Recent Progress in Regge Calculus Ruth M Williamsab ∗ Adamtp, Silver Street, Cambridge CB3 9EW, United Kingdom Bgirton College, Cambridge CB3 0JG, United Kingdom

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Recent Progress in Regge Calculus Ruth M Williamsab ∗ aDAMTP, Silver Street, Cambridge CB3 9EW, United Kingdom bGirton College, Cambridge CB3 0JG, United Kingdom

While there has been some advance in the use of Regge calculus as a tool in numerical relativity, the main progress in Regge calculus recently has been in quantum gravity. After a brief discussion of this progress, attention is focussed on two particular, related aspects. Firstly, the possible definitions of diffeomorphisms or gauge transformations in Regge calculus are examined and examples are given. Secondly, an investigation of the signature of the simplicial supermetric is described. This is the Lund–Regge metric on simplicial configuration space and defines the distance between simplicial three–geometries. Information on its signature can be used to extend the rather limited results on the signature of the supermetric in the continuum case. This information is obtained by a combination of analytic and numerical techniques. For the three–sphere and the three–torus, the numerical results agree with the analytic ones and show the existence of degeneracy and signature change. Some “vertical” directions in simplicial configuration space, corresponding to simplicial metrics related by gauge transformations, are found for the three–torus.

This article is dedicated to Tullio Regge on the occasion of his sixty–ﬁfth birthday.

