacetime 112
2.5. The third quantized simplicial gravity action 113
2.6. The partition function and its perturbative expansion 114
2.7. GFT definition of the canonical inner product 115
2.8. Summary: GFT as a general framework for quantum gravity 116
3. An example: 3d Riemannian quantum gravity 117
4. Assorted questions for the present, but especially for the future 120
Acknowledgements 124
References 125
An Essay on the Spectral Action and its Relation to Quantum Gravity ......127
Mario Paschke
1. Introduction 127
2. Classical spectral triples 130
3. On the meaning of noncommutativity 134
4. NC description of the standard model: the physical intuition behind it 136
4.1. The intuitive idea: an picture of quantum spacetime at low energies 136
4.2. The postulates 138
4.3. How such a noncommutative spacetime would appear to us 139
5. Remarks and open questions 140
5.1. Remarks 140
5.2. Open problems, perspectives, more speculations 141
5.3. Comparision: intuitive picture/other approaches to Quantum Gravity143
6. Towards a quantum equivalence principle 145
6.1. Globally hyperbolic spectral triples 145
6.2. Generally covariant quantum theories over spectral geometries 147
References 149
Towards a Background Independent Formulation of Perturbative Quantum Gravity
..........................................................................151
Romeo Brunetti and Klaus Fredenhagen
1. Problems of perturbative Quantum Gravity 151
2. Locally covariant quantum field theory 152
3. Locally covariant fields 155
4. Quantization of the background 158