Approaches to Quantum Gravity

New directions in background independent Quantum Gravity 137Definition 4 Background independence I (BI-I): a theory is background independentif its basic quantities and concepts do not presuppose the existence of a givenbackground spacetime metric.All well-developed background independent approaches to Quantum Gravitysuch as loop quantum gravity [33][34], causal sets [6], spin foams [31][32][26][3][29] and Oriti, this volume, causal dynamical triangulations [1][2], or quantumRegge calculus [30], implement background independence as a special case ofthe above by quantum analogy to the classical theory:Definition 5 Background independence II (BI-II): a background independent theoryof quantum geometry is characterized by (a) quantum geometric microscopicdegrees of freedom or a regularization of the microscopic geometry and (b) a quantumsum-over-histories of the allowed microscopic causal histories (or equivalenthistories in the Riemannian approaches).Recently, new approaches to Quantum Gravity have been proposed that satisfyBI-I but not BI-II: the computational universe [19], internal relativity ([8]and Dreyer, this volume) and quantum graphity [14]. More specifically, Dreyeradvocates the following.Definition 6 Background independence (Dreyer): a theory is background independentif all observations are internal, i.e. made by observers inside the system.Note that this is a natural condition for a cosmological theory as has also beenpointed out in [23].In summary, what constitutes a background independent theory is a questionthat is currently being revisited and new, on occasion radical, suggestions havebeen offered. These are opening up new exciting avenues in Quantum Gravityresearch and will be our focus in this chapter. In order to discuss them in somedetail, however, we shall give examples of each in the unifying context of QCH.9.4 QCH as a discrete Quantum Field TheoryThere is substantial literature in Quantum Gravity and high energy physics thatpostulates that in a finite region of the universe there should be only a finite numberof degrees of freedom, unlike standard Quantum Field Theory where we have aninfinite number of degrees of freedom at each spacetime point. This is supported byBekenstein’s argument, the black hole calculations in both string theory and loopquantum gravity and is related to holographic ideas.It has been suggested that such a locally finite version of Quantum Field Theoryshould be implemented by a many-Hilbert space theory (as opposed to the single