PHYSICAL REVIEW D 084012 (2013) Wheeler …aeneas.ps.uci.edu/hamber/wdw4_prd.pdfHamiltonian lattice...

Wheeler-DeWitt equation in 3 þ 1 dimensions

Herbert W. Hamber*

Max Planck Institute for Gravitational Physics (Albert Einstein Institute), D-14476 Potsdam, Germany

Reiko Toriumi†

Department of Physics and Astronomy, University of California, Irvine, California 92697-4575, USA

Ruth M. Williams‡

Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA,United Kingdom and Girton College, University of Cambridge, Cambridge CB3 0JG, United Kingdom

(Received 14 December 2012; published 10 October 2013)

Physical properties of the quantum gravitational vacuum state are explored by solving a lattice version

of the Wheeler–DeWitt equation. The constraint of diffeomorphism invariance is strong enough to

uniquely determine part of the structure of the vacuum wave functional in the limit of infinitely fine

triangulations of the three-sphere. In the large fluctuation regime, the nature of the wave function solution

is such that a physically acceptable ground state emerges, with a finite nonperturbative correlation length

naturally cutting off any infrared divergences. The location of the critical point in Newton’s constant Gc,

separating the weak from the strong coupling phase, is obtained, and it is inferred from the general

structure of the wave functional that fluctuations in the curvatures become unbounded at this point.

Investigations of the vacuum wave functional further suggest that for weak enough coupling, G<Gc, a

pathological ground state with no continuum limit appears, where configurations with small curvature

have vanishingly small probability. One would then be lead to the conclusion that the weak coupling,

perturbative ground state of quantum gravity is nonperturbatively unstable and that gravitational screening

cannot be physically realized in the lattice theory. The results we find tend to be in general agreement with

the Euclidean lattice gravity results and would suggest that the Lorentzian and Euclidean lattice

formulations of gravity ultimately describe the same underlying nonperturbative physics.

DOI: 10.1103/PhysRevD.88.084012 PACS numbers: 04.60.�m, 04.60.Gw, 04.60.Nc, 98.80.Qc

I. INTRODUCTION

We have argued in previous work that the correct iden-tification of the true ground state for quantum gravitationnecessarily requires the introduction of a consistent non-perturbative cutoff, followed by the construction of thecontinuum limit in accordance with the methods of therenormalization group. To this day the only known way tointroduce such a nonperturbative cutoff reliably in quan-tum field theory is via the lattice formulation. A wealth ofresults has been obtained over the years using theEuclidean lattice formulation, which allows the identifica-tion of the physical ground state and the accurate calcu-lations of gravitational scaling dimensions, relevant for thescale dependence of Newton’s constant in the universalscaling limit.

In this work we will focus instead on the Hamiltonianapproach to gravity, which assumes from the very begin-ning a metric with Lorentzian signature. Recently aHamiltonian lattice formulation was written down basedon the Wheeler–DeWitt equation, where the gravity

Hamiltonian is expressed in the metric-space representa-tion. Specifically, in Refs. [1,2] a general discreteWheeler–DeWitt equation was given for pure gravity,based on the simplicial lattice transcription of gravityformulated by Regge and Wheeler. Here we extend thework initiated in Refs. [1,2] to the physical case of 3þ 1dimensions and show how nonperturbative vacuum solu-tions to the lattice Wheeler–DeWitt equations can be ob-tained for arbitrary values of Newton’s constant G. Theprocedure we follow is similar to what was done earlier in2þ 1 dimensions. We solve the lattice equations exactlyfor several finite and regular triangulations of the three-sphere and then extend the result to an arbitrarily largenumber of tetrahedra. We then argue that for large enoughvolumes, the exact lattice wave functional is expected todepend on geometric quantities only, such as the totalvolumes and the total integrated curvature. In this process,the regularity condition on the solutions of the wave equa-tion at small volumes plays an essential role in constrainingthe form of the vacuum wave functional. A key ingredientin the derivation of the results is of course the local diffeo-morphism invariance of the Regge–Wheeler latticeformulation.From the structure of the resulting wave function, a

number of potentially useful physical results can be

*[email protected]†[email protected]‡[email protected]

PHYSICAL REVIEW D 88, 084012 (2013)

1550-7998=2013=88(8)=084012(23) 084012-1 � 2013 American Physical Society

http://dx.doi.org/10.1103/PhysRevD.88.084012

obtained. First, one observes that the nonperturbative cor-relation length is found to be finite for sufficiently large G.At the critical point G ¼ Gc, which we determine exactlyfrom the structure of the wave function, fluctuations in thecurvature become unbounded, thus signaling a divergencein the nonperturbative gravitational correlation length. Weargue that such a result can be viewed as consistent with theexistence of a nontrivial ultraviolet fixed point (or a phasetransition in statistical field theory language) in G.Furthermore, the behavior of the theory in the vicinity ofsuch a fixed point is expected to determine, through stan-dard renormalization group arguments, the scale depen-dence of the gravitational coupling in the vicinity of theultraviolet fixed point.

An outline of the paper is as follows. In Sec. II, as abackground to the rest of the paper, we briefly summarizethe formalism of canonical gravity. At this stage the con-tinuumWheeler–DeWitt equation with its invariance prop-erties are introduced. We then briefly outline the generalproperties of the lattice Wheeler–DeWitt equation pre-sented in our previous work, and in Sec. III, we makeexplicit various quantities that appear in it. Here we alsoemphasize the important role of continuous lattice diffeo-morphism invariance in the Regge theory, as it applies tothe case of 3þ 1-dimensional gravity. Section IV focuseson basic scaling properties of the lattice equations anduseful choices for the lattice coupling constants, with theaim of giving a more transparent form to the results ob-tained later. Section V presents an outline of the method ofsolution for the lattice equations, which are later discussedin some detail for a number of regular triangulations of thethree-sphere. Then a general form of the wave function isgiven that covers all previous discrete cases and thus allowsa study of the infinite volume limit. Section VI discussesthe issue of how to define an average volume and thus anaverage lattice spacing, an essential ingredient in the in-terpretation of the results given later. Section VII discussesmodifications of the wave function solution obtained whenthe explicit curvature term in the Wheeler–DeWitt equa-tion is added. Later, a partial differential equation for thewave function is derived in the curvature and volumevariables. General properties of the solution to this equa-tion are discussed in Sec. VIII. Section IX contains a briefsummary of the results obtained so far.

II. CONTINUUM AND DISCRETEWHEELER–DEWITT EQUATION

Our work deals with the canonical quantization of grav-ity, and we begin here therefore with a very brief summaryof the classical canonical formalism [3] as formulated byArnowitt, Deser, and Misner [4]. Many of the results foundin this section are not new, but nevertheless it will beuseful, in view of later applications, to recall here themain results and provide suitable references for expres-sions used in the following sections. Here xi (i ¼ 1, 2, 3)

will be coordinates on a three-dimensional manifold, andindices will be raised and lowered with gijðxÞ (i, j ¼ 1, 2,

3), the three-metric on the given spacelike hypersurface.As usual, gij denotes the inverse of the matrix gij. Our

conventions are such that the spacetime metric has signa-ture �þþþ , that 4R is non-negative in a spacetimecontaining normal matter, and that 3R is positive in athree-space of positive curvature.One goes from the classical to the quantum description

of gravity by promoting the metric gij, the conjugate

momenta �ij, the Hamiltonian density H, and the momen-tum density Hi to quantum operators, with gij and �ij

satisfying canonical commutation relations. Then the clas-sical constraints select physical states j�i, such that in theabsence of sources

Hj�i ¼ 0 Hij�i ¼ 0; (1)

whereas in the presence of sources, one has more generally

Tj�i ¼ 0 Tij�i ¼ 0; (2)

with T and Ti describing matter contributions that can be

added to H and Hi. As is the case in nonrelativisticquantum mechanics, one can choose different representa-tions for the canonically conjugate operators gij and �ij.

In the functional metric representation, one sets

g ijðxÞ ! gijðxÞ �ijðxÞ ! �iℏ � 16�G � �

�gijðxÞ :(3)

Then quantum states becomewave functionals of the three-metric gijðxÞ,

j�i ! �½gijðxÞ�: (4)

The constraint equations in Eq. (2) then become theWheeler–DeWitt equation [5,6],��16�G � Gij;kl

�2

�gij�gkl� 1

16�G

ffiffiffig

p ð3R� 2�Þ

þ H�

��½gijðxÞ� ¼ 0; (5)

and the momentum constraint equation listed below. InEq. (5), Gij;kl is the inverse of the DeWitt supermetric,

Gij;kl ¼ 1

2g�1=2ðgikgjl þ gilgjk � gijgklÞ: (6)

The three-dimensional DeWitt supermetric itself isgiven by

Gij;kl ¼ 1

2

ffiffiffig

p ðgikgjl þ gilgjk � 2gijgklÞ: (7)

In the metric representation, the diffeomorphism constraintreads

HAMBER, TORIUMI, AND WILLIAMS PHYSICAL REVIEW D 88, 084012 (2013)

084012-2

�2igijrk

�

�gjkþ H�

i

��½gijðxÞ� ¼ 0; (8)

where H� and H�i again are possible matter contributions.

