SPIN FOAM MODELS OF RIEMANNIAN QUANTUM GRAVITY JOHN C. BAEZ, J. DANIEL CHRISTENSEN, THOMAS R. HALFORD, AND DAVID C. TSANG Abstract. Using numerical calculations, we compare three versions of the Barrett- Crane model of 4-dimensional Riemannian quantum gravity. In the version with face and edge amplitudes as described by De Pietri, Freidel, Krasnov, and Rovelli, we show the partition function diverges very rapidly for many triangulated 4-manifolds. In the version with modified face and edge amplitudes due to Perez and Rovelli, we show the partition function converges so rapidly that the sum is dominated by spin foams where all the spins labelling faces are zero except for small, widely separated islands of higher spin. We also describe a new version which appears to have a convergent partition function without drastic spin-zero dominance. Finally, after a general discussion of how to extract physics from spin foam models, we discuss the implications of convergence or divergence of the partition function for other aspects of a spin foam model. 1. Introduction Despite increasing interest in spin foam models of 4-dimensional quantum gravity [6, 29], most work so far has focused on the formal properties of these models, rather than the crucial question of whether they yield reasonable physics at experimentally accessible length scales. Apart from the predilections of the researchers working in this field, there are two main reasons for this. First, it is not obvious which computations would settle this issue. Second, it is difficult to do any sort of computation with these models. If the discreteness of a typical spin foam occurs at the Planck scale, a brute-force simulation of a region of space the size of a proton for the time it takes light to cross this region would require summing over spin foams having roughly 1080 vertices. Of course, it would be interesting to simulate even a much smaller spin foam. However, in the Barrett-Crane model of 4-dimensional quantum gravity we must compute a quantity called the 10j symbol for each spin foam vertex [11 , 12]. In the Lorentzian versions of this theory, no efficient way to compute the 10j symbol is known so far: the only existing methods are Monte Carlo calculations that sometimes require over 1010 samples to achieve reasonable accuracy [8]. This makes even very small spin foams difficult to deal with. The situation is a bit better for Riemannian versions of the Barrett-Crane model. An efficient algorithm has been developed which computes the Riemannian 10j symbols in O(j5) time using O(j2) memory, where j is the average of the ten spins involved [17 ]. As an example, on a 1GHz Pentium III CPU, this algorithm takes about 5 milliseconds to compute the 10j symbol with all spins equal to 5_2, and about 2.5 seconds to compute the 10j symbol with all spins equal to 10. This makes it feasible to compare the quali- tative behavior of different versions of the Riemannian Barrett-Crane model by means of computer calculations for small spin foams. As it turns out, the results are dramatic and enlightening. ______________ Date: July 21, 2002. 1 2 JOHN C. BAEZ, J. DANIEL CHRISTENSEN, THOMAS R. HALFORD, AND DAVID C. TSANG In what follows, we start with a quick review of the existing spin foam models of Riemannian quantum gravity. Then we study three versions of the Barrett-Crane model of 4-dimensional Riemannian quantum gravity. In all three versions the spin foam vertex amplitudes are given by the Riemannian 10j symbols; they differ only in their formulas for edge and face amplitudes. In Section 3 we show that in the model due to De Pietri, Freidel, Krasnov and Rovelli [19 ], the partition function diverges very rapidly for the simplest triangulation of the 4-sphere, and probably for many other triangulated 4-manifolds as well. In Section 4 we turn to the model due to Perez and Rovelli [32 ]. Here it is already known that the partition function converges for any nondegenerate triangulation of any compact 4-manifold [30 , 31]. We show that in fact the partition function converges so fast that the sum over spin foams is dominated by those where almost all the spins labelling faces are zero. In Section 5 we describe a new model with intermediate behavior. This model seems to have a convergent partition function without drastic spin-zero dominance. In Section 6 we discuss the implications of our results. 2. Review Spin foam models are an attempt to describe the geometry of spacetime in a way that takes quantum theory into account from the very start. A spin foam is a 2-dimensional analogue of a Feynman diagram. Abstractly, a Feynman diagram can be thought of as a graph with edges labelled by group representations and vertices labelled by intertwining operators. Similarly, a spin foam is a 2-dimensional cell complex with polygonal faces labelled by representations and edges labelled by intertwining operators. Like Feynman diagrams, spin foams serve as a basis of `quantum histories': the actual time evolution of the system is described by a linear combination of these quantum histories, weighted by certain amplitudes. Feynman diagrams are 1-dimensional because they describe quantum histories of collections of point particles; spin foams are 2-dimensional because in loop quantum gravity, the gravitational field is described not in terms of point particles but 1-dimensional `spin networks'. An ordinary quantum field theory provides a recipe for computing the amplitude for any Feynman diagram in terms of amplitudes for edges and vertices. Similarly, a spin foam model consists of a recipe to compute an amplitude for any spin foam as a product of face, edge and vertex amplitudes. The partition function in a spin foam model is computed as a sum or integral of these spin foam amplitudes. Using suitably weighted sums and normalizing by dividing by the partition function, one can also compute expectation values of observables. A number of spin foam models have been developed for both Lorentzian and Riemannian quantum gravity. By `Lorentzian quantum gravity', we mean any quantum theory whose partition function is, at least morally speaking, given by Z eiS, where S is the Einstein-Hilbert action for a Lorentzian metric on spacetime, or some closely related action. If all goes well, a theory of this sort should reduce in a suitable limit to the classical Einstein equations for Lorentzian metrics. `Riemannian quantum gravity' is the same sort of thing, but for Riemannian metrics. It is important not to confuse Riemannian quantum gravity with `Euclidean quantum gravity', which also uses the Einstein-Hilbert action for a Riemannian metric, but where the partition function is SPIN FOAM MODELS OF RIEMANNIAN QUANTUM GRAVITY 3 given by Z e-S . With the help of analytic continuation to imaginary times, Euclidean quantum gravity is a widely used (though controversial) tool for studying Lorentzian quantum gravity. Riemannian quantum gravity is a wholly different theory, which at best will reduce in some limit to the classical Einstein equations for Riemannian metrics. Thus the large body of conventional wisdom about Euclidean quantum gravity may not apply to spin foam models of Riemannian quantum gravity _ only further work can decide this. Riemannian quantum gravity seems to have limited relevance to real-world physics. Nonetheless, spin foam models of Riemannian quantum gravity have proved to be a useful warmup for work on spin foam models of Lorentzian quantum gravity. The Riemannian models are simpler because the rotation group is compact, unlike the Lorentz group. For a compact group, the irreducible unitary representations are finite-dimensional and indexed by discrete rather than continuous parameters. This means that in Riemannian spin foam models there is no difficulty showing the convergence of a single spin foam amplitude, and the partition function is computed as a sum rather than an integral over spin foams. In retrospect, the very first spin foam model was the Ponzano-Regge model of 3- dimensional Riemannian quantum gravity [34 ]. InRthis model, one triangulates a given 3-manifold and expresses the partition function eiS as a sum over spin foams lying in the dual 2-skeleton of the triangulation. The gauge group in this theory is the double cover Spin(3) = SU (2) of the 3d rotation group. A heuristic argument suggested that the result was actually independent of the triangulation. In fact the sum diverges, and contrary to Ponzano and Regge's original expectations, the naive way of regularizing it does not give triangulation independent results. Much later, Turaev and Viro [39 ] discovered that one could regularize the Ponzano- Regge model by replacing SU (2) with the corresponding quantum group, SU q(2). In this q-deformed model the partition function converges for