Approaches to Quantum Gravity

Questions and answers 333the holes can be seen only when probing the geometry with regions which are“smaller” than the scale of the graph. Macroscopically the geometry thereforeremains non degenerate because a semiclassical state necessarily is based onvery “fine” graphs.In conclusion, there are absolutely no problems in LQG associated with thattype of question.• Q - R. Percacci - to T. Thiemann:LQG can be seen as an attempt to directly “quantize Einstein’s theory”. As discussedin Burgess’ contribution, Einstein’s theory can be seen as a low energyeffective field theory and one would expect that the gravitational dynamics getsmodified at very high energies. For example, higher derivative terms couldappear in the action. To what extent could one hope to generalize the resultsof LQG for these more general actions?– A - T. Thiemann:The semiclassical limit of LQG is the Einstein–Hilbert term. The correctionterms of higher power in or rather l 2 Pcan indeed be interpreted as higherderivative terms of the type that Burgess is discussing. The important point isthat this interpretation holds only when using the equations of motion of theEinstein–Hilbert term. This is necessary in order to substitute the canonicalmomenta of the canonical theory by the covariantly defined extrinsic curvaturewhich supplies the higher covariant derivatives. The real question is whyone does not quantize higher derivative actions directly. The answer is verysimple: one could, but unless the additional terms are topological, i.e. are atleast on shell equal to total derivatives, one changes the number of degreesof freedom of the theory. Let us discuss a simple example, an R 2 term. Evenafter performing an integration by parts, this term will depend on time derivativesof the spatial metric up to third if not fourth order. Thus, in order to solvethe equations of motion, one needs to specify initial data involving the spatialmetric together with its velocity, acceleration and possibly time derivatives ofthird order. Thus, even at linear order the theory does not only have the familiartwo polarization degrees of freedom of gravitational waves but in factmore. Notice that this is a purely classical observation and in the literature iswell known as generalized Ostrogadsky method. See e.g. the book by Tuytinon constrained systems or recent papers by Woodard. Hence, as in Yang–Mills theories, higher derivative effective actions are never to be thought ofas classical starting points for quantization but rather as effective tools orvehicles in order to do calculations such as only computing tree diagramsof the effective theory rather than doing all loop orders of the fundamentaltheory. This is the same in the Lagrangian and in the Hamiltonian approach.In summary, there is total agreement in the two approaches.• Q - R. Percacci - to T. Thiemann: