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 Point104Jump To The Next Citation Point81Jump To The Next Citation Point]. This approach goes back to Regge [174Jump To The Next Citation Point], 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 [168187Jump To The Next Citation Point104Jump To The Next Citation Point124Jump To The Next Citation Point106Jump To The Next Citation Point197190Jump To The Next Citation Point8Jump To The Next Citation Point]; various quantum aspects are reviewed in [106Jump To The Next Citation Point107112110116Jump To The Next Citation Point13340Jump To The Next Citation Point160190Jump To The Next Citation Point198].

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

∫ SRegge[l2] = ∑ V (λ − k Ab-δb) ∼ d4x √g--(λ − k-R), (12 ) i b Vb 2 bonesb
where k = 81πG-, Ab is the area of a triangular bone, ∑ δb = 2π − σ⊃b πœƒ(σ,b) the deficit angle there, and Vb a local four-volume element. πœƒ(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 (12View Equation) to the case of manifolds with boundary. The boundary contribution to the action is given by
∑ ∫ ∘ ---- Abψb ∼ d3x g(3) K, (13 ) b⊂boundary
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
∫ − SRegge[l2] Z (k,λ ) = π’Ÿl e i (14 )
with ∫ π’Ÿl representing the discrete analogue of the sum over all metrics. A crucial input in (14View Equation) is the choice of an appropriate measure π’Ÿl. In general, a cutoff is required for both short and long edge lengths to make the functional integral convergent [104Jump To The Next Citation Point81Jump To The Next Citation Point18]. 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 [181Jump To The Next Citation Point183Jump To The Next Citation Point]. 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] 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] (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.

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