In the following, we shall set both of these to zero as wewill focus here almost exclusively on the pure gravitationalcase. Then the last constraint represents the necessary andsufficient condition that the wave functional c ½g� be aninvariant under coordinate transformations [7].

We note here that in the continuum one expects thecommutator of two Hamiltonian constraints to be relatedto the diffeomorphism constraint. In the following, we will,for the time being, overlook this rather delicate issue andfocus our efforts instead mainly on the solution of theexplicit (lattice) Hamiltonian constraint of Eq. (15). Itshould nevertheless be possible to revisit this importantissue at a later stage, once an exact, or approximate,candidate expression for the wave functional is found.The key issue at that stage will then be if the lattice wavefunctional satisfies all physical requirements, including themomentum constraint, in a suitable lattice scaling limitwherein the (average) lattice spacing is much smaller thana suitable physical scale, such as the scale of the localcurvature, or some other sort of agreeable physical corre-lation length. For a more in-depth discussion of the analo-gous problem in 2þ 1 dimensions, we refer the reader toour previous work [2], where an explicit form for thecandidate wave functional was eventually given in termsof manifestly invariant quantities such as areas andcurvatures.

We should also mention here that a number of ratherbasic issues needs to be considered before one can gaina consistent understanding of the full content of thetheory (see, for example, Refs. [8–12]). These includepotential problems with operator ordering and the speci-fication of a suitable Hilbert space, which entails achoice for the norm of wave functionals, for example,in the Schrodinger form

k�k2 ¼Z

d�½g��½gij��½gij�; (9)

where d�½g� is the appropriate (DeWitt) functional mea-sure over the three-metric gij. In this work, we will

attempt to address some of those issues, as they willcome up within the relevant calculations.

In this paper, the starting point will be the Wheeler–DeWitt equation for pure gravity in the absence ofmatter, Eq. (5),

��ð16�GÞ2Gij;klðxÞ �2

�gijðxÞ�gklðxÞ�ffiffiffiffiffiffiffiffiffiffigðxÞ

qð3RðxÞ� 2�Þ

��½gijðxÞ� ¼ 0; (10)

combined with the diffeomorphism constraint of Eq. (8),

�2igijðxÞrkðxÞ �

�gjkðxÞ��½gijðxÞ� ¼ 0: (11)

Both of these equations express a constraint on the statej�i at every x. It is then natural to view Eq. (10) as madeup of three terms, the first one identified as the kinetic termfor the metric degrees of freedom, the second one involving� ffiffiffi

gp 3R and thus seen as a potential energy contribution (of

either sign, due to the nature of the three-curvature 3R), andfinally the cosmological constant term proportional toþ�

ffiffiffig

pacting as a masslike term. The kinetic term contains

a Laplace–Beltrami-type operator acting on the six-dimensional Riemannian manifold of positive definite met-rics gij, withGij;kl acting as its contravariant metric [7]. As

shown in the quoted reference, the manifold in question hashyperbolic signature �þþþþþ , with pure dilationsof gij corresponding to timelike displacements within this

manifold of metrics.Next we turn to the lattice theory. Here we will generally

follow the procedure outlined in Ref. [1] and discretize thecontinuumWheeler–DeWitt equation directly, a procedurethat makes sense in the lattice formulation, as these equa-tions are still formulated in terms of geometric objects, forwhich the Regge theory is very well suited. It is known thaton a simplicial lattice [13–19] (see, for example, Ref. [20]for a more detailed presentation of the Regge–Wheelerlattice formulation) deformations of the squared edgelengths are linearly related to deformations of the inducedmetric. In a given simplex �, take coordinates based at avertex 0, with axes along the edges emanating from 0. Thenthe other vertices are each at unit coordinate distance from0 (see Fig. 1 as an example of this labeling for a tetrahe-dron). With this choice of coordinates, the metric within agiven simplex is

gijð�Þ ¼ 1

2ðl20i þ l20j � l2ijÞ: (12)

We note that in the following discussion only edges andvolumes along the spatial directions are involved. Then byvarying the squared volume of a given simplex � in ddimensions to quadratic order in the metric (in the contin-uum), or in the squared edge lengths belonging to thatsimplex (on the lattice), one is led to the identification[21,22]

01

2

l02

l01

l12

3

l03

l23

l13

FIG. 1. A tetrahedron with labels.

WHEELER-DEWITT EQUATION IN 3þ 1 DIMENSIONS PHYSICAL REVIEW D 88, 084012 (2013)

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Gijðl2Þ ¼ �d!X�

1

Vð�Þ@2V2ð�Þ@l2i @l

2j

; (13)

where the quantity Gijðl2Þ is local, since the sum over �only extends over those simplices which contain either thei or the j edge. In the formulation of Ref. [1], it will beadequate to limit the sum in Eq. (13) to a single tetrahedronand define the quantity Gij for that tetrahedron. Then, inschematic terms, the lattice Wheeler–DeWitt equation forpure gravity takes on the form��ð16�GÞ2Gijðl2Þ @2

@l2i @l2j

�ffiffiffiffiffiffiffiffiffiffigðl2Þ

q½3Rðl2Þ�2��

��½l2�¼0;

(14)

with Gijðl2Þ the inverse of the matrix Gijðl2Þ given above.

The range of summation over the indices i and j and theappropriate expression for the scalar curvature will bemade explicit later in Eq. (15).

It is clear that Eqs. (5) or (14) express a constraint ateach ‘‘point’’ in space. Indeed, the first term in Eq. (14)contains derivatives with respect to edges i and j connectedby a matrix element Gij, which is nonzero only if i and j

are close to each other and thus nearest neighbor.1 Oneexpects therefore that the first term can be represented by asum of edge contributions, all from within one (d� 1)-simplex � (a tetrahedron in three dimensions). The secondterm containing 3Rðl2Þ in Eq. (14) is also local in the edgelengths: it only involves those edge lengths which enterinto the definition of areas, volumes, and angles around thepoint x. The latter is therefore described, through thedeficit angle �h, by a sum over contributions over all(d� 3)-dimensional hinges (edges in 3þ 1 dimensions)

h attached to the simplex �. This then leads in threedimensions to a more explicit form of Eq. (14):�

�ð16�GÞ2 Xi;j��

Gijð�Þ @2

@l2i @l2j

� 2n�hXh��

lh�h

þ 2�V�

��½l2� ¼ 0: (15)

In the above expression, �h is the deficit angle at the hinge

(edge) h, lh the corresponding edge length, and V� ¼ffiffiffiffiffiffiffiffiffiffigð�Þp

the volume of the simplex (tetrahedron in threespatial dimensions) labeled by �. The matrix Gijð�Þ is

obtained either from Eq. (13) or from the lattice transcrip-tion of Eq. (6),

Gij;klð�Þ ¼ 1

2g�1=2ð�Þ½gikð�Þgjlð�Þ þ gilð�Þgjkð�Þ

� gijð�Þgklð�Þ�; (16)

with the induced metric gijð�Þ within a simplex � given in

Eq. (12). Note that the combinatorial factor n�h gives thecorrect normalization for the curvature term, since thelatter has to give the lattice version of

R ffiffiffig

p 3R ¼2P

h�hlh when summed over all simplices �. One cansee then that the inverse of n�h counts the number of timesthe same hinge appears in various neighboring simplicesand depends therefore on the specific choice of underlyinglattice structure. The lattice Wheeler–DeWitt equationgiven in Eq. (15) was the main result of a previous paper[1] and was studied extensively in 2þ 1 dimensions in aprevious work [2].

III. EXPLICIT SETUP FOR THE LATTICEWHEELER–DEWITT EQUATION

In the following, we will now focus on athree-dimensional lattice made up of a large number oftetrahedra, with squared edge lengths considered as thefundamental degrees of freedom. For ease of notation, wedefine l201 ¼ a, l212 ¼ b, l202 ¼ c, l203 ¼ d, l213 ¼ e, l223 ¼ f.For the tetrahedron labeled as in Fig. 1, we have

g11 ¼ a; g22 ¼ c; g33 ¼ d; (17)

g12 ¼ 1

2ðaþ c� bÞ; g13 ¼ 1

2ðaþ d� eÞ;

g23 ¼ 1

2ðcþ d� fÞ;

(18)

and its volume V is given by

V2 ¼ 1

144½afð�a� fþ bþ cþ dþ eÞ

þ bdð�b� dþ aþ cþ eþ fÞþ ceð�c� eþ aþ bþ dþ fÞ� abc� ade� bef� cdf�: (19)

1In Regge gravity, spacetime diffeomorphisms correspond tomovements of the vertices which leave the local geometryunchanged (see, for example, Refs. [13,19,23] and further refer-ences therein). In the present case, the lattice Hamiltonianconstraints can be naturally viewed as generating local deforma-tions of the spatial lattice hypersurface. One would thereforeexpect the Hamiltonian constraint to be based here on the latticevertices as well. But this seems nearly impossible to implement,as the definition of the local lattice supermetric Gijðl2Þ based onEq. (12) clearly requires the consideration of a full tetrahedron,as do the derivatives with respect to the edges, and finally thevery definition of the curvature and volume terms. One couldpossibly still insist on defining the Hamiltonian constraint on avertex by averaging over contributions from many neighboringtetrahedra, but this would make the lattice problem intractablefrom a practical point of view. How this choice will ultimatelyaffect the counting of degrees of freedom is unclear at this stage,for two reasons. The first one is that in the Regge theory there isin general a certain redundancy of degrees of freedom [13], withunwanted ones either decoupling or acquiring a mass of the orderof the ultraviolet cutoff. Furthermore, as will be shown later forexample in Eq. (44), the detailed relationship between thenumber of lattice vertices and tetrahedra clearly depends onthe chosen lattice structure, and more specifically on the locallattice coordination number.


