Approaches to Quantum Gravity

Spacetime symmetries in histories canonical gravity 71which is a time-ordered sequence of properties of the physical system, each onerepresented by a single-time projection operator on the standard Hilbert space. Theemphasis is given on histories rather than states at a single time.The probabilities and the dynamics are contained in the decoherence functional,a complex-valued function on the space of historieswhere ρ t0 is the initial quantum state and whered H,ρ (α, β) = tr( ˜C † α ρ t 0C β ), (5.2)˜C α := U(t 0 , t 1 ) ˆα t1 U(t 1 , t 2 )...U(t n−1 , t n ) ˆα tn U(t n , t 0 ) (5.3)is the class operator that represents the history α.When a set of histories satisfies a decoherence condition, d H,ρ (α , β) = 0thenα,βare in the consistent set, which means that we have zero interference betweendifferent histories, and then it is possible to consistently assign probabilities to eachhistory in that set; it is called a consistent set.Then we can assign probabilities to each history in the consistent setd H,ρ (α , α) = Prob(α; ρ t0 ) = tr( ˜C † α ρ t 0C α ). (5.4)One of the aims of the histories formalism is to provide a generalised quantummechanics definition, so that, one may deal with systems possessing a non-trivialcausal structure, including perhaps Quantum Gravity. In particular, Hartle has providedexamples of how this procedure would work, based mainly on a path integralexpression of the decoherence functional [10].5.2.2 HPO formalism – basicsIn the History Projection Operator (HPO) approach to consistent histories theorythe emphasis is given on the temporal quantum logic. Thus it offers the possibilityof handling the ideas of space and time in a significantly new way within thequantum theory.A history is represented by a tensor product of projection operatorsˆα := ˆα t1 ⊗ˆα t2 ⊗ ... ⊗ˆα tn , (5.5)each operator ˆα ti being defined on a copy of the single-time Hilbert space H ti at thattime t i and corresponding to some property of the system at the same time indicatedby the t-label. Therefore – in contrast to the sum over histories formalism – ahistory is itself a genuine projection operator defined on the history Hilbert spaceV n , which is a tensor product of the single-time Hilbert spacesV n := H t1 ⊗ H t2 ⊗ ... ⊗ H tn . (5.6)