Quantum Gravity

5QUANTUM GEOMETRODYNAMICS5.1 The programme of canonical quantizationGiven a classical theory, one cannot derive a unique ‘quantum theory’ from it.The only possibility is to ‘guess’ such a theory and to test it by experiment. Forthis purpose, one has devised sets of ‘quantization rules’ which turned out tobe successful in the construction of quantum theories, for example, of quantumelectrodynamics. Strictly speaking, the task is to construct a quantum theoryfrom its classical limit.In Chapter 4, we have developed a Hamiltonian formulation of GR. This isthe appropriate starting point for a canonical quantization, which requires thedefinition of a configuration variable and its conjugate momentum. A special featureof GR is the fact, as is the case in all reparametrization-invariant systems,that the dynamics is entirely generated by constraints: the total Hamiltonianeither vanishes as a constraint (for the spatially compact case) or solely consistsof surfaces terms (in the asymptotically flat case). The central difficulty isthus, both conceptually and technically, the correct treatment of the quantumconstraints, that is, the quantum version of the constraints (4.69) and (4.70) ortheir versions in the connection or loop representation.In Chapter 3, we presented a general procedure for the quantization of constrainedsystems. Following Dirac (1964), a classical constraint is turned into arestriction on physically allowed wave functionals, see Section 3.1,G a ≈ 0 −→ ĜaΨ =0. (5.1)At this stage, such a transition is only a heuristic recipe which has to be mademore precise. Following Ashtekar (1991) and Kuchař (1993) we shall divide the‘programme of canonical quantization’ into six steps which will shortly be presentedhere and then implemented (or attempted to be implemented) in thefollowing sections.The first step consists in the identification of configuration variables andtheir momenta. In the language of geometric quantization (Woodhouse 1992), itis the choice of polarization. Together with the unit operator, these variables arecalled the ‘fundamental variables’ V i . The implementation of Dirac’s procedureis the translation of Poisson brackets into commutators for the fundamentalvariables, that is,V 3 = {V 1 ,V 2 }−→ ˆV 3 = − i [ ˆV 1 , ˆV 2 ] . (5.2)In the geometrodynamical formulation of GR (see Sections 4.1 and 4.2) thefundamental variables are, apart from the unit operator, the three-metric h ab (x)133