Approaches to Quantum Gravity

Consistent discretizations as a road to Quantum Gravity 389The transformation has to be such that it implements the evolution equationsas operatorial relations acting on the space of wavefunctions in the Heisenbergrepresentation, whereU(q|q ′ ) =〈n + 1, q ′ |n, q〉, (20.11)and where |n + 1, q ′ 〉 and |n, q〉 are the eigenvectors of the configuration operatorsˆq in the Heisenberg representation at levels n + 1andn respectively. The evolutionequations take the form,〈n + 1, q|ˆq n+1 − f ( ˆq n , ˆp n )|n, q ′ 〉 = 0, (20.12)〈n + 1, q|ˆp n+1 − g( ˆq n , ˆp n )|n, q ′ 〉 = 0, (20.13)with f , g the quantum evolution equations, which are chosen to be self-adjoint inorder for the transformation to be unitary. Explicit examples of this constructionfor cosmological models can be seen in [17]. If at the end of this process one hasconstructed a transformation that is truly unitary the quantization is complete inthe discrete space and one has a well defined framework to rigorously compute theconditional probabilities that arise when one uses a relational time to describe thephysical system. This is a major advantage over attempts to construct the relationalpicture with systems where one has constraints. There are some caveats tothis construction that are worth pointing out. As we mentioned, our constructiongenerically yields discrete theories that are constraint-free. To be more precise, thetheories do not have the constraints associated with space-time diffeomorphisms.If the theory under consideration has other symmetries (for instance the Gausslaw of Yang–Mills theory or gravity written in the new variable formulation), suchsymmetries may be preserved upon discretization (we worked this out explicitlyfor Yang–Mills and BF theory in [4]). The resulting discrete theory therefore willhave some constraints. If this is the case, the above construction starts by consideringas wavefunctions states that are gauge invariant and endowed with a Hilbertspace structure given by a gauge invariant inner product. The resulting theory hastrue (free) Lagrange multipliers associated with the remaining constraints. The unitarytransformation will depend on such parameters. An alternative is to work in arepresentation where the constraints are solved automatically (like the loop representationfor the Gauss law). There one has no constraints left and the inner productis the kinematical one in the loop representation and the unitary transformationdoes not depend on free parameters. Other issues that may arise have to do withthe fact that in many situations canonical transformation do not correspond quantummechanically to unitary transformations. This problem has been discussed,for instance, by Anderson [1]. He noted that the only canonical transformationsthat can be implemented as unitary transformations are those that correspond to an