PHYSICAL REVIEW D 81, 084048 (2010) Gravitational Wilson loop in discrete quantum gravity
H. W. Hamber Department of Physics and Astronomy, University of California Irvine, California 92717, USA
R. M. Williams Department of Applied Mathematics and Theoretical Physics, Wilberforce Road, Cambridge CB3 0WA, United Kingdom (Received 9 July 2009; revised manuscript received 2 March 2010; published 27 April 2010) Results for the gravitational Wilson loop, in particular, the area law for large loops in the strong coupling region, and the argument for an effective positive cosmological constant, discussed in a previous paper, are extended to other proposed theories of discrete Euclidean quantum gravity in the strong coupling limit. We argue that the area law is a generic feature of almost all nonperturbative Euclidean lattice formulations, for sufficiently strong gravitational coupling. The effects on gravitational Wilson loops of the inclusion of various types of light matter coupled to lattice quantum gravity are discussed as well. One finds that significant modifications to the area law can only arise from extremely light matter particles. The paper ends with some general comments on possible, physically observable, consequences.
DOI: 10.1103/PhysRevD.81.084048 PACS numbers: 04.60. m, 04.60.Gw, 04.60.Nc, 98.80.Qc
I. INTRODUCTION short-distance fluctuations of the underlying geometry, with the key assumption being the use of a Haar integration The identification of possible observables is an impor- measure for the local rotations at strong coupling.1 A tant part of formulating a theory of quantum gravity. In general result then emerges, at least for the Euclidean general it is expected that these quantum observables will theory, which is that the Wilson loop generically exhibits be represented by expectation values of operators which an area law for sufficiently strong gravitational coupling have physical interpretations in the context of a manifestly (large G) and near-planar loops [3,4]. It should be noted covariant formulation. In this paper, we focus on the gravi- here that in contrast to gauge theories, the Wilson loop in tational analog of the Wilson loop [1,2], which provides quantum gravity [6] does not provide useful information on physical information about the parallel transport of vec- the static potential, which is obtained instead from the tors, and therefore on the effective curvature, around large, correlation between particle world-lines [7,8]. Instru- near-planar loops. We will extend the analysis of earlier mental in deriving the results of [3] was the first-order work [3,4] to more general theories of discrete quantum Regge lattice [9] formulation of gravity, discussed origi- gravity. A recent complementary discussion of the signifi- nally in [10]. cance of physical observables in a quantum theory of Furthermore, from a semiclassical point of view, a vec- gravity can be found, for example, in [5]. tor’s rotation around a large macroscopic loop is expected In classical gravity the parallel transport of a coordinate vector around a closed loop is described by a rotation, 1 which is a given function of the affine connection along In the following we will be dealing almost exclusively, unless the space-time path. Then the total rotation matrix UðCÞ is stated otherwise, with the Euclidean theory. Thus, for example, P we will be considering Oð4Þ rotations and not Oð3; 1Þ rotations, given by the path-ordered ( ) exponential of the integral of for which convergence issues can arise when employing the Haar the affine connection via measure for the lattice theory at strong coupling. We note that in the context of the field-theoretic 2 þ expansion for gravity in I the continuum, as well as in the renormalizable higher derivative P U ðCÞ¼ exp dx : (1) formulation in four dimensions, no differences appear in the path C relevant beta functions for gravity between the Lorentzian and Euclidean case, to all orders in the relevant expansion parame- The gravitational Wilson loop then represents naturally a ters. In the continuum a physical difference between the two quantum average of a suitable trace (or contraction) of the cases would then have to originate from nonanalytic terms in the above nonlocal operator, as described in detail in [3]. Its beta functions, possibly due to nontrivial saddle points in the Euclidean theory. Also, the nonperturbative treatment of the large distance (i.e. for loops whose size is very large lattice Lorentzian case by numerical methods generally involves compared to the lattice cutoff) behavior can be estimated, complex weights expðiSÞ, which are known to be very difficult to provided one makes some suitable assumptions about the deal with reliably by statistical means.
1550-7998=2010=81(8)=084048(21) 084048-1 Ó 2010 The American Physical Society H. W. HAMBER AND R. M. WILLIAMS PHYSICAL REVIEW D 81, 084048 (2010) to be directly related, by Stoke’s theorem, to some sort of The fundamental renormalization group invariant quan- average curvature enclosed by the loop. In this semiclas- tity appearing in the textbook result of Eq. (5)2 represents sical picture one would write for the rotation matrix U the gauge field correlation length, defined, for example, from the exponential decay of connected Euclidean corre- Z 1 lations of two infinitesimal loops separated by a distance U ðCÞ exp R AC ; (2) 2 SðCÞ jxj, I 0 0 where AC is the usual area bivector associated with the GhðxÞ¼ TrP exp ig A ðx Þdx ðxÞ C0 loop in question, I 00 00 I TrP exp ig A ðx Þdx ð0Þ : (6) 1 00 A ¼ dx x : (3) C c C 2 Here the C ’s are two infinitesimal loops centered around x The use of semiclassical arguments in relating the above and 0 respectively, suitably defined on the lattice as ele- rotation matrix UðCÞ to the surface integral of the Riemann mentary square loops, and for which one has at sufficiently tensor assumes (as is usual in the classical context) that the large separations curvature is slowly varying on the scale of the very large GhðxÞ expð jxj= Þ: (7) loop. Then, in such a semiclassical description of the jxj!1 parallel transport process, one can reexpress the connection in terms of a suitable coarse-grained, or semiclassical, Thus the inverse of the correlation length is seen to Riemann tensor, and thus relate the quantum Wilson loop correspond, via the Lehmann representation, to the lowest gauge invariant mass excitation in the gauge theory, the expectation value discussed previously to an observable 3 large-scale curvature. The latter is represented phenom- scalar glueball. enologically by the long distance, observed cosmological Through the renormalization group is related to the - function of Yang-Mills theories, with the renormalization constant obs. It is important in this context to note, as an underlying group invariant obtained from integrating the Callan- theme, the close analogy between the Wilson loop in Symanzik -function, Z gravity and the one in gauge theories, both theories involv- g dg0 1ðgÞ¼const exp ; (8) ing a connection as a fundamental entity. Furthermore, a lot ðg0Þ is known about the behavior of the Wilson loop in non- Abelian gauge theories at strong coupling, some of it from with the ultraviolet cutoff, so that is then identified analytical estimates and some from large-scale numerical with the invariant gauge correlation length appearing in simulations. Let us recall that in non-Abelian gauge theo- Eqs. (5) and (7). ries, the Wilson loop expectation value for a closed planar In an earlier paper [3], we adapted the gauge definition loop C is defined by [1] of the Wilson loop to the gravitational case, specifically to the case of lattice gravity, and in the context of the discre- I tization scheme due to Regge [9]. On the lattice, with each WðCÞ¼ TrP exp ig A ðxÞdx ; (4) C neighboring pair of simplices s, s þ 1 one can associate a Lorentz transformation U ðs; s þ 1Þ, which describes a with A taA and the ta’s the group generators of how a given vector V transforms between the local coor- SUðNÞ in the fundamental representation. In the pure dinate systems in these two simplices. This transformation gauge theory at strong coupling [1,2], it is easy to show is directly related to the continuum path-ordered (P) ex- that the leading contribution to the Wilson loop follows an ponential of the integral of the local affine connection, with area law for sufficiently large loops the connection here having support only on the common interface between two simplices. The lattice action itself 2 hWðCÞi expð AC= Þ (5) only contains contributions from infinitesimal loops, but A!1 more generally one might want to consider near-planar, but noninfinitesimal, closed loops C (see Fig. 1). Along this where AC is the minimal area spanned by the planar loop C and the gauge field correlation length. Furthermore, it closed loop the overall rotation matrix will be given by can be shown that the area law is fairly universal at strong coupling, in the sense that it is not too sensitive to specific 2See, for example, Peskin and Schroeder, An Introduction to ð Þ Quantum Field Theory, p. 783, Eq. (22.3). short-distance details of the SU N -invariant lattice action. 3 Indeed one expects the result of Eq. (5) to have universal We do not distinguish here, for the sake of simplicity, between the square root of the string tension and the mass validity in the lattice continuum limit, the latter being taken gap. In SUðNÞ Yang-Mill theories, and QCD, in particular, these in the vicinity of the ultraviolet fixed point at gauge cou- represent nearly the same mass scale, up to a constant of order pling g ¼ 0. one.
