Approaches to Quantum Gravity

Consistent discretizations as a road to Quantum Gravity 379variables of the discrete theories whose values are determined by the equations ofmotion. We call this approach in the context of constrained theories “consistentdiscretizations”.The consistently discretized theories are both puzzling and attractive. On theone hand, it is puzzling that the Lagrange multipliers get fixed by the theory. Don’tthe Lagrange multipliers represent the gauge freedom of general relativity? Theanswer is what is expected: the discretization breaks the freedom and solutionsto the discrete theory that are different correspond, in the continuum limit, to thesame solution of the continuum theory. Hence the discrete theory has more degreesof freedom. On the other hand, the lack of constraints make the consistently discretizedtheories extremely promising at the time of quantization. Most of the hardconceptual questions of Quantum Gravity are related to the presence of constraintsin the theory. In comparison, the consistently discretized theories are free of theseconceptual problems and can be straightforwardly quantized (to make matters evensimpler, as all discrete theories, they have a finite number of degrees of freedom).In addition, they provide a framework to connect the path integral and canonicalapproaches to Quantum Gravity since the central element is a unitary evolutionoperator. In particular they may help reconcile the spin foam and canonical looprepresentation approaches. They also provide a natural canonical formulation forRegge calculus [20].In this chapter we would like to briefly review the status of the consistentdiscretization approach, both in its application as a classical approximation togravitational theories and as a tool for their quantization. Other brief reviews withdifferent emphasis can be seen in [18; 19]. The organization of this chapter is asfollows. In section 20.2 we consider the application of the technique to a simple,yet conceptually challenging mechanical model and discuss how features thatone observes in the model are actually present in more realistic situations involvinggeneral relativity. In section 20.3 we outline various applications of the framework.In section 20.4 we discuss in detail the quantization of the discrete theories and insection 20.5 we outline how one can define the quantum continuum limit. We endwith a summary and outlook.20.2 Consistent discretizationsTo introduce and illustrate the method in a simple – yet challenging – modelwe consider the model analyzed in detail by Rovelli [27] in the context of theproblem of time in canonical Quantum Gravity: two harmonic oscillators withconstant energy sum. We have already discussed this model in some detail in[19] but we would like to revisit it here to frame the discussion with a differentemphasis.