084012-4

The matrix Gij is then given by

Gij ¼ � 1

24V

�2f eþ f� b bþ f� e dþ f� c cþ f� d p

eþ f� b �2e bþ e� f dþ e� a q aþ e� d

bþ f� e bþ e� f �2b r bþ c� a aþ b� c

dþ f� c dþ e� a r �2d cþ d� f aþ d� e

cþ f� d q bþ c� a cþ d� f �2c aþ c� b

p aþ e� d aþ b� c aþ d� e aþ c� b �2a

0BBBBBBBBBBB@

1CCCCCCCCCCCA; (20)

where the three quantities p, q, and r are defined as

p ¼ �2a� 2fþ bþ cþ dþ e;

q ¼ �2c� 2eþ aþ bþ dþ f;

r ¼ �2b� 2dþ aþ cþ eþ f:

(21)

To obtain Gij, one can then either invert the above expres-sion or evaluate

Gij;kl ¼ 1

2ffiffiffig

p ðgikgjl þ gilgjk � gijgklÞ; (22)

and later replace derivatives with respect to the metric byderivatives with respect to the squared edge lengths, as in@

@g11¼ @

@a þ @@b þ @

@e , etc. One finds [1] that the matrix rep-

resenting the coefficients of the derivatives with respect to

the squared edge lengths is the same as the inverse ofGij, aresult that provides a nontrivial confirmation of the correct-ness of the Lund–Regge result of Eq. (13). Then in 3þ 1dimensions, the discrete Wheeler–DeWitt equation is��ð16�GÞ2Gij

@2

@si@sj�2n�h

Xh

ffiffiffiffiffish

p�hþ2�V

��½s�¼0;

(23)

where the sum is over hinges (edges) h in the tetrahedronand V is the volume of the given tetrahedron. Note that theabove represents one equation for every tetrahedron on thelattice. Thus, if the lattice containsN3 tetrahedra, there willbeN3 coupled equations that will need to be solved in orderto determine �½s�. Note also the mild nonlocality of theequation in that the curvature term, through the deficitangles, involves edge lengths from neighboring tetrahedra.Of course, in the continuum, the derivatives also give somevery mild nonlocality. Figure 2 gives a pictorial represen-tation of lattices that can be used for numerical studies ofquantum gravity in 3þ 1 dimensions.

In the following, we will be concerned at some pointwith various discrete, but generally regular, triangulationsof the three-sphere [24,25]. These were already studied insome detail within the framework of the Regge theory inRef. [17], where in particular the 5-cell �4, the 16-cell �4,and the 600-cell regular polytopes (as well as a few others)were considered in some detail. For a very early

application of these regular triangulations to general rela-tivity, see Ref. [26].We shall not dwell here on a well-known key aspect of

the Regge–Wheeler theory, which is the presence of acontinuous, local lattice diffeomorphism invariance, ofwhich the main aspects in regards to its relevance for the3þ 1 formulation of gravity were already addressed insome detail in various works, both in the framework ofthe lattice weak field expansion [1,13] and beyond it[19,23]. Here we will limit ourselves to some brief remarkson how this local invariance manifests itself in the 3þ 1formulation and, in particular, in the case of the discretetriangulations of the sphere studied later on in this paper. Ingeneral, lattice diffeomorphisms in the Regge–Wheelertheory can be viewed as corresponding to local deforma-tions of the edge lengths about a vertex, which leave thelocal geometry physically unchanged, the latter being de-scribed by the values of local lattice operators correspond-ing to local volumes and curvatures [13,19,23]. The case offlat space (curvature locally equal to zero) or near-flatspace (curvature locally small) is obviously the simplestto analyze [23]: by moving the location of the verticesaround on a smooth manifold, one can find different as-signments of edge lengths representing locally the sameflat, or near-flat, geometry. It is then easy to show that oneobtains a d � N0-parameter family of local transformations

FIG. 2. A small section of a suitable spatial lattice for quantumgravity in 3þ 1 dimensions.


084012-5

for the edge lengths, as expected for lattice diffeomor-phisms. For the present case, the relevant latticediffeomorphisms are the ones that apply to the three-dimensional, spatial theory. The reader is referred toRef. [27] and, more recently, Ref. [1] for their explicitform within the framework of the lattice weak fieldexpansion.

With these observations in mind, we can now turn to adiscussion of the solution method for the lattice Wheeler–DeWitt equation in 3þ 1 dimensions. One item that needsto be brought up at this point is the proper normalization ofvarious terms (kinetic, cosmological, and curvature)appearing in the lattice equation of Eqs. (15) and (23).For the lattice gravity action in d dimensions, one hasgenerally the following well-understood correspondence:

Zddx

ffiffiffig

p $ X�

V�; (24)

where V� is the volume of a simplex; in three dimensions,it is simply the volume of a tetrahedron. The curvature terminvolves deficit angles in the discrete case,

1

2

Zddx

ffiffiffig

pR $ X

h

Vh�h; (25)

where �h is the deficit angle at the hinge h, and Vh is theassociated ‘‘volume of a hinge’’ [28]. In four dimensions,the latter is the area of a triangle (usually denoted by Ah),whereas in three dimensions, it is simply given by thelength lh of the edge labeled by h. In this work, we willfocus almost exclusively on the case of 3þ 1 dimensions;consequently the relevant formulas will be Eqs. (24) and(25) for dimension d ¼ 3.

The continuum Wheeler–DeWitt equation is local, ascan be seen from Eq. (10). One can integrate the Wheeler–DeWitt operator over all space and obtain��ð16�GÞ2

Zd3x�ðgÞþ2�

Zd3x

ffiffiffig

p �Zd3x

ffiffiffig

pR

��¼0;

(26)

with the super-Laplacian on metrics defined as

�ðgÞ � Gij;klðxÞ �2

�gijðxÞ�gklðxÞ : (27)

We have seen before that in the discrete case one has onelocal Wheeler–DeWitt equation for each tetrahedron [seeEqs. (14) and (15)], which can be written as�

�ð16�GÞ2�ðl2Þ � Xh��

�hlh þ 2�V�

�� ¼ 0; (28)

where now �ðl2Þ is the lattice version of the super-Laplacian, and we have set for convenience ¼ 2n�h.As we shall see below, for a regular lattice of fixed coor-dination number, is a constant and does not depend on

the location on the lattice. In the above expression,�ðl2Þ isa discretized form of the covariant super-Laplacian, actinglocally on the space of s ¼ l2 variables,

�ðl2Þ � Gij

@2

@si@sj; (29)

with the matrix Gij given explicitly in Eq. (20). Note thatthe curvature term involves six deficit angles �h, associatedwith the six edges of a tetrahedron.Now, the local lattice Wheeler–DeWitt equation of

Eq. (23) applies to a single given tetrahedron (labeledhere by �), with one equation to be satisfied at eachtetrahedron on the lattice. At this point, some simple addi-tional checks can be performed. For example, one can alsoconstruct the total Hamiltonian by simply summing overall tetrahedra, which leads to��ð16�GÞ2X

�

�ðl2Þ þ 2�X�

V� � X�

Xh��

lh�h

�� ¼ 0:

(30)

The above expression represents therefore an integral overHamiltonian constraints with unit density weights. Notethat indeed the second term involves the total lattice vol-ume (the lattice analog of

Rd3x

ffiffiffig

p), and the third one

contains, as expected, the total lattice curvature (the latticeanalog of

Rd3x

ffiffiffig

pR) [28].

Summing over all tetrahedra (�) is different fromsumming over all hinges (h), and the above equation isequivalent to��ð16�GÞ2X

�

�ðl2Þþ2�X�

V��qXh

lh�h

��¼0; (31)

where q here is the lattice coordination number. The latteris determined by how the lattice is put together (whichvertices are neighbors to each other, or, equivalently, by theso-called incidence matrix). Here q is therefore the numberof neighboring simplices that share a given hinge (edge).For a flat triangular lattice in 2d q ¼ 6, whereas for theregular triangulations of S3 we will be considering below,one has q ¼ 3, 4, 5. For more general, irregular triangu-lations, q might change locally throughout the lattice. Inthis case, it is more meaningful to talk about an averagelattice coordination number hqi [16]. For proper normal-ization in Eq. (30), one requires the three-dimensionalversion of Eqs. (24) and (25), which fixes the overallnormalization of the curvature term

� 2n�h ¼ 2

q; (32)

thus determining the relative weight of the local volumeand curvature terms.2 At this point, it seems worth

2For more general, irregular triangulations, q might changelocally throughout the lattice. Then it will be more meaningful totalk about an average lattice coordination number hqi [16].


