Approaches to Quantum Gravity

80 N. Savvidou5.4 A spacetime approach to Quantum Gravity theory5.4.1 MotivationThe histories approach to General Relativity suggests a new, spacetime-focussed,approach to Quantum Gravity, characterized by two features that are not implementedin the existing Quantum Gravity schemes.First, the Lorentzian metric is quantized, analogous to the ‘external’ quantumfield in the history approach to scalar quantum field theory [22; 14]. This contrastswith conventional canonical Quantum Gravity where only a spatial three-metricis quantised. Second, the history scheme incorporates general covariance via amanifest representation of the spacetime diffeomorphism group.The canonical quantisation scheme was originally developed with the hope ofproviding a background independent formulation of Quantum Gravity. The generalprocedure involves (i) a 3 + 1 splitting of spacetime; (ii) the construction of asuitable Hilbert space to accommodate the basic kinematical quantities of the theory;and (iii) the definition of self-adjoint operators that represent the Hamiltonianconstraints. The imposition of the constraints on the state vectors then projects outthe physical degrees of freedom.The canonical treatment of Quantum Gravity introduces a spacelike foliationthat enters the quantum description. However, the physical predictions should beindependent of the choice of this foliation. This is part of the famous ‘problem oftime’, as are attempts to understand the spacetime diffeomorphism group in thiscontext. These issues are significantly addressed by the histories formulation withits genuine spacetime description of physical quantities.The definition of the history group provides the HPO formalism with a quantisationscheme that follows the general lore of canonical quantisation, providinghowever a fully covariant description – see for example the quantum treatment ofminisuperspace models in [3].The obvious technical problem in a histories-based quantisation is the rigorousimplementation of the dynamics by a history analogue of the Hamiltonianconstraint operator. As in standard canonical theory, the classical expression isnon-quadratic – indeed non-polynomial – in the field variables, and so the constructionof an operator for the Hamiltonian constraint seems a hopeless task usingconventional methods. For this reason, we intend to exploit the basic ideas of loopquantum gravity, which has been a promising approach for the construction thisoperator.5.4.2 Towards a histories analogue of loop quantum gravityLoop quantum gravity is a successful canonical theory in many respects. The basicalgebra is defined with reference to objects that have support on loops in the threedimensionalsurface .