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

Loop Quantum Cosmology 391states and basic operators, while the Hamiltonian constraint can be obtainedwith constructions analogous to those in the full theory. Betweenthe dynamics of models and the full theory there is thus no complete linkyet and not all ingredients of models have been derived so far. But the formulationof quantum gravity in a background independent manner impliescharacteristic properties which are also realized in models. This allows usto reconsider the singularity problem, now with methods from full quantumgravity. In fact, symmetric models present a class of systems whichcan often be treated explicitly while still being representative for generalphenomena. For instance, the prime examples of singular situations in gravity,and some of the most widely studied physical applications, are alreadyobtained in isotropic or spherically symmetric systems, which allow accessto cosmology and black holes.4.1. RepresentationBefore discussing the quantum level we reformulate isotropic cosmology inconnection and triad variables instead of a. The role of the scale factoris now played by the triad component p with |p| = a 2 whose canonicalmomentum is the isotropic connection component c = − 1 2ȧ with {c, p} =8πG/3. The main difference to metric variables is the fact that p, unlikea, can take both signs with sgnp being the orientation of space. This is aconsequence of having to use triad variables which not only know aboutthe size of space but also its orientation (depending on whether the set oforthonormal vectors is left or right handed).States in the full theory are usually written in the connection representationas functions of holonomies. Following the reduction procedure for anisotropic symmetry group leads to orthonormal states which are functionsof the isotropic connection component c and given by 36〈c|µ〉 = e iµc/2 µ ∈ R . (7)On these states the basic variables p and c are represented bywith the properties:ˆp|µ〉 = 1 6 l2 P µ|µ〉 (8)ê iµ′ c/2|µ〉 = |µ + µ ′ 〉 (9)(i) [êiµ′ c/2, ˆp] = − 1 6 l2 P µ′ e ̂ −iµ′ c/2= i({e iµ′ c/2 , p}) ∧ ,(ii) ˆp has a discrete spectrum and