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3 Quantum Gravity in 2+1 Dimensions

The reader may well have decided that for an author reviewing quantum gravity, I have spent an inordinate amount of time on the classical theory. There is a good reason for this, though: Each of the approaches described in the preceding sections leads very naturally to an approach to quantization, which is now - with a few twists - fairly straightforward. Indeed, the main reason that 2+1 dimensions offer such an attractive setting for quantum gravity is that the classical solutions can be completely described by a finite set of parameters. Such a description effectively reduces quantum gravity to quantum mechanics, allowing us to evade the complications of quantum field theory. This is not to imply that all approaches to quantum gravity simplify - the Wheeler-DeWitt equation, for example, apparently does not - but it allows us to explore at least a few approaches in depth.

  3.1 Reduced phase space quantization
  3.2 Chern-Simons quantization
  3.3 Covariant canonical quantization
  3.4 A digression: Observables and the problem of time
  3.5 “Quantum geometry”
  3.6 Lattice methods I: Ponzano-Regge and spin foams
  3.7 Lattice methods II: Dynamical triangulations
  3.8 Other lattice approaches
  3.9 The Wheeler-DeWitt equation
  3.10 Lorentzian path integrals
  3.11 Euclidean path integrals and quantum cosmology

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