1. INTRODUCTION in numerical relativity, in studies of evolution of model universes and calculations of gravita- 1996 is a year in which we celebrate not only the tional radiation, providing predictions for mea- sixty–fifth birthday of Tullio Regge, but also the surements by LIGO. Other classical work includes thirty–fifth birthday of one of his brain–children, an investigation of the use of area variables in Regge calculus [1], a discretization of general rel- four–dimensional Regge calculus but this issue is ativity which has provided an amazing source of not fully resolved as yet. fascination and fun for those who have worked on A very interesting and promising development it, as well as the occasional moment of frustra- in the last five years has been an understanding of tion! the connection between the work of Ponzano and Some of the recent progress in Regge calcu- Regge [8] and the state sum or manifold invari- lus is described elsewhere in this volume [2–4]. ant of Turaev and Viro [9], which uses 6j–symbols Rather than attempting a general review of other for quantum groups to provide a regularized ver- advances, I shall mention a few areas and then sion of the Ponzano–Regge invariant. Just as the concentrate on one, that of gauge transformations semi–classical limit of the Ponzano–Regge invari- and the simplicial supermetric [5,6]. ant is related to the Feynman path integral for Although much of the current work in Regge three–dimensional gravity with the Regge calcu- calculus involves attempts at setting up a quan- lus action, the Turaev–Viro invariant is related tum theory of gravity, there has also been some to Chern–Simons gravity [10]. Thus it is possible progress on the classical front. An algorithm for to write down a finite theory of quantum gravity, the parallel evolution of sets of disconnected ver- with cosmological constant, in three dimensions tices in a spacelike hypersurface has been for- [11]. Barrett and Crane [12] have shown recently mulated [7] and should prove an invaluable tool that the Ponzano–Regge wave function satisfies the Wheeler–DeWitt equation. There have been ∗This work was supported in part by the UK Particle Physics and Astronomy Research Council many attempts at extending these ideas to four 2 dimensions (see for example [13] and references those wishing to use results from lattice gauge in [14]) but there are still a number of open ques- theories, is transformations of the edge–lengths tions and unresolved issues. which leave the action invariant. Some consider The other main area of activity in quantum the difference between the two a matter of se- Regge calculus is in numerical simulations in two, mantics for, after all, how would one check that three and four dimensions [15–17] In particular the discrete geometry is left invariant other than there has been extensive investigation of the rôle by considering the change in the action? It is of the measure (see also [3]) and the inclusion of not quite as simple as that, and depends to some matter represented by a scalar field. There is still extent on whether one thinks of the lattice as some controversy over the relationship between being embedded in some continuum manifold or the Regge calculus simulations and the alterna- whether one regards the lattice itself as existing tive method known as dynamical triangulations independently of any background embedding. of random surfaces (see eg [18]). If one adopts the “invariance of the geometry” Numerical simulations are best guided and definition, then a sensible implementation is to re- complemented by analytical results. One of the quire that all local curvatures (and hence deficit main tools here is the weak field expansion about angles) be unchanged under the transformation of flat space or some other classical solution. This edge–lengths. This is a very strong requirement has been used recently to study gauge transfor- and will not be met in general. The only excep- mations in simplicial gravity [5] and this work will tion is flat space where, in general, an infinite be described more fully in section 2. number of choices of edge–length will correspond The existence of gauge transformations or dif- to the same flat geometry. In the neighbour- feomorphisms is crucially important in another hood of flat space there will be approximate dif- area of current research, the study of the simpli- feomorphisms [19] where, when the edge–lengths cial supermetric. This is the metric on the space change by order , the deficit angles will change of simplicial three geometries and a knowledge of by smaller than order . its signature is crucial for determining spacelike On the other hand, in the “invariance of the hypersurfaces in superspace, important in some action” definition, it is easy to imagine changes formulations of quantum gravity. There are so– in the edge–lengths which could decrease the called “vertical directions” in simplicial configu- deficit angles in one region and compensatingly ration space or superspace, which correspond to increase them in another region, producing no metrics which are related by diffeomorphisms and overall change in the action. This invariance one objective of current work is to identify these could even be local in the sense that changes directions. Sections 3, 4.1 and 4.2 consist of dis- in the lengths of edges meeting at one vertex cussions of the known results on the signature of could be made so that the action (restricted to the supermetric in the continuum, and analytic scalar curvature, curvature–squared and volume and numerical results in the discrete case. contributions, say) could be unchanged locally, ie: when evaluated over the simplices meeting 2. GAUGE TRANSFORMATIONS OR at that vertex. Before mentioning recent results DIFFEOMORPHISMS? demonstrating precisely this locality property, let us consider in more detail why local gauge in- The first point that needs to be understood variance is important in lattice gravity [20] . In here is that there are differing points of view ordinary gauge theories, local gauge invariance about how to define gauge transformations or dif- plays a central rôle as it gives rise to the Taylor– feomorphisms in discrete gravity. One definition, Slavnov identities, which ensure for example that which tends to be adopted by those approach- the gauge bosons in the theory remain massless ing the subject from classical relativity, is trans- to all orders of perturbation theory. Similar re- formations of the edge–lengths which leave the sults hold for lattice regularized versions of these geometry invariant. The other, favoured more by theories. A small gauge breaking in such a lat- 3 tice theory would invalidate its usefulness as a To define the geometry of superspace one needs representation of the original quantum theory; the notion of distance. This is defined on the in general small gauge breakings are amplified larger space of three–metrics by the DeWitt su- by loop corrections which pick up contributions permetric [23], the properties of which are well– from all momenta and, in particular, short dis- understood. Now the “points” of superspace tance artifacts in the lattice model. In contin- are classes of diffeomorphically equivalent three– uum quantum gravity, identities analagous to the metrics and the DeWitt supermetric induces a Taylor–Slavnov identities can be written down by supermetric on this superspace. The properties exploiting the local invariance properties of the of this induced supermetric are only partially un- functional integral and, clearly, it is desirable that dertood, as we shall see. lattice analogues exist here as for other gauge the- Consider a fixed three–manifold M and let ories. Since it is the lattice action which appears µ(M) be the space of three–metrics on M,with in the functional integral, invariance of the action typical element hab(x). The distance in µ be- is the relevant consideration here. tween two metrics separated by an infinitesimal In the weak field expansion of Regge gravity displacement δhab is defined by about flat space, gauge transformations, exact 2 3 abcd zero modes, were found in four dimensions in an δSµ = d xN(x)G (x)δhab(x)δhcd(x)(1) investigations of the lattice propagator [21]. Sim- ZM ilar results have been found in three [22] and in abcd two [5] dimensions. The form of these zero modes where N(x) is the lapse function and G the is precisely the discretized form of the diffeomor- inverse DeWitt supermetric given by phisms in the continuum theory [5,22]. The weak