084012-6

emphasizing that from now on wewill focus exclusively onthe set of coupled local lattice Wheeler–DeWitt equations,given explicitly in Eq. (23) or (28), with one equation foreach lattice tetrahedron.

IV. CHOICE OF COUPLING CONSTANTS

We will find it convenient, in analogy to what is com-monly done in the Euclidean lattice theory of gravity, tofactor out an overall irrelevant length scale from the prob-lem and set the (unscaled) cosmological constant equal to 1[17]. Indeed, recall that the Euclidean path integral statis-tical weight always contains a factor PðVÞ / exp ð��0VÞ,where V ¼ R ffiffiffi

gp

is the total volume on the lattice, and �0

is the unscaled cosmological constant. A simple globalrescaling of the metric (or edge lengths) then allows oneto entirely reabsorb this �0 into the local volume term. Thechoice �0 ¼ 1 then trivially fixes this overall scale onceand for all. Since �0 ¼ 2�=16�G, one then has � ¼ 8�Gin this system of units. In the following, we will also find it

convenient to introduce a scaled coupling ~� defined as

~� � �

2

�1

16�G

�2: (33)

Then for �0 ¼ 1 (in units of the UV cutoff or, equivalently,in units of the fundamental lattice spacing), one has~� ¼ 1=64�G.Two further notational simplifications will be useful in

the following. The first one is introduced in order to avoidlots of factors of 16� in many of the formulas. So from nowon, we shall write G as a shorthand for 16�G,

16�G ! G: (34)

In this new notation, one has � ¼ G=2 and ~� ¼ 1=4G. Theabove notational choices then lead to a more streamlinedrepresentation of the Wheeler–DeWitt equation, namely,�

��þ 1

G

ffiffiffig

p � 1

G2

ffiffiffig

p 3R

�� ¼ 0: (35)

Note that we have arranged things so that now the kineticterm (the term involving the Laplacian) has a unit coeffi-cient. Then in the extreme strong coupling limit (G ! 1),the kinetic term is the dominant one, followed by thevolume (cosmological constant) term (using the facts

about ~� given above) and, finally, by the curvature term.Consequently, at least in a first approximation, the curva-ture R term can be neglected compared to the other twoterms, in this limit.

A second notational choice will later be dictated by thestructure of the wave function solutions, which often

involve numerous factors offfiffiffiffiG

p. It will therefore be useful

to define a new coupling g as

g � ffiffiffiffiG

p(36)

so that ~� ¼ 4=g2 (the latter g should not be confused withthe square root of the determinant of the metric).

V. OUTLINE OF THE GENERALMETHOD OF SOLUTION

The previous discussion shows that in the strong cou-pling limit (large G) one can, at least in a first approxima-tion, neglect the curvature term, which will then beincluded at a later stage. This simplifies the problem con-siderably, as it is the curvature term that introducescomplicated interactions between neighboring simplices.Here the general procedure for finding a solution will be

rather similar to what was done in 2þ 1 dimensions, as theformal issues in obtaining a solution are not dramaticallydifferent. First, an exact solution is found for equilateraledge lengths s. Later, this solution is extended to determinewhether it is consistent to higher order in the weak fieldexpansion, where one writes for the squared edge lengthsthe expansion

l2ij ¼ sð1þ hijÞ; (37)

with a small expansion parameter. The resulting solutionfor the wave function can then be obtained as a suitablepower series in the h variables, combined with the standardFrobenius method, appropriate for the study of quantummechanical wave equations for suitably well-behavedpotentials. In this method, one first determines the correctasymptotic behavior of the solution for small and largearguments and later constructs a full solution by writing theremainder as a power series or polynomial in the relevantvariable. While this last method is rather time consuming,we have found nevertheless that in some cases (such as thesingle triangle in 2þ 1 dimensions and the single tetrahe-dron in 3þ 1 dimensions, described in Ref. [1] and alsobelow), one is lucky enough to find immediately an exactsolution, without having to rely in any way on the weakfield expansion.More importantly, in Ref. [2], it was found that already

in 2þ 1 dimensions this rather laborious weak fieldexpansion of the solution is not really necessary, for thefollowing reason. Diffeomorphism invariance (on the lat-tice and in the continuum) of the theory severely restrictsthe form of the Wheeler–DeWitt wave function to a func-tion of invariants only, such as total three-volumes andcurvatures, or powers thereof. In other words, the wavefunction is found to be a function of invariants such asRddx

ffiffiffig

por

Rddx

ffiffiffig

pRn, etc. (these will be listed in more

detail below for the specific case of 3þ 1 dimensions,where one has d ¼ 3 in the above expressions).For concreteness and computational expedience, in the

following, we will look at a variety of three-dimensionalsimplicial lattices, including regular triangulations of thethree-sphere S3 constructed as convex four-polytopes, thelatter describing closed and connected figures composed oflower dimensional simplices. Here these will include the


084012-7

five-cell four-simplex or hypertetrahedron (Schlafli symbolf3; 3; 3g) with 5 vertices, 10 edges, and 5 tetrahedra; the16-cell hyperoctahedron (Schlafli symbol f3; 3; 4g) with 8vertices, 24 edges, and 16 tetrahedra; and the 600-cellhypericosahedron (Schlafli symbol f3; 3; 5g) with 120 ver-tices, 720 edges, and 600 tetrahedra [24,25]. Note that theEuler characteristic for all four-polytopes that are topo-logical three-spheres is zero, � ¼ N0 � N1 þ N2 � N3 ¼0, where Nd is the number of simplices of dimension d. Wealso note here that there are no known regular equilateraltriangulations of the flat three-torus in three dimensions,although very useful slightly irregular (but periodic) trian-gulations are easily constructed by subdividing cubes on asquare lattice into tetrahedra [27].

In the following, we will also recognize that there arenatural sets of variables for displaying the results. One ofthem is the scaled total volume x, defined as

x � 4ffiffiffiffiffiffi2�

pqG

X�

V� ¼ 4ffiffiffiffiffiffi2�

pqG

Vtot: (38)

Later on, we will be interested in making contact withcontinuum manifolds, by taking the infinite volume (orthermodynamic) limit, defined in the usual way as

N� ! 1; Vtot ! 1;Vtot

N�

! const; (39)

with N� � N3 here the total number of tetrahedra. Itshould be clear that this last ratio can be used to definea fundamental lattice spacing a0, for example, via

Vtot=N� � V� ¼ a30=6ffiffiffi2

p.

The full solution of the quantum mechanical problemwill, in general, require that the wave functions be properlynormalized, as in Eq. (9). This will introduce at some stagewave function normalization factors N , which will laterbe fixed by the standard rules of quantum mechanics. If thewave function were to depend on the total volume Vtot only(which is the case in 2þ 1 dimensions, but not in 3þ 1),then the relevant requirement would simply be

k�k2 �Z

d�½g� � j�½gij�j2

¼Z 1

0dVtot � Vm

tot � j�ðVtotÞj2 ¼ 1; (40)

where d�½g� is the appropriate functional measure over thethree-metric gij and m a positive real number representing

the correct entropy weighting. But, not unexpectedly, in3þ 1 dimensions, the total curvature also plays a role, sothe above can only be regarded as roughly correct in thestrong coupling limit (large G), where the curvature con-tribution to the Wheeler–DeWitt equation can safely beneglected. As in nonrelativistic quantum mechanics, thenormalization condition in Eqs. (9) and (40) plays a crucialrole in selecting out of the two solutions the one that is

regular and therefore satisfies the required wave functionnormalizability condition.To proceed further, it will be necessary to discuss each

lattice separately in some detail. For each lattice geometry,we will break down the presentation into two separatediscussions. The first part will deal with the case of noexplicit curvature term in the Wheeler–DeWitt equation.Each regular triangulation of the three-sphere will be firstanalyzed separately and subjected to the required regular-ity conditions. Here a solution is first obtained in theequilateral case and later promoted on the basis of latticediffeomorphism invariance to the case of arbitrary edgelengths, as was done in Ref. [2]. Later, a single generalsolution will be written down, involving the parameter q,which covers all previous triangulation cases, and therebyallows a first study of the infinite volume limit. The secondpart deals with the extension of the previous solutions tothe case when the curvature term in the Wheeler–DeWittequation is included. This case is more challenging to treatanalytically, and the only results we have obtained so fardeal with the large volume limit, for which the solution isnevertheless expected to be exact (as was the case in 2þ 1dimensions [2]).