any compact 3-manifold. Even better, it turns out to be triangulation independent. By now it is clear that this partition function is that of 3-dimensional Riemannian quantum gravity with a positive cosmological constant; the deformation parameter q is related to the cosmological constant by a simple formula [3, 26]. In 1997, one week before the concept of spin foam was formalized [5], Barrett and Crane proposed a spin foam model of 4-dimensional Riemannian quantum gravity [11 ]. Here the partition function is computed as a sum over spin foams lying in the dual 2-skeleton of a triangulated 4-manifold. The spin foam faces are all labelled by `balanced' representations of Spin (4) = SU (2) x SU (2), that is, those of the form j j. The edges are all labelled by a specific intertwiner called the Barrett-Crane intertwiner. The idea behind these choices was to express Riemannian quantum gravity as a constrained version of the spin foam model for topological gravity due to Turaev, Ooguri, Crane and Yetter [18 , 28, 38]. This idea also motivated using the 10j symbols for the vertex amplitudes. Barrett and Crane wisely refrained from giving formulas for edge and face amplitudes, which later turned out to be the most controversial aspect of the whole theory. Unfor- tunately, without these, their model was incomplete. In particular, it was impossible to say whether or not the partition function converges. They did note that one can q-deform their model, obtaining a model based on the quantum group SU q(2) x SU __q(2). In this q-deformed version the partition function becomes a finite sum, so it converges regard- less of the choice of edge and face amplitudes. However, nobody has been able to find a 4 JOHN C. BAEZ, J. DANIEL CHRISTENSEN, THOMAS R. HALFORD, AND DAVID C. TSANG nontrivial choice of edge and face amplitudes that makes the result triangulation indepen- dent. By analogy with the 3-dimensional case, it is widely expected that the q-deformed Barrett-Crane model is related to 4-dimensional Riemannian quantum gravity with posi- tive cosmological constant. Since this is not a topological field theory, there is actually no reason to expect a triangulation-independent partition function. Further progress was made by De Pietri, Freidel, Krasnov and Rovelli [19 ], who showed that the Barrett-Crane model naturally arises from a quantum field theory on a product of 4 copies of the 3-sphere, thought of as a homogeneous space of the group SU (2) x SU (2). Feynman diagrams in this `group field theory' correspond precisely to the spin foams appearing in the Barrett-Crane model, and the vertex amplitudes are the same as well. There are two important differences, however. First, the group field theory approach gives specific formulas for edge and face amplitudes! Second, instead of summing over spin foams lying in the dual 2-skeleton of a fixed triangulation of a fixed 4-manifold, one computes the partition function by summing over spin foams lying in the dual 2- skeleta of all triangulations of all compact 4-manifolds _ and even a more general class of well-behaved `pseudomanifolds', namely spaces made by gluing finitely many 4-simplices together pairwise along their tetrahedral faces. In short, the DFKR approach naturally extends the Barrett-Crane model to incorporate a sum over triangulations and even a sum over topologies. This sidesteps the awkward need for an arbitrary choice of triangulation, but makes the convergence of the partition function even less likely. Later, Perez and Rovelli [32 ] modified the DFKR proposal, describing a group field theory that corresponds to a version of the Barrett-Crane model with modified edge and face amplitudes. Their goal was to eliminate divergences from the model, and they made substantial progress: Perez [30 , 31] was able to prove that in this modified model, the sum of spin foam amplitudes converges if we restrict to spin foams lying in the dual 2-skeleton of a given well-behaved pseudomanifold, so long as each triangle of this pseudomanifold lies in at least three 4-simplices. The issue of pseudomanifolds not satisfying this condition remains a challenge, as does the sum over pseudomanifolds. As we shall see in Section 6, it is not completely obvious that one needs a convergent partition function for a well-behaved spin foam model. After all, for physics we need to compute, not the partition function, but expectation values of observables. Only after we can compute these can we tackle the important question of whether a given spin foam model reduces to general relativity (possibly coupled to matter) in the large-scale limit. Nonetheless, it appears that convergence or divergence of the partition function is closely tied to other important qualitative features of a spin foam model. This makes it worthwhile to study the convergence issue. In the next three sections, we do this for three versions of the Riemannian Barrett-Crane model: the DFKR version, the Perez-Rovelli version, and a new version. We only consider the convergence issue for one pseudomanifold at a time, not the sum over pseudomanifolds. 3. The De Pietri-Freidel-Krasnov-Rovelli model We begin by recalling the general idea of the Riemannian Barrett-Crane model. As mentioned already, this model can be defined for any simplicial complex formed by taking a finite set of 4-simplices and attaching distinct ones pairwise along their tetrahedral faces until all faces are paired. In what follows we shall restrict attention to manifolds, but only to simplify the terminology; everything generalizes painlessly to these well-behaved pseudomanifolds. SPIN FOAM MODELS OF RIEMANNIAN QUANTUM GRAVITY 5 Let M be a triangulated compact 4-manifold and let n be the set of n-simplices in the triangulation. By definition, 4-simplices, 3-simplices and 2-simplices correspond to vertices, edges and faces of the dual 2-skeleton, respectively. In all versions of the Riemannian Barrett-Crane model, a spin foam F simply amounts to a labelling of each face f 2 2 by a spin j(f ) 2 {0, 1_2, 1, . .}.. Note that there are four faces incident to each edge in the dual 2-skeleton. We require that the spins j1, . .,.j4 labelling the faces incident to any edge be `admissible', meaning that there exists a nonzero SU (2) intertwining operator f :j1 j2 ! j3 j4. In all versions of the Riemannian Barrett-Crane model, the amplitude of a spin foam is computed by a formula of this sort: Y Y Y (1) Z(F ) = A(f ) A(e) A(v) f2 2 e2 3 v2 4 and the partition function is given by X (2) Z(M ) = Z(F ). F Here the complex numbers A(f ), A(e) and A(v) are called face, edge and vertex amplitudes, respectively. Each face amplitude is computed only using the spin j(f ) labelling that face. Each edge amplitude is computed using the spins labelling the 4 faces incident to that edge; we call these spins j1(e), . .,.j4(e). Finally, each vertex amplitude is computed using the spins labelling the 10 faces incident to that vertex; we call these spins j1(v), . .,.j10(v). To give formulas for these amplitudes we use the standard graphical notation for SU (2) spin networks [16 , 22 ]. Normalization issues are crucial here. We call a triple of spins j1, j2, j3 `admissible' if there exists a nonzero intertwining operator f :j1 j2 ! j3; this happens precisely when these spins satisfy the triangle inequality and sum to an integer. Given an admissible triple of spins, we normalize the canonical intertwining operator f :j1 j2 ! j3 so that _j1__________________________________ ____________________________________________________________ ___________________________________________________________* *_j2 o____________________________________________________o= 1. ___________________________________________________________* *_______ __________________________________________________________ j3 As this normalization sometimes requires dividing by the square root of a negative number, it introduces a potential sign ambiguity. Luckily, in our calculations trivalent vertices always come in matching pairs, so these signs cancel. Actually, in what follows almost all our diagrams will be balanced spin networks [11 , 40]. In such a spin network, labelling an edge by the spin j really means that it is labelled by the irreducible representation j j of the group Spin(4) = SU (2)xSU (2). Such representations are called `balanced'. Also, in a balanced spin network, an unlabelled 4-valent vertex is really labelled by the Barrett-Crane intertwiner. This is defined in terms of SU (2) spin 6 JOHN C. BAEZ, J. DANIEL CHRISTENSEN, THOMAS R. HALFORD, AND DAVID C. TSANG networks by: ??? j2 """ ???j1? j2""" ? " ??j1?? """ ??? """ ??j1?? j2""" ?o"" ?o"" ???"""" X 2k | | (3) ""o?? = (-1) (2k + 1) k| |k , "" ??? k | | """j3" j4 ??? ""o"???? ""o??? ?? """ ??? """"j3" j4 ?? """j3 j4 ?? where we sum over