field expansion about flat space is quite relevant abcd 11 ac bd G¯ (x)= h2(x)[h (x)h (x)+ for the full quantum theory, because in the neigh- 2 bourhood of the ultraviolet fixed point found in had(x)hbc(x) − 2hab(x)hcd(x)] (2) three dimensions [22] , the average curvature in zero and locally the curvature is small on the scale This has signature (−, +, +, +, +, +, )andso of the lattice spacing. the signature of the metric on µ has an infinite Recent work on edge–length variations about number of both negative and positive signs. non–flat background lattices [5] indicates that the Denote by Riem(M) the superspace of three– local gauge invariance is still there. It holds also geometries on M. A supermetric on Riem(M)is in the presence of a scalar field. induced from the DeWitt supermetric on µ(M) In the remainder of this paper the main empha- by choosing δhab to represent the displacement sis will be on transformations of the edge–lengths between nearby three–geometries. Since each leaving the geometry invariant. These will be dis- three–geometry can be represented by different cussed in the context of the simplicial supermet- three–metrics, δhab is not unique: for any vector ric. A much more detailed account of this work ξa(x), can be found in [6] (see also references therein). 0 δhab(x)=δhab(x)+D(aξb)(x), (3) 3. THE CONTINUUM SUPERMETRIC represents the same displacement in superspace as δh . Thus there are so–called “vertical” direc- The superspace of three–geometries on a ab tions in µ(M), pure gauge directions, fixed three–manifold is important in various ap- proaches to quantum gravity. For example, in vertical kab (x)=D(aξb)(x)(4) Dirac quantization the states are represented by wave functions on superspace and, in quantum and “horizontal” directions, which are orthog- cosmology, such wave functions define initial and onal to all of the vertical ones, in the metric on final conditions. µ(M). 4

The non–uniqueness of δhab means that there specifying a metric with signature (+++); this is are different notions of distance between three– done by assigning (a) values to the n1 squared geometries in Riem(M). The conventional choice edge–lengths; these must be positive and sat- is to define δS2 to be the minimum value of isfy the triangle and tetrahedral inequalities, and 2 δSµ for all δhab representing the displacement (b) a flat metric, consistent with these values, between metrics on the nearly three–geometries. to the interior of each tetrahedron. The region This means that distance is measured in “horizon- of Rn1 , with axes the squared edge–lengths ti, tal” directions in superspace, which is equivalent i =1,2, ..., n1, where the inequalities mentioned to choosing a gauge specified by the conditions in (a) are satisfied, is called simplicial configuration space, K(M). Distinct points in K(M), b c D (kab − habkc)=0. (5) corresponding to different assignments of edge– lengths in M will, in general, correspond to dis- In order to define spacelike hypersurfaces in su- tinct three–geometries. As discussed in section perspace it is necessary to know the signature of 2, this is not always true; the displacements of the supermetric. This is a non–trivial problem vertices of a flat geometry in a flat embedding because superspace is infinite–dimensional. The space give a new assignment of edge–lengths cor- known results [24,25] include the following: responding to the same flat geometry. These (i) there is at least one negative direction at are the simplicial analogues of diffeomorphisms each point, corresponding to constant conformal and there are also approximate simplicial diffeo- displacements morphisms where the geometry is almost flat locally [19]. Thus the continuum limit of K(M)is k = δΩ2h (x) . (6) ab ab the space of three–metrics, not the superspace of three–geometries, which is why K(M) has been This is horizontal because it satisfies the gauge called simplicial configuration space rather than choice above; simplicial superspace. (ii) if M is the three–sphere, then in the neigh- In order to define “vertical” and “horizontal” bourhood of the round metric, the signature has directions in K(M) it is necessary to define a met- just one negative direction, the conformal one, ric: and all other directions are positive; (iii) every manifold admits geometries with 2 ` m n δS = Gmn(t )δt δt , (7) negative Ricci curvatures; in the open region of superspace defined by such geometries the sig- where δS2 is the infinitesimal displacement be- nature has infinite numbers of both positive and tween two points in K(M ). The supermetric Gmn negative signs. This means that, for the sphere, can be induced from the DeWitt supermetric on there must be points in superspace where the met- the space of continuum three–metrics as follows. ric is degenerate. A simplicial geometry in K(M)canberepre- These results cover only a small part of su- sented (not uniquely) by a point in µ(M )cor- perspace, so the procedure is now to try to ob- responding to a piecework flat metric, and a dis- tain more complete information on the signature placement δtm in K(M) can be represented by a by looking at simplicial approximations to super- perturbation δhab in µ(M). Then we define space based on Regge calculus.

