3.2 Higher-derivative terms3 Quantum Regge Calculus3 Quantum Regge Calculus

3.1 Path integral for Regge calculus

A path-integral quantization of 4d Regge calculus was first considered in the early eighties [181Jump To The Next Citation Point In The Article, 104Jump To The Next Citation Point In The Article, 81Jump To The Next Citation Point In The Article]. This approach goes back to Regge [174Jump To The Next Citation Point In The Article], 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 tex2html_wrap_inline2427 . The metric tensor associated with each simplex is expressed as a function of the squared edge lengths tex2html_wrap_inline2429 of tex2html_wrap_inline2427, which are the dynamical variables of this model. For introductory material on classical Regge calculus and simplicial manifolds, see [168, 187Jump To The Next Citation Point In The Article, 104Jump To The Next Citation Point In The Article, 124Jump To The Next Citation Point In The Article, 106Jump To The Next Citation Point In The Article, 197, 190Jump To The Next Citation Point In The Article, 8Jump To The Next Citation Point In The Article]; various quantum aspects are reviewed in [106Jump To The Next Citation Point In The Article, 107, 112, 110, 116Jump To The Next Citation Point In The Article, 133, 40Jump To The Next Citation Point In The Article, 160, 190Jump To The Next Citation Point In The Article, 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 tex2html_wrap_inline2427, and singular curvature assignments to its two-simplices b (the bones or hinges).

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

  equation289

where tex2html_wrap_inline2437, tex2html_wrap_inline2439 is the area of a triangular bone, tex2html_wrap_inline2441 the deficit angle there, and tex2html_wrap_inline2443 a local four-volume element. tex2html_wrap_inline2445 is the angle between the two 3-simplices of tex2html_wrap_inline2427 (tex2html_wrap_inline2449 minus the angle between their inward normals) intersecting in b . Hartle and Sorkin [128] have generalized (12Popup Equation) to the case of manifolds with boundary. The boundary contribution to the action is given by

  equation312

where tex2html_wrap_inline2453 is the angle between the normals of the two three-simplices meeting at b . The Euclidean path integral on a finite simplicial complex of fixed connectivity takes the form

  equation321

with tex2html_wrap_inline2457 representing the discrete analogue of the sum over all metrics. A crucial input in (14Popup Equation) is the choice of an appropriate measure tex2html_wrap_inline2459 . In general, a cutoff is required for both short and long edge lengths to make the functional integral convergent [104Jump To The Next Citation Point In The Article, 81Jump To The Next Citation Point In The Article, 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 Rocek and Williams [181Jump To The Next Citation Point In The Article, 183Jump To The Next Citation Point In The Article]. 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 [101Jump To The Next Citation Point In The Article] 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 [183Jump To The Next Citation Point In The Article] (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.



3.2 Higher-derivative terms3 Quantum Regge Calculus3 Quantum Regge Calculus

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