spins k such that the triples j1, j2, k and j3, j4, k are both admissible. In terms of balanced spin networks, the face, edge and vertex amplitudes of the DFKR model are given as follows: _________________________________________________* *______________ A(f ) = j(f)____o____________________________________________* *___________________________________________@ A(e) = ______1______j1(e) __________________________________________________* *__ ___________________________________________________* *_____________________j2(e) ____________________________________________________* *___________________________________________@ o____________________________________________________* *___________________________________________@ ____________________________________________________* *___________________________________________@ __________________________________________________* *___________________________________j3(e) j4(e) (4) vvoHHHHv~~))) j2(v)vvvv~~)) HHHj1(v)H vvv ~~ )) HHH vvv ~j~(v))) HHH o)v)_________6_____________o~_~~HHHvv)) ))HHj7(v)~~H )j10(vvv~~v)) A(v) = ))) HH~~H )vvv) ~~ . ~HHH vvv) ~ j3(v)))) ~~ HH vv )) j5(v)~~ ~j8(v)HHHj9(v)vv~v) ))~~ vvv HHH )) ~~ ))~~vvv HHH~~)) o______j4(v)_____o~~v Here the intertwiner in the first spin network is just the identity operator, so the face amplitude A(f ) is just the dimension of the representation j(f ) j(f ), that is, (2j(f )+1)2. The `4j symbol' j1___________________________________ ________________________________________________________* *____j _________________________________________________________* *_____2_____________________________________@ o_________________________________________________________* *___________________________________________@ _________________________________________________________* *_________j_________________________________@ _______________________________________________________* *___ 3 j4 equals the dimension of the space of SU (2) intertwining operators f :j1 j2 ! j3 j4. This in turn equals the number of spins k such that the triples j1, j2, k and j3, j4, k are both admissible. Finally, the vertex amplitude A(v) is called the `10j symbol'. There is no simple formula for this, so to compute it we shall need the algorithm developed by Christensen and Egan [17 ]. SPIN FOAM MODELS OF RIEMANNIAN QUANTUM GRAVITY 7 At this point some comments might be helpful. The above formulas were first derived by De Pietri, Freidel, Krasnov and Rovelli [19 ] using the group field theory approach. How- ever, they also arise naturally from the idea that the Barrett-Crane model is a constrained version of the SU (2) x SU (2) Turaev-Ooguri-Crane-Yetter model. In the latter model one works with spin foams where faces are labelled by arbitrary irreducible representations of SU (2) x SU (2) and edges are labelled by arbitrary intertwiners chosen from an orthonor- mal basis. Here one uses the fact that intertwiners f :H ! H0 between finite-dimensional unitary group representations naturally form a Hilbert space with = tr(f *g). To get the above version of the Barrett-Crane model, one restricts the Turaev-Ooguri- Crane-Yetter formulas to spin foams where faces are labelled by balanced intertwiners and edges are labelled by the normalized Barrett-Crane intertwiner. The Barrett-Crane intertwiner in equation (3) is not normalized; instead, its inner product with itself is j1___________________________________ ________________________________________________________* *____j _________________________________________________________* *_____2_____________________________________@ o_________________________________________________________* *___________________________________________@ _________________________________________________________* *_________j_________________________________@ ________________________________________________________* *__ 3 j4 so to normalize it we must divide by the square root of this quantity. However, since each Barrett-Crane intertwiner in the 10j symbols appears twice in the formula for the Z(F ) _ once for each of the two 4-simplices incident to a given 3-simplex _ we obtain a factor of ______1______ j1(e)___________________________________________, _________________________________________________________* *________________________j2(e) __________________________________________________________* *___________________________________________@ o__________________________________________________________* *___________________________________________@ __________________________________________________________* *___________________________________________@ ________________________________________________________* *_____________________________j3(e) j4(e) which gives the edge amplitude A(e). We can study the convergence of the partition function (2) by imposing a cutoff on the spins labelling spin foam faces. Since these spins determine the areas of the corresponding 2-simplices in the triangulation of the spacetime manifold, we can think of this as a sort of `infrared cutoff' which rules out large areas. Let us write |F | for the maximum of the spins labelling the faces of the spin foam F . Imposing a spin cutoff |F | J, the partition function becomes a finite sum X (5) ZJ (M ) = Z(F ). |F | J For a simple but interesting example, we can take M to be a 4-sphere triangulated as the boundary of a 5-simplex. In Table 1 we show the results of computing ZJ (M ) in this case for various low values of J. 8 JOHN C. BAEZ, J. DANIEL CHRISTENSEN, THOMAS R. HALFORD, AND DAVID C. TSANG _________________ |__J__|ZJ(M)___|_0 |__0_1|.000.10__|_5 |_1=23|.722.10__|_9 |__1_7|.812.10__|_13 |_3=22|.128.10__|_16 |__2_1|.345.10__|_ Table 1: S4 partition function _ DFKR model with spin cutoff J It seems that ZJ (M ) grows at a spectacular rate as J increases. We can begin to understand this by estimating the face, edge and vertex amplitudes in equation (4). In the limit of large spins, the face amplitudes clearly grow as O(j2) where j is the spin labelling the face in question. For the edge amplitudes, we can use the fact that j1___________________________________ ________________________________________________________* *____j _________________________________________________________* *_____2_____________________________________@ o_________________________________________________________* *___________________________________________@ _________________________________________________________* *_________j_________________________________@ _______________________________________________________* *___ 3 j4 equals the number of spins k such that both j1, j2, k and j3, j4, k are admissible triples. In general, if j1, . .,.j4 are admissible and of order j, the number of such spins k is also of order j, so the edge amplitudes grow as O(j) when all spins are rescaled by the same factor. The only exception occurs when j1, . .,.j4 lie at the `border of admissibility', that is, when _j1__________________________________ ___________________________________________________________* *_j ___________________________________________________________* *__2________________________________________@ o____________________________________________________________* *___________________________________________@ ___________________________________________________________* *_______j___________________________________@ __________________________________________________________* * 3 j4 In this case the 4j symbol remains equal to one as all four spins are rescaled. The asymptotic behavior of the vertex amplitude is much more subtle. Starting from the integral formula for the 10j symbols and doing a stationary phase approximation, Barrett and Williams [13 ] computed the asymptotics of the 10j symbols as all ten spins are multiplied by some factor j which approaches infinity. This calculation yields a factor of j-9=2 times an oscillating function of j. Unfortunately, computer calculations show a different rate of decay and no significant oscillatory behavior [9]. It now seems clear that the stationary phase points do not dominate the integral. In general, it appears that the 10j symbols decay as O(j-2 ) as all spins are rescaled by the same factor. The only exception occurs when the four spins labelling edges incident to some vertex lie at the border of admissibility; then the 10j symbols decay more rapidly. The more vertices lie at the border of admissibility, the more rapid the decay. The triangulation of the 4-sphere as the boundary of a 5-simplex gives spin foams with 20 faces, 15 edges and 6 vertices. Thus, for a spin foam F with all faces labelled by spins of order j, with no vertices lying on the border of admissibility, the amplitude is Z(F ) = O(j2.20j-1.15j-2.6) = O(j13). Together with the fact that Z(F ) is always nonnegative [8], so that no cancellations are