084048-2 GRAVITATIONAL WILSON LOOP IN DISCRETE QUANTUM ... PHYSICAL REVIEW D 81, 084048 (2010) Y unit matrix and U the product of rotation matrices relating ð Þ¼ h U C Us;sþ1 : (9) the coordinate frames in the 4-simplices around the hinge. s C The motivation for this second choice was that analytical In analogy with the infinitesimal loop case, one would like calculations could then be performed more easily in the to state that for the overall rotation matrix one has strong coupling regime, using methods analogous to the
ðCÞBðCÞÞ ones used successfully for gauge theories [1,2]. Indeed it U ðCÞ ½e ; (10) can be shown [3] that this second action contribution is equal to where B ðCÞ is now an area bivector perpendicular to the loop and ðCÞ the corresponding deficit angle, which will Sh ¼ kAh sinð hÞ; (14) work only if the loop is close to planar so that B can be taken to be approximately constant along the path C.Bya independently of the parameter , where h is the deficit near-planar loop around the point P, we mean one that is angle at the hinge. For small deficit angles one expects this constructed by drawing outgoing geodesics on a plane to be a good approximation to the standard Regge action, through P. and general universality arguments would suggest that the The matrix U ðCÞ in Eq. (9) then describes the parallel lattice continuum be the same in the two theories. The transport of a vector round the loop C. If that is true, then expectation values of gravitational Wilson loops were then one can define an appropriate coordinate scalar by con- defined by either tracting the above rotation matrix UðCÞ with an appropriate bivector, namely hWðCÞi ¼ htrðU1U2 ...UnÞi; (15)
WðCÞ¼! ðCÞU ðCÞ (11) or where the bivector, ! ðCÞ, is intended as being represen- hWðCÞi ¼ htr½ðBC þ I4ÞU1U2 ...... Un i; (16) tative of the overall geometric features of the loop (for example, it can be taken as an average of the hinge bivector where the Uis are the rotation matrices along the path, and, ! ðhÞ along the loop). Finally, in the quantum theory one in the second expression, BC is a suitable average direction is interested in the quantum average or vacuum expectation bivector for the loop C, which is assumed to be near-planar. value of the above loop operator WðCÞ, as in the gauge The values of hWi in the strong coupling regime (i.e. for theory expression of Eq. (4). small k) can then be calculated for a number of loops, The next step is to relate the so defined, and computed, including some containing internal plaquettes. It was found quantum average to physical observable properties of the that for large near-planar loops around n hinges, to lowest manifold. Indeed for any continuum manifold one can nontrivial order (i.e. corresponding to a tiling of the interior define locally the parallel transport of a vector around a of the loop by a minimal surface), near-planar loop C. Then parallel transporting a vector kA n around a closed loop represents a suitable operational hWi ½p þ q 2 ; (17) way of detecting curvature locally. Thus a direct calcula- 16 tion of the vacuum expectation of the quantum Wilson loop provides a way of determining an effective curvature at where þ ¼ n, and A is the average area of the pla- large distance scales, even in the case where short distance quettes. Then using n ¼ AC=A, where AC is the area of the fluctuations in the metric may be significant. loop, the area-dependent first factor can be written as For calculational convenience, the actual computation of 2 the quantum gravitational Wilson loop in [3] was achieved exp½ðAC=AÞ logðkA=16Þ ¼ expð AC= Þ (18) by using a slight variant of Regge calculus, where the 1=2 contribution to the action from the hinge h is given not where we have set ¼½A=j logðkA=16Þj . Recall that by the original Regge expression for strong coupling, k ! 0,so is real, and that the quantity is in principle defined independently of the expectation value of the Wilson loop, through the correla- Sh ¼ kAh h; (12) tion of suitable local invariant operators at a fixed geodesic with k ¼ 1=8 G, but instead by the modified form distance. In the following we shall assume, in analogy to what is k known to happen in non-Abelian gauge theories, that even S ¼ A tr½ðB þ I ÞðU U 1Þ ; (13) h 4 h h 4 h h though the above form for the Wilson loop was derived in the extreme strong coupling limit, it will remain valid where Ah is the area of the triangular hinge where the throughout the whole strong coupling phase and all the curvature is located, Bh (called Uh in [3,4]) is a bivector way up to the nontrivial ultraviolet fixed point, with the orthogonal to the hinge, is an arbitrary multiple of the correlation length !1the only relevant and universal
084048-3 H. W. HAMBER AND R. M. WILLIAMS PHYSICAL REVIEW D 81, 084048 (2010) length scale in the vicinity of the fixed point. The evidence effect of matter couplings on the gravitational Wilson loop for the existence of such a fixed point comes from three results, and Secs. VI, VII, VIII, and IX contain systematic different sources, which have recently been reviewed, for discussions of scalar matter, fermions, gauge fields and the example, in Ref. [4], and references therein. The first lattice gravitino. Regarding these matter fields, the main source is the 2 þ expansion for gravity, which exhibits conclusion is that the previous results are not affected, such a fixed point in G to one and two loops, shows that unless there are near massless spin 1=2 and spin 3=2 only one relevant direction exists to all orders in this particles (i.e. whose mass is comparable to the exceedingly expansion, and provides a quantitative estimate for the small gravitational scale 1). Section XI consists of some critical exponent at the nontrivial ultraviolet fixed point. conclusions. The second source is the lattice gravity theory in d ¼ 4 based on Regge’s simplicial formulation, which also ex- II. GAUGE-THEORETICAL TREATMENT OF hibits a phase transition, with a single calculable nontrivial EINSTEIN GRAVITY ON A FLAT BACKGROUND relevant exponent . The third source is the Einstein- LATTICE Hilbert truncation renormalization method in the contin- uum, which, although approximate in nature, provides a We will first look at formulations of Einstein gravity as a third rough independent estimate for the exponent at the gauge theory on a flat hypercubical background lattice, nontrivial ultraviolet fixed point. and, in particular, expand on the work of Mannion and The next step was to interpret the result in semiclassical Taylor [11] and of Kondo [12]. In these cases, the standard terms. By the use of Stokes’s theorem, the parallel trans- machinery for calculating Wilson loops in lattice gauge port of a vector round a large loop depends on the expo- theories [2] can be taken over without too many modifica- nential of a suitably-coarse-grained Riemann tensor over tions. Although such formulations were not the first chro- the loop. So by comparing linear terms in the expansion of nologically of those we consider in this paper, we treat this expression with the corresponding term in the expres- them first because they are, in many respects, the simplest. sion of the area law, one can show [3] that the average The idea is to write Einstein gravity in four dimensions, without cosmological constant, on a flat background lat- curvature is of order 1= 2, at least in the strong coupling tice, treating it as a gauge theory with gauge group limit. Since the scaled cosmological constant is a measure SLð2;CÞ, and relating it to the Einstein-Cartan formalism. of the intrinsic curvature of the vacuum, this also suggests In fact, for simplicity, we shall consider an Euclidean that the cosmological constant is positive, and that the version, replacing SLð2;CÞ by SOð4Þ. The Minkowskian manifold is de Sitter at large distances. formulation presents new problems due to the noncom- The question now arises as to whether these results are pactness of the group, which will not be addressed here; peculiar to the particular formulation of discrete gravity basically the group-theoretic methods used below cannot used. This led to a study of other proposed formulations, be applied in the same fashion, and new convergence issues most of which were written down more than 20 years ago. arise due to the different nature of the Haar measure. In this work we will show that where it seems possible to In the following nearest neighbor sites are labeled by n define and calculate gravitational Wilson loops, the same and n þ , and their frames are related by area law emerges, and automatically implies a positive cosmological constant. U ðnÞ¼expðiA ðnÞÞ ¼ U ðn þ Þ 1; (19) Another key question we will address is whether these results are affected in any way by the presence of matter. where After all the Universe is not devoid of matter, and the pure gravity results should only be considered as a first-order 1 ab approximation to the full quantum theory (in a spirit simi- A ðnÞ¼ aA ðnÞSab; (20) 2 lar to the quenched approximation in non-Abelian gauge theories). This will be discussed here again in the context with a the lattice spacing and Sab the Oð4Þ generators, of the Regge formulation of discrete gravity used in [3], represented by the 4 4 matrices using the methods of coupling matter to gravity reviewed, for example, in [4]. i S ¼ ½ ; ; (21) An outline of the paper is as follows. In Sec. II,we ab 4 a b describe formulations of Einstein gravity as a gauge theory on a flat background lattice, and in Sec. III, the with the Euclidean gamma matrices, a, satisfying MacDowell-Mansouri description of de Sitter gravity, as y transcribed onto a flat background lattice by Smolin. More f a; bg¼2 ab; a ¼ a;a¼ 1; ...; 4: (22) recent developments of discrete gravity, spin foam models, are discussed briefly in Sec. IV, and Sec. V contains The curvature round an elementary plaquette spanned by mention of other relevant theories. We then turn to the the and directions is given as usual by
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U ðnÞ¼U ðnÞU ðn þ ÞU ðn þ þ ÞU ðn þ Þ 1 1 ¼ U ðnÞU ðn þ ÞU ðn þ Þ U ðnÞ ; (23) γ and it can be shown that in the limit of small lattice spacing, n δ 2 U ðnÞ expðia R Þ; (24) FIG. 2 (color online). A parallel transport loop, spanned by the where and directions, with four oriented links on the boundary. The parallel transport matrices U along the links, represented here by arrows, appear in pairs and are sequentially integrated over using R ¼ @ A @ A þ i½A ;A : (25) the uniform measure. One notices that the usual lattice gauge theory type action, consisting of sums of U terms, would give an R R term in the limit of small a, so terms involving the vierbein Z a Z ¼ ½dA ½dE expð SÞ; (28) e ðnÞ and the matrix 5 ¼ 1 2 3 4 have to be intro- duced. One defines Q Q where ½dA ¼ d U ðnÞ, ½dE ¼ dE ðnÞ, and 1 X n; H n; S ¼ tr½ E ðnÞU ðnÞE ðnÞ (26) d U is the Haar measure on SOð4Þ. 16 2 5 H n; ; ; ; Our interest here is in the definition and evaluation of Wilson loops in the strong coupling expansion. The authors a where E ðnÞ¼ae a and is the Planck length in suit- of Ref. [11] define the loop around one plaquette, spanned able units. It can then be shown that by the and directions (see Fig. 2), by Y a4 X ab c d 6 W ¼ Tr½E ðnÞU ðnÞE ðnÞ ; (29) S ¼ abcdR ðnÞe ðnÞe ðnÞþOða Þ; 2 ; 4 n; ; ; ; (27) and so which is the Einstein action in first-order form [13]. Z Y Furthermore by construction the action is invariant under 1 hWi¼ ½dA ½dE Tr½E ðnÞU ðnÞE ðnÞ local Oð4Þ rotations. For reasons which will become ap- Z ; parent, we shall consider a symmetrized form of the action: expð SÞ: (30) for each plaquette, rather than having the E 5E term inserted only at the base point, we shall consider the average of its insertion at all vertices of the plaquette. They go on to show that in the strong coupling expansion, In the following the partition function is defined by the the dominant term is proportional to usual path integral expression Z s t a b hWi¼ ½dE abste e e e ; (31)
where there is no sum over and . Now suppose that ¼ 1, ¼ 2. Then the sum over and leads to
s t a b abste3e4e3e4; (32)
which is zero on symmetry grounds. Therefore their defi- nition needs some modification, or one has to go to higher orders in the strong coupling expansion. In the latter case, FIG. 1 (color online). Illustration of the gravitational analog of the Wilson loop. A vector is parallel-transported along the larger it is possible to get a nonzero contribution by going to order 6 outer loop. The enclosed minimal surface is tiled with parallel 1=k , but here we concentrate on the first possibility. transport polygons, here chosen to be triangles for illustrative Omitting the Es from W also gives zero for hWi, so the purposes. For each link of the dual lattice, the elementary modification we make is to insert a 5 into W. The lowest parallel transport matrices Uðs; s0Þ are represented by arrows. order contribution is then
084048-5 H. W. HAMBER AND R. M. WILLIAMS PHYSICAL REVIEW D 81, 084048 (2010) Z X X 1 0 0 0 2 ½dA ½dE Tr½ 5E ðnÞU ðnÞE ðnÞ Tr½ 5E ðn ÞU ðn ÞE ðn Þ 16 ; n0; ; ; ; a4 Z X ¼ ½dA ½dE Tr½ U ðnÞU ðn þ ÞU ðn þ Þ 1U ðnÞ 1 16 2 5 s t X ; 0 0 0 1 0 1 s t a 0 b 0 Tr½ 5 aU ðn ÞU ðn þ ÞU ðn þ Þ U ðn Þ b e ðnÞe ðnÞe ðn Þe ðn Þ: (33) n; ; ; ;
The integration over the As is equivalent to the integration Us of the expression over U’s in SOð4Þ with the Haar measure: ðU Þ A ðU Þ ðU 1Þ B ðU Þ ðU 1Þ C ðU Þ Z 1 ab bc 2 cd 2 ef fg 3 gh 3 ij jk 4 kl 1 1 1 1 d UU U ¼ ; (34) ðU ÞmnDnoðU Þop (41) H ij kl 4 il jk 4 1 gives and we obtain 1 Z TrðABCDÞ : (42) a4 X 4 de hi lm pa ½dE Tr½ 4 2 d 5 c b 5 a 64 We see that the effect of the integration is to give a factor of a b c d 1 for each U, and to give the trace of the product of factors e ðnÞe ðnÞe ðnÞe ðnÞ: (35) 4 at each vertex. For a vertex with no insertion, we obtain the Now we compute trace of the identity matrix, 4, and for one insertion of E 5E, the value is zero since it is traceless. For two 2 Tr ½ d 5 c b 5 a ¼4ð ab cd ac bd ad bcÞ; insertions of E 5E, we obtain 12= , where the integration (36) over the Es has been done. Recall that there is also a factor of 1=16 2 for each plaquette, corresponding to the rele- and, using vant terms in the expansion of the exponential of minus the action. This means that within our loop, if it is to have a 1 2 a b 2 a b nonzero value, every vertex must have either no E E g ¼ a Trð a bÞe e ¼ a abe e ; (37) 5 4 factors or two of them. This is why we took the average we obtain of insertions at all vertices of the plaquettes in the action; it would be impossible to get nonzero contribution from the Z X 1 internal plaquettes otherwise. ½dE ðg g g g g g Þ: 2 Before proceeding with the calculations, let us mention 16 an alternative to the procedure of averaging the contribu- (38) tion of the action from a plaquette over its vertices, a possibility, similar to the procedure in [3]. If we replace Suppose that ¼ 1, ¼ 2, then the sum over the indices in the ’s and g’s gives −1 U 2 U 2 2 4ðg34 g33g44Þ: (39) We expand the metrics in terms of the vierbeins and define the measure of integration to include a damping factor ð a2= Þ8 exp½ a2 ðeb Þ2 at each point, with Re > A B b; U 0 [14], obtaining U1 3 − 3 −1 U 1 : (40) U1 D C 3 4 2 2 (Note that we are ignoring a possible factor of the deter- minant of the vierbein in the measure.) Before considering larger loops, let us obtain an algo- −1 U 4 U 4 rithm which simplifies the calculations considerably. Consider a vertex with the matrices A, B, C, D attached FIG. 3 (color online). A vertex where various parallel transport to it, and U-matrices attached to the lines entering and matrices enter and leave, and where there are insertions on their leaving the vertex, as shown in Fig. 3. Integration over the paths.
