Approaches to Quantum Gravity

Quantum Regge calculus 363group. We conclude that it is only infinitesimal conformal transformations whichare well-defined so far in Regge calculus.A final point in this introductory section is the existence of a continuum limit.Cheeger, Müller and Schrader [14; 15] showed rigorously that the Regge actionconverges to the continuum action, in the sense of measures, provided that certainconditions on the flatness of the simplices are satisfied. From the opposite perspective,Friedberg and Lee [22] derived the Regge action from the continuum in acertain limit. Rather than considering the action, Barrett [4; 5] explored the relationshipbetween the Regge variational equations and Einstein’s equations, and setup a criterion [6] for solutions of the linearised Regge equations to converge toanalytic solutions of the linearised Einstein equations.Regge calculus has been used in many aspects of classical general relativity, butthat is not our concern here. We now consider various ways in which it has beenused in Quantum Gravity. Most use the sum over histories approach to calculatethe partition function or transition amplitude, although of course it is also possibleto use the canonical approach, as will be seen in the penultimate section.19.2 The earliest quantum Regge calculus: the Ponzano–Regge modelThe first application of Regge calculus to Quantum Gravity came about in a ratherunexpected way. In a paper on the asymptotic behaviour of 6 j-symbols, Ponzanoand Regge [58] formulated the following model. Triangulate a 3-manifold, andlabel each{edge with a}representation of SU(2), j i ={0, 1/2, 1, ...}. Assign a 6 j-j1 jsymbol, 2 j 3(a generalised Clebsch–Gordan coefficient, which relatesj 4 j 5 j 6bases of states when three angular momenta are added) to each tetrahedron. Formthe state sumZ = ∑ ∏∏(2 j i + 1)(−1) χ {6 j}, (19.5)j i itetrahedrawhere the χ in the phase factor is a function of the j i . This sum is infinite in manycases, but it has some very interesting properties. In particular, the semi-classicallimit exhibits a connection with Quantum Gravity. The edge lengths can be thoughtof as ( j i + 1/2), and the limit is obtained by keeping these quantities finite while tends to zero and j i tends to infinity. Ponzano and Regge showed that, for largej i , the asymptotic behaviour of the 6 j-symbol is{ }( )j1 j 2 j 3∼ 1 ∑√ cos jj 4 j 5 j i θ i + π/4 , (19.6)6 12π Vi