Higher-derivative terms

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 $V b$ denoting the local Voronoi four-volume at $b$ . The fact that (15

) should not be identified with $∫ 4 √ -- 2 d x g R$ is less surprising in light of the classical result [81

, 82

], that Regge’s expression $Ab δb$ for the scalar curvature (for $d > 2$ ) 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 $ℝN$ ).

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 $δb$ , as well as terms containing powers of the solid angle $δv$ at a vertex. The introduction of local vierbeins and parallel transporters is also necessary if one considers fermion coupling [104 , 176 ].