` m n 3 abcd 4. THE SIMPLICIAL SUPERMETRIC Gmn(t )δt δt = d xN(x)G¯ (x) ZM 4.1. Analytic Results on the Signature ×δhab(x)δhcd(x)(8) A simplicial three–manifold M is constructed abcd from tetrahedra joined together so that the neigh- where G is evaluated at the the piecewise bourhood of each point is homeomorphic to a re- flat metric. To make this definition meaningful gion of R3. A simplicial geometry is “fixed” by we need to specify the following: 5

(i) the value of N(x). Take N(x)=1; leads, at ﬁrst order, to (ii) the gauge inside each tetrahedron. We call 2 this choice the “Regge gauge freedom” and a nat- a 1 ∂V (τ) m δha(τ)= 2 m δt , (16) ural choice is to take δhab to be constant inside V ∂t each tetrahedron: and, at second order, to

2 Dcδhab(x)=0, inside each τ ∈ Σ3 , (9) 1 ∂V (τ) G (t`)=− . (17) mn V(τ) ∂tm∂tn τ∈Σ3 where 3 is the set of tetrahedra in K(M). X Of course, there may still be variations of the P This is the expression written down by Lund edge–lengths which preserve the geometry, so this and Regge [27] for the metric on K(M). It can type of gauge freedom remains. be shown that this gives the shortest distance be- Evaluation of the integral above leads to the tween simplicial three–metric among all choices expression of Regge gauge which vanish on the triangles. It is not exactly ”horizontal” in the sense of the continuum because of the possibility of simplicial dif- G (t`)δtmδtn = V (τ){δh (τ)δhab(τ) mn ab feomorphisms. τ∈Σ3 P a 2 We are particularly interested in the signature − [δha(τ)] } (10) of the Lund–Regge supermetric Gmn and we give where V (τ) is the volume of tetrahedron τ. here several analytic results which give limited information about it. The relation between hab and the squared (i) The conformal direction is always timelike. edge–lengths tab, linking vertices a and b,maybe obtained by picking a vertex 0 in a tetrahedron Suppose that and taking basis vectors along the edges from that δtm = δΩ2 tm . (18) vertex. Then Now V 2 is a homogeneous polynomial of de- 1 l hab(τ)= (t0a+t0b−tab) (11) gree three in t , so Euler’s theorem applied to the 2 expression for Gmn above leads to and hence ` m n ` Gmn(t )t t = −6VTOT(t ) < 0 . (19) 1 δh (τ)= (δt + δt − δt ) . (12) ab 2 0a ab 0b Also, the conformal direction is orthogonal to any gauge direction δt2 because This ﬁxes the Regge gauge for each tetrahedron. ∂V G = tmδtn = −4 TOTδtn = −4δV (20) We now use the expression [26] mn ∂tn TOT 1 which is zero for a gauge transformation (by V 2(τ)= det[h (τ)] . (13) (3!)2 ab which we mean here any change in edge–lengths which does not change the geometry). l and consider perturbations δt corresponding (ii) There are at least n1 − n3 spacelike direc- to δhab. Expansion of tions where n3 is the number of tetrahedra in K(M). This follows from showing that 1 V 2(tl + δtl)= det(h + δh ) (14) (3!)2 ab ab 1 ∂V (τ) ∂V (τ) G δtmδtn = −4 δtmδtn via mn V (τ) ∂tm ∂tn τ X 1 ab 2 l l + V (T)δhab(τ)δh (τ) (21) V (t + δt )= 2 exp[Trlog(hab + δhab)] (15) (3!) τ X 6

