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The topic of this lecture is "quantum gravity", one of the most difficult and fundamental problems in theoretical physics. In this lecture, we cover why we expect a theory of quantum gravity to be necessary for a consistent formulation of gravity and quantum matter and outline various routes that could lead to this goal. In particular, we focus on so-called background independent approaches that do not rely on a fixed background space-time in their construction. Our plan for this lecture is as follows:
- Brief recap of general relativity
- Failure of perturbative quantization and possible resolutions
- Canonical quantization of gravity (ADM formalism, loop quantum gravity)
- Path integral quantization (Regge calculus, causal dynamical triangulations, causal set theory, spin foam quantum gravity, group field theory)
- Random geometries (Matrix models, tensor models)
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