A. Nature of solutions in 3 þ 1 dimensions

In this work, we will be concerned with the solution ofthe Wheeler–DeWitt equation for discrete triangulations ofthe three-sphere S3. In general, for an arbitrary triangula-tion of a smooth closed manifold in three dimensions, onecan write down the Euler equation

N0 � N1 þ N2 � N3 ¼ 0 (41)

and the Dehn–Sommerville relation

N2 ¼ 2N3: (42)

The latter follows from the fact that each triangle is sharedby two tetrahedra and each tetrahedron has four triangles,thus 2N2 ¼ 4N3. In addition, for the regular triangulationsof the three-sphere we will be considering here, one has theadditional identity

N1 ¼ 6

qN3; (43)

where q is the local coordination number, defined as thenumber of tetrahedra meeting at an edge. For the threeregular triangulations of the three-sphere, we will look atone that has q ¼ 3, 4, 5. The above relations then allow usto relate the number of sites (N0) to the number of tetrahe-dra (N3),

N0 ¼ N3

�6

q� 1

�: (44)

It will also turn out to be convenient to collect here anumber of useful definitions, results, and identities thatapply to the regular triangulations of the three-sphere, valid


084012-8

strictly when all edge lengths take on the same identicalvalue l ¼ ffiffiffi

sp

. For the total volume

Vtot �X�

V� $Z

d3xffiffiffig

p; (45)

one has

Vtot ¼ N3V� ¼ s3=2

6ffiffiffi2

p N3; (46)

whereas the total curvature

Rtot � 2Xh

�hlh $Z

d3xffiffiffig

pR (47)

is given by

Rtot ¼ 12ffiffiffis

pq

�2�� qcos�1

�1

3

��N3: (48)

The latter relationship can be inverted to give the para-meter q as a function of the curvature

q ¼ q0

�1� Rtot

Rtot þ 24�ffiffis

pq0

N3

�(49)

and its inverse as

q�1 ¼ q�10 þ Rtot

24�ffiffiffis

pN3

; (50)

so that this last quantity is just linear in Rtot. A very specialvalue for q corresponds to the choice q ¼ q0 for whichRtot ¼ 0. For this case, one has

q0 � 2�

cos�1ð13Þ¼ 5:1043: (51)

We emphasize here again that the relationships just givenabove apply to the rather special case of an equilateraltriangulation.

Then, summarizing all the previous discussions, thediscretized Wheeler–DeWitt equation one wants to solvehere in the most general case is the one given in Eq. (23)or (28),��G2

Xi;j��

Gijð�Þ @2

@l2i @l2j

� Xh��

lh�h þ 2�V�

�c ½l2� ¼ 0;

(52)

with parameter given by

¼ 2

q: (53)

Note that Eq. (52) still represents one equation per latticetetrahedron. Thus, if the lattice is made up of N3 tetrahe-dra, the problem will in general still require the solution ofN3 coupled equations of the type given in Eq. (52), involv-ing in the most general case N1 edge lengths. As will bediscussed further below, the proposed method of solution

will be quite similar to what was used earlier in 2þ 1dimensions [2], namely, a combination of the weak fieldexpansion and the Frobenius method, which in Ref. [2]gave the exact solution for the wave function for eachlattice in the limit of large areas. If the reader is notinterested here in the details of the solution for eachindividual lattice, then (s)he can skip the followingsections and move on directly to Sec. V F.

B. One-cell complex (single tetrahedron)

As a first case, we consider here the quantum-mechanical problem of a single tetrahedron. One hasN0 ¼ 4, N1 ¼ 6, N2 ¼ 4, N3 ¼ 1, and q ¼ 1 [note thatthese do not satisfy the Euler and Dehn–Sommervillerelations; only the relation between N1, N3, and q,Eq. (44), is satisfied for a single tetrahedron]. The singletetrahedron problem is relevant for the strong coupling(large G) limit. In this limit, one can neglect the curvatureterm, which couples different tetrahedra to each other, andone is left with the local degrees of freedom, involving asingle tetrahedron.The Wheeler–DeWitt equation for a single tetrahedron

with a constant curvature density term R reads��ð16�GÞ2Gij

@2

@si@sjþ ð2�� RÞV

��½s� ¼ 0; (54)

where now the squared edge lengths s1 . . . s6 are all part ofthe same tetrahedron, and Gij is given by a rather compli-

cated, but explicit, 6� 6 matrix given earlier.As in the 2þ 1 case previously discussed in Ref. [2],

here, too, it is found that, when acting on functions of thetetrahedron volume, the Laplacian term still returns someother function of the volume only, which makes it possibleto readily obtain a full solution for the wave function. Interms of the volume of the tetrahedron V�, one has theequivalent equation for �½s� ¼ �ðV�Þ (note that we havenow replaced for notational convenience 16�G ! G),

c 00ðV�Þ þ 7

V�

c 0ðV�Þ þ 32�

G2c ðV�Þ ¼ 0; (55)

with primes indicating derivatives with respect to V�. Fromnow on, wewill set the constant curvature densityR ¼ 0. Ifone introduces the dimensionless (scaled volume) variable


pG

Vtot; (56)

where Vtot � V� is the volume of the tetrahedron, thenthe differential equation for a single tetrahedron becomessimply

c 00ðxÞ þ 7

xc 0ðxÞ þ c ðxÞ ¼ 0: (57)

Solutions to Eq. (55) or (57) are Bessel functions Jm or Ym

with m ¼ 3,


084012-9

c RðVtotÞ ¼ const: J3

�4

ffiffiffiffiffiffi2�

pG

Vtot

�=V3

tot; (58)

or

c SðVtotÞ ¼ const: Y3

�4


pG

Vtot

�=V3

tot: (59)

Only JmðxÞ is regular as x ! 0, JmðxÞ � �ðmþ 1Þ�1 �ðx=2Þm. In terms of the variable x, the regular solution istherefore

c ðVtotÞ / J3ðxÞx3

/J3�4ffiffiffiffi2�

pG Vtot

V3tot

; (60)

and the only physically acceptable wave function is

�ða; b; . . . fÞ ¼ �ðVtotÞ ¼ NJ3�4ffiffiffiffi2�

pG Vtot

V3tot

; (61)

with normalization constant

N ¼ 45ffiffiffiffiffiffiffiffiffi77�

p

1024 23=4

�Gffiffiffiffi�

p�5=2

: (62)

The latter is obtained from the wave function normali-zation requirementZ 1

0dVtotj�ðVtotÞj2 ¼ 1: (63)

Note that the solution given in Eq. (60) is exact and afunction of the volume of the tetrahedron only; its onlydependence on the values of the edge lengths of the tetra-hedron [or, equivalently, on the metric; see Eq. (12)] isthrough the total volume. It is worth stressing here that inorder to find the exact solution for the wave function itwould have been enough to in fact just consider the equi-lateral case. The complete solution would then be read offimmediately from this special case, if one were to assume(as one should) that the exact wave functional is expectedto be a function of invariants only, and therefore gaugeindependent.

One can compute the average volume of the singletetrahedron, which is given by

hVtoti �Z 1

0dVtot � Vtot � j�ðVtotÞj2 ¼ 31185�G

262144ffiffiffiffiffiffi2�

p

¼ 0:2643Gffiffiffiffi�

p : (64)

This last result allows us to define an average latticespacing by comparing it to the value for an equilateral

tetrahedron for which Vtot ¼ ð1=6 ffiffiffi2

p Þa30. One obtains

a0 ¼ 1:3089

�Gffiffiffiffi�

p�1=3

: (65)

In terms of the parameter ~� defined in Eq. (33) one hasffiffiffiffi�

p=G ¼

ffiffiffiffiffiffi2~�

p. With the notation of Eq. (36) one has as

well G=ffiffiffiffi�

p ¼ ffiffiffiffiffiffiffi2G

p ¼ ffiffiffi2

pg. Then for a single tetrahedron

one has hVtoti � hV�i ¼ 0:3738g.The single tetrahedron problem is clearly quite relevant

for the limit of strong gravitational coupling, 1=G ! 0. Inthis limit lattice quantum gravity has a finite correlationlength, comparable to one lattice spacing,

�� a0: (66)

This last result is seen here simply as a reflection of the factthat for large G the edge lengths, and therefore the metric,fluctuate more or less independently in different spatialregions, due to the absence of the curvature term in theWheeler–DeWitt equation. This is of course true also inthe Euclidean lattice theory, in the same limit [17]. It is theinclusion of the curvature term that later leads to a couplingbetween fluctuations in different spatial regions, an essen-tial ingredient of the full theory.

