Approaches to Quantum Gravity

22 G. ’t HooftThis means that there is one very special state where all energies are zero: the vacuumstate. Identifying the vacuum state is particularly difficult in our theory, but itseems that the vacuum also poses problems in other approaches. In Loop QuantumGravity, it is notoriously difficult to say exactly what the vacuum state is in termsof the fundamental loop states that were introduced there. In superstring theory,there are many candidates for the vacuum, all being distinctly characterized by theboundary conditions and the fluxes present in the compactified part of space-time.String theory ends up leaving an entire ‘landscape’ of vacuum states with no furtherindication as to which of these to pick. It is of crucial importance in any viabletheory of Planck length physics to identify and describe in detail the vacuum state.It appears to be associated with very special fluctuations and correlations of the virtualparticles and fields that one wishes to use to describe physical excited states,and the particles in it.There exists an important piece of information telling us that the vacuum is notjust the state with lowest energy. There must exist an additional criterion to identifythe vacuum: it is flat – or nearly so. In perturbative gravity, this cannot beunderstood. The cosmological constant should receive a large finite renormalizationcounter term from all virtual interactions in the very high energy domain. Asuperior theory in which the cosmological constant vanishes naturally (or is limitedto extremely tiny values) has not yet been found or agreed upon [27]. This shouldbe a natural property of the vacuum state. To see most clearly how strange thissituation is, consider the Einstein–Hilbert action,∫ √−g (S =116πG R + ), (2.10)8πGHere, the first term describes the response of the total action to any deformationcausing curvature. This response is huge, since Newton’s constant, which is tiny,occurs in the denominator. In contrast, the second term describes the response ofthe total action upon scaling. This response is very tiny, since the cosmologicalconstant is extremely small – indeed it was thought to vanish altogether untilrecently.In Fig. 2.1, a piece of fabric is sketched with similar properties in ordinary3-space. Globally, this material allows for stretching and squeezing with relativelylittle resistance, but changing the ratios of the sides of the large triangle, or itsangles, requires much more force. One could build more elaborate structures fromthese basic triangular units, such that their shapes are fixed, but their sizes not. Anengineer would observe, however, that even if the hinges and the rods were madeextremely strong and sturdy, resistance against changes of shape would still berather weak. In the limit where the sizes of the structures are very large comparedto those of the hinges, resistance against changes of shape would dwindle.