Approaches to Quantum Gravity

56 J. Stachelproblem on a spacelike hypersurface (see [12]), the ten field equations split intofour constraints and six evolution equations. The ten components of the pseudometricprovide a very redundant description of the field, which as noted earlierhas only two degrees of freedom per S-T point. Isolation of these “true” degreesof freedom of the field is a highly non-trivial problem. One approach is to findsome kinematical structure, such that they may be identified with components ofthe metric tensor in a coordinate system adapted to this structure (see, e.g., the discussionin Section 4.6 of the conformal two-structure). Apart from some simplemodels (see Section 4.7), their complete isolation has not been achieved; but theprogram is still being pursued, especially using the Feynman approach (see, e.g.,[20]). Quantization of the theory may be attempted either after or before isolationof the true observables. In quantization methods before isolation, as in loop QuantumGravity, superfluous degrees of freedom are first quantized and then eliminatedvia the quantized constraints (see, e.g., [2]).Classical GR initial value problems can serve to determine various ways of definingcomplete (but generally redundant) sets of dynamical variables. Each problemrequires introduction of some non-dynamical structures for the definition of sucha set, which suggests the need to develop corresponding measurement procedures.The results also provide important clues about possible choices of variables forQG. These questions have been extensively studied for canonical quantization.One can use initial value formulations as a method of defining ensembles of classicalparticle trajectories, based on specification of half the maximal classical initialdata set at an initial (or final) time. The analogy between the probability of someoutcome of a process for such an ensemble and the corresponding Feynman probabilityamplitude (see, e.g., [31]) suggests a similar approach to field theories. InSection 4.2, this possibility was discussed for the loop formulation of electromagnetictheory. The possibility of a direct Feynman-type formulation of QG hasbeen suggested (see, e.g., [6; 7; 20]); and it has been investigated for connectionformulations of the theory, in particular for the Ashtekar loop variables. Reisenbergerand Rovelli [22; 23] maintain that: “Spin foam models are the path-integralcounterparts to loop-quantized canonical theories”. 21 These canonical methods ofcarrying out the transition from classical to quantum theory are based on Cauchyor spacelike hypersurface initial value problems (see Section 4.6.1). Another possiblestarting point for canonical quantization is the null-hypersurface initial valueproblem (see Section 4.6.1). Whether analogous canonical methods could bebased on two-plus-two initial value problems (see Section 4.6.2) remains to bestudied.21 See [3] for the analogy between spin foams in GR and processes in quantum theory: both are examples ofcobordisms.