C. Five-cell complex (configuration of five tetrahedra)

The first regular triangulation of S3 we will consider isthe five-cell complex, sometimes referred to as the hyper-tetrahedron. Here one has N0 ¼ 5, N1 ¼ 10, N2 ¼ 10,N3 ¼ 5, and q ¼ 3, since there are three tetrahedra meet-ing on each edge. Then for the parameter appearing inEq. (52), one has

¼ 2

3: (67)

First, we will consider the case of no curvature term in thelattice Wheeler–DeWitt equation of Eq. (52). The curva-ture term will be reintroduced at a later stage (see Sec. VII),as its presence considerably complicates the solution of thelattice equations.Solving the lattice equations directly (by brute force, one

might say) in terms of the edge length variables is a ratherdifficult task, since many edge lengths are involved, in-creasingly more so for finer triangulations. Nevertheless,it can be done, to some extent, in 2þ 1 dimensions [2], andpossibly even in 3þ 1 dimensions, analytically for somespecial cases or numerically for more general cases. Toobtain a full solution to the lattice equations, we rely hereinstead on a simpler procedure, already employed success-fully (and checked explicitly) in 2þ 1 dimensions.First, an exact wave function solution to the lattice

Wheeler–DeWitt equations is obtained for the equilateralcase, where all edges in the simplicial complex areassumed to have the same length. This is achieved (as inRef. [2]) by utilizing a combination of the weak fieldexpansion of Eq. (37) and the Frobenius (or power seriesexpansion method) in order to obtain a solution to Eq. (52).In order to obtain such a solution, one first looks at the limitof large and small volumes, from which the asymptoticbehavior of the solution is determined. Note that one hasone Wheeler–DeWitt equation per lattice tetrahedron,which implies that one is seeking a solution to N3 coupled


084012-10

equations, involving a single wave function for which thearguments are the N1 edge lengths. Nevertheless, since oneis dealing here with a regular triangulation of the sphere,all equations will have exactly the same form due to thesymmetry of the problem. It will therefore be adequate,because of this symmetry, to focus on a given singletetrahedron and on how the associated local latticeWheeler–DeWitt operator acts on the total wave function.As stated previously, the latter will in general involve alllattice edge lengths. But a further simplification arisesbecause of the locality of the lattice Wheeler–DeWittequation, which restricts interactions to edge lengths thatare not too far apart. As a consequence, when determiningthe structure of the wave function solution, it will beadequate to only consider terms (local volume contribu-tions, for example) that involve edges which are directlyaffected by the derivative terms in the local Wheeler–DeWitt operator of Eqs. (28) and (52). Nevertheless, theproblem is, in spite of the above-mentioned simplifica-tions, still of considerable algebraic complexity in viewof the many edges that still are affected by the action of thelocal Wheeler–DeWitt operator. These generally includeall the edges within the given tetrahedron, as well as arather considerable number of edges located in the neigh-boring tetrahedra. For a given candidate solution (writtenin terms of invariants, such as the total volume and thecurvature), the task is then to determine if such a solutionindeed satisfies the local Wheeler–DeWitt equation, mean-ing that the rhs of Eq. (52) can be made to vanish, forexample, by a suitable choice of wave function parameters.Again this can be a challenging task (due to the largenumber of variables involved) unless further simplifica-tions are invoked in order to reduce the complexity of theproblem. An additional step at this stage is therefore toconstrain the solution by expanding the rhs of the locallattice Wheeler–DeWitt equation [Eq. (52)] according tothe weak field expansion of Eq. (37).

Then, in the next step, the diffeomorphism invariance ofthe simplicial lattice theory is used to promote the previ-ously obtained expression for the wave function to itspresumably unique general coordinate invariant form, in-volving various geometric volume and curvature terms. Itis a nontrivial consequence of the invariance properties ofthe theory that such an invariant expression can beobtained, without any further ambiguity, at least in somesuitable limits to be discussed further below (essentially,the large volume and small curvature limit). Note that as aresult of this procedure the wave function is ultimately notnecessarily assumed to depend on a single, global mode;instead, it is still regarded as a function of all lattice metricdegrees of freedom, as will be discussed, and used, furtherbelow [see, for example, the expressions given later inEqs. (115) and (117)]. In a number of instances, such aprocedure can be checked explicitly and systematicallywithin the framework of the weak field expansion and

used to show that the form of the relevant wave functionsolution is indeed, as expected, strongly constrained bydiffeomorphism invariance [2]. In this respect, the proce-dure we will follow here is quite different from the oneused for minisuperspace models, where the infinitely manymetric degrees of freedom of the continuum are condensed,from the very beginning and therefore already in the origi-nal Wheeler–DeWitt equation, to one or two single modes,such as the scale factor and the vacuum expectation of ascalar field.3

In the case of the five-cell complex, and for now withoutan explicit curvature term in the Wheeler–DeWittequation, one obtains the differential equation

c 00ðVtotÞ þ 95

9Vtot

c 0ðVtotÞ þ 32�

9G2c ðVtotÞ ¼ 0 (68)

for a wave function that, for now, depends only on the totalvolume, c ¼ c ðVtotÞ. To obtain this result, it is assumed atfirst that the simplicial complex is built out of equilateraltetrahedra; in accordance with the previous discussion,this constraint will be removed below. In terms of thedimensionless variable x defined as


p3G

Vtot; (69)

one has the equivalent form for Eq. (68):

c 00ðxÞ þ 95

9xc 0ðxÞ þ c ðxÞ ¼ 0: (70)

This last equation can then be solved immediately, and thesolution is

c ðVtotÞ /J43

9ðxÞx439

/J43

9

�4ffiffiffiffi2�

p3G Vtot

Vtot

439

; (71)

up to an overall wave function normalization constant. Asin the previously discussed tetrahedron case, and also as in2þ 1 dimensions, one discards the Bessel function of thesecond kind (Y) solution, since it is singular at the origin.

D. 16-cell complex (configuration of 16 tetrahedra)

The next regular triangulation of S3 we will consideris the 16-cell complex, sometimes referred to as thehyperoctahedron. One has in this case N0 ¼ 8, N1 ¼ 24,

3We should recall that in 2þ 1 dimensions, an exact wavefunctional was obtained for the three regular triangulations of thesphere (the tetrahedron, octahedron, and icosahedrons), for ar-bitrary edge length assignments, in addition to the other twocases of a single triangle and of a regularly triangulated two-torus. In all the above instances, it was found that the exact wavefunction solution could be described by a single function of thetotal area, of the Bessel type for strong coupling and of theconfluent hypergeometric type in the more general case [2]. As isthe case here in 3þ 1 dimensions, the Bessel function index nthere was found to be linearly related to the total number oflattice triangles N2.


084012-11

N2 ¼ 32, N3 ¼ 16, and q ¼ 4, since there are four tetra-hedra meeting on each edge. For the parameter inEq. (52), one has

¼ 2

4: (72)

In the case of the 16-cell complex (again for now withoutan explicit curvature term in the Wheeler–DeWittequation), one obtains the differential equation

c 00ðVtotÞ þ 47

2Vtot

c 0ðVtotÞ þ 2�

G2c ðVtotÞ ¼ 0 (73)

for a wave function that depends only on the total volume,c ¼ c ðVtotÞ. In terms of the variable

x �ffiffiffiffiffiffi2�

pG

Vtot; (74)

one has an equivalent form for Eq. (73):

c 00ðxÞ þ 47

2xc 0ðxÞ þ c ðxÞ ¼ 0: (75)

The correct wave function solution is now

c ðVtotÞ /J45

4ðxÞx454

/J45

4

� ffiffiffiffi2�

pG Vtot

Vtot

454

; (76)

up to an overall wave function normalization constant.Again, we discarded the Bessel function of the secondkind (Y) solution, since it is singular at the origin.

E. 600-cell complex (configurationof 600 tetrahedra)

The last, and densest, regular triangulation of S3 we willconsider here is the 600-cell complex, often called thehypericosahedron. For this lattice, one has N0 ¼ 120,N1 ¼ 720, N2 ¼ 1200, N3 ¼ 600, and q ¼ 5, since thereare now five tetrahedra meeting at each edge. For theparameter in Eq. (52), one has

¼ 2

5: (77)

For this 600-cell complex (again for now without anexplicit curvature term in the Wheeler–DeWitt equation),one obtains the differential equation

c 00ðVtotÞ þ 672

Vtot

c 0ðVtotÞ þ 32�

25G2c ðVtotÞ ¼ 0 (78)

for a wave function that depends only on the total volume,c ¼ c ðVtotÞ. In terms of the variable


p5G

Vtot; (79)

one has an equivalent form for Eq. (78):

c 00ðxÞ þ 672


Then the solution of the Wheeler–DeWitt equation withouta curvature term is

c ðVtotÞ /J671

2ðxÞ

x6712

/J671

2


p5G Vtot

Vtot

6712

; (81)

again up to an overall wave function normalization con-stant. As in previous cases, we discard the Bessel functionof the second kind (Y) solution, since it is singular at theorigin.

