Approaches to Quantum Gravity

136 F. Markopoulou9.2.2 The meaning of ƔAt this stage we have said nothing about the physical interpretation of Ɣ or theindividual quantum systems A(x) on its vertices. While Ɣ has the same propertiesas a causal set, 4 i.e. the discrete analog of a Lorentzian spacetime, it does not haveto be one. For example, in the circuit model of quantum computation, a circuit,that is, a collection of gates and wires also has the same properties as Ɣ and simplyrepresents a sequence of information transfer which may or may not be connectedto spatiotemporal motions (see [27], p. 129).We shall use this flexibility of the QCH to illustrate both the difference betweena background dependent and a background independent system as well as the distinctionbetween background independent theories of quantum geometry and a newset of pre-geometric theories that have been recently proposed. In what follows, weshall see that three different interpretations of Ɣ and the A(x)s give three differentsystems. (1) A discrete version of algebraic Quantum Field Theory, when Ɣ isa discretization of a Lorentzian spacetime and A(x) is matter on it. (2) A causalspin foam, i.e. a background independent theory of quantum geometry. Here Ɣ is alocally finite analog of a Lorentzian spacetime and the A(x) contain further quantumgeometric degrees of freedom. Such a theory is background independent whenwe consider a quantum superposition of all Ɣs. (3) A pre-geometric backgroundindependent theory, when neither Ɣ nor the A(x)s have geometric information. Thepossibility that such a system, with a single underlying graph Ɣ may be backgroundindependent has only recently been raised and explored.We shall discuss each of these three possibilities in detail in the rest of thischapter, starting with the necessary definitions of background independence, next.9.3 Background independenceBackground independence (BI) is thought to be an important part of a quantumtheory of gravity since it is an important part of the classical theory. 5 Backgroundindependence in General Relativity is the fact that physical quantities are invariantunder spacetime diffeomorphisms. There is no definite agreement on the form thatBI takes in Quantum Gravity. Stachel gives the most concise statement of backgroundindependence: “In a background independent theory there is no kinematicsindependent of dynamics.”In this chapter, we shall need to discuss specific aspects of background independenceand to aid clarity we give the following definitions that we shall use.4 See [6].5 See [7; 36; 35].