which is non–negative if We now describe the results for two manifolds, S3 and T 3. ∂V (τ) δtm =0 forallτ. (22) ∂τm 4.2.1. The three–sphere We chose the simplest triangulation of S3,the Therefore variations of the edge–lengths leav- surface of a four–simplex, which has 5 vertices, ing the volumes of all tetrahedra unchanged cor- 10 edges, 10 triangles and 5 tetrahedra. Thus, respond to spacelike directions. There are n3 con- in this case, simplicial configuration space is 10– ditions on n1 edge–length changes so there are at dimensional, each point in it corresponding to an least n1 − n3 independent spacelike directions. assignment of 10 squared edge–lengths. (iii) Diffeomorphism modes: consider a flat 3 Since the signature is scale invariant, the simplicial geometry embedded locally in R ,with longest edge can be fixed at limit length. To x vertices at xA,A =1,2, ..., n0. Displacements of sample the whole space by dividing the limit in- these vertices through δxxA produce changes in the m terval into 10, it looks at first as though there edge–lengths δt which correspond to gauge di- will be 1010 points to investigate. However, the rections since the flat geometry is unchanged. It triangle and tetrahedral inequalities reduce this can be shown (see [6] for details) that to 102,160 points. The results on the signature are the same everywhere; the signature is

2 ∂VTOT 2 (−, +, +, +, +, +, +, +, +, +). It is easy to see δS =2 δxxCD ∂tCD that this is consistent with the analytic results. 2 The results on the eigenvalues of Gmn are also 2 ∂V (τ) − x ·δxx (23) consistent with symmetry considerations. The V (τ) ∂t CD CD τ CD ! symmetry group of the triangulation is S5,the X permutation group on five vertices. For the case Although the second term gives a negative con- of equal edge–lengths in the triangulation, the tribution and the first does not appear to have a eigenvalues of Gmn maybeclassifiedbytheir- definite sign, I conjecture that gauge modes are reducible representations of S5 and their degen- positive (see further comments on this in the nu- eracies given by the dimensions of those repre- merical section). sentations. This is because a permutation of the 4.2. Numerical Results on the Signature vertices corresponds to a matrix acting on the 10– The limited analytic information on the signa- dimensional space of edges. These matrices give ture of the simplicial supermetric described above a 10–dimensional reducible representation of S5 may be checked and supplemented by numerical which can be decomposed into irreducible repre- evaluation of the supermetric over the whole of sentations as simplicial configuration space or at least a non– 10=1+4+5. (24) trivial region of it. The general method employed here is as follows: This can be compared with the numerical re- (i) choose a simple triangulation of a chosen sults when all the ti are one. The eigenvalues of manifold; √ √ Gmn are −1/ 2 with√ degeneracy 1, 1/3 2with (ii) assign edge–lengths (consistent with the tri- degeneracy 4, and 5/6 2 with degeneracy 5. In angle and tetrahedral inequalities); general, these degeneracies are broken by depar- (iii) calculate the Lund–Regge supermetric ture from the completely symmetric assignment Gmn; of edge–lengths. (iv) find the eigenvalues of the matrix Gmn: (v) count the numbers of positive, negative and 4.2.2. The three–torus zero eigenvalues to determine the signature; To obtain a triangulation of T 3,wetakealat- (vi) update the edge–lengths and repeat the tice of cubes, with nx,ny and nz cubes in the procedure. x−,y−and z−directions respectively. Each cube 7 is divided into 6 tetrahedra by drawing in face di- two multiplicities of 3 are interpreted as 2 + 1. agonals and a body diagonal [21]. Then there are We do not understand such accidental degener- n0 = nxnynz vertices, 7n0 edges, 12n0 triangles acy, but it has been observed elsewhere, for ex- 2 and 6n0 tetrahedra, and simplicial configuration ample for some triangulations of CP [28,29]. space is 7n0-dimensional. We considered two par- To investigate the existence of gauge modes, ticular flat triangulations, one with right–angled the edges are varied in the directions of the eigen- tetrahedra and one with isosceles tetrahedra (see vectors as already described and the new deficit [6] for details). The computations ranged from angles calculated to see whether the geometry re- 3 × 3 × 3 vertices and 189 edges to 6 × 6 × 7 mains flat. For the isosceles tetrahedral lattice, vertices and 1764 edges. Unlike the three–sphere the results are that the eigenvectors with eigen- 1 case, the numbers of edges, even in the smallest values λ = 2 appear to be approximate diffeo- case, mean that it is impossible to sample the morphisms, since the deficit angles are of order 3 whole of simplicial configuration space, so the in- and, for an n0 = n × n × n lattice, there are vestigation was done in two ways, firstly by mak- 6n − 4 eigenvectors corresponding to this eigen- ing random variations in the edge–lengths about value. flat space and secondly by making perturbations The conjecture about positive eigenvalues for along the flat–space eigenvectors vj : the gauge modes has not been proved but has