F. Summary and general case for zero curvature

In this section, we summarize and extend the previousresults for the wave functions, obtained so far for the threeseparate cases of the 5-cell, 16-cell, and 600-cell triangu-lation of the three-sphere S3. The single tetrahedron case issomewhat special (it cannot contain a curvature term) andwill be left aside for the moment. Also, all the previousresults so far apply to the case of no explicit curvature termin the Wheeler–DeWitt equation of Eq. (52); the inclusionof the curvature term will be discussed later. Consequently,the following discussion still focuses on the strongcoupling limit, G ! 1.For the following discussion, the relevant Wheeler–

DeWitt equation is the one in Eq. (52),��G2

Xi;j��

Gijð�Þ @2

@l2i @l2j

� Xh��

lh�h þ 2�V�

�c ½l2� ¼ 0;

(82)

which depends on the parameter

¼ 2

q; (83)

where q represents the number of tetrahedra meeting at anedge. The above equation is quite general and not approxi-mate in any way. Nevertheless, it depends on the locallattice coordination number q (how the edges are con-nected to each other, or, in other words, on the incidencematrix).Now, all previous differential equations for the wave

function as a function of the total volume Vtot [Eqs. (68),(73), and (78)] can be summarized as a single equation:

c 00ðVtotÞ þ ð11þ 9qÞ2q2

N3

Vtot

c 0ðVtotÞ þ 32

q2�

G2c ðVtotÞ ¼ 0:

(84)

Equivalently, in terms of the scaled volume variabledefined as


084012-12


pqG

Vtot; (85)

one can summarize the results of Eqs. (70), (75), and (80)through the single equation

c 00ðxÞ þ ð11þ 9qÞ2q2

N3


It will be convenient here to define the (Bessel function)index n as

n � 11þ 9q

4q2N3 � 1

2; (87)

so that for the 5-cell, 16-cell, and 600-cell, one has

2nþ 1 ¼ 95

9ðq ¼ 3; N3 ¼ 5Þ;

¼ 47

2ðq ¼ 4; N3 ¼ 16Þ;

¼ 672ðq ¼ 5; N3 ¼ 600Þ; (88)

respectively, and in the general case,

2nþ 1 ¼ ð11þ 9qÞ2q2

N3; (89)

thus reproducing n ¼ 43=9, 45=4 and 671=2, respectively,in the three cases. Then Eq. (86) is just

c 00ðxÞ þ 2nþ 1


Consequently, the wave function solutions are

c / JnðxÞxn

/Jn�4ffiffiffiffi2�

pqG Vtot


pqG Vtot

n; (91)

up to an overall wave function normalization constant, thussummarizing all the results so far for the individual regulartriangulations [Eqs. (71), (76), and (81)]. A more explicit,but less transparent, form for the wave function solution is

c ðVtotÞ ¼ N � V12�

N3ð11þ9qÞ4q2

tot � J�12þ

N3ð11þ9qÞ4qq2

�4


pqG

Vtot

�; (92)

with N an overall wave function normalization constant.Its large volume behavior is completely determined by theasymptotic expansion of the Bessel J function,

c ðxÞ’JnðxÞxn

�x!1 x�n

ffiffiffiffiffiffiffi2

�x

ssin

�xþ�

4�n�

2

�þO

�1

xnþ32

�:

(93)

It is also easy to see that the argument of the Besselfunction solution J in Eqs. (91) and (92) has the followingexpansion for large volumes:

x ¼ 4ffiffiffiffiffiffi2�

pq0G

Vtot þ a2036

ffiffiffi2

p�


pG

Rtot; (94)

with a0 (a30 � 6ffiffiffi2

pV=N3) representing here the average

lattice spacing. Thus, the second correction is of order

ðV=N3Þ2=3Rtot. Note that nothing particularly interestingis happening in the structure of the wave function so far.Similarly, the index n of the Bessel function solution inEqs. (91) and (92) has the following expansion for largevolumes and small curvatures:

n ¼ ð11þ 9q0Þ4q20

N3 � 1

2þ ð22þ 9q0Þ

96�q0a0Rtot þOðR2Þ; (95)

with a0 again defined as above. Note here that the second

correction is of order ðN3=VÞ1=3Rtot. It follows that theasymptotic behavior for the exponent of the fundamentalwave function solutions for large volume and smallcurvature is given by

i

�4


pq0G

Vtot þ a2036

ffiffiffi2

p�


pG

Rtot þOðR2Þ�

��11þ 9q0

4q20N3 þ 22þ 9q0

96�q0a0Rtot þOðR2Þ

�lnVtot:

(96)

Let us make here some additional comments. One mightwonder what concrete lattices correspond to values of ngreater that 671=2, which is after all the highest valueattained for a regular triangulation of the three-sphere,namely, the 600-cell complex. For each of the three regulartriangulations of S3 with N0 sites, one has for the number

of edges N1 ¼ 66�q N0, for the number of triangles N2 ¼

2q6�q N0, and for the number of tetrahedra N3 ¼ q

6�q N0,

where q is the number of tetrahedra meeting at an edge(the local coordination number). In the three cases exam-ined previously, q was of course an integer between threeand five; in two dimensions, it is possible to have one moreinteger value of q corresponding to the regularly triangu-lated torus, but this is not possible here. In any case, onealways has for a given triangulation of the three-sphere theEuler relationN0 � N1 þ N2 � N3 ¼ 0. The interpretationof other, even noninteger, values of q is then clear.Additional triangulations of the three-sphere can be con-structed by considering irregular triangulations, where theparameter q is now seen as an average coordination num-ber. Of course the simplest example is what could bedescribed as a semiregular lattice, with Na edges withcoordination number qa and Nb edges with coordinationnumber qb, such that Na þ Nb ¼ N1. Various irregular andrandom lattices were considered in detail some time ago inRef. [16], and we refer the reader to this work for a clearexposition of the properties of these kind of lattices. In thefollowing, we will assume that such constructions are


084012-13

generally possible, so that even noninteger values of q aremeaningful and are worth considering.

VI. AVERAGE VOLUME ANDAVERAGE LATTICE SPACING

At this stage, it will be useful to examine the question ofwhat values are allowed for the average volume. The latterwill be needed later on to give meaning to the notion of anaverage lattice spacing. In general, the average volume isdefined as

hVtoti � h�jVtotj�ih�j�i ¼

Rd�½g� � Vtot½gij� � j�½gij�j2R

d�½g� � j�½gij�j2;

(97)

where d�½g� is the appropriate (DeWitt) functional mea-sure over three-metrics gij.

Now consider the wave function obtained given inEq. (91), with n defined in Eq. (87). This wave functionis relevant for the strong coupling limit, where the explicitcurvature term in the Wheeler–DeWitt equation can beneglected. In this limit, one can then compute the averagetotal volume

hVtoti ¼R10 dVtot � Vtot � jc ðVtotÞj2R1

0 dVtot � jc ðVtotÞj2: (98)

One then obtains immediately for the average volume of atetrahedron

hV�i ¼2�3

2�2n�ðn� 12Þ�ð2nþ 1

2Þ�ðnÞ3N3

� qGffiffiffiffi�

p : (99)

If the whole lattice is just a single tetrahedron, then one hasn ¼ 3 and consequently

hV�i ¼ 31185�G

262144ffiffiffi2

p ffiffiffiffi�

p ¼ 0:2643Gffiffiffiffi�

p ; (100)

from which one can define an average lattice spacing a0 via

hV�i ¼ a30=6ffiffiffi2

p. For large N3, one has

a30 ¼3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi11þ 9q

p2

ffiffiffiffiffiffiffi2�

pN3

Gffiffiffiffi�

p : (101)

But, in general, one cannot assume a trivial entropy factorfrom the functional measure, and one should evaluateinstead

hVtoti ¼R10 dVtot � Vm

tot � Vtot � jc ðVtotÞj2R10 dVtot � Vm

tot � jc ðVtotÞj2; (102)

with some powerm ¼ c0N3 and c0 a real positive constant.One then obtains for the average volume of a singletetrahedron

hV�i ¼ 1

N3

hVtoti ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic0½11þ q0ð9� c0q0Þ�

q G


p ;

(103)

which is finite as N3 ! 1. Note that in order for theabove expression to make sense, one requires c0 < ð11þ9q0Þ=q20 ’ 2:185. If the exponent in the entropy factor is

too large, the integrals diverge. One then finds that thecorresponding lattice spacing is given by

a30 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic0½11þ q0ð9� c0q0Þ�

q 3G

4ffiffiffiffi�

p : (104)

The lesson learned from this exercise is that in gravity thelattice spacing a0 (the fundamental length scale, or theultraviolet cutoff if one wishes) is itself dynamical andthus set by the bare values of G and �. In a system of units

for which �0 ¼ 1, one then has a0 � g1=3. Either way, thechoice for a0 has no immediate direct physical meaningand has to be viewed instead in the context of a subsequentconsistent renormalization procedure. In the following, itwill be safe to assume, based on the results of Eqs. (65) and(104), that

a30 ¼ f3Gffiffiffiffi�

p ; (105)

in units of the UV cutoff, where f is a numerical constantof order 1 (for concreteness, in the single tetrahedron case,one has f 1:3089).