084048-6 GRAVITATIONAL WILSON LOOP IN DISCRETE QUANTUM ... PHYSICAL REVIEW D 81, 084048 (2010)
γ
n δ
FIG. 4 (color online). A parallel transport loop around two plaquettes, with insertions of E 5E on the loop shown by large dots.
5 by 5 þ I4 for some arbitrary parameter , the con- tinuum limit of the action acquires a term proportional to R , which is zero because of the symmetries of the Riemann tensor, so the action is unaltered. However, the value of the Wilson loop is still zero, not because of the traces but because of factors of the Kronecker delta, which give zero on symmetry grounds. FIG. 5 (color online). A larger parallel transport loop with 12 For a loop around two plaquettes, we find that we obtain oriented links on the boundary. As before, the parallel transport a nonzero value only if the E 5E is inserted at the place matrices along the links appear in pairs and are sequentially where the loop meets the second plaquette (see Fig. 4). The integrated over using the uniform measure. The new ingredient value of hWi is then in this configuration is an elementary loop at the center not touching the boundary. As in Fig. 4, the insertions of E 5E on 1 3 2 the loop are shown by large dots. : (43) 4 4 2 2 X 1 For a loop around many plaquettes, we choose to insert S ¼ 2 Tr½ 5U ðnÞH ðnÞH ðnÞ ; 4 n; ; ; ; the factors of E 5E in the loop wherever the loop meets a new plaquette. For a loop with internal plaquettes, there (45) has to be an even number of internal plaquettes as the insertions need to be paired between them. (For example, where see the loop around nine plaquettes in Fig. 5; there is no H ðnÞ¼exp½iaea ðnÞ : (46) way the insertions on the one internal plaquette can give a a nonzero value.) This means that we obtain nonzero values This has the consequence that the action is bounded. (The only for Wilson loops surrounding an even number of minus sign, which appears different from the sign in the plaquettes; the simplest case, with 12 plaquettes, is shown formalism of [11], is because of the different relative in Fig. 6. There are two ways of getting nonzero values from the internal plaquettes, corresponding to pairings of the insertions at the two vertices they have in common, which gives a factor of 2 in the answer, which, when integrated over the vierbeins, is 2 3 12 : (44) 4116 4 2 2
Larger loops can then be treated in a similar way. From the results obtained so far, we deduce that, as the authors of [11] claimed, there is indeed an area law for large Wilson loops. The physical interpretation is of course very differ- ent, as discussed in the IIntroduction, and later in the Conclusion. We now consider briefly the work of Kondo [12]. His basic formalism is very similar to that of [11], except that FIG. 6 (color online). A Wilson loop around 12 plaquettes, of rather than introducing the vierbeins into the action di- which two are internal, with the insertions of E 5E on the loop rectly, he introduces exponentials of them, with the action shown by large dots as before.
084048-7 H. W. HAMBER AND R. M. WILLIAMS PHYSICAL REVIEW D 81, 084048 (2010) AB position of the 5 factor.) In practice, in calculations, it is where R is the curvature associated with an Oð4; 1Þ impossible to work out traces without expanding the ex- (minus sign) or Oð3; 2Þ (plus sign) gauge connection, ponentials and retaining the lowest order terms in the p ABCDffiffiffiffiffiffi 5 is the totally antisymmetric 5-tensor and g ¼ lattice spacing, so the formalism reduces to that of [11] 32 =l a dimensionless coupling constant. The parallel in this respect, and the same values are obtained for the transport operators along the links of the lattice are defined Wilson loops. (We have checked that the lowest order by contribution comes from the product of the linear terms Z nþ in the expansions of the exponentials.) However, Kondo 1 AB U ðnÞ¼P exp g dx A ðxÞTAB ; (51) also aims to set up a formalism which has reflection 2 n positivity, so his action contains sums over reflections, where the TAB are matrix representations of the relevant and if this full action is used, it is very complicated to Lie algebra. Then the curvature around a plaquette on a evaluate Wilson loops. hypercubic lattice, U ðnÞ, is identical to the definition of Note that this method of averaging the action contribu- Mannion and Taylor [11] [Eq. (23)], and this is related to tion of each plaquette over the vertices of the plaquette the curvature by needs to be used here, and could also be used in [3], 1 eliminating the necessity for introducing the parameter . ½U ðnÞ ¼ a2gRABðT Þ þ Oða3Þ: (52) 2 ij AB ij III. LATTICE FORMULATION OF MACDOWELL- The continuum action is then transcribed onto the lattice as MANSOURI GRAVITY X 1 ijkl5 S ¼ 2 ½U ðnÞ ij½U ðnÞ kl ABCD5: An earlier version of lattice gravity was given by Smolin g n [15], who transcribed the MacDowell and Mansouri [16] (53) formulation of general relativity onto a flat background lattice. MacDowell and Mansouri built a gauge theory by It involves a sum over contributions from perpendicular defining ten (antisymmetric) gauge potentials by plaquettes at each lattice vertex, in analogy to the construc- tion of the FF~ term in non-Abelian gauge theories. In order 1 ab ab 5a a to maintain the discrete symmetries of the lattice (reflec- A ¼ ! ;A ¼ e ; (47) l tions and rotations through multiples of ), this is extended ab a where ! and e are the usual gravitational connection to a sum over all orientations of the dual plaquettes and vierbein, and l is a lattice spacing. The curvature and 1 X X S ¼ ijkl5½UO ðnÞ torsion are defined in terms of the gauge potentials, and the 16g2 ij action is of the form n O;O0 O0 Z ½U ðnÞ kl ABCD5: (54) 4 ab cd S ¼ d x abcdR R ; (48) The partition function is then given by where Rab is the Riemann tensor for Oð3; 2Þ or Oð4; 1Þ. Z Z ¼ ½dU expðiSÞ; (55) This can be shown to be equivalent, after multiplication by 1=32l2= 2, with the bare Planck length, to where we take [dU] to be the normalized Haar measure. Z l2 1 We restrict the integration to Oð5Þ, rather than considering S ¼ d4x R0abR0cd þ eR0 32 2 abcd 2 2 also Oð3; 2Þ and Oð4; 1Þ as in [15], since for the noncom- 2 pact groups one has to define the measure by dividing 2 2 e ; (49) through by the (infinite) volume of the gauge group. For l the five-dimensional representations used, the relevant in- 0ab where R is the usual Riemann curvature tensor. The first tegrals are term is a topological invariant, the Gauss-Bonnet term, Z while the second and third are obviously the Einstein ½dHU ¼1; (56) term and the cosmological constant term, respectively, with a scaled cosmological constant ¼ 2= 2l2. Note Z that in this formulation the relative coefficients of various ½dHU ½U ðnÞ ij ¼ 0; (57) action contributions are fixed in terms of the bare parame- ter and l. Z 0 1 ½d U ½U ðnÞ ½U ðn Þ ¼ 0 : (58) Then the starting point in [15] is the continuum action H ij kl 5 il jk nn Z 1 4 AB CD The structure of the action, based on pairs of dual pla- S ¼ d x R R ABCD5; (50) g2 quettes, means that the calculations are somewhat different
084048-8 GRAVITATIONAL WILSON LOOP IN DISCRETE QUANTUM ... PHYSICAL REVIEW D 81, 084048 (2010)
(ρ,σ ) (ρ,σ )
n n (µ,ν )
µ ν FIG. 8 (color online). Two pairs of dual plaquettes joined ( , ) together, with the ones in the ð ; Þ-plane sharing only the vertex n, and the ones in the ð ; Þ-plane back to back. FIG. 7 (color online). Two pairs of dual plaquettes joined together, with the ones in the ð ; Þ-plane lying side-by-side, and the ones in the ð ; Þ-plane back to back. each pair lies adjacent to the other in the plane or they meet at one point, and the other two are joined back-to-back (see from the case in [11]. In particular, since we want even- Figs. 7 and 8). We then calculate the contribution from this U tually to evaluate Wilson loops for planar surfaces, we can configuration in both cases, when integration over the s take as our basic building block a combination of two pairs on the back-to-back faces is performed. of dual plaquettes, put together so that one plaquette from The quantity to evaluate in the first case is
