Approaches to Quantum Gravity

Spacetime symmetries in histories canonical gravity 79A function on the full state space represents a physical observable if it is projectableinto a function on red . Hence, it is necessary and sufficient that itcommutes with the constraints on the constraint surface.Contrary to the canonical treatments of parameterised systems, the classicalequations of motion of the histories theory are explicitly realised on the reducedstate space red . Indeed, the equations of motion are the paths on the phase spacethat remain invariant under the symplectic transformations generated by the actionfunctional projected on red{ ˜S, F t } (γ cl ) = 0,where F t is a functional of the field variables and it is constant in t. The path γ cl is asolution of the equations of motion, therefore it corresponds to a spacetime metricthat is a solution of the Einstein equations.The canonical action functional S is also diffeomorphic-invariant{V W , S} =0. (5.14)This is a significant result: it leads to the conclusion that the dynamics of thehistories theory is invariant under the group of spacetime diffeomorphisms.The parameter with respect to which the orbits of the constraints are defined,is not in any sense identified with the physical time t. In particular, one candistinguish the paths corresponding to the equations of motion by the condition{F, Ṽ } γcl = 0.In standard canonical theory, the elements of the reduced state space are all solutionsto the classical equations of motion. In histories canonical theory, however, anelement of the reduced state space is a solution to the classical equations of motiononly if it also satisfies the above condition. The reason for this is that the historiesreduced state space red contains a much larger number of paths, essentiallyall paths on Ɣ red . For this reason, histories theory may naturally describe observablesthat commute with the constraints but which are not solutions to the classicalequations of motion.This last point should be particularly emphasised because of its possible correspondingquantum analogue. We know that in quantum theory, paths may berealised that are not solutions to the equations of motion. The histories formalism,in effect, distinguishes between instantaneous laws [16] (namely constraints), anddynamical laws (equations of motion). Hence, it is possible to have a quantum theoryfor which the instantaneous laws are satisfied, while the classical dynamicallaws are not. This distinction is present, for example, in the history theory of thequantised electromagnetic field [7], where all physical states satisfy the Gauss lawexactly; however, electromagnetism field histories are possible which do not satisfythe dynamical equations.