possible, this is already enough to show that ZJ (M ) ! +1 as J ! +1. In fact, just by summing over spin foams where all faces are labelled by the same spin, we already see SPIN FOAM MODELS OF RIEMANNIAN QUANTUM GRAVITY 9 that ZJ (M ) must tend to infinity at a rate no slower than J14. However, this is a drastic underestimate. In fact, doing a least squares fit to a log-log plot of the above tabulated values of ZJ (M ), we estimate that ZJ (M ) grows as approximately J23. From these considerations it is clear that the partition function in the DFKR model will diverge, not just in this example, but for many triangulated 4-manifolds. Since the divergence is mainly due to rapid growth of the face amplitudes with increasing spin, it seems the partition function only has a chance of converging if there are few spin foam faces compared to spin foam edges and vertices. Spin foams of this sort come from triangulations where there are few triangles compared to tetrahedra and 4-simplices. Since the number of tetrahedra in a nondegenerate triangulation is always 5_2times the number of 4-simplices, this occurs when the average number of 4-simplices meeting along each triangle is high. As explained more carefully in Section 6, we can try to extract physical information from spin foam models by computing expectation values of observables. These are weighted averages of functions assigning to each spin foam a real number, where the weight asso- ciated to each spin foam is its amplitude. The Metropolis algorithm [25 ] is a powerful tool for numerically computing weighted averages, but only when the weights are nonneg- ative. Quantum amplitudes are usually complex, so the Metropolis algorithm is normally applicable to quantum theory problems only after one has converted them into statis- tical mechanics problems via Wick rotation. Luckily, the amplitude for a spin foam in the DFKR model is always nonnegative! In fact, this is true for all the versions of the Riemannian Barrett-Crane model that we consider here [8]. This allows us to apply the Metropolis algorithm without becoming enmeshed in the subtleties of Wick rotation. As we shall see, this algorithm provides great insight into which spin foams dominate the partition function. For readers unfamiliar with the Metropolis algorithm, let us briefly describe how it works in general before turning to our application here. This algorithm is a random walk technique for sampling configurations from some finite set with desired relative frequencies given by some function p : ! [0, 1). To use the algorithm one must first choose a finite set of `moves' fj: ! . Starting at an arbitrary configuration x0 2 , one then generates a random sequence of configurations xi as follows. For each i, choose a move fj uniformly at random. Let xi+1 = fj(xi) with probability p(fj(xi))=p(xi) and otherwise let xi+1 = xi. (If p(fj(xi))=p(xi) > 1, always set xi+1 = fj(xi).) If the moves are chosen appropriately, the distribution of the entries xi in the sequence will tend to the desired distribution p as i tends to infinity. To get the right limiting distribution, the moves must be chosen `large enough' so that the algorithm is ergodic, i.e., so that any configuration x with p(x) > 0 has a nonzero chance of occurring. However, to get reasonably fast convergence to the limiting distri- bution, the moves must be chosen `small enough' so that the algorithm spends much of its time moving between highly weighted configurations, rather than spinning its wheels rejecting configurations with small p. Choosing the moves to balance these opposing needs is a bit of an art. In practice, one determines experimentally whether one's moves work well. To use the Metropolis algorithm to compute expectation values of observables in some version of the Riemannian Barrett-Crane model, we want to sample spin foams with relative frequencies given by their amplitudes. To do this we take to be some finite set of spin foams and let p : ! [0, 1) assign to each spin foam its amplitude. We also need to choose a set of moves for going from one spin foam to another. For our example of the 4-sphere triangulated as the boundary of a 5-simplex, we chose to use moves that consist 10 JOHN C. BAEZ, J. DANIEL CHRISTENSEN, THOMAS R. HALFORD, AND DAVID C. TSANG of picking a tetrahedron in the dual 2-skeleton and adding or subtracting 1_2to each of the spins labelling the four faces of this tetrahedron. The 10j symbols vanish unless the sum of the spins labelling faces incident to each edge is an integer. Our moves preserve this constraint. If subtracting 1_2from a spin would make it negative, we leave all the spins unchanged, but still count this process as a move. In our experiments, this collection of moves produces very fast convergence of the Metropolis algorithm, and the predicted answer is accurate in all cases in which we have been able to compute the exact answer by other means. It is interesting to compare the behavior of this algorithm for various versions of the Barrett-Crane model. Unfortunately, in the DFKR version, the divergence of the partition function means that the random walk will drift toward spin foams with ever larger spins, since these have the largest amplitudes and there are many of them. Table 2 shows a small portion of a typical run of the Metropolis algorithm for this version, with a spin cutoff of J = 5_2. The first column is the iteration number. In steps that are not shown, the program stayed at the same labelling. The second column displays the twenty spins labelling faces, each multiplied by two. The third column shows the amplitude of the corresponding spin foam. One can see that the sum over spin foams is dominated by those with many spins close to the cutoff. _________________________________________ |_iteration|________F___________|_Z(F_)__|9 |__335291_3|4234252435354544545_|4.4.10_|_9 |__335296_3|4234152435454545555_|3.1.10_|_9 |__335302_3|4244142345454545555_|1.9.10_|_8 |__335303_3|4344043335454545555_|5.6.10_|_9 |__335304_3|4444043335555555555_|1.0.10_|_9 |__335310_2|4443133335555555555_|3.4.10_|_9 |__335312_2|3444132235555555555_|2.0.10_|_9 |__335320_1|3443242235555555555_|1.1.10_|_8 |__335321_0|4533242235555555555_|2.5.10_|_8 |__335323_0|4543252345555555555_|4.6.10_|_8 |__335324_0|4544252344555554455_|3.9.10_|_7 |__335327_0|5545251244555554455_|9.9.10_|_7 |__335328_0|5445150254555554455_|1.5.10_|_7 |__335351_0|5445250254455555445_|1.4.10_|_ Table 2: sample Metropolis labellings _ DFKR model with spin cutoff 5_2 SPIN FOAM MODELS OF RIEMANNIAN QUANTUM GRAVITY 11 4. The Perez-Rovelli Model In the Perez-Rovelli model, the face, edge and vertex amplitudes are as follows: _________________________________________________* *______________ A(f ) = j(f)____o____________________________________________* *___________________________________________@ j1(e)___________________________________________ ___________________________________________________* *______________________________j2(e) ____________________________________________________* *___________________________________________@ o____________________________________________________* *___________________________________________@ ____________________________________________________* *___________________________________________@ __________________________________________________* *___________________________________j3(e) A(e) = ___________________j4(e)_____________________________________________* *___________________________________________@ j1(e)___o___________________________________________________________* *________________________________oj2(e)_____@ (6) voHH) vvv~~)HH) j2(v)vvv ~~)) HHHj1(v)H vvv ~~ )) HHH vvv ~j~(v))) HHH o)v)_________6_____________o~_~~HHHvv)) ))HHj7(v)~~H )j10(vvv~~v)) A(v) = ))) HH~~H )vvv) ~~ . ~HHH vvv) ~ j3(v)))) ~~ HH vv )) j5(v)~~ ~j8(v)HHHj9(v)vv~v) ))~~ vvv HHH )) ~~ ))~~vvv HHH~~)) o______j4(v)_____o~~v Here again a comment is in order: the original papers by Perez and Rovelli [30 , 32] give a different formula for the edge amplitudes, but that formula does not really follow from their group field theory. Above we use the corrected formula which appears in a forth- coming review article by Perez [31 ]; we have carefully translated from his normalization conventions to our own. With these formulas, Perez has shown that the partition function converges for any well-behaved pseudomanifold satisfying the condition that each triangle lies in at least three 4-simplices. This includes the triangulation of S4 as the boundary of a 5-simplex. Nonetheless it is illuminating to compute the cutoff partition function ZJ (M ) in this example with various choices of the spin cutoff. The results appear in Table 3. _____________________ |__J__|__ZJ(M)_____|_ |__0_1|.000000000000_| |_1=21|.000014319178_| |__1_1|.000014323656_| |_3=21|.000014323670_| |__2_1|.000014323670_| Table 