### 3.1 Path integral for Regge calculus

A path-integral quantization of 4d Regge calculus was first considered in the early eighties [181, 104, 81].
This approach goes back to Regge [174], who proposed approximating Einstein’s continuum theory by a
simplicial discretization of the metric space-time manifold and the gravitational action. Its
local building blocks are four-simplices . The metric tensor associated with each simplex
is expressed as a function of the squared edge lengths of , which are the dynamical
variables of this model. For introductory material on classical Regge calculus and simplicial
manifolds, see [168, 187, 104, 124, 106, 197, 190, 8]; various quantum aspects are reviewed
in [106, 107, 112, 110, 116, 133, 40, 160, 190, 198].
One may regard a Regge geometry as a special case of a continuum Riemannian manifold, a so-called
piecewise flat manifold, with a flat metric in the interior of its 4-simplices , and singular curvature
assignments to its two-simplices (the bones or hinges).

The Einstein action with cosmological term in the Regge approach is given by

where , is the area of a triangular bone, the deficit angle there, and
a local four-volume element. is the angle between the two 3-simplices of ( minus the
angle between their inward normals) intersecting in . Hartle and Sorkin [128] have generalized (12)
to the case of manifolds with boundary. The boundary contribution to the action is given by
where is the angle between the normals of the two three-simplices meeting at . The Euclidean path
integral on a finite simplicial complex of fixed connectivity takes the form
with representing the discrete analogue of the sum over all metrics. A crucial input in (14) is the
choice of an appropriate measure . In general, a cutoff is required for both short and long edge lengths
to make the functional integral convergent [104, 81, 18]. One is interested in the behaviour
of expectation values of local observables as the simplicial complex becomes large, and the
existence of critical points and long-range correlations, in a scaling limit and as the cutoffs are
removed.
A first implementation of these ideas was given by RoΔek and Williams [181, 183]. They obtained a
simplicial lattice geometry by subdividing each unit cell of a hypercubic lattice into simplices. Their main
result was to rederive the continuum free propagator (see also [101] for related results) in the
limit of weak perturbations about flat space. This calculation can be repeated for Lorentzian
signature [196].

Some non-perturbative aspects of the path integral were investigated in [183] (see also [182]). In this
work, discrete analogues of space-time diffeomorphisms are defined as the local link length transformations
which leave the action invariant, and go over to translations in the flat case. It is argued that
an approximate invariance should exist in 4d. One may define analogues of local conformal
transformations on a simplicial complex by multiplication with a positive scale factor at each vertex,
but the global group property is incompatible with the existence of the generalized triangle
inequalities.