Approaches to Quantum Gravity

70 N. Savvidouof time. The development of the Histories Projection Operator (HPO) approach inparticular, showed that it is characterised by two distinctive features. First, a historyis a temporally extended object that it is represented quantum mechanically by asingle projection operator, on a suitably constructed Hilbert space [12]. Second,the theory possesses a novel temporal structure [21], since time is implemented bytwo distinct parameters, one of which refers to the kinematics of the theory, whilethe other refers to its dynamical behavior. At the classical level the two parameterscoincide for all histories that correspond to the solutions of the equations of motion.Hence, the HPO theory is endowed with a rich kinematical structure. In the caseof General Relativity this results in the fact that different ‘canonical’ descriptionsof the theory, corresponding to different choices of spacelike foliation, coexist inthe space of histories and may be related by a properly defined transformation.This allows the preservation of the spacetime description of the theory, even if onechooses to work with canonical variables.The General Relativity histories theory suggests a quantum mechanical treatmentof the full Lorentzian metric. Other programmes also put emphasis on thespacetime description, namely the causal set approach [6], and the Lorentziandynamical triangulations [1]. The twistor programme has the same avowed aims.The HPO formalism, however, allows the incorporation of other theories, enrichingthem with a spacetime kinematical description, while preserving the main featuresof their dynamical behavior.A histories-based quantisation of General Relativity, like the canonical theory,has to address the issue of defining an appropriate Hamiltonian constraint operator.Loop quantum gravity has made the greatest progress so far in the construction ofsuch an operator, therefore it would be very interesting to exploit a histories versionof this theory.5.2 History Projection Operator theory5.2.1 Consistent histories theoryThe consistent histories formalism was originally developed by Griffiths [9] andOmnés [17; 18], as an interpretation of quantum theory for closed systems.Gell-Mann and Hartle [8] elaborated this scheme in the case of quantum cosmology– the Universe being regarded as a closed system. They emphasised inparticular that a theory of Quantum Gravity that is expected to preserve the spacetimecharacter of General Relativity would need a quantum formalism in which theirreducible elements are temporally extended objects, namely histories.The basic object in the consistent histories approach is a historyα := ( ˆα t1 , ˆα t2 , ..., ˆα tn ), (5.1)