3: S4 partition function _ Perez-Rovelli model with spin cutoff J This time it appears that the convergence is spectacularly rapid. But again, this is easy to understand: it is mainly due to the denominator of the edge amplitude in formula (6). Since each triangle in this triangulation of the 4-sphere is the face of 3 tetrahedra, the 12 JOHN C. BAEZ, J. DANIEL CHRISTENSEN, THOMAS R. HALFORD, AND DAVID C. TSANG factors of ________________________________________________ j(f)____o_________________________________________________________* *___________________________________________@ appearing in the edge and face amplitudes combine to give j1(e)_______________________________ _________________________________________* *________________________________ X Y Y ___________________________________________* *______________________________________j2(e)@ ZJ (M ) = (2j(f ) + 1)-4 o___________________________________________* *___________________________________________@ |F | Jf2 2 e2 3 __________________________________________* *___________________________________________@ j4(e) The factors of (2j(f ) + 1)-4 strongly suppress faces labelled by nonzero spins. The 10j symbols also tend to suppress nonzero spins. While the 4j symbols grow with increasing spin, they do so too slowly to make much of a difference. In particular, Z0(M ) = 1 because there is one spin foam with all faces labelled by spin 0, and every balanced spin network with all edges labelled by spin 0 evaluates to 1. In computing Z1=2(M ), we must also consider spin foams where some faces are labelled by spin 1_2. At each spin foam edge, if one of the incident faces is labelled by a nonzero spin, then at least one other must be as well, or else the Barrett-Crane intertwiner there will vanish. This is a powerful constraint. For the 4-sphere triangulated as the boundary of a 5-simplex, it implies that if there is one spin foam face labelled by a nonzero spin, then there must be at least four. When four of the faces are labelled by spin 1_2, the factors of (2j(f ) + 1)-4 multiply to give 2-16 . Spin foams with more nonzero spins, or spins greater than 1_2, will be suppressed even further. In fact, it is instructive to work out by hand the contribution to the partition function given by spin foams with four spin foam faces labelled by 1_2and the rest zero. To give a nonzero result, the four faces must form a tetrahedron in the dual 2-skeleton. This results in a triangular spin- 1_2loop in four of the 10j symbols. Since 1_2 * * @ ____________________________________________________________________* *___________________________1_ @ ____________________________________________________________2_______* *___________________________________________@ o____________________________________________________________________* *___________________________________________@ __________________________________________________________________0_* *______________________________________ @ __________________________________________________________ * * @ 0 * * @ vvoHHH~)) voHHHv~)) 0vvvv~~))HHH0H) 1_2vvvv~~)HHH1_2)) vvv ~~ ) HHH vvv ~~ ) HHH ovv)_______0_________H_~~H)) )vv_______1_2_______HH_~~H)) )HHH ~~ )) vvo~v o)HHH ~~ )) vvo~v )) HHH0~~ 0vvv))~~= 1, and )) HH0~~H )0vvv)~~= 1_, )) ~~HHH vvv)) ~~ )) ~~HHH vvv)) ~~ 2 0 )) ~ 0 HHH0vv) 0~ 0)) ~ 0HHHv0v ) ~0 ) ~~ vvvHH )) ~~ ) ~~ vvvHH )) ~~ ))~~vvvv HHHH~~)) ))~~vvvv HHHH~~)) o______0_____o_~~v o_____0______o~~v each spin foam of this sort contributes an amplitude of 2-16 . ( 1_2)4 = 2-20 . There are 6 4 = 15 spin foams of this sort, so their total contribution to the partition function is 15 . 2-20 ~=.0000143051. Glancing at Table 3, we see that together with the spin foam having all faces labelled by spin zero, this accounts for the first seven decimal places of the partition function. As a result, when we use the Metropolis algorithm to randomly walk through spin foams in this example, it focuses attention on spin foams where almost all spins are zero. Table 4 shows a complete run of the algorithm with a cutoff of 5=2 and 5 million iterations. As in SPIN FOAM MODELS OF RIEMANNIAN QUANTUM GRAVITY 13 Table 2, the first column is the iteration number. In steps that are not shown, the program stayed at the same labelling. The second column displays the twenty spins labelling faces, each multiplied by two. The third column shows the amplitude of the corresponding spin foam. One can see that that after 256 steps the initial spin foam has randomly walked to the spin foam with all faces labelled by by spin zero. Except for a brief foray to a spin foam with four spin- 1_2faces between moves 611050 and moves 611136, the algorithm spends all the rest of its time at the spin foam with all spin-zero faces. _____________________________________________ |_iteration|________F___________|___Z(F_)____|-14 |________00|0110000010111110011_|1.421.10____|-12 |________10|0100000010101020011_|2.874.10____|-7 |______2560|0000000010000010011_|9.537.10____|0 |______4580|0000000000000000000_|1.000.10____|-7 |__611050_|00100101010000000000_|9.537.10____|0 |__611136_|00000000000000000000_|1.000.10____| Table 4: sample Metropolis labellings _ Perez-Rovelli model with spin cutoff 5_2 Generalizing from this example, we can easily guess the behavior of the Perez-Rovelli model on an arbitrary triangulated 4-manifold. The partition function will be dominated by spin foams having mostly spin-zero faces, and a low density of small islands of faces with higher spin. In fact, we can use the `dilute gas' approximation [7] to estimate the density of a particular sort of island in the spin foams that would most often be sampled by the Metropolis algorithm. For example, if we consider tetrahedra in the dual 2-skeleton, most of them will have all faces labelled by spin 0. About one in 220 will have all four faces labelled by spin 1_2, and an even smaller fraction will have faces labelled by higher spins. We discuss the implications of this `spin-zero dominance' in Section 6. 5. A New Model Since the partition function of the DFKR model diverges rapidly, while that of the Perez-Rovelli model converges so rapidly that the sum is dominated by spin foams with almost all faces labelled with spin zero, it seems worthwhile to seek a model with interme- diate behavior. It would be nice to derive this model from a group field theory. However, one can also take an exploratory attitude and simply seek face, edge and vertex amplitudes that give partition functions `near the brink of convergence', but on the convergent side. 14 JOHN C. BAEZ, J. DANIEL CHRISTENSEN, THOMAS R. HALFORD, AND DAVID C. TSANG Comparing various candidates, we found this model to be the most promising: A(f ) = 1 A(e) = ______1______j1(e) __________________________________________________* *__ ___________________________________________________* *_____________________j2(e) ____________________________________________________* *___________________________________________@ o____________________________________________________* *___________________________________________@ ____________________________________________________* *___________________________________________@ __________________________________________________* *___________________________________j3(e) j4(e) ) (7) vvvoHHHHv~~)) j2(v)vvv ~~)) HHHj1(v)H vvv ~~ )) HHH vvv ~j~(v))) HHH o)v)_________6_____________o~_~~HHHvv)) ))HHj7(v)~~H )j10(vvv~~v)) A(v) = ))) HH~~H )vvv) ~~ . ~HHH vvv) ~ j3(v)))) ~~ HH vv )) j5(v)~~ ~j8(v)HHHj9(v)vv~v) ))~~ vvv HHH )) ~~ ))~~vvv HHH~~)) o______j4(v)_____o~~v The first thing to note is this model's simplicity. As in the DFKR model, the edge amplitudes arise naturally from normalizing the Barrett-Crane intertwiners in the 10j symbol. But unlike the DFKR model, this new model has trivial face amplitudes. Thus the only real ingredient of this model is the 10j symbol built from normalized Barrett- Crane intertwiners. The absence of loops ________________________________________________________* *____ j____o___________________________________________________* *________________________________________ in the above formulas is the main reason the model lies near the brink of convergence. These loops grow rapidly as a function of j, so they tend to make the partition func- tion diverge or converge very quickly, depending on whether more of them appear in the numerator or denominator in the partition function. Table 5 shows the partition function of our new model for the triangulation of S4 as the boundary of a 5-simplex, as a function of the spin cutoff J. Though the partition function appears to be converging, it is hard to be sure from this limited data. Unfortunately, the calculation of Z5=2(M ) already involved a sum over approximately 3.6 trillion spin foams. (There are a total of 620 ways to label all twenty faces with spins from 0 to 5_2, but of these, only 620=210 ~=3.6 . 