Approaches to Quantum Gravity

Emergent relativity 103It is here that a discrete form of background time makes its appearance. The individualU l s appear in a definite order given by the parameter l. We will see thoughthat this time is not related to the time as perceived by an observer in the model.Without restriction we can assume that the U l s have the formU l = e −iθ l P(7.9)= (1 − P) + e −iθ lP, (7.10)for a projection operator P = P 2 . We can represent such U l s as in figure 7.3A.Thetwo parts of equation 7.10 can be given a physical interpretation. In the subsystemthat P projects onto the two qubits that U is acting on scatter. This results in aphase shift of θ l . In the orthogonal subspace the two qubits do not scatter. Herethere is no phase shift. The whole unitary U can now be written as followsn∏U = ((1 − P l ) + e −iθ lP l ) (7.11)=l=1∑b 1 ,...,b n ∈{0,1}e −i ∑ nl=1 b l θ lP n (b n ) ···P 1 (b 1 ), (7.12)A(1–P) + e –iθ l P+BCθ 5θ 6θ lθ 4θθ 3θ 1 θ 2Fig. 7.3. A The unitary U l consists of two parts. On the right the two qubits scatteroff each other, giving rise to a phase θ l . On the right the qubits miss each other.B The different U l , l = 1,...,n, giveriseto2 n different possible computationalhistories. Each history consists of a causal set and a set of phases θ l . C The twoincoming and outgoing qubits give four directions on the light cone at a node ofthe causal set. These four null directions determine four of the ten components ofthe metric.