Z 1 0 0 0 0 S ¼ ½d U ð Þ2 ijkl5 i j k l 5½U ðnÞU ðn þ ÞU 1ðn þ ÞU 1ðnÞ ½U ðnÞU ðn þ ÞU 1ðn þ ÞU 1ðnÞ 16g2 H ij kl 1 1 1 1 ½U ðnÞU ðn þ ÞU ðn ÞU ðn Þ i0j0 ½U ðnÞU ðn þ ÞU ðn þ ÞU ðnÞ k0l0 ; (59)
(with no summation over , ). Integration over the U s and U sgives 2 1 2 ijkl5 jj0kl5 1 1 1 ð Þ ½U ðnÞU ðn þ ÞU ðn þ ÞU ðn þ ÞU ðn ÞU ðn Þ 0 : (60) 16g2 ij
Z Now 1 Y hwi¼ ½dHUi W expðiSÞ: (65) 0 0 0 Z ijkl5 jj kl5 ¼ 2ð 6 ij 6 jj 6 ij 6 jj Þ (61) i where As explained in [15], calculations are done in this formal- ism on the assumption that one can ignore the zero-torsion 6 ik ¼ ik i5 k5; (62) constraint; the basis for this is that the torsion is suppressed 1 so the final contribution, including a factor of 4 from the by a factor of l , where l is large. As a result, one only needs summation over and ,is to integrate over the U’s, and there is no need to integrate over the vierbeins in this formalism. Note that because of 2 1 24 ijkl5 jj0kl5 1 the structure of the basic building blocks, we can define ½U ðnÞU ðn þ ÞU ðn þ Þ 16g2 25 Wilson loops only around paths which contain an even 1 1 number of plaquettes. The simplest of these is shown in U ðn þ ÞU ðn ÞU ðn Þ ij0 6 ij0 : (63) Fig. 4, and the area can be tiled by only one of the two In the second case, the calculation proceeds in a similar possible building blocks, giving the value ð1=ð16g2Þ2Þ way, to give ð192=125Þ. 1 2 8 The next most simple cases are shown in Fig. 9. The first ð 6 0 6 0 6 0 6 0 Þ½U ðnÞU ðn þ Þ of these can be tiled in four possible ways with the first of 16g2 5 ii jj ij ji the building blocks, giving ð1=ð16g2Þ4Þð21232=57Þ, while in 1 1 1 1 U ðn þ ÞU ðnÞ ij½U ðn ÞU ðn Þ the second, which can be tiled in eight ways with the first building block and in one way with the second, the final U ðn ÞU ðn Þ 0 0 : (64) i j contribution is ð1=ð16g2Þ4Þð28321=57Þ. For the simplest We now define a Wilson loop as the product of the U configuration with internal plaquettes, a loop surrounding factors around the given path, with no extra factors in 12 plaquettes (see Fig. 6), there are many (1072) different this case, and we calculate its expectation value as usual: ways of tiling it, so we need to add the contributions from
084048-9 H. W. HAMBER AND R. M. WILLIAMS PHYSICAL REVIEW D 81, 084048 (2010) The square root in the action makes it almost impossible to do any general analytical calculations. To put the results of Wilson loop calculations in this and the previous section, together with the results of [3], into context, it is interesting to make comparisons by relating the coupling constants to that of the continuum action. The of Mannion and Taylor [11] and Kondo [12], and the g of Smolin [15] are related to the k ¼ 1=8 G of [4]by 1 k 1 l2k ¼ ; ¼ : (66) 2 2a4 g2 32 Making a further normalization of the constants involved by equating the results for the smallest loop results, the answers of [11,12] agree with those of [13] until the loops contain internal plaquettes, and then, for example, the 12- FIG. 9 (color online). Two arrangements for Wilson loops plaquette results differ by a factor of 2=6. The results of around four plaquettes. [15] are of the same order of magnitude as those of [3].
IV. SPIN FOAM MODELS all the different ways. The tiling shown in Fig. 6 Spin foam models grew out of a combination of ideas 2 12 26 4 21 gives ð1=ð16g Þ Þð2 3 =5 Þ, and then combining this from the Ponzano-Regge model of three-dimensional dis- 2 12 with the other contributions, we obtain ð1=ð16g Þ Þ crete Lorentzian quantum gravity, and from loop quantum 30 4 23 2 ð2 3 9481=5 Þ. Notice the dependence on 1=g in the gravity. In loop quantization, the fundamental excitations various cases evaluated. Again, larger loops can then be are loops created by Wilson loop operators analogous to treated in a similar way although the calculations become the ones used in gauge theories [19], and one assumes that increasingly tedious. This indicates the usual area law for states can be written as power series in spatial Wilson loops the gravitational Wilson loop. We note here that the authors of the connection [20]. What does this intimate connection of Ref. [17] have performed numerical simulations using between Wilson loops and spin foam models mean in the the action from [15], with an SOð4Þ invariant action and a context of this paper? Haar measure over the group SOð5Þ, considering then both In the three-dimensional formulation of Turaev and Viro the weak and the strong coupling regimes. [21], which is a regularized version of the Ponzano-Regge We should state at this point that in this paper we have model, it has been shown [22] that the graph invariant chosen to focus almost exclusively on the strong coupling defined by Turaev [23] coincides, in the semiclassical limit of various models of lattice gravity, and, in particular, limit, with the expectation value of a Wilson loop. This is on the emergence of the area law for the Wilson loop. New a consequence of the asymptotic behavior of 6j-symbols, problems can arise when approaching the lattice contin- with certain arguments fixed, involving rotation matrices uum limit in the vicinity of the critical point, if one exists. which combine to give parallel transport operators along As an example, in some lattice models the transition ap- the graph. The extension of this result to graph invariants in pears to be first order [17], which would mean that either discrete four-manifolds has not been made (as far as we the lattice action has to be modified by adding second know) and it is not clear anyway whether an area law could neighbor terms, or that the critical exponents have to be be obtained for large loops since the concept of a planar obtained by analytic continuation from the strong coupling loop is not well-defined. phase, approaching in this way the fixed point located in One way of obtaining a spin foam model is from BF the metastable phase. Within the limited framework of this theory [24]. In four dimensions, representation labels are work we shall not address these additional technical issues, assigned to triangles and group elements to sections of the and assume instead that a number of lattice theories exam- dual loop around each triangular hinge. The integral of the ined here describe to some extent correctly at least the group elements around the dual loop gives the holonomy, physics of the strong coupling phase of gravity. which is a measure of the curvature, F. Thus an evaluation A formalism related to that of Smolin is described of Wilson loops is a basic ingredient in calculating the by Das, Kaku and Townsend [18]. They transcribe West’s action, which is then conventionally expressed in terms of de Sitter invariant formulation of Einstein gravity onto the sums over amplitudes for the vertices, edges and faces of lattice, obtaining an action with plaquette contribution the spin foams. Alternatively, in group field theories, the proportional to the square root of the trace of a square action involves the integral over products of functions of involving Smolin’s action. They showed that their theory the group variables, corresponding to a kinetic term and an agrees with the one in [15] in the lattice continuum limit. interaction term. Here the evaluation of Wilson loops is
084048-10 GRAVITATIONAL WILSON LOOP IN DISCRETE QUANTUM ... PHYSICAL REVIEW D 81, 084048 (2010) somewhat similar to the way matter is inserted; certain using edges are picked out (to form the loop) and are then treated P Z ðTrU þ c:c:Þ differently in the summation process [25]. dU exp pl pl DJ ðUyÞ The authors of Ref. [26] have shown that there is an pl 2Tr1 m1m2 exact duality transformation mapping the strong coupling 2 ¼ I ð ÞT ð Þ; (71) regime of a non-Abelian gauge theory to the weak coupling m1;m2 1 J regime of a system of spin foams defined on the lattice.
