Approaches to Quantum Gravity

Spacetime symmetries in histories canonical gravity 71
which is a time-ordered sequence of properties of the physical system, each one
represented by a single-time projection operator on the standard Hilbert space. The
emphasis 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 histories
where ρ t0 is the initial quantum state and where
d H,ρ (α, β) = tr( ˜C † α ρ t 0
C 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) = 0then
α,βare in the consistent set, which means that we have zero interference between
different histories, and then it is possible to consistently assign probabilities to each
history in that set; it is called a consistent set.
Then we can assign probabilities to each history in the consistent set
d H,ρ (α , α) = Prob(α; ρ t0 ) = tr( ˜C † α ρ t 0
C α ). (5.4)
One of the aims of the histories formalism is to provide a generalised quantum
mechanics definition, so that, one may deal with systems possessing a non-trivial
causal structure, including perhaps Quantum Gravity. In particular, Hartle has provided
examples of how this procedure would work, based mainly on a path integral
expression of the decoherence functional [10].
5.2.2 HPO formalism – basics
In the History Projection Operator (HPO) approach to consistent histories theory
the emphasis is given on the temporal quantum logic. Thus it offers the possibility
of handling the ideas of space and time in a significantly new way within the
quantum theory.
A history is represented by a tensor product of projection operators
ˆα := 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 that
time t i and corresponding to some property of the system at the same time indicated
by the t-label. Therefore – in contrast to the sum over histories formalism – a
history is itself a genuine projection operator defined on the history Hilbert space
V n , which is a tensor product of the single-time Hilbert spaces
V n := H t1 ⊗ H t2 ⊗ ... ⊗ H tn . (5.6)