Approaches to Quantum Gravity

Spacetime symmetries in histories canonical gravity 69determined by its interaction with matter, and he tried to determine its equations ofmotion. Remarkably, these turned out to be almost uniquely fixed by a symmetryrequirement, which became known as the principle of general covariance: the equationsof motion ought to retain their form in any coordinate system associated to themanifold. In modern language, one would say that the equations of motion ought tobe invariant under the action of the group Di f f (M) of passive diffeomorphisms onthe spacetime manifold M. If, however, the equations of motion are invariant underthe action of Di f f (M), they cannot contain any non-dynamical fields, for the latterdo not remain invariant under the action of the Di f f (M) group. Hence, generalcovariance implies that the theory of gravity ought to be background independent,i.e. no fixed externally imposed structures are to be used in the formulation of thetheory’s laws of motion.The standard quantization procedures applied to General Relativity seem to contradictits basic principles. Quantum theory is fundamentally canonical: the Hilbertspace refers to the properties of a system at a single moment of time, hence,manifest covariance is lost at the first step.More importantly, the canonical commutation relations are defined on a ‘spacelike’surface, however, a surface is spacelike with respect to some particularspacetime metric g – which is itself a quantum observable that is expected to fluctuate.The prior definability of the canonical commutation relations is not merelya mathematical requirement. In a generic quantum field theory they implementthe principle of microcausality: namely that field observables that are defined inspacelike separated regions commute. However, if the notion of spacelikeness isalso dynamical, it is not clear in what way this relation will persist.The canonical treatment of Quantum Gravity introduces a spacelike foliation thatenters the quantum description. However, the physical predictions should be independentof the choice of this foliation. This is part of the famous ‘problem of time’,as are attempts to understand the spacetime diffeomorphism group in this context.In one or another form the aforementioned problems persist in the major programmestowards Quantum Gravity, namely canonical quantization and spacetime(perturbative) quantization – see [12] for a related discussion.The histories framework is motivated by the belief that it would be prudentto preserve the basic principles of General Relativity in our attempts to quantizegravity. This reason conveys the importance of a genuine spacetime description ofphysical events.5.1.2 The histories theory programmeThe fundamental entity of the theory is the notion of a history: it corresponds tothe specification of information about the state of a system, at different moments