They obtain an expression for the expectation value of a where TJð Þ I2Jþ1ð Þ=I1ð Þ [2,29] is the ‘‘Fourier non-Abelian Wilson loop (or spin network) in terms of transform’’ of the Wilson action and the In are modified integrals of expressions involving finite-dimensional uni- Bessel functions. The partition function is then tary representations, intertwiners and characters of the 2 no: of plaquettesX Y gauge group, together with a gauge constraint factor for Z ¼ I ð Þ ð2J þ 1ÞT ð Þ each lattice point. The integrals are done explicitly, leaving 1 P JP JP pl complicated products and sums over intertwiners, projec- Y Z JP JP JP tors and the character decomposition of the exponential of dUlDm1m2 ðU1ÞDm2m3 ðU2ÞDm3m4 ðU3Þ the action. Their calculation is very general, and to evalu- links l ate a Wilson loop in the usual sense, considerable simpli- JP Dm m ðU Þ: (72) fication can be made. The links which form the Wilson 4 1 4 loop can all be labeled with the same representation, and, In two dimensions, each link is shared by two plaquettes as the loop has no multivalent vertices, the intertwiners all and the integration over Ds gives Kronecker deltas, become trivial. Even so, the calculation is very compli- whereas in three dimensions, each link is shared by four cated for a general gauge group, plaquettes and the integration over Ds gives 6j-symbols as To illustrate the ideas behind the work of these authors, in the Ponzano-Regge model. To compute the expectation we will describe the corresponding calculations in lower value of a Wilson loop in representation js, a factor of dimensions and with gauge group SUð2Þ [27,28]. We shall Djs ðUÞ must be inserted for each link on the loop. In two summarize the description in [28]. The partition function dimensions, use of the asymptotics of TJð Þ leads to the for gauge theory on a cubic lattice is written as usual as an area law at large (strong coupling) [28]. In three dimen- integral over link variables Ul, with the action being a sum sions, the extra Ds along the link give rise to 9j-symbols, over plaquettes contributions and the asymptotic behavior of the Wilson loop has not P Z Y been calculated explicitly. 1 plðTrUpl þ c:c:Þ Z ¼ dU exp ; (67) The formulation of spin foam models which seems the l 2Tr1 links most tractable for the calculation of Wilson loops is the one with being the dimensionless inverse coupling. The of Ref. [30]. (Their expressions are essentially identical to those written down earlier by Caselle, D’Adda and Magnea matrix Upl is the standard product of four link matrices U around the plaquette. The idea of the duality trans- [3,10] (see also [31]). In the absence of a boundary, their l action can be written as (see Eq. (13)) formation is to make a Fourier transform in the plaquette X variables, by first inserting unity for each plaquette into the S ¼ Tr½BfðtÞUfðtÞ ; (73) partition function, in the form f Y Z 1 ¼ dUpl ðUpl;U1U2U3U4Þ; (68) where the sum is over triangular hinges, f, UfðtÞ is the pl product of rotation matrices linking the coordinate frames of the tetrahedra and four-simplices around the hinge and where U are the link variables around the plaquette. 1...4 B ðtÞ is a bivector for the hinge, defined as the dual of The -function can be realized by products of Wigner f ð Þ D-functions f t . This in turn is the integral over the triangle f of the X two-form ðtÞ¼eðtÞ^eðtÞ, formed from the vierbein in ð Þ¼ ð þ Þ J ð yÞ J ð Þ U; V 2J 1 Dm1m2 U Dm2m1 V : (69) tetrahedron t. The action is independent of which tetrahe- J¼0;1=2;1;... dron is regarded as the initial one in the path around the hinge. There is a slight subtlety in the definition of U ðtÞ,as The unity is then inserted into the partition function in the f form the basic rotation variables are taken to be Vtv, which Y Z X relates the frame in tetrahedron t to that in 4-simplex v, ¼ ð þ Þ J ð yÞ J ð Þ of which t is a face, which is crossed in the path around 1 dUpl 2J 1 Dm1m2 U Dm2m3 U1 pl J hinge f. Then J ð Þ J ð Þ J ð Þ U ðtÞ¼V V ...V : (74) Dm3m4 U2 Dm4m5 U3 Dm5m1 U4 : (70) f tv1 v1t1 vnt The integration over the plaquette matrices is performed The action is sufficiently similar to that used by us in an
084048-11 H. W. HAMBER AND R. M. WILLIAMS PHYSICAL REVIEW D 81, 084048 (2010) earlier paper [3] that we may take over the formalism for fields. After including reflections in order to preserve dis- calculating Wilson loops from there. The integration over crete rotation and reflection symmetry on the lattice, he the Vs, which are elements of SOð4Þ in the Euclidean case, squares and then takes a square root, to ensure scalar, rather proceeds exactly as in [3], and the same problem arises than pseudoscalar, properties in the continuum limit, as in with the unmodified action of [30], as the bivector B is [18]. A torsion constraint is also necessary here. As in traceless. Therefore the definition of the action and of the formulations discussed earlier, these features make calcu- gravitational Wilson loop has to be modified by an addition lations very complicated. of I4,asin[3], which again does not affect the value of the Finally the authors of Ref. [14] have presented a unified action. The results obtained are equivalent to those in our treatment of Poincare´, de Sitter and conformal gravity on earlier paper, which indicates that the area law also holds the lattice. This shares many features with the formulations for this formulation of spin foam models. already described, so we will not discuss it further here. The main difference is that the lattice vierbein field is defined on the lattice links rather than at the vertices. The V. OTHER DISCRETE MODELS OF QUANTUM formulation is reflection positive, but the mode doubling GRAVITY problem seems to persist, as seen form the expansion about We now consider very briefly various other approaches a flat background. to discrete quantum gravity and the possibility of evaluat- Causal dynamical triangulations [34] are based on the ing the expectation values of gravitational Wilson loops in action of Regge Calculus, but the approach differs in that them. Kaku [32] has proposed a lattice version of confor- all simplices have identical spacelike edges and identical mal gravity, with action timelike edges, and the discrete path integral involves X summing over triangulations. In this case it is not clear S ¼ Tr½ 5P ðnÞP ðnÞ ; (75) how to use the methods discussed here and in [3], which n are based on the invariant Haar measure for continuous rotation matrices, since this formulation does not contain where P ðnÞ gives the curvature round a plaquette and is explicitly continuous degrees of freedom which could be related to the Us in our previous equations, with U ðnÞ used for such purpose. given in terms of the Oð4; 2Þ generators. The strong cou- The proposed formulation of Weingarten [35], based on pling expansion of the partition function is given by squares, cubes and hypercubes, rather than simplices, in- Z volves six-index complex variables corresponding to X 1 Z ¼ ½dU ½d cubes, so although it is possible to define a large planar m m! loop, it is not clear how to evaluate a Wilson loop, except in 1 X m the special case when the parameter (the coefficient of Tr½ P ðnÞP ðnÞ 5 the term in the action which gives the contribution from the n boundaries of the 4-cells) is set equal to zero, which seems a a expfi Tr½ ð1 þ 5ÞP ðnÞ g; (76) to correspond to the unphysical case of infinite cosmologi- cal constant. where the last term is included to impose the zero-torsion A more radical approach to discrete quantum gravity, in constraint. The analytic calculation of Wilson loops is which the ingredients are a set of points and the causal complicated considerably by the presence of this con- ordering between them, is known as causal sets. Recent straint. If it is ignored, and Wilson loops defined as a progress includes a calculation of particle propagators product of Us round the loop as usual, then comparison from discrete path integrals [36]. In this formulation, it is with other calculations suggests that an area law will be not clear how to define a (closed) Wilson loop connecting obtained. (The calculations are very similar to those of points which are not causally related, and defining a near Ref. [15] if one assumes a form for the Oð4; 2Þ integrals planar loop is also a problem here. as in his paper. The 5s disappear in the process of eval- uating the basic building blocks.) Again the caveats men- VI. EFFECTS OF SCALAR MATTER FIELDS tioned at the beginning of the paper in comparing the Lorentzian to the compact (Euclidean) case, and the ensu- In the next four sections, we consider whether the pres- ing differences in the group theoretic structures as they ence of matter affects the area law behavior of gravitational relate to the Haar measure, apply here as well. Wilson loops in the strong coupling limit. For each type of Rather than considering conformal gravity, Tomboulis matter, we first describe briefly its transcription to the [33] has formulated a lattice version of the general higher lattice [4]. derivative gravitational action in order to prove unitarity. A scalar field can be introduced as the simplest type of He uses the gauge group Oð4Þ and considers vierbeins dynamical matter that can be coupled invariantly to grav- coupled as ‘‘additional matter fields,’’ as in Mannion and ity. In the continuum the scalar action for a single compo- Taylor [11] and Kondo [12], together with further auxiliary nent field ðxÞ is usually written as
084048-12 GRAVITATIONAL WILSON LOOP IN DISCRETE QUANTUM ... PHYSICAL REVIEW D 81, 084048 (2010) Z 1 pffiffiffi for the new lattice. Mass and curvature terms such as the I½g; ¼ dx g½g @ @ þðm2 þ RÞ 2 þ... 2 ones appearing in Eq. (77) can be added to the action, so that a more general lattice action is of the form (77) X X 1 i j 2 1 where the dots denote scalar self-interaction terms. Thus, I ¼ VðdÞ þ VðdÞðm2 þ R Þ 2 (84) 2 ij l 2 i i i for example, a scalar field potential Uð Þ could be added hiji ij i containing quartic field terms, whose effects could then be where the term containing the discrete analog of the scalar of interest in the context of cosmological models where curvature involves spontaneously broken symmetries play an important role. X pffiffiffi ðdÞ ðd 2Þ The dimensionless coupling is arbitrary; two special Vi Ri hV gR: (85) 1 h cases are the minimal ( ¼ 0) and the conformal ( ¼ 6 ) h i coupling case. In the following we shall mostly consider In the expression for the scalar action, VðdÞ is the (dual) the case ¼ 0. It is straightforward to extend the treatment i volume associated with the site i, and the deficit angle to the case of an N -component scalar field a with a ¼ h s on the hinge h. The lattice scalar action contains a mass 1; ...;N . s parameter m, which has to be tuned to zero in lattice units One way to proceed is to introduce a lattice scalar i defined at the vertices of the simplices. The corresponding to achieve the lattice continuum limit for scalar lattice action can then be obtained through a procedure by correlations. which the original continuum metric is replaced by the When considering whether the gravitational Wilson loop induced lattice metric. Within each n-simplex one defines a area law holds for large loops in the strong coupling limit, metric the matter considered must be almost massless, otherwise its effects will not propagate over large distances and so gijðsÞ¼ei ej; (78) cannot change the large Wilson loop result found in the pure gravity case. In fact, since the lattice Lagrangian for with 1 i, j n, and which in the Euclidean case is a b the scalar matter involves only factors related to the lattice positive definite. In components one has gij ¼ abei ej . metric (functions of the edge lengths) and not the connec- In terms of the edge lengths lij ¼jei ejj, the metric is tion (provided the parameter ¼ 0), the integration over given by the connections, which is what gives the area law, is 1 unaffected. g ðsÞ¼ ðl2 þ l2 l2 Þ: (79) ij 2 0i 0j ij The volume of a general n-simplex is then given by VII. EFFECTS OF LATTICE FERMIONS qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c c 1 On a simplicial manifold spinor fields s and s are V ðsÞ¼ detg ðsÞ: (80) most naturally placed at the center of each d-simplex s.In n n! ij the following we will restrict our discussion for simplicity To construct the lattice action for the scalar field, one then to the four-dimensional case, and largely follow the origi- performs the replacement nal discussion given in [40,41]. As in the continuum, the g ðxÞ!g ðsÞ @ @ ! (81) construction of a suitable lattice action requires the intro- ij i j duction of the Lorentz group and its associated tetrad fields a with the scalar field derivatives replaced by finite differ- e ðsÞ within each simplex labeled by s. Within each sim- ences plex one can choose a representation of the Dirac gamma matrices, denoted here by ðsÞ, such that in the local @ !ð Þ ¼ ; (82) i iþ i coordinate basis where the index labels the possible directions in which f ðsÞ; ðsÞg ¼ 2g ðsÞ: (86) one can move away from a vertex within a given simplex. After some rearrangements one finds a lattice expression These in turn are related to the ordinary Dirac gamma for the action of a massless scalar field [37,38] matrices a, which obey X 2 f a; bg¼2 ab; (87) 2 1 ðdÞ i j Iðl ; Þ¼ Vij : (83) ab 2 hiji lij with the flat metric, by a ðdÞ ðsÞ¼ea ðsÞ ; (88) Here Vij is the dual (Voronoi) volume [39] associated with a the edge ij, and the sum is over all links on the lattice. so that within each simplex the tetrads e ðsÞ satisfy the Other choices for the lattice subdivision will lead to a usual relation similar formula for the lattice action, with the Voronoi ab dual volumes replaced by their appropriate counterparts ea ðsÞebðsÞ ¼ g ðsÞ: (89)
