Approaches to Quantum Gravity

Spacetime symmetries in histories canonical gravity 73therefore the definition of the time-averaged energy operator H is crucial for theformalism.5.2.3 Time evolution – the action operatorThe introduction of the history group allowed the definition of continuous-timehistories; however, any notion of dynamics was lost and the theory was put onhold. The situation changed after the introduction of a new idea concerning thenotion of time: the distinction between dynamics and kinematics corresponds to themathematical distinction between the notion of ‘time evolution’ from that of ‘timeordering’ or ‘temporal logic time’. The distinction proved very fruitful, especiallyfor the histories General Relativity theory.The crucial step in the identification of the temporal structure was the definitionof the action operator S [21], a quantum analogue of the Hamilton–Jacobifunctional, written for the case of a one-dimensional simple harmonic oscillator asS κ :=∫ +∞−∞dt (p t ẋ t − κ(t)H t ), (5.11)where κ(t) is an appropriate test function. The results can be generalised appropriatelyfor other systems.The first term of the action operator S κ is identical to the kinematical part ofthe classical phase space action functional. This ‘Liouville’ operator is formallywritten asV =∫ ∞−∞so that S κ = V − H κ . The ‘average-energy’ operatorĤ κ =∫ ∞−∞dt (p t ẋ t ) (5.12)dt κ(t)Ĥ t ; H t = p2 t2m + mω22 x 2 tis also smeared in time by smearing functions κ(t). The Hamiltonian operatormay be employed to define Heisenberg picture operators for the smeared operatorslike x fˆx f (s) := e i s Ĥ ˆx f e − i s Ĥwhere f = f (t) is a smearing function. Hence Ĥ κ generates transformations withrespect to the Heisenberg picture parameter s, therefore, s is the time label as itappears in the implementation of dynamical lawse i τ Ĥ ˆx f (s) e − i τ Ĥ =ˆx f (s+τ).