### 3.2 Higher-derivative terms

Simplicial analogues of higher-derivative terms were introduced in [117, 119]. In the continuum, with an
appropriate choice of coupling constants, their inclusion makes the path integral less ill-behaved. The
simplest higher-derivative term in Regge calculus is given by
with denoting the local Voronoi four-volume at . The fact that (15) should not be identified with
is less surprising in light of the classical result [81, 82], that Regge’s expression
for the scalar curvature (for ) converges to its continuum counterpart not pointwise,
but only after integration, i.e., “in the sense of measures” (see also [103, 101], where similar
convergence properties were studied by using an imbedding into a sufficiently large vector space
).
More complicated higher-curvature terms can in principle be constructed, using a simplicial analogue of
the Riemann tensor (see, for example, [174, 106, 119, 55]), but have up to now not been used in numerical
simulations. A related proposal by Ambjørn et al. [19] is to include terms in the action that depend on
higher powers of the deficit angle , as well as terms containing powers of the solid angle at a
vertex. The introduction of local vierbeins and parallel transporters is also necessary if one considers
fermion coupling [104, 176].