084048-13 H. W. HAMBER AND R. M. WILLIAMS PHYSICAL REVIEW D 81, 084048 (2010) 0 1 In general the tetrads are not fixed uniquely within a 0 010 B C simplex, being invariant under local Lorentz transforma- B 0 000C ¼ B C: (96) tions. In the following it will be preferable to discuss the 13 @ 1000A Euclidean case, for which ab ¼ ab. 0 000 In the continuum the action for a massless spinor field is given by For fermions the corresponding spin rotation matrix is then obtained from Z pffiffiffi c c I ¼ dx g ðxÞ D ðxÞ (90) i S ¼ exp (97) 4 1 ab where D ¼ @ þ ! ab is the spinorial covariant 2 with generators ¼ 1 ½ ; . Taking appropriate derivative containing the spin connection ! ab. In the 2i absence of torsion, one can use a matrix Uðs0;sÞ to describe traces, one can obtain a direct relationship between the U 0 the parallel transport of any vector from simplex s to a original rotation matrix ðs; s Þ and the corresponding spin S 0 neighboring simplex s0, rotation matrix ðs; s Þ y 0 0 U ¼ TrðS S Þ=Tr1 (98) ðs Þ¼U ðs ;sÞ ðsÞ: (91) U therefore describes a lattice version of the connection. which determines the spin rotation matrix up to a sign. Indeed in the continuum such a rotation would be de- Now, if one assigns two spinors in two different contiguous simplices s1 and s2, one cannot in general assume that the scribed by the matrix tetrads ea ðs1Þ and ea ðs2Þ in the two simplices coincide. dx U U ¼ðe Þ (92) They will in fact be related by a matrix ðs2;s1Þ such that with the affine connection. The coordinate increment ea ðs2Þ¼U ðs2;s1Þeaðs1Þ (99) dx is interpreted as joining the center of s to the center of S 0 0 and whose spinorial representation is given in Eq. (98). s , thereby intersecting the face fðs; s Þ. On the other hand, S 0 Such a matrix ðs2;s1Þ is now needed to additionally in terms of the Lorentz frames ðsÞ and ðs Þ defined parallel transport the spinor c from a simplex s to the within the two adjacent simplices, the rotation matrix is 1 neighboring simplex s2. The invariant lattice action for a given instead by massless spinor takes therefore the form a 0 a 0 0 X U bðs ;sÞ¼e ðs Þe bðsÞU ðs ;sÞ (93) 1 0 0 0 0 I ¼ Vðfðs; s ÞÞc sSðUðs; s ÞÞ ðs Þn ðs; s Þc s0 (this last matrix reduces to the identity if the two ortho- 2 faces fðss0Þ normal bases ðsÞ and ðs0Þ are chosen to be the same, in 0 (100) which case the connection is simply given by Uðs ;sÞ ¼ 0 e ae ). Note that it is possible to choose coordinates so where the sum extends over all interfaces fðs; s Þ connect- a 0 0 that Uðs; s0Þ is the unit matrix for one pair of simplices, but ing one simplex s to a neighboring simplex s , n ðs; s Þ is 0 0 it will not then be unity for all other pairs in the presence of the unit normal to fðs; s Þ and Vðfðs; s ÞÞ its volume. The curvature. above spinorial action can be considered closely analogous One important new ingredient is the need to introduce to the lattice fermion action proposed originally by Wilson lattice spin rotations. Given, in d dimensions, the above [1] for non-Abelian gauge theories. It is possible that it still rotation matrix Uðs0;sÞ, the spin connection Sðs; s0Þ be- suffers from the fermion doubling problem, although the tween two neighboring simplices s and s0 is defined as situation is less clear for a dynamical lattice [42]. follows. Consider S to be an element of the 2 -dimensional It is clear that the situation with gravitational Wilson representation of the covering group of SOðdÞ, SpinðdÞ, loops is a bit more complicated than in the scalar field case, with d ¼ 2 or d ¼ 2 þ 1, and for which S is a matrix of since the action now contains the spin connection matrix, dimension 2 2 . Then U can be written in general as which is a function of the matrices U which play the role of the connection. What is more, the generators of the spin 1 U ¼ exp (94) rotation matrices are in a different representation from the 2 generators of the rotation matrices, and it seems impossible to obtain, to lowest order, a spin zero object out of the where is an antisymmetric matrix. The ’s are 1 dðd 2 combination of two objects of spin one-half (S) and spin Þ d d SOðdÞ 1 matrices, generators of the Lorentz group ( one (U), unless one applies the fermion contribution twice SOðd ; Þ in the Euclidean case, and 1 1 in the Lorentzian to each link, in which case a nonzero contribution can case), whose explicit form is arise. We note here that if the Wilson loop were to contain a perimeter contribution, it would be of the form ½ ¼ (95) LðCÞ so that, for example, WðCÞ constðkmÞ exp½ mpLðCÞ (101)
084048-14 GRAVITATIONAL WILSON LOOP IN DISCRETE QUANTUM ... PHYSICAL REVIEW D 81, 084048 (2010)
with 1 the unit matrix, V the 4-volume associated with the plaquette , A the area of the triangle (plaquette) , and c a numerical constant of order one. If one denotes by ¼ cV =A the d 2-volume of the dual to the plaquette , then the quantity
V ¼ c (105) A 2 A is simply the ratio of this dual volume to the plaquettes FIG. 10 (color online). Illustration on how a perimeter contri- area. The edge lengths lij and therefore the metric enter the bution to the gravitational Wilson loop arises from matter field lattice gauge field action through these volumes and areas. contributions. Note that now the arrows representing rotation One important property of the gauge lattice action of matrices reside in principle in different representations. Eq. (104) is its local invariance under gauge rotations gi defined at the lattice vertices., One can further show that where LðCÞ is the length of the perimeter of the near-planar the discrete action of Eq. (104) goes over in the lattice loop C, m the particle’s mass, equal here to m ¼jlnk j continuum limit to the correct Yang-Mills action for mani- p p m folds that are smooth and close to flat. for small k , with k the weight of the single link con- m m Regarding the effects of gauge fields on the gravitational tribution from the matter particle (sometimes referred to as Wilson loop one can make the following observation. the hopping parameter). Area and perimeter contributions Since the gauge action contains no factors related to the to the near-planar Wilson loop would then become com- lattice connection, the Wilson loop area law for large parable only for exceedingly small particle masses, m P gravitational loops will remain unaffected. In particular LðCÞ= 2, i.e. for Compton wavelengths comparable to a this will be true for the photon (which in principle could macroscopic loop size (taking AðCÞ LðCÞ2=4 ). have led to important long-distance effects, since it is To demonstrate the perimeter behavior (see Fig. 10), one massless). would need to show that the matrix S on the face between simplices s and s0 would have a term proportional to the corresponding Uðs; s0Þ, with a coefficient composed of IX. EFFECTS FROM A LATTICE GRAVITINO -matrices, thereby possibly giving a nonzero contribution Supergravity in four dimensions naturally contains a to the U-integration. (This does not seem to be true in the spin-3=2 gravitino, the supersymmetric partner of the infinitesimal case to lowest order, where, for example, graviton. In the case of N ¼ 1 supergravity these are Sð Þ¼I þ 1 ½U ð Þ U ð Þ .) 34 4 2 4 13 34 24 34 the only 2 degrees of freedom present. The action contains, besides the Einstein-Hilbert action for the gravitational VIII. EFFECTS OF GAUGE FIELDS degrees of freedom, the Rarita-Schwinger action for the gravitino, as well as a number of additional terms (and In the continuum a locally gauge invariant action cou- fields) required to make the action manifestly supersym- pling an SUðNÞ gauge field to gravity is metric off-shell. Z ffiffiffi 1 4 p a a A spin-3=2 Majorana fermion in four dimensions corre- Igauge ¼ d x gg g F F (102) c 4g2 sponds to self-conjugate Dirac spinors , where the Lorentz index ¼ 1...4. In flat space the action for a a a abc b c with F ¼r A r A þ gf A A and a; b; c ¼ such a field is given by the Rarita-Schwinger term 1; ...;N2 1. On the lattice one can follow a procedure 1 analogous to Wilson’s construction on a hypercubic lattice, L c T c RS ¼ C 5 @ (106) with the main difference that the lattice is now possibly 2 simplicial. Given a link ij on the lattice one assigns group where C is the charge conjugation matrix. Locally the elements Uij, with each U an N N unitary matrix with action is invariant under the gauge transformation 1 determinant equal to one, and such that Uji ¼ U . Then ij c ðxÞ!c ðxÞþ@ ðxÞ (107) with each triangle (plaquette) , labeled by the three vertices ijk, one associates a product of three U matrices, where ðxÞ is an arbitrary local Majorana spinor. U U ¼ U U U : (103) The construction of a suitable lattice action for the ijk ij jk ki spin-3=2 particle proceeds in a way that is rather similar The discrete action is then given by [37] to what one does in the spin-1=2 case. On a simplicial manifold the Rarita-Schwinger spinor fields c ðsÞ and 1 X c I ¼ V Re½Trð1 U Þ (104) c ðsÞ are most naturally placed at the center of each gauge g2 A2 d-simplex s. Like the spin-1=2 case, the construction of a
084048-15 H. W. HAMBER AND R. M. WILLIAMS PHYSICAL REVIEW D 81, 084048 (2010) 0 0 0 suitable lattice action requires the introduction of the c ðsÞSðUðs; s ÞÞ ðs Þc ðs Þ Lorentz group and its associated vierbein fields ea ðsÞ c ð Þ ðUð 0ÞÞ ð 0Þc ð 0Þ within each simplex labeled by s, together with represen- s S s; s s s (113) P tations of the Dirac gamma matrices (see the previous 0 discussion of Dirac fields). and the sum faces fðss0Þ extends over all interfaces fðs; s Þ 0 Now in the presence of gravity the continuum action for connecting one simplex s to a neighboring simplex s . a massless spin-3=2 field is given by When compared to the spin-1=2 case, the most important modification is the second rotation matrix U ðs; s0Þ, Z 1 pffiffiffi c c which describes the parallel transport of the fermionic I3=2 ¼ dx g ðxÞ 5 D ðxÞ (108) 0 2 vector c from the site s to the site s , which is required in order to obtain locally a Lorentz scalar contribution to with the Rarita-Schwinger field subject to the Majorana the action. c c T constraint ¼ C ðxÞ . Here the covariant derivative is In this case again one expects the Wilson loop to follow defined as a perimeter law, as in the spin one-half case of Eq. (101), because the matter action explicitly contains factors of U 1 ab D c ¼ @ c c þ ! ab c (109) which will contribute when the Us and Ss around the loop 2 are integrated over, which of course requires that one also take into account the spin connection matrices. These add and involves both the standard affine connection ,as well as the vierbein connection complexity but are not expected, due to the nature of the interaction, to change the answer. The same general con- 1 siderations then apply as in the spin-1=2 case: the perime- ! ¼ ½e ð@ e @ e Þþe e ð@ e Þec ab 2 a b b a b c ter contribution to the gravitational Wilson loop can ða $ bÞ (110) significantly modify the area law result only if the corre- sponding particle mass is exceedingly small. 