1012 satisfy the constraint that the spins labelling faces incident to any edge sum to an integer; only these can give a nonzero result, so we only summed over these.) This calculation occupied 28 CPUs for 23 hours. Going further with this brute-force approach would require much longer. SPIN FOAM MODELS OF RIEMANNIAN QUANTUM GRAVITY 15 _____________________ |__J__|__ZJ(M)_____|_ |__0_1|.000000000000_| |_1=22|.342658607645_| |__1_3|.378038633798_| |_3=23|.966290480574_| |__2_4|.293589340364_| |_5=24|.480621474940_| Table 5: S4 partition function _ the new model with spin cutoff J In Section 6 we describe an indirect method which gives stronger evidence that the partition converges for this triangulation of S4. Of course, one would really like a mathe- matical proof that the partition function converges _ and not just in this case, but more generally. Numerical calculations show that it diverges for the well-behaved pseudomani- fold formed by taking two 4-simplices and gluing them together along all their tetrahedral faces. However, this leaves open the possibility that the partition function converges for the class of well-behaved pseudomanifolds considered by Perez _ namely, those where each triangle lies in at least three 4-simplices. Proving this would require good bounds on the 10j symbol. Numerical computations suggest that the following bound holds: fifi vvoHH~)) fi fifi j2vvvv~~))HHHj1H fifi fifivvvv ~~~ ))) HHHH fifi fifiov))___j6_________Ho~_~~~HHHHvvv)))fifi10Y fifi)))j7~~HHHH vvvvj1~~0))fifi C1 (2ji + 1)- 1_5. fifij3)))~~~HHHHvvvv)j5~~~))fifi i=1 fifi))~~~j8vHHHHj9v~~~vv)))fifi fifi o)____________~~~~HHHvvvv)fifi j4 o fi This is consistent with our previous observation that the 10j symbols decay as O(j-2 ) as all spins are rescaled by the same factor, but it gives more information when some spins are much larger than others. We can use this bound to sketch a rough argument that the partition function converges for well-behaved pseudomanifolds in which each triangle lies in at least three 4-simplices. Far from the border of admissibility, it is easy to prove that the 4j symbols satisfy fifij fi fifi_1_______________________________________________________________* *_________________________________________jf@ fifio________________________________________________________________* *__________________________________________2@ fifi_________________________________________________________________* *___________________________________________@ fi _j4____________________________________________________fifi If we could ignore the border of admissibility, this estimate and the above bound on the 10j symbols would imply a bound on the cutoff partition function: X Y 9_ ZJ (M ) C3 (2j(f ) + 1)- 20n(f) |F | Jf2 2 Y X 9_ C3 (2j(f ) + 1)- 20n(f). f2 2 j(f)2{0, 1_2,...,J} 16 JOHN C. BAEZ, J. DANIEL CHRISTENSEN, THOMAS R. HALFORD, AND DAVID C. TSANG Here n(f ) is the number of vertices (or equivalently, edges) of the face f , which is the same as the number of 4-simplices containing the triangle dual to f . The curious number 9_ - 1_5 20 comes from the fact that each vertex of1the face f gives a factor of (2j(f ) + 1) , while each edge gives a factor of (2j(f ) + 1)- _4, and 1_5+ 1_4= _9_20. Since X 9_ (2j + 1)- 20n j2{0, 1_2,...} converges when n 3, this bound would imply convergence of ZJ (M ) as J ! 1 whenever each triangle is contained in at least three 4-simplices. Unfortunately, this argument neglects the border of admissibility, where the 4j symbols grow more slowly. Luckily, the 10j symbols decay more rapidly near the border of admissibility! We are therefore optimistic that this hole in the argument can be fixed. As with other versions of the Barrett-Crane model, we can get a qualitative feel for the new model using the Metropolis algorithm. Table 6 shows a small portion of a typical run of this algorithm, again using the triangulation of S4 as the boundary of a 5-simplex and imposing a spin cutoff of 5_2. This table is organized just like Tables 2 and 4. Note that in the new model, both low spins and spins near the cutoff show up frequently, but with a predominance of low spins. _____________________________________________ |_iteration|________F___________|___Z(F_)____|-7 |_4995398_|03103000002104313300_|4.768.10____|-7 |_4995458_|03104000001104314400_|1.953.10____|-7 |_4995513_|04104000000103414400_|1.600.10____|-8 |_4995517_|04104001000102413401_|6.250.10____|-8 |_4995520_|04114001000112303401_|2.441.10____|-8 |_4995529_|04104001000102413401_|6.250.10____|-7 |_4995534_|04104001100102313300_|1.526.10____|-8 |_4995542_|04104001200102213201_|6.028.10____|-8 |_4995547_|04104002200101212202_|1.191.10____|-10 |_4995554_|14105112200101212202_|4.961.10____|-10 |_4995565_|05215112200101212202_|2.297.10____|-9 |_4995576_|05215113200100211201_|9.303.10____|-9 |_4995577_|05215113100100311302_|2.943.10____|-10 |_4995582_|05215013100200312312_|3.489.10____|-10 |_4995587_|05215013110200322301_|4.361.10____|-10 |_4995596_|04215013111201222301_|2.224.10____|-11 |_4995601_|04215013211201122202_|8.954.10____|-9 |_4995610_|04215013311201022101_|1.608.10____|-9 |_4995620_|04115013311102012101_|6.441.10____|-7 |_4995626_|04114013310102011001_|1.031.10____| Table 6: sample Metropolis labellings _ the new model with spin cutoff 5_2 It is interesting to see the frequencies with which faces are labelled by various spins. We show this in Table 7, based on a Metropolis run with spin cutoff J = 50 and half a billion iterations. The results obtained are very similar to results obtained with a spin cutoff of J = 25_2, indicating that they are not just an artifact of the cutoff. We only show results SPIN FOAM MODELS OF RIEMANNIAN QUANTUM GRAVITY 17 up to j = 5, but higher spins were seen as well, with smoothly declining frequencies. The most important thing to note is that while spin zero is the most likely spin to occur, there is still a substantial fraction of faces labelled by other spins. ________________ |_spin|frequency|_ |__0__|69.548%_|_ |_1=2_|18.733%_|_ |__1__|6.2878%_|_ |_3=2_|2.5510%_|_ |__2__|1.1958%_|_ |_5=2_|.61995%_|_ |__3__|.34893%_|_ |_7=2_|.21243%_|_ |__4__|.13535%_|_ |_9=2_|.08989%_|_ |__5__|.06252%_|_ Table 7: spin frequencies in S4 _ the new model 6. Implications Our computation of partition functions is only relevant to the `physics' of the models being studied to the extent that it sheds light on the behavior of observables. The issue of observables in quantum gravity is a thorny one, but we really need to confront it here. One approach would be to follow the ideas of canonical quantum gravity and use a sum over open spin foams to compute the projection onto the space of physical states [4, 35, 37]. Observables would then be described as operators on this Hilbert space. While conceptually well-motivated, this approach will take a great deal of work to implement. One of the goals of the spin foam program is to develop a `sum over histories' approach to quantum gravity that has a chance of making more rapid progress [33 ]. Ideally this approach would be compatible with the canonical approach, but not require an explicit computation of the projection onto physical states. In this section we attempt to interpret our computations using this `sum over histories' approach. To begin with, let us tentatively call any function O from spin foams to real numbers an `observable'. Fixing a spin foam model, we can try to compute the expectation value of O as follows: P O(F )Z(F ) (8) = ___F____________P, F Z(F ) where Z(F ) is the amplitude our model assigns to the spin foam F , and the sum is taken over all spin foams. The denominator of this fraction is the partition function. Formulas like equation (8) are familiar in quantum field theory on a fixed background spacetime, but we must reevaluate their meaning in the current context. Exactly what sort of `expectation value' is this quantity ? In quantum field theory a formula of this sort is used to compute vacuum expectation values. However, in the context of quantum gravity, the notion of energy and thus the whole concept of `vacuum' becomes problematic. We thus propose to interpret as the average of O over all histories. This is consistent with the interpretation of a spin foam as a `quantum history' [6]. 