Approaches to Quantum Gravity

496 J. Kowalski-Glikmanas it is believed, the correct limit of (Quantum) Gravity in the case when spacetimeis flat? From the perspective of gravity, flat Minkowski spacetime is someparticular configuration of the gravitational field, and as such is to be described bythe theory of gravity. It corresponds to configurations of the gravitational field inwhich this field vanishes. However, equations governing the gravitational field aredifferential equations and thus describe the solutions only locally. In the case ofMinkowski space particle kinematics we have to deal not only with a (flat) gravitationalfield but also with particles themselves. The particles are, of course, thesources of the gravitational field and even in the flat space limit the trace of theparticles’ back reaction on spacetime might remain in the form of some globalinformation, even if locally, away from the locations of the particle, the spacetimeis flat. Of course we know that in general relativity the energy-momentumof matter curves spacetime, and the strength of this effect is proportional to gravitationalcoupling (Newton’s constant.) Thus we are interested in the situation inwhich the transition from general relativity to Special Relativity corresponds tosmooth switching off the couplings. In principle two situations are possible (infour dimensions):(i)weak gravity, semiclassical limit of Quantum GravityG, → 0,√G= κ remains finite; (25.2)(ii) weak gravity, small cosmological constant limit of Quantum Gravity → 0, κ remains finite. (25.3)The idea is therefore to devise a controllable transition from the full (Quantum)Gravity coupled to point particles to the regime, in which all local degrees of freedomof gravity are switched off. Then it is expected that locally, away from theparticles’ worldlines, gravity will take the form of Minkowski (for = 0) or (anti)de Sitter space, depending on the sign of . Thus it is expected that DSR arises asa limit of general relativity coupled to point particles in the topological field theorylimit. To be more explicit, consider the formulation of gravity as the constrainedtopological field theory, proposed in [10]:∫S =(B IJ ∧ F IJ − α 4 B IJ ∧ B KL ɛ IJKL5 − β )2 B IJ ∧ B IJ . (25.4)Here F IJ is the curvature of SO(4, 1) connection A IJ ,andB IJ is a two-form valuedin the algebra SO(4, 1). The dimensionless parameters α and β are related togravitational and cosmological constants, and the Immirzi parameter. The α termbreaks the symmetry, and for α ̸= 0 this theory is equivalent to general relativity.