Path integral for Regge calculus

3.1 Path integral for Regge calculus

A path-integral quantization of 4d Regge calculus was first considered in the early eighties [181

, 104

, 81

]. This approach goes back to Regge [174

], 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 $σ$ . The metric tensor associated with each simplex is expressed as a function of the squared edge lengths $l2$ of $σ$ , which are the dynamical variables of this model. For introductory material on classical Regge calculus and simplicial manifolds, see [168 , 187

, 104

, 124

, 106

, 197 , 190

, 8

]; various quantum aspects are reviewed in [106

, 107 , 112 , 110 , 116

, 133 , 40

, 160 , 190

, 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 $σ$ , 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

where $k = 81πG-$ , $Ab$ is the area of a triangular bone, $∑ δb = 2π − σ⊃b 𝜃(σ,b)$ the deficit angle there, and $Vb$ a local four-volume element. $𝜃(σ,b)$ is the angle between the two 3-simplices of $σ$ ( $π$ minus the angle between their inward normals) intersecting in $b$ . Hartle and Sorkin [128 ] have generalized (12

) to the case of manifolds with boundary. The boundary contribution to the action is given by

where $ψb$ 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

with $∫ 𝒟l$ representing the discrete analogue of the sum over all metrics. A crucial input in (14

) is the choice of an appropriate measure $𝒟l$ . In general, a cutoff is required for both short and long edge lengths to make the functional integral convergent [104

, 81

, 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 Roček and Williams [181 , 183 ]. 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 [101 ] 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 [183 ] (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.