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Quantum Gravity

Quantum gravity is the study of theories which incorporate known gravitational and quantum phenomena. This usually involves the assumption that gravity is itself a quantum phenomenon.

This can be studied by considering the more general class of quantum field theories which exhibit general covariance. This subject is called topological quantum field theory, or TQFT.

Quantum gravity was originally studied, by Dirac and others, as the problem of quantising general relativity, considered as a dynamical system. This approach has many difficulties, detailed by Isham in

Quantum gravity, an Oxford Symposium, 1975, OUP
The newer approach consists of abandoning the idea of general relativity as a dynamical system, and considering instead the quantum field theory of manifolds. In particle physics, the ideal is that each Lorentzian or Riemannian manifold gives rise to a law of propagation of quantum fields, once the internal structure is specified (gauge group, fields, interactions). The isometries of the space become symmetries of the quantum field theory.

In a TQFT, each manifold gives rise to a quantum propagation law. The mappings of manifolds are symmetries of this theory. In this way, quantum gravity is related to manifold topology.

Without examples, this theory would be pure wishful thinking. There are many examples of TQFTs in 2+1=3 dimensions, some of which are related to quantum gravity in 3 dimensions .

This framework is now leading to suggestions for quantum gravity in 4 dimensions . This is based on the experience with 3 dimensions but is rather more speculative. © 1997 John Barrett