Approaches to Quantum Gravity

74 N. SavvidouThe novel feature in this construction is the definition of the ‘Liouville’ operatorˆV , which generates transformations with respect to the time label t as it appears inthe history algebra, hence, t is the label of temporal logic or the label of kinematicse i τ ˆV ˆx f (s)e − i τ ˆV =ˆx f ′(s),f ′ (t) = f (t + τ).We must emphasise the distinction between the notion of time evolution fromthat of logical time-ordering. The latter refers to the temporal ordering of logicalpropositions in the consistent histories formalism. The corresponding parameter tdoes not coincide with the notion of physical time – as it is measured for instanceby a clock. It is an abstraction, which keeps from physical time only its orderingproperties, namely that it designates the sequence at which different events happen– the same property that it is kept by the notion of a time-ordered product inquantum field theory. Making this distinction about time, it is natural to assumethat in the HPO histories one may not use the same label for the time evolutionof physical systems and the time-ordering of events. The former concept incorporatesalso the notion of a clock, namely it includes a measure of time duration, assomething distinct from temporal ordering.The realisation of this idea on the notion of time was possible in this particularframework because of the logical structure of the theory, as it was originallyintroduced in the consistent histories formalism and as it was later recovered astemporal logic in the HPO scheme. One may say then that the definition of thesetwo operators, V and H, implementing time translations, signifies the distinctionbetween the kinematics and the dynamics of the theory.However, a crucial result of the theory is that Ŝ κ is the physical generator of thetime translations in histories theory, as we can see from the way it appears in thedecoherence functional and hence the physical predictions of the theory.Relativistic quantum field theoryIn the classical histories theory, the basic mathematical entity is the space of differentiablepaths ={γ | γ : R → Ɣ}, taking their value in the space Ɣ of classicalstates. The key idea in this new approach to classical histories is contained in thesymplectic structure on this space of temporal paths. In analogy to the quantumcase, there are generators for two types of time transformation: one associated withclassical temporal logic, and one with classical dynamics. One significant featureis that the paths corresponding to solutions of the classical equations of motionare determined by the requirement that they remain invariant under the symplectictransformations generated by the action.Starting from the field theory analogue of the Eq. (5.7), the relativistic analogueof the two types of time translation in a non-relativistic history theory is the existenceof two distinct Poincaré groups. The ‘internal’ Poincaré group is analogous