Approaches to Quantum Gravity

558 L. Smolinthe Ashtekar formulation, which are the SU(2) L connection A i amomenta Ẽia ≈ e ∧ e this becomesand its conjugateF i ab + 3 ɛ abcẼc i = 0. (28.13)One can solve this with a Hamilton–Jacobi function on configuration space [51;47], which is a function S(A i ) such that Ẽi a = δS(Ai ). This leads to the equationδ A i aFab i + 3 ɛ δS(A i )abc = 0. (28.14)δ A i aThere is also the Gauss’s law constraint which requires thatThese have a unique solutionD a Ẽ a i = D aδS(A i )δ A i aS(A i ) =− k4π∫= 0. (28.15)Y CS (A i ) (28.16)where Y (A i ) is the Chern–Simons invariant. Thus we can consider the Chern–Simons invariant to be a time functional on the Euclidean configuration space.If we choose = S 3 we find that there is a periodicity due to the property thatunder large gauge transformations with winding number n∫∫Y CS (A i ) → Y CS (A i ) + 8π 2 n (28.17)This means that the Euclidean configuration space is a cylinder which furtherimplies that all correlation functions, for any fields in the theory are periodic inan imaginary time variable given by the Chern–Simons functional S(A i ). But bythe KMS theorem, this means that the theory is at a finite temperature. If one worksout the periodicity one finds precisely the temperature (28.12).This applies to the full Quantum Gravity theory because it means that any quantumstate on the full configuration space of the theory will be periodic in imaginarytime. Thus, with very little effort we greatly extend the significance of the de Sittertemperature. This is an example of the power of seeing General Relativity in termsof connection variables and it is also an example of the importance of topologicalfield theory to the physics of Quantum Gravity.28.4 The problem of the emergence of classical spacetimeWe have just seen that LQG gets several things about gravitational physics right,including the entropy of horizons and the temperature of de Sitter spacetime. There