3.2 Higher-derivative terms

Simplicial analogues of higher-derivative terms were introduced in [117119Jump To The Next Citation Point]. 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
∫ ∑ A2bδ2b 4 √-- μνρσ -V---∼ d x g R μνρσR , (15 ) b b
with V b denoting the local Voronoi four-volume at b. The fact that (15View Equation) should not be identified with ∫ 4 √ -- 2 d x g R is less surprising in light of the classical result [81Jump To The Next Citation Point82Jump To The Next Citation Point], 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 [103101], 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, [174106Jump To The Next Citation Point119Jump To The Next Citation Point55]), but have up to now not been used in numerical simulations. A related proposal by Ambjørn et al. [19Jump To The Next Citation Point] 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 [104Jump To The Next Citation Point176].


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