i i j support from a number of special cases for the tnew = tflat + v , (25) right–tetrahedral lattice, where it reduces to i j for all t and each v in turn, with small. 2 The results described here are for the 189– 1 δS2 = δxx2 − x ·δxx (27) dimensional case. For flat space, the signature CD 12 CD CD CD τ " C, D # has 13 negative signs and 176 positive ones. This X X X τ changes away from flat space and there are even Thus δS2 is positive in the following cases: some zero eigenvalues which are presumably finite (i) if just one vertex moves through δrr, analogues of the infinite number of non-gauge directions predicted by Giulini [24] for open regions δS2 =8δrr2 (28) with negative Ricci curvature. To understand the degeneracy of the eigenval- (ii) if the edges in only one co–ordinate direc- ues, we need to look at the symmetry group of the tion change, lattice. For the 3×3×3 case, this group has a subgroup of translations (mod 3) complemented by 2 1 2 δS ≥ δxxCD (29) a subgroup Z2 × S3 (corresponding to “parity” 2 transformations and permutations of the axes). X (iii) if all the δxx ’s have the same magniture The 189–dimensional reducible representation of CD δL, this group decomposes as 3 δSn > (δL)2(n − n ) > 0 (30) 1 4 3 189 = 3 × (11)+2×(21)+3×(22)

