100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

362 A. Ashtekarwithout any reference to a background classical geometry. It is not rigidlytied to general relativity. However, for reasons explained in section 2.1, sofar most of the detailed work on quantum dynamics is restricted to generalrelativity, where it provides tools to write down quantum Einstein’s equationsin the Hamiltonian approach and calculate transition amplitudes inthe path integral approach. Until recently, effort was focussed primarily onHamiltonian methods. However, over the last five years or so, path integrals—called spin foams— have drawn a great deal of attention. This work hasled to fascinating results suggesting that, thanks to the fundamental discretenessof quantum geometry, path integrals defining quantum generalrelativity may be finite. A summary of these developments can be found inreferences[20,25]. In this Section, I will summarize the status of the Hamiltonianapproach. For brevity, I will focus on source-free general relativity,although there has been considerable work also on matter couplings. 24,25,38For simplicity, let me suppose that the ‘spatial’ 3-manifold M is compact.Then, in any theory without background fields, Hamiltonian dynamicsis governed by constraints. Roughly this is because in these theories diffeomorphismscorrespond to gauge in the sense of Dirac. Recall that, on theMaxwell phase space, gauge transformations are generated by the functionalD a E a which is constrained to vanish on physical states due to Gausslaw. Similarly, on phase spaces of background independent theories, diffeomorphismsare generated by Hamiltonians which are constrained to vanishon physical states.In the case of general relativity, there are three sets of constraints. Thefirst set consists of the three Gauss equationsG i := D a E a i= 0, (2)which, as in Yang-Mills theories, generates internal SU(2) rotations on theconnection and the triad fields. The second set consists of a co-vector (ordiffeomorphism) constraintC b := E a i F i ab = 0, (3)which generates spatial diffeomorphism on M (modulo internal rotationsgenerated by G i ). Finally, there is the key scalar (or Hamiltonian) constraintS := ɛ ijk E a i Eb j F ab k + . . . = 0 (4)which generates time-evolutions. (The . . . are extrinsic curvature terms, expressibleas Poisson brackets of the connection, the total volume constructedfrom triads and the first term in the expression of S given above. We will