Approaches to Quantum Gravity

New directions in background independent Quantum Gravity 135A 1=A 2=(9.7)A 3=Fig. 9.1. The three generators of evolution on the network space H. Theyarecalled expansion, contraction and exchange moves.where the sum is over all topologically distinct embeddings of all such networks in with the natural inner product 〈S i |S i ′〉=δ Si S i ′ .Local dynamics on H can be defined by excising pieces of S and replacing themwith new ones with the same boundary [21; 26]. The generators of such dynamicsare given graphically in Fig. 9.1. Given a network S, application of A i results in i |S〉 = ∑ α|S αi ′ 〉, (9.8)where S ′ αiare all the networks obtained from S by an application of one move oftype i (i = 1, 2, 3). Together with the identity 1, these moves generate the evolutionalgebraA evol = {1, A i } , i = 1, 2, 3 (9.9)on H.Finally, changing the network S by the above local moves produces a directedgraph Ɣ. The vertices of S are also the vertices of Ɣ. The generator moves correspondto complete pairs and hence unitary operators, however, the operatorsbetween individual vertices are CP and the resulting system of locally evolvingnetworks is a Quantum Causal History. For example, in this change of S to S ′we have operated with A 3 between complete pair sets ξ and ζ and with A 1 betweencomplete pairs η and ɛ.Themapfromx to y is a CP map.