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

388 M. Bojowaldt. The latter fact is a consequence of general covariance: the Hamiltonianis a constraint equation restricting allowed states ψ(a, φ), rather than aHamiltonian generating evolution in coordinate time. Nevertheless, one caninterpret equation (4) as an evolution equation in the scale factor a, whichis then called internal time. The left hand side thus becomes a second ordertime derivative, and it means that the evolution of matter is measuredrelationally with respect to the expansion or contraction of the universe,rather than absolutely in coordinate time.Straightforward quantization thus gives us a quantum evolution equation,and we can now check what this implies for the singularity. If we lookat the equation for a = 0, we notice first that the matter Hamiltonian stillleads to diverging energy densities. If we quantize (2), we replace p φ by aderivative, but the singular dependence on a does not change; a −3 wouldsimply become a multiplication operator acting on the wave function. Moreover,a = 0 remains a singular point of the quantum evolution equation ininternal time. There is nothing from the theory which tells us what physicallyhappens at the singularity or beyond (baring intuitive pictures whichhave been developed from this perspective 12,13 ).So one has to ask what went wrong with our expectations that quantizinggravity should help. The answer is that quantum theory itself did notnecessarily fail, but only our simple implementation. Indeed, what we usedwas just quantum mechanics, while quantum gravity has many consistencyconditions to be fulfilled which makes constructing it so complicated. At thetime when this formalism was first applied there was in fact no correspondingfull quantum theory of gravity which could have guided developments.In such a simple case as isotropic cosmology, most of these consistency conditionstrivialize and one can easily overlook important issues. There aremany choices in quantizing an unknown system, and tacitly making onechoice can easily lead in a wrong direction.Fortunately, the situation has changed with the development of strongcandidates for quantum gravity. This then allows us to reconsider the singularityand other problems from the point of view of the full theory, makingsure that also in a simpler cosmological context only those steps are undertakenthat have an analog in the full theory.3.3. Loop quantum gravitySingularities are physically extreme and require special properties of anytheory aimed at tackling them. First, there are always strong fields (clas-