Approaches to Quantum Gravity

134 F. MarkopoulouNote that completely positive maps between algebras go in the reverse directionto the edges of the graph. This is as usual for maps between states (forward) andbetween operators (pullbacks).The above axioms ensure that the actual relations between the vertices of a givengraph are reflected in the operators of the QCH. 3 Furthermore, as shown in [11], ifwe are given the CP maps on the edges, these axioms mean that unitary operationswill be found at the right places: interpolating between complete pairs. Whenξand ζ are a complete pair, we can regard the subgraph that interpolates between ξand ζ as the evolution of an isolated quantum system. We would expect that in thiscase the composite of the individual maps between ξ and ζ is unitary and indeedthe above axioms ensure that this is the case.9.2.1 Example: locally evolving networks of quantum systemsPossibly the most common objects that appear in background independent theoriesare networks. Network-based, instead of metric-based, theories are attractiveimplementations of the relational content of diffeomorphism invariance: it is theconnectivity of the network (relations between the constituents of the universe)that matter, not their distances or metric attributes. We shall use a very simplenetwork-based system as a concrete example of a QCH.We start with a network S of n = 1,...,N nodes, each with three edges attachedto it, embedded in a topological three-dimensional space (no metric on ). Thenetwork S is not to be confused with the graph Ɣ, it is changes of S that will giverise to Ɣ. A map from S to a quantum system can be made by associating a finitedimensionalstate space H n to each minimal piece of S, namely, one node and threeopen edges:H n = (9.4)Two such pieces of S with no overlap are unrelated and thus the state space of theentire network S is the tensor product over all the constituents,H S = ⊗ n∈SH n , (9.5)and the state space of the theory isH = ⊕ S iH Si , (9.6)3 Very interesting recent results of Livine and Terno [18] further analyze and constrain the allowed graphstructure to take into account the quantum nature of the physical information flow represented.