1 with Dirac spin matrices ab ¼ 2i ½ a; b , and the usual Levi-Civita tensor, such that ¼ g . X. POSSIBLE PHYSICAL CONSEQUENCES It is easiest to just consider two neighboring simplices s 1 In the previous sections we presented evidence for an and s , covered by a common coordinate system x . When 2 area law behavior for a variety of different lattice discre- the two vierbeins in s and s are made to coincide, one can 1 2 tizations of gravity, all studied in the strong coupling limit. then use a common set of gamma matrices within both We have not pursued yet the computation of higher order simplices. Then (in the absence of torsion) the covariant terms in the strong coupling expansion, which could be derivative D in Eq. (108) reduces to just an ordinary done. But we believe that the basic result, which we expect c ð Þ derivative. The fermion field x within the two sim- to be geometric in character, could be further tested by plices can then be suitably interpolated, and one obtains a numerical means throughout the whole strong coupling lattice action expression very similar to the spinor case. phase. If the analogy with non-Abelian gauge theories One can then relax the condition that the vierbeins ea ðs1Þ and the concept of universal critical behavior continues and ea ðs2Þ in the two simplices coincide. If they do not, to hold in Euclidean gravity, then one would expect that the U then they will be related by a matrix ðs2;s1Þ such that area law result would hold not just at strong coupling but instead throughout the whole strong coupling region, up e ðs Þ¼U ðs ;s Þe ðs Þ (111) a 2 2 1 a 1 the nontrivial ultraviolet fixed point, if one can be found in and whose spinorial representation S was given previously the relevant lattice regularized theory, of which we have in Eq. (98). But the new ingredient in the spin-3=2 case is given here a few examples. Furthermore the SOð4Þ lattice S model of Sec. II is one example where the analogy with that, besides requiring a spin rotation matrix ðs2;s1Þ,now 0 Wilson’s non-Abelian gauge theory on the lattice is clearly one also needs the matrix U ðs; s Þ describing the corre- sponding parallel transport of the Lorentz vector c ðsÞ seen as more than just superficial resemblance. The evi- dence for an ultraviolet fixed point for gravity has recently from a simplex s to the neighboring simplex s . The 1 2 been reviewed in [4] and will not be repeated here. Our invariant lattice action for a massless spin-3=2 particle results and similar related lattice results could then be takes therefore the form tested further in the case of gravity, for example, by nu- 1 X merical means, regarding their universal character and I ¼ Vðfðs; s0ÞÞ c ðsÞSðUðs; s0ÞÞ ðs0Þ 2 scaling behavior in the vicinity of the nontrivial fixed point. faces fðss0Þ In this section we wish to briefly discuss instead a 0 0 0 n ðs; s ÞU ðs; s Þc ðs Þ (112) possible physical interpretation of the Euclidean gravita- tional Wilson loop result, along the lines of the proposal in with Refs. [3,4], and thus in terms of its relationship to a large-
084048-16 GRAVITATIONAL WILSON LOOP IN DISCRETE QUANTUM ... PHYSICAL REVIEW D 81, 084048 (2010) scale average curvature. Note that contrary to some earlier tion. In above quoted expression, is therefore intended to statements in the literature, the Wilson loop in gravity does be the renormalization group invariant quantity obtained not provide any useful information about the static gravi- by integrating the -function for the Newtonian coupling tational potential [6–8]. The arguments presented below G, should therefore be taken with some clear caveats, namely, Z G dG0 that (i) the results have been derived from the Euclidean 1 ðGÞ¼const exp 0 (115) theory, whose relationship to the Lorentzian case remains ðG Þ to be explored, that (ii) they assume concepts of universal- with the ultraviolet cutoff (and thus analogous to Eq. (8) ity of critical behavior which nevertheless are known to for gauge theories). In the vicinity of the ultraviolet fixed apply to just about any other quantum field theory except point at Gc possibly gravity, and finally (iii) that it is assumed that the phase structure of Euclidean lattice gravity is such that a @ 0 ðGÞ Gð Þ ðGcÞðG GcÞþ...; (116) nontrivial fixed point can be found (which is not obvious at @ G!Gc this point for some of the lattice models discussed previ- which gives ously in this paper). 1 Having then ascertained with some degree of confidence ðGÞ/ jðG GcÞ=Gcj ; (117) that in a number of different, and quite unrelated, 0 Euclidean lattice discretizations of gravity the gravitational with a correlation length exponent ¼ 1= ðGcÞ.In loop follows an area law at least for sufficiently strong particular the correlation length ðGÞ is related to the coupling G, which we choose to write here as bare Newtonian coupling G, and diverges, in units of the cutoff , as one approaches the fixed point at Gc. Thus for 2 1 hWðCÞi expð AC= Þ (114) a bare G very close to Gc the two scales, and can be A!1 vastly different. Furthermore the result of Eq. (114) was with determined by scaling and dimensional arguments derived from the lattice theory of gravity in the strong to be the unique nonperturbative gravitational correlation coupling limit G !1. But one would expect, based on length, let us now turn to a possible physical interpretation general scaling arguments and the analogy with non- of the result. Here the formula of Eq. (114), inspired by the Abelian gauge theories, see Eq. (5), that such a behavior analogy to gauge theories which gives Eq. (5) and by the would persist throughout the whole strong coupling phase well-established universality of critical behavior, is ex- G>Gc, all the way up to the nontrivial ultraviolet fixed pected to summarize, at least for the purpose of our argu- point at Gc. This is indeed what happens in non-Abelian ment, the behavior of the gravitational Wilson loop gauge theories and spin systems such as the nonlinear throughout the whole strong coupling domain. In the sigma model: the only scale determining the nontrivial same way that the analogous textbook result, Eq. (5), in a scaling properties in the vicinity of the fixed point is ; sense summarizes the long-distance behavior of the Wilson the corresponding behavior is known as universal renor- loop for non-Abelian gauge theories in terms of the only malization group scaling. admissible renormalization group invariant scale. Here we As discussed at the beginning of this paper and in will therefore explore some possible ramifications of the Refs. [3,4], the rotation matrix appearing in the gravita- above Ansatz in the context of the nontrivial fixed point in tional Wilson loop can be related classically to a well- G, or asymptotic safety, scenario for quantum gravity, defined classical physical process, one in which a vector is recently reviewed, for example, in Ref. [4]. This is perhaps parallel transported around a large loop, and at the end is not the only possible scenario, but it is the one we are most compared to its original orientation. Then the vector’s familiar with, and in our view also the most credible one at rotation is directly related to some sort of average curva- U this point, supported by the 2 þ expansion for gravity, by ture enclosed by the loop; the total rotation matrix ðCÞ is given by a path-ordered (P ) exponential of the integral of the nonperturbative Regge lattice calculations, and by the analogy with the much simpler but very well understood the affine connection , as in Eq. (1). In a semiclassical perturbatively nonrenormalizable nonlinear sigma model. description of the parallel transport process of a vector In particular, we intend to explore here briefly, following around a very large loop, one can reexpresses the connec- closely the arguments of Ref. [3], the connection of the tion in terms of a suitable coarse-grained, semiclassical lattice result of Eq. (114) to a semiclassical picture, de- slowly varying Riemann tensor, as in Eq. (2). Since the scribing the properties of curvature on very large, macro- rotation is small for weak curvatures, one has for a macro- scopic distance scales. The procedure followed here and in scopic observer [3] is simple and quite analogous to the original procedure Z 1 proposed by Wilson for gauge theories [1]: the quantum U ðCÞ 1 þ R AC þ ... : (118) 2 SðCÞ Wilson loop average is computed in the full theory, and the answer is then compared to the result obtained when the At this stage one can compare the above semiclassical path integral is dominated by a single classical configura- expression to the quantum result of Eqs. (44), (66), and
084048-17 H. W. HAMBER AND R. M. WILLIAMS PHYSICAL REVIEW D 81, 084048 (2010) (114), and, in particular, one would like to relate the Let us explore this last point further. At first it would coefficients of the area terms. Since one expression seem, from the nontrivial ultraviolet fixed point, or asymp- [Eq. (118)] is a matrix and the other [Eq. (114)] is a scalar, totic safety, scenario point of view,4 that in principle the one needs to take the trace after first contracting the scale could take any value, including very small ones, rotation matrix with (BC þ I4), as in our second definition based on the naive estimate lP, where lP is the Planck of the Wilson loop of Eq. (16), giving length whose magnitude is comparable to the (inverse of Z the) ultraviolet cutoff . The last choice would of course 1 preclude any observable quantum effects in the foreseeable WðCÞ Tr ðBC þ I4Þ exp R AC : (119) 2 SðCÞ future. But the relationship between and large-scale Next, following Ref. [3], it is advantageous to consider the curvature, or more precisely between and obs, arising lattice analog of a background classical manifold with out of the specific properties of the gravitational Wilson constant or near-constant large-scale curvature, loop as proposed in Eqs. (122) and (123), opens up a new possibility. Namely a very large, cosmological value for 1 1028 cm, given the present observational bounds on R ¼ ðg g g g Þ (120) 3 obs. Closely related possibilities exist, such as an identi- fication of with the Hubble constant as measured today, so that here one can set for the curvature tensor ’ 1=H0; since this quantity is presumably time- 1 R ¼ RB B ; (121) dependent, a possible scenario is one in which ¼ 2 1 H1 ¼ limt!1HðtÞ, with H1 ¼ 3 obs. This in turn would where R is some average curvature over the loop, and the suggest a number of other related observations, such as the area bivectors B here will be taken to coincide with BC. fact that for distances r one still resides in the short- The trace of the product of (BC þ I4) with this expression distance regime, where correlations are still expected to 2 then gives TrðRBCACÞ¼ 2RAC. This is to be compared behave as power laws; significant deviations from classical with the linear term from the other exponential expression, gravity would then arise only for distance comparable or 2 AC= . Thus the average curvature is computed to be of greater than . the order Finally we note that another physical consequencepffiffiffiffiffiffiffiffiffi arises from the tentative identification of with 1= :asin 2 obs R þ1= (122) gauge theories, one expects to determine the scale de- at least in the small k ¼ 1=8 G limit. An equivalent way pendence of the effective Newton’s constant Gð Þ appear- of phrasing the last result makes use of the classical field ing in the field equations, where the latter is obtained, for equations in the absence of matter R ¼ 4 . Then the rather example, from solving the renormalization group equa- surprising result emerges that 1= 2 should be identified, up tions for G, Eqs. (115) and (116). As discussed in [43], a to a constant of proportionality of order one, with the running of the gravitational constant of the type discussed observed scaled cosmological constant , in [7] is best expressed in a fully covariant formulation, obs such as an effective classical, but nonlocal, set of field 1 equations of the type obs ’þ 2 : (123) 1 R g R þ g ¼ 8 GðhÞT (124) The latter can then be regarded either as a measure of the 2 vacuum energy, or of the intrinsic curvature of the vacuum. with ’ 1= 2, and GðhÞ the running Newton’s constant It would seem therefore that a direct calculation of the gravitational Wilson loop, within the boundaries of our G ! GðhÞ (125) limited strong coupling results, could provide a direct in- with the running given by sight into whether the manifold is de Sitter or anti-de Sitter at large distances. Moreover, in the case of lattice gravity 1 1=2 GðhÞ¼G 1 þ a þ ... ; (126) at strong coupling, as has been shown in this work, it seems c 0 2h virtually impossible to obtain a negative sign in Eqs. (122) or (123), which would then suggest that Euclidean quan- and a0 ’ 42 > 0 and ’ 1=3 [44]. Gc in the above ex- tum gravity can only give a positive cosmological constant pression should be identified to a first approximation with at large distances. (Again, the analogy with non-Abelian gauge theories comes to mind, where one has for the non- 4The existence of a nontrivial ultraviolet fixed point in fact 2 4 perturbative gluon condensate hF i 1= , where is implies the existence of such a new nonperturbative scale, which the nonperturbative QCD correlation length, 1 ; arises as an integration constant from the Callan-Symanzik QCD MS renormalization group equations close to the UV fixed point the analog of the vacuum condensate in non-Abelian field [4], in the same way that a similar scale arises out of the gauge theories is then naturally seen here as the vacuum renormalization group equations for asymptotically free Yang- expectation value of the curvature). Mills theories.
