Approaches to Quantum Gravity

Quantum Regge calculus 363
group. We conclude that it is only infinitesimal conformal transformations which
are 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 action
converges to the continuum action, in the sense of measures, provided that certain
conditions on the flatness of the simplices are satisfied. From the opposite perspective,
Friedberg and Lee [22] derived the Regge action from the continuum in a
certain limit. Rather than considering the action, Barrett [4; 5] explored the relationship
between the Regge variational equations and Einstein’s equations, and set
up a criterion [6] for solutions of the linearised Regge equations to converge to
analytic solutions of the linearised Einstein equations.
Regge calculus has been used in many aspects of classical general relativity, but
that is not our concern here. We now consider various ways in which it has been
used in Quantum Gravity. Most use the sum over histories approach to calculate
the partition function or transition amplitude, although of course it is also possible
to use the canonical approach, as will be seen in the penultimate section.
19.2 The earliest quantum Regge calculus: the Ponzano–Regge model
The first application of Regge calculus to Quantum Gravity came about in a rather
unexpected way. In a paper on the asymptotic behaviour of 6 j-symbols, Ponzano
and Regge [58] formulated the following model. Triangulate a 3-manifold, and
label each
{
edge with a
}
representation of SU(2), j i ={0, 1/2, 1, ...}. Assign a 6 j-
j1 j
symbol, 2 j 3
(a generalised Clebsch–Gordan coefficient, which relates
j 4 j 5 j 6
bases of states when three angular momenta are added) to each tetrahedron. Form
the state sum
Z = ∑ ∏
∏
(2 j i + 1)(−1) χ {6 j}, (19.5)
j i i
tetrahedra
where the χ in the phase factor is a function of the j i . This sum is infinite in many
cases, but it has some very interesting properties. In particular, the semi-classical
limit exhibits a connection with Quantum Gravity. The edge lengths can be thought
of 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 large
j i , the asymptotic behaviour of the 6 j-symbol is
{ }
( )
j1 j 2 j 3
∼ 1 ∑
√ cos j
j 4 j 5 j i θ i + π/4 , (19.6)
6 12π V
i