+2×(4) + 5 × (61 +62 +63 +64) since n1 >n3 . Let us briefly summarize the numerical results +2×(65 +66 +67 +68) (26) of this section. For triangulations of both mani- where 21 is the first irreducible representation folds there are negative modes, including the con- of dimension 2, and so on. The multiplicities of formal direction. For the five–cell triangulation of 3 the eigenvalues of Gmn for the right–tetrahedron S there is a single negative mode, but prelimi- lattice agree exactly with this complicated de- nary results on the 600–cell triangulation indicate composition provided that the two apparent mul- a total of 92 [30]. In all the cases studied there tiplicities of 8 are interpreted as 6 + 2, and the are at least n1 − n3 positive modes. The gauge 8 modes found for T 3 are positive but we have no ume. proof that this is generally true. The results for 5. H.W. Hamber and R.M. Williams, Gauge In- T 3 show that the supermetric can become degen- variance in Simplicial Gravity, to appear in erate and the signature changes as simplicial con- Nucl. Phys. B, hep–th/9607145. figuration space is explored. 6. J.B. Hartle, W.A. Miller and R.M. Williams, The main advantage of the discrete approach to The Signature of the Simplicial Supermetric, the supermetric is that the continuum infinite di- gr–qc/9609028. mensional superspace is reduced to simplicial con- 7. J.W. Barrett, M. Galassi, W.A. Miller, R.D. figuration space which is finite dimensional but Sorkin, P.A. Tuckey and R.M. Williams, A preserves elements of both the physical degrees parallelizable implicit evolution scheme for of freedom and the diffeomorphisms. For larger Regge Calculus, to appear in Mod. Phys. A and larger triangulations, more and more aspects 8. G. Ponzano and T. Regge in F.Block et of both should be recovered. al.(eds.), Semiclassical Limit of Racah Coeffi- One obvious line of further study is to under- cients in Spectroscopic and Group Theoreti- stand the vertical and horizontal directions in cal Methods in Physics, North–Holland, Am- simplicial configuration space and relate them to sterdam, 1968. the vertical and horizontal directions in the con- 9. V.G. Turaev and O.Y. Viro, Topology 31 tinuum. Another avenue is to extend the work (1992) 865. on S3 to a triangulation with an arbitrarily large 10. E. Witten, Commun. Math. Phys. 121 (1989) number of vertices, which would then encode all 351. the dynamic and gauge degrees of freedom, unlike 11. J.W. Barrett, J. Math. Phys. 36 (1995) 6161. the five vertex model studied here, which is too 12. J W. Barrett and L. Crane, An algebraic small to exhibit the expected number of (approx- interpretation of the Wheeler–DeWitt equa- imate) gauge modes (the number of edges 10 is tion, gr–qc/9609030. less than 3n0 in this case). Of course, the meth- 13. M. Carfora, M. Martellini and A. Marzuoli, ods described here could also be applied to trian- Four–dimensional Combinatorial Gravity gulations of other manifolds. from the Recoupling of Angular Momenta, preprint FNT/T 95/31. 5. ACKNOWLEDGEMENTS 14. R.M. Williams, J. Math. Phys. 36 (1995) 6276. I would like to thank the organizers of the 15. H.W. Hamber and R.M. Williams, Nucl. Second Meeting on Constrained Dynamics and Phys. B415 (1994) 463, Nucl. Phys. B451 Quantum Gravity, Santa Margharita Ligure, (1995) 305. Italy, September 1996 for carrying out their tasks 16. W. Bock and J. Vink, Nucl. Phys. B438 with such wisdom and efficiency. I would also (1995) 320. like to thank my collaborators Herbert Hamber, 17. C. Holm and W. Janke, Phys. Lett. B335 James Hartle and Warner Miller for their major (1994) 143. contributions to the work described here. 18. J. Ambjorn, Nucl. Phys. B (Proc. Suppl.) 42 (1995) 3. REFERENCES 19. P. Morse, Class. Quant. Grav. 9 (1992) 2489. 20. H.W. Hamber, private communication. 1. T. Regge, Nuovo Cimento 19 (1961) 558 21. M. Ro˘cek and R.M. Williams, Z. Phys. C21 2. G. Immirzi, Quantizing Regge Calculus, this (1984) 371. volume. 22. H.W. Hamber and R.M. Williams, Phys. Rev. 3. P. Menotti, Functional Integration on Two- D47 (1993) 510. Dimensional Regge Geometries, this volume. 23. B. DeWitt, Phys. Rev. 160 (1967) 1113. 4. G.K. Savvidy, Regge Calculus and Alterna- 24. D. Giulini, Phys. Rev. D51 (1995) 5630. tive Actions for Quantum Gravity, this vol- 25. J.L. Friedman and A. Higuchi, Phys. Rev. 9

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