3.3 First simulations3 Quantum Regge Calculus3.1 Path integral for Regge

3.2 Higher-derivative terms

Simplicial analogues of higher-derivative terms were introduced in [117, 119Jump To The Next Citation Point In The Article]. 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

  equation339

with tex2html_wrap_inline2443 denoting the local Voronoi four-volume at b . The fact that (15Popup Equation) should not be identified with tex2html_wrap_inline2465 is less surprising in light of the classical result [81Jump To The Next Citation Point In The Article, 82Jump To The Next Citation Point In The Article], that Regge's expression tex2html_wrap_inline2467 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 tex2html_wrap_inline2471).

More complicated higher-curvature terms can in principle be constructed, using a simplicial analogue of the Riemann tensor (see, for example, [174, 106Jump To The Next Citation Point In The Article, 119Jump To The Next Citation Point In The Article, 55]), but have up to now not been used in numerical simulations. A related proposal by Ambjørn et al [19Jump To The Next Citation Point In The Article] is to include terms in the action that depend on higher powers of the deficit angle tex2html_wrap_inline2473, as well as terms containing powers of the solid angle tex2html_wrap_inline2475 at a vertex. The introduction of local vierbeins and parallel transporters is also necessary if one considers fermion coupling [104Jump To The Next Citation Point In The Article, 176].



3.3 First simulations3 Quantum Regge Calculus3.1 Path integral for Regge

image Discrete Approaches to Quantum Gravity in Four Dimensions
Renate Loll
http://www.livingreviews.org/lrr-1998-13
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