18 JOHN C. BAEZ, J. DANIEL CHRISTENSEN, THOMAS R. HALFORD, AND DAVID C. TSANG Now, for many functions O we do not expect to be well-defined. For example, if O(F ) is some measure of the total 4-volume of the spacetime corresponding to F , there is no obvious reason why should converge. This is not bad; it just means that the question "what is the expected 4-volume of spacetime?" is ill-posed when one is given no further information about the history in question. There is no reason a theory should be able to answer this sort of question. However, one can often make ill-posed expectation values well-posed by `conditioning' them. In other words, instead of asking "what is the expected value of O?" one asks "what is the expected value of O, given that...?" In physics this conditioning is usually done by specifying either a state or the value or expectation value of some observables. For example, in the path integral approach to quantum mechanics, one can compute the expected value of the position of a particle at t = 1, given its position at t = 0, by restricting the path integral to paths that start at this given position at t = 0. We can even condition further by specifying information about its position at some other times; this has been extensively studied in the theory of consistent histories [20 , 21, 27]. A path represents a classical history, but we can also do conditioning when we compute expectation values by summing over quantum histories. The most familiar example of a quantum history is a Feynman diagram. Using Feynman diagrams we can compute the expectation value of some observable measured in the future, given information about the incoming particles in the past, by restricting the sum over Feynman diagrams to those with certain specified incoming edges. We can even condition on properties of the internal edges of a Feynman diagram, e.g. computing the probability that two electrons scatter given that they have exchanged a specific number of virtual photons. One must be careful when working with probabilities of this sort, since they can fail to satisfy the classical rules, thanks to interference effects. However, the theory of consistent histories provides a framework for correctly dealing with them. Spin foams are close analogs of Feynman diagrams, and indeed they are Feynman diagrams in the group field theory approach [36 ]. This means that, as with Feynman diagram theories, in spin foam models we can condition any expectation value by limiting the class of spin foams to be summed over, or weighting them with a suitable factor. This amounts to replacing the formula for Z(F ) by a modified formula which takes this conditioning into account. The simplest example consists of setting Z(F ) to zero when F fails to lie in the dual 2- skeleton of a fixed triangulated 4-manifold. While doing this simplifies many calculations, and we have implicitly done so throughout this paper, it would only be physically well- motivated if we knew spacetime were equipped with a specific triangulation. A more realistic example would arise if we were trying to use a Lorentzian spin foam model to make predictions about the collision of gravitational waves. Here we would need to restrict the integral over spin foams to those having a spacelike slice in which incoming gravitational waves of a specified sort were present. If we were studying these waves in a bounded region of spacetime, we might also restrict to open spin foams of a certain `size'. Of course, we are far from being able to do this at present. Having modified Z(F ) to take the conditioning into account, there are still some further subtleties about the convergence of formula (8). If the sums in both numerator and denominator converge, is well-defined. However, we can make do with less: if both sums diverge, we can impose a cutoff on both, take the ratio of the two, and then try to take a limit as the cutoff is removed. It follows that the convergence of the conditioned partition function is neither necessary nor sufficient for computing . SPIN FOAM MODELS OF RIEMANNIAN QUANTUM GRAVITY 19 What does it mean if we need to impose a cutoff and then take a limit as the cutoff is removed to compute expectation values of observables? The answer depends on the nature of the cutoff, so it is good to focus on a concrete example. We have seen one example in Section 3, where we considered the DFKR model on a fixed triangulation of the 4-sphere. Since the partition function diverged, we found it useful to insert a cutoff on the spins labelling spin foam faces, or equivalently, triangles in the triangulation. In this model we can define the area of a triangle labelled by the spin j to be p ________ (9) area= 8ssfl`2P j(j + 1), where `P is the Planck length and fl is an ad hoc constant roughly analogous to the Barbero-Immirzi parameter [15 , 23 ]. A spin cutoff then amounts to a kind of `infrared cutoff', since it rules out spacetime geometries containing triangles of large area. Without any further conditioning, the expectation value of an observable in this theory would be given by P O(F )Z(F ) (10) = Jlim!1__|F_|_J__________P. |F | JZ(F ) What does it mean when this limit exists but both numerator and denominator diverge as J ! 1? It means that the dominant contribution to the expectation value of the observable O comes from spacetime geometries containing triangles of arbitrarily large area! This seems physically unrealistic, since in our world spacetime discreteness exists, if at all, only on short length scales. One possible objection is that this does not take into account the conditioning needed to phrase a sensible physical question. Perhaps conditioning automatically damps the contribution of spin foams with large triangles. This seems most plausible if we are asking questions about a bounded region of spacetime and restrict the sum over spin foams to those of a certain `size'. Another way out might be to take a limit where we let the Barbero-Immirzi parameter go to zero as we let J ! 1: in other words, a kind of `continuum limit', where spacetime discreteness gets pushed to ever smaller distance scales. A limit of this sort has already been discussed by Bojowald [14 ] in the context of Lorentzian quantum gravity, working with the real Ashtekar variables. He shows that in this limit, loop quantum cosmology reduces to ordinary quantum cosmology. One can imagine taking a similar limit in Rie- mannian quantum gravity. However, a great deal more work would be required to see if this is a viable strategy. In short, the divergence of the partition function as we remove the spin cutoff is not necessarily a disaster for the DFKR model. However, it seems one would need some sophisticated maneuvers to extract interesting results from this model. Anyone interested in this might be wise to start by reexamining the Ponzano-Regge model of 3-dimensional Riemannian quantum gravity, which exhibits a similar divergence. We can avoid or at least postpone facing these subtleties by working with the Perez- Rovelli model, where the partition function converges for a fixed triangulation. Of course, a divergence may still arise in the sum over triangulations. But still more pressing, in our opinion, is the task of understanding spin-zero dominance. While this phenomenon has no obvious analogue in the Lorentzian Barrett-Crane model, it is still worth pondering. What does it mean when the partition function is dominated by spin foams whose faces are mostly labelled by spin zero? If we believe equation (9), these correspond to triangles of area zero! Perhaps this model is a theory of highly degenerate quantum geometries where most of the 4-simplices are shrunk to nothing, vaguely reminiscent of the `crumpled 20 JOHN C. BAEZ, J. DANIEL CHRISTENSEN, THOMAS R. HALFORD, AND DAVID C. TSANG phase' in Euclidean quantum gravity [2, 24]. Perhaps suitable conditioning will remove this effect. Perhaps spin-zero faces should be ignored for some reason. Or perhaps Alekseev, Polychronakos, and Smedb"ack [1] are right, and the correct area formula is given not by equation (9) but by area = 8ssfl`2P(j + 1_2). This would drastically affect our interpretation of spin-zero faces. At present, all we can say for sure is that a theory with drastic spin-zero dominance raises as many difficult issues for our approach to computing expectation values as one where the partition function diverges as the spin cutoff is removed. The new model described in this paper seems to avoid these issues; here we can compute expectation values of observables and get some interesting results, at least if we fix a triangulation. The simplest observable in the Riemannian Barrett-Crane model is the average area of a triangle. Ignoring a factor of 8ssfl`2P, this is given by X p _______________ O(F ) = __1__| j(f )(j(f ) + 1) f22| 2 where | 2| is the number of triangles in the triangulation. We use J to stand for the expectation value of this observable with a spin cutoff of J: P O(F )Z(F ) J = ___|F_|_J__________P |F | JZ(F ) where for a simple calculation we sum over spin foams in the dual 2-skeleton of the trian- gulation of S4 as the boundary of a 5-simplex. Some results for this quantity are shown in Table 7. The results