084048-18 GRAVITATIONAL WILSON LOOP IN DISCRETE QUANTUM ... PHYSICAL REVIEW D 81, 084048 (2010) ffiffiffiffiffiffi p 33 the laboratory scale value Gc 1:6 10 cm [4,43]. coupled to gravity is less clear-cut, although it is only The running of G can then be worked out in detail for massless or almost massless matter which propagates to specific coordinate choices, and in the static isotropic case sufficiently large distances to affect large gravitational one finds a gradual slow increase in G with distance, in Wilson loops. In that case, scalar matter and gauge fields accordance with the formula (in particular the photon) do not affect the area law. For very low mass fermions (e.g. neutrinos), it is possible that a0 1 G ! GðrÞ¼G 1 þ m3r3 ln þ ... (127) the coupling gives rise to a perimeter contribution, which 3 m2r2 could replace the area law for suitable ratios of coupling in the regime r 2MG, where 2MG is the horizon radius, constants, but this seems unlikely. Similarly, the lattice and m 1= . The results of Eqs. (122) and (123) then gravitino could produce a perimeter law. These possibil- open up a new possibility, and would suggest that the scale ities will be investigated in future work. Numerical simu- entering the quantum scale dependence of GðrÞ is not of the lations of simplicial lattice gravity could provide vital clues order of the Planck length, but instead a very large scale, here [44]. Numerical simulations in general require a comparablepffiffiffiffiffiffiffiffiffi to the observed cosmological constant, ¼ general definition of the Wilson loop applicable to any 1= obs. geometry, a subject which has been discussed previously in a number of places, and which we will repeat here for XI. CONCLUSIONS completeness. The argument relating the quantum vacuum expectation From our study of Wilson loops, where defined and values of a gravitational Wilson loop to the corresponding calculable, in all theories of Euclidean discrete gravity classical quantity, namely, the amount of rotation a vector that we have found, it seems that the area law holds for experiences when parallel transported around a closed loop large loops in the strong coupling domain. This would in a given classical background geometry, requires, as in suggest that one can infer, as in [3], that a universal ordinary gauge theories, that a connection be made be- prediction of strongly coupled Euclidean gravity without tween the full quantum domain dominated by large short- matter is that the scaled cosmological constant is positive. distance field fluctuations on the one hand, and the semi- We have argued that the basic result, which appears to be classical domain of smooth fields at large distances on the geometric in character as in the better understood case of other. Originally it was thought that the gravitational non-Abelian gauge theories, could be further tested by Wilson loop, as computed in most of the original papers numerical means throughout the whole strong coupling on hypercubic lattice gravity referred to in this work, phase. If the analogy with non-Abelian gauge theories would give information about the static potential, but this and the concept of universal critical behavior continues to hold in Euclidean gravity, then one would expect that the was shown later by Modanese to be incorrect [6]. Instead, area law result would hold not just at strong coupling but the gravitational Wilson loop is now understood to provide instead throughout the whole strong coupling region, up physical information about the large-scale curvature of the the nontrivial ultraviolet fixed point, if one can be found. fluctuating geometry in question [7,8]. But we wish to emphasize here again that the arguments Initially the discussion of the gravitational Wilson loop connecting the area law result in the Euclidean theory to in the quoted papers focused on the weak field case, where the physical scaled cosmological constant should be taken the expectation of the loop is clearly well-defined. The with some clear caveats, namely, that they have been calculation is then technically quite similar to the pertur- derived from the Euclidean theory, that they assume con- bative calculation of a square Wilson loop in non-Abelian cepts of universality of critical behavior, and finally that gauge theories. A flat or near-flat background geometry is they assume that the phase structure of various Euclidean allowed to fluctuate locally, and a vector is parallel trans- lattice gravity models is such that a nontrivial fixed point ported around a circular loop. An integration over the can be found in all of them. Nevertheless we believe the fluctuating part of the metric then yields an explicit and value of our results might lie in the fact that they open the well-defined expression for the gravitational Wilson loop, possibility of (a) providing a set of explicit, unambiguous suitably defined as a trace of the holonomy of the Levi- and presumably universal predictions which could be Civita connection. The limiting factor for such a calcula- tested by numerical means, and (b) suggesting a new tion, already recognized at the time by the quoted author, is physical connection between two at first seemingly unre- of course the fact that higher order radiative corrections are lated quantities, namely, the scale for the running of the just as important as the leading contribution, due to the coupling G in the asymptotic safety scenario and the perturbative nonrenormalizability in four dimensions. cosmological constant , leading possibly to a number of Nevertheless the gravitational Wilson loop for a regu- testable cosmological and astrophysical predictions. lated closed circular loop in flat space, or in a given near- We wish to make here a number of additional comments flat background geometry (such as one that would arise relating to the interpretation of the Euclidean lattice re- from having to satisfy the classical field equations with a sults. The effect on the Wilson loop of adding matter nonvanishing small classical cosmological constant term)
084048-19 H. W. HAMBER AND R. M. WILLIAMS PHYSICAL REVIEW D 81, 084048 (2010) is a completely well-defined object, to all orders in the antipode (the point most distant, within the submanifold, weak field expansion, and in any dimension d>2. from the chosen point). Its dimension will be d 2, mak- Similarly, one can argue based on semiclassical argu- ing it suitable in three dimensions as a parallel transport ments, such as the ones advocated, for example, by Hartle path, with a given calculable length L, thus giving a useful [45,46] in conjunction with the emergence of a classical and unambiguous definition of the gravitational Wilson domain out of an underlying fluctuating geometry, that all loop in three dimensions. In dimensions higher than three which is required to define a gravitational Wilson loop is the above geometric procedure needs to be iterated a the existence of a smooth near-classical (and four- sufficient number of times until the desired maximal dimensional) geometry at very large distances, for which near-planar one-dimensional path is obtained. Thus in the parallel transport of a vector around a circular loop is four dimensions a point and its antipode need to be picked well-defined according to classical general relativity (one again within the compact submanifold of dimensions d could of course define loops of arbitrary sizes and shapes, 2, resulting in a one-dimensional Wilson loop path span- but for the present argument a large circular loop of length ning the resulting equator (again defined as the locus of the L will suffice). Clearly such a definition breaks down if the points equidistant from the point picked and its antipode). notion of a circular loop cannot be stated, in which case It is clear from the construction that many equivalent though the geometry is not near flat at large distances, and loops can be defined locally in this way. Of course in two no physically acceptable theory of gravity is recovered in dimensions only one such loop, centered at P and of size R, this regime, making the whole exercise rather pointless. It exists for a given fluctuating manifold. In three dimen- would seem therefore that the computation of a Wilson sions, given an origin P, there is on the other hand a two- loop in the lattice theory of gravity only makes sense if a parameter family of near-planar loops of size R associated semiclassical space-time is recovered at large distances, with the center point in question, and in accordance with making a definition of a circular loop meaningful. the loop’s possible orientation. This would be the set of all Nevertheless, irrespective of whether semiclassical great circles on a 2-sphere, parametrized by two angles. space-time is recovered at large distances, such a gravita- Then in four dimensions the corresponding statement is tional Wilson loop can still be defined in rather general that given a point P, a three-parameter family of near- terms. One way to proceed is to focus on a set point P planar loops of size R centered at P can be constructed located on a given fluctuating manifold, and consider a in the way described above. one-parameter family of geodesics emanating from that Finally we should point out that if a timelike coordinate point, all lying in a given 2d plane sited at the point in can somehow be defined, then the consideration of the question. Following the geodesics out to a distance R one gravitational Wilson could in principle be restricted, for obtains a suitable path over which to evaluate the trace of example, to spacelike loops only, thus effectively reducing the holonomies; repeating the same procedure for many the dimension of the geometrical problem by one. points and many field configurations one then would obtain a quantum average for the same quantity. The extent to ACKNOWLEDGMENTS which the corresponding loops are flat can then be deter- mined by comparing the radius R with the length of the The authors wish to thank Hermann Nicolai and the Max loop perimeter L; in a near-flat geometry at very large Planck Institut fu¨r Gravitationsphysik (Albert-Einstein- distances one would expect for large R and L the relation- Institut) in Potsdam for very warm hospitality. The work ship R L=ð2 Þ. described in this paper was done while both authors were A slightly more general way of defining a planar Wilson visitors at the AEI. One of the authors (H. W. H.) also loop can be given as follows. Consider a point P on a wishes to thank Sergio Caracciolo for pointing out addi- d-dimensional manifold, and construct the d 1 dimen- tional references on the gravitational Wilson loop, and for sional surface around the point P defined as the locus of all helpful correspondence. The work of H. W. H. was sup- points situated at a fixed geodesic R distance from P.Next ported in part by the Max Planck Gesellschaft zur consider the equator on this submanifold, defined as the set Fo¨rderung der Wissenschaften, and by the University of of all points equidistant from the point in question and its California. The work of R. M. W. was supported in part by the UK Science and Technology Facilities Council.
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