Approaches to Quantum Gravity

100 O. DreyerIsspacetimefundamental?AreNo Uses No Einstein's NobackgroundEquationstimeused ??InternalRelativityYes Yes YesStrings,LQG, etc.VolovikLloydFig. 7.1. Choices on the road to Quantum Gravity.uses the Einstein equations to formulate the theory. The other possibility is to arguefor why the Einstein equations hold true. In section 7.3 we will show how such anargument can be made. We call this approach Internal Relativity.7.2 Two views of timeIn this section we review two approaches to Quantum Gravity that differ in theway they view time. The first approach comes from solid state physics; the secondcomes from quantum information theory.7.2.1 Fermi pointsIn this section we are interested in the low energy behavior of quantum mechanicalFermi liquids. It turns out that this behavior does not depend on the details of themodel but is rather described by a small number of universality classes. Whichuniversality class a given model falls into is determined by the topology of theenergy spectrum in momentum space. The best known class is that of a simpleFermi surface (see figure 7.2Aa). In an ideal free Fermi gas the Fermi surface isthe boundary in momentum space between the occupied and unoccupied states. Ifp F is the corresponding momentum then the energy spectrum is given byE( p) = v F (| p|−p F ). (7.1)In addition to these fermionic degrees of freedom there are also bosonic excitationsgiven by oscillations of the Fermi surface itself. The dynamics of the fermionic andbosonic degrees of freedom is described by the Landau theory of Fermi liquids.The other well known situation is that of a fully gapped system (see figure7.2Ab). In this case the next available energy level above the Fermi surface is everywhereseparated from it by a non-zero amount . This situation is encountered insuperfluids and superconductors.