for J 5_2are exact, while the results for higher cutoffs are ap- proximate, obtained using the Metropolis algorithm. The last data point was run using no cutoff; spins larger than 54 never occurred in our runs, even after more than 11 billion iterations. It appears that the limit = Jlim!1J exists. _______________ |__J__|_J___| |__0__|0.000000_| |_1=2_|0.121987_| |__1__|0.210441_| |_3=2_|0.265911_| |__2__|0.302153_| |_5=2_|0.326524_|_ |_15=20|.381160_| |_25=20|.396701_| |__50_0|.399991_| |__1__|0.400005_| Table 7: Expected average area of a triangle in S4 _ the new model with spin cutoff J SPIN FOAM MODELS OF RIEMANNIAN QUANTUM GRAVITY 21 Now, an interesting thing about the average triangle area is that for this observable, the limit Jlim!1J can only exist if the partition function converges. This means our numerical evidence that J converges is also numerical evidence that the partition function converges! To see why this is true, suppose the partition function diverges. Since in this model the spin foam amplitudes Z(F ) are always nonnegative [8], the cutoff partition functions ZJ (M ) must approach +1 as we remove the spin cutoff. This implies that for any spin J we have X X 2 Z(F ) Z(F ) |F | J |F | J0 for all sufficiently large spins J0. This implies X X Z(F ) 2 Z(F ). |F | J0 J<|F | J0 Using this, we see that for all sufficiently large J0, P 0O(F )Z(F ) J0 = __|F_|_J____________P |F | J0Z(F ) P 0O(F )Z(F ) __J<|F_|_J_____________P |F | J0Z(F ) P 0O(F )Z(F ) __J<|F_|_J_____________2 P J<|F | J0Z(F ) p _________ J(J + 1) ____________2| 2| sincepin_any_spin foam with J < |F |, there is at least one triangle with area at least J(J + 1) , so the average triangle area is at least this quantity divided by the number of triangles. Since J can be chosen as large as we like, we see that J0lim!1J0 = +1. This is a nice example of how the convergence of the partition function is intimately linked to the behavior of observables. In conclusion, we wish to emphasize that while the interpretation of spin foam models raises many difficult issues, this is only to be expected given the novelty of the whole setup. We expect rapid progress, especially if the more traditional tools of theoretical physics are supplemented by computer calculations. Hard numbers have a marvelous way of making problems more concrete. Acknowledgements We thank Greg Egan and Alejandro Perez for extremely valuable discussions. We also thank SHARCNet for providing the supercomputer at the University of Western Ontario on which we computed Z5=2(M ) in our new model. The second author was supported by a grant from NSERC, and the third and fourth authors were supported by NSERC and SHARCNet. 22 JOHN C. BAEZ, J. DANIEL CHRISTENSEN, THOMAS R. HALFORD, AND DAVID C. TSANG References [1]A. Alekseev, A. P. Polychronakos and M. Smedb"ack, On area and entropy of a black hole, availab* *le as hep-th/0004036. [2]J. Ambjorn, B. Durhuus, and T. Jonsson, Quantum Geometry: A Statistical Field Theory Approach, Cambridge U. Press, Cambridge, 1997. [3]F. Archer and R. M. Williams, The Turaev-Viro state sum model and three-dimensional quantum gravity, Phys. Lett. B273 (1991), 438-444. [4]M. Arnsdorf, Relating covariant and canonical approaches to triangulated models of quantum grav* *ity, available as gr-qc/0110026. [5]J. C. Baez, Spin foam models, Class. Quantum Grav. 15 (1998), 1827-1858. Available as gr- qc/9709052. [6]J. C. Baez, An introduction to spin foam models of quantum gravity and BF theory, in Geometry and Quantum Physics, eds. Helmut Gausterer and Harald Grosse, Springer, Berlin, 2000. Available as gr-qc/9905087. [7]J. C. Baez, Spin foam perturbation theory, available as gr-qc/9910050. [8]J. C. Baez and J. D. Christensen, Positivity of spin foam amplitudes, Class. Quant. Grav. 19 (2* *002), 2291-2305. Version with additional corrections available as gr-qc/0110044. [9]J. C. Baez, J. D. Christensen and G. Egan, Asymptotics of 10j symbols, in preparation. [10]J. W. Barrett, The classical evaluation of relativistic spin networks, Adv. Theor. Math. Phys. 2 (1998), 593-600. Available as math.QA/9803063. [11]J. W. Barrett and L. Crane, Relativistic spin networks and quantum gravity, Jour. Math. Phys. 39 (1998), 3296-3302. Available as gr-qc/9709028. [12]J. W. Barrett and L. Crane, A Lorentzian signature model for quantum general relativity, Class. Quantum Grav. 17 (2000), 3101-3118. Available as gr-qc/9904025. [13]J. W. Barrett and R. M. Williams, The asymptotics of an amplitude for the 4-simplex, Adv. Theor. Math. Phys. 3 (1999), 209_215. Available as gr-qc/9809032. [14]M. Bojowald, The semiclassical limit of loop quantum cosmology, Class. Quant. Grav. 18 (2001), L109-L116. Available as gr-qc/0105113. [15]R. Capovilla, M. Montesinos, V. A. Prieto and E. Rojas, BF gravity and the Immirzi parameter, C* *lass. Quant. Grav. 18 (2001), L49-L52. Erratum: ibid. 18 (2001), 1157. Available as gr-qc/0102073. [16]J. Scott Carter, Daniel E. Flath and Masahico Saito, The Classical and Quantum 6j-Symbols, Prin* *ce- ton U. Press, Princeton, 1995. [17]J. D. Christensen and G. Egan, An efficient algorithm for the Riemannian 10j symbols, Class. Qu* *ant. Grav. 19 (2002), 1184-1193. Available as gr-qc/0110045. Code for computing 10j symbols available at http://jdc.math.uwo.ca/spin-foams/. [18]L. Crane and D. Yetter, A categorical construction of 4d TQFTs, in Quantum Topology, eds. L. Ka* *uff- man and R. Baadhio, World Scientific, Singapore, 1993, pp. 120-130. Available as hep-th/9301062. [19]R. De Pietri, L. Freidel, K. Krasnov and C. Rovelli, Barrett-Crane model from a Boulatov-Ooguri* * field theory over a homogeneous space, Nucl. Phys. B574 (2000), 785-806. Available as hep-th/9907154. [20]M. Gell-Mann and J. B. Hartle, Classical equations for quantum systems, Phys. Rev. D47 (1993), 3345-3382. Available as gr-qc/9210010. [21]R. B. Griffiths, Consistent Quantum Mechanics, Cambridge U. Press, Cambridge, 2002. [22]L. Kauffman and S. Lins, Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds, Prince* *ton U. Press, Princeton, New Jersey, 1994. [23]R. Livine, Immirzi parameter in the Barrett-Crane model?, available as gr-qc/0103081. [24]R. Loll, Discrete approaches to quantum gravity in four dimensions, Living Reviews in Relativit* *y 1 (1998). Available at http://www.livingreviews.org/. [25]N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, Equation of state calculations by fast computing machines, J. Chem. Phys. 21 (1953), 1087-1092. [26]S. Mizoguchi and T. Tada, 3-dimensional gravity from the Turaev-Viro invariant, Phys. Rev. Lett. 68 (1992), 1795-1798. Available as hep-th/9110057. [27]R. Omn`es, Understanding Quantum Mechanics, Princeton U. Press, Princeton, 1999. [28]H. Ooguri, Topological lattice models in four dimensions, Mod. Phys. Lett. A7 (1992), 2799-2810. Available as hep-th/9205090. [29]D. Oriti, Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravit* *y, available as gr-qc/0106091. SPIN FOAM MODELS OF RIEMANNIAN QUANTUM GRAVITY 23 [30]A. Perez, Finiteness of a spinfoam model for Euclidean quantum general relativity, Nucl. Phys. * *B599 (2001) 427-434. Available as gr-qc/0011058. [31]A. Perez, Group quantum field theories and spin foam models for quantum gravity, in preparation. [32]A. Perez and C. Rovelli, A spin foam model without bubble divergences, Nucl. Phys. B599 (2001) 255-282. Available as gr-qc/0006107. [33]A. Perez and C. Rovelli, Observables in quantum gravity, available as gr-qc/0104034. Available * *as gr-qc/0104034. [34]G. Ponzano and T. Regge, Semiclassical limit of Racah coefficients, in Spectroscopic and Group Theoretical Methods in Physics, ed. F. Bloch, North-Holland, New York, 1968. [35]M. Reisenberger and C. Rovelli, "Sum over surfaces" form of loop quantum gravity, Phys. Rev. D56 (1997), 3490-3508. Available as gr-qc/9612035. [36]M. Reisenberger and C. Rovelli, Spin foams as Feynman diagrams, available as gr-qc/0002083. Ava* *il- able as gr-qc/0002083. [37]C. Rovelli, The projector on physical states in loop quantum gravity, Phys. Rev. D59 (1999), 10* *4015. Available as gr-qc/9806121. [38]V. Turaev, Quantum invariants of 3-manifolds and a glimpse of shadow topology, in Quantum Group* *s, Springer Lecture Notes in Mathematics 1510, Springer, Berlin, 1992, pp. 363-366. [39]V. Turaev and O. Viro, State sum invariants of 3-manifolds and quantum 6j symbols, Topology 31 (1992), 865-902. [40]D. N. Yetter, Generalized Barrett-Crane vertices and invariants of embedded graphs, J. Knot The* *ory Ramifications 8 (1999), 815-829. Department of Mathematics, University of California, Riverside, California 92521 USA E-mail address: baez@math.ucr.edu Department of Mathematics, University of Western Ontario, London, ON N6A 5B7 Canada E-mail address: jdc@uwo.ca E-mail address: thalford@ieee.org E-mail address: dtsang@physics.ubc.ca