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Loop quantum gravity provides a general framework for diffeomorphism invariant quantum field theories, within which there is a complete quantization of Einstein's theory of general relativity. There is both a canonical and path integral formalism. In the canonical formalism the Hilbert space of spatially diffeomorphism invariant states has been constructed in closed form. Regularization procedures have been developed which result in finite, diffeomorphsim invariant or covariant operators which represent certain classical diffeomororphism invariant or covariant functions on the phase space. Among these are the area and volume operators, which turn out to have finite, computable spectra. These spectra have been computed in detail.
The Hamiltonian constraint and Hamiltonian in certain fixed gauges are also among the operators that have been constructed using these regularization methods, they are represented by finite, diffeomorphism covariant operators. An infinite dimensional space of solutions to the Hamiltonian constraint has been constructed. Some physical operators (those that commute with all the constraints) have been constructed, using several different methods. These include using path integral methods to construct the projection operator onto physical states, making use of matter fields as physical coordinates or explicit operator constructions. Among those that can be so represented are the area and volume operators, thus their spectra represent genuine physical predictions of the theory.
The path integral form of loop quantum gravity has been constructed and its relationship to the canonical theory is understood. In the Euclidean case the path integral provides a projection operator onto the space of solutions of the Hamiltonian constraints. This can be used to construct explicit expressions for the expectation values of physical (commuting with all the constraints) observables. These have been computed explicitly by series and numerical methods in several 1+1 dimensional examples.
All these results hold for both the Euclidean and Lorentzian signature cases, as well as for extensions of general relativity such as supergravity, and coupling to all usual forms of matter is understood. There are also results in other dimensions such as 1+1 and 2+1. Many of these results have been obtained through several different methods, and the results of these different methods agree up to operator ordering terms, that are by now well understood. Many of the results were first gotten by adopting regularization procedures for products of operators from QCD, and were later verified by being proved as rigorous theorems in mathematical quantum field theory. This applies both to the geometrical operators such as area and volume and the hamiltonian constraint. There remain open questions about the physical interpretation of the solutions to the hamiltonion constraint, these are the subject of much current work and will be discussed in detail below (see website).
(Lee Smolin, http://www.qgravity.org/loop/)
Editor: Haselhurst
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