VII. LARGE VOLUME SOLUTION FORNONZERO CURVATURE

The next task in line is to determine the form of the wavefunction when the curvature term in the Wheeler–DeWittequation of Eq. (52) is not zero. In particular, we will beinterested in the changes to the wave function given inEqs. (91) and (92), with argument x in Eq. (94) andparameter n in Eq. (95). We define here the total integratedcurvature Rtot as in Eq. (47), which is of course differentfrom the local curvature appearing in the lattice Wheeler–DeWitt equation of Eq. (52),

R� � Xh��

�hlh: (106)

In order to establish the structure of the solutions for largevolumes Vtot, we will assume, based in part on the results ofthe previous sections and on the analogous calculation in2þ 1 dimensions [2], that the fundamental wave functionsolutions for large volumes have the form

exp

�i

��Z

d3xffiffiffig

p þ �Z

d3xffiffiffig

pRþ

Zd3x

ffiffiffig

pR2

þ �Z

d3xffiffiffig

pR��R

�� þ � � ��: (107)

Note here that the structure of the above expression, andthe nature of the terms that enter into it, are basicallydictated by the requirement of diffeomorphism invarianceas it applies to the argument of the wave functional. Apartfrom the cosmological term, allowed terms are all the ones


084012-14

that can be constructed from the Riemann tensor and itscovariant derivatives, for a fixed topology of three-space.Clearly, at large distances (infrared limit), the most impor-tant terms will be the Einstein and cosmological terms,with coefficients � and �, respectively. In three dimen-sions, the Riemann and Ricci tensor have the same numberof algebraically independent components (6) and arerelated to each other by

R�� ¼ ��

�R�

� 1

2��

�: (108)

The Weyl tensor vanishes identically, and one has

R��R�� 4R��R

�� 3R2 ¼ 0

C��C�� ¼ 0:

(109)

As a consequence, there is in fact only one local curvaturesquared term one can write down in three spatial dimen-sions. Nevertheless, higher derivative terms will onlybecome relevant at very short distances, comparable or

smaller than the Planck lengthffiffiffiffiG

p; in the scaling limit, it

is expected that these can be safely neglected.When expressed in lattice language, the above form

translates to an ansatz of the form

exp fiðc0Vtot þ c1RmtotÞg; (110)

with m assumed to be an integer. In addition, from thestudies of lattice gravity 2þ 1 dimensions, one expects alnVtot term as well in the argument of the exponential [2].This suggests a slightly more general ansatz,

exp fiðc0Vtot þ c1RmtotÞ þ c2 lnVtot þ c3 lnRtotg: (111)

The next step is to insert the above expression into thelattice Wheeler–DeWitt equation, Eq. (52), and determinethe values of the five constants c0 . . . c3, m. This can bedone consistently just to leading order in the weak fieldexpansion of Eq. (37), which is entirely adequate here, as itwill provide enough information to uniquely determinethe coefficients. Here we will just give the result of thisexercise. For the five-cell complex (q ¼ 3), one obtains

c � exp

�i

�4

ffiffiffi2

p ffiffiffiffi�

p3G

Vtot �ffiffiffi2

p

Gffiffiffiffi�

p Rtot

�� 95

18lnVtot

�;

(112)

whereas for 16-cell complex (q ¼ 4), one finds

c � exp

�i

� ffiffiffi2

p ffiffiffiffi�

pG

Vtot � 3ffiffiffi2

p

4Gffiffiffiffi�

p Rtot

�� 47

4lnVtot

�;

(113)

and finally for 600-cell complex (q ¼ 5),

c � exp

�i

�4

ffiffiffi2

p ffiffiffiffi�

p5G


p

5Gffiffiffiffi�

p Rtot

�� 336 lnVtot

�:

(114)

These expressions allow us again to identify the answer forgeneral q as

c � exp

�i

�4


pqG


p

qGffiffiffiffi�

p Rtot

�

� ð11þ 9qÞN3

4q2lnVtot

�: (115)

Note that in deriving the above results, we considered thelarge volume limit V ! 1, treating the number of tetrahe-draN3 as a fixed parameter. In writing down this last result,we have used the fact that such a q dependence of thecurvature term is expected on the basis of Eq. (32), andsimilarly for the volume term in view of Eq. (38). Inaddition, the log term is expected on general grounds tohave a coefficient proportional to the number of latticetetrahedra N3, as it does (exactly) in 2þ 1 dimensions[2]. Note that later the effect of the log term will be inpart compensated by the measure (or entropy) contributionof Eq. (102).4

Then from the previous expression, we can now read offthe values for the various coefficients, namely,

c0 ¼ 4ffiffiffiffiffiffi2�

pqG

c1 ¼ � 3ffiffiffi2

p

qGffiffiffiffi�

p

c2 ¼ �ð11þ 9qÞN3

4q2c3 ¼ 0;

(116)

with the only possible value m ¼ 1.In order to make contact with the strong coupling result

for the wave function derived in the previous sections[Eqs. (92) and (94)–(96)], one needs to again expand theabove answer for small curvatures. One obtains for theexponent of the wave function the following expression:

i

�4


pq0G

Vtotþ�

a2036

ffiffiffi2

p�


pG

� 6

q0Gffiffiffiffiffiffi2�

p�RtotþOðR2Þ

�

��11þ9q0

4q20N3þ 22þ9q0

96�q0a0RtotþOðR2Þ

�lnVtot; (117)

4A rather similar procedure was successfully used earlier in2þ 1 dimensions, where it was found that the three regulartriangulations of the sphere, the single triangle, and the regulartriangulation of the torus were all described, for large areas andto all orders in the weak field expansion, by a single wavefunctional involving confluent hypergeometric functions, withthe total area and total curvature serving as arguments. Theresulting extrapolation to the infinite volume limit yielded exactgravitational scaling exponents [2] in rough agreement (to about6%) with results obtained earlier by numerical integration in theEuclidean lattice theory of gravity [27].


084012-15

with a0 again representing the average lattice spacing,

a30 � 6ffiffiffi2

pV=N3. This finally determines uniquely the

coefficients � and � appearing in Eq. (107),

�¼ 4

q0�


pG

�¼ a2036

ffiffiffi2

p��


pG

� 6

q0� 1

Gffiffiffiffiffiffi2�

p : (118)

The most important result so far is the appearance of twocontributions of opposite sign in �, signaling the appear-ance of a critical value for G where � vanishes.

This critical point is located at �c ¼ 108ffiffiffi2

p�=q0a

20 or,

in a system of units where � ¼ G=2 (see Sec. IV),5

Gc ¼ 216ffiffiffi2

p�

q0� 1

a20: (119)

But since the average lattice spacing a0 is itself a functionof G and � [see Eqs. (65), (104), and (105)], one obtains inthe same system of units

Gc ¼ 36 23=831=4�3=4

f3=2q3=40

’ 28:512; (120)

using the value of f for the single tetrahedron, or equiv-alently gc ’ 5:340, a rather large value. Nevertheless, weshould keep in mind that in this paper we are also using asystem of units where we set 16�G ! G. So, in a conven-tional system of units, one has the more reasonable resultGc 0:5672 in units of the fundamental UV cutoff.6

Evidence for a phase transition in lattice gravity in 3þ 1dimensions was also seen earlier from an application of the

variational method, using Jastrow–Slater correlatedproduct trial wave functions [1]. Note that the resultsof Eqs. (117) and (118) imply a dependence of the funda-mental wave function on the curvature, of the type

c ðRÞ � eiRtot=R0 ; (121)

with R0 a characteristic scale for the total, integratedcurvature. Thus, R0 � 1=ðg� gcÞ with Gc, and thereforegc ¼

ffiffiffiffiffiffiGc

p, given in Eq. (119). Therefore, at the critical

point, fluctuations in the curvature become unbounded, justas is the case for the fluctuations in a scalar field when therenormalized mass approaches zero.7 Note that since at thecritical point Gc the curvature term vanishes further inves-tigations there would require the retention of curvaturesquared terms, which in general are not expected to bezero. One would then expect, based again on invariancearguments, that the leading contribution there should comefrom the TT mode contribution, which is indeed quadraticin the curvature.At this stage, one can start to compare with the results

obtained previously without the explicit curvature term inthe Wheeler–DeWitt equation, Eqs. (94) and (95). Themain change is that here one would be led to identify

x ¼ 4ffiffiffiffiffiffi2�

pq0G

Vtot þ�

a2036

ffiffiffi2

p��


pG

� 6

q0� 1

Gffiffiffiffiffiffi2�

p�Rtot;

(122)

so that the Bessel function argument x [see Eq. (94)] nowcontains a new contribution, of opposite sign, in the cur-vature term. Its origin can be traced back to the newcurvature contribution c1 in Eq. (116), which in turn arisesbecause of the explicit curvature term now present in thefull Wheeler–DeWitt equation. On the other hand, as isalready clear from the result for c2 in Eq. (116), the index nof the Bessel function solution in Eqs. (91) and (92) is leftunchanged,

n ¼ 11þ 9q04q20

N3 � 1

2þ 22þ 9q0

96�q0a0Rtot þOðR2

totÞ; (123)

with again an average lattice spacing a0 defined as before.But there is a better way to derive correctly the modified

form of the wave function. From the asymptotic solutionfor the wave function of Eq. (115), it is possible to first

5As in the Euclidean lattice gravity case, one does not expectthe critical coupling Gc to represent a universal quantity; itsvalue will still reflect specific choices made in defining theunderlying lattice discretization and therefore more generallyin specifying a suitable ultraviolet cutoff (this fact is known infield theory language as scheme dependence). These circum-stances can be seen here already when looking at the simplestregular lattices enumerated previously in this work and whichare clearly not unique choices even for a fixed number of sites. Inaddition, one expects a further dependence of Gc on the choiceof functional measure and therefore on the supermetric [see, forexample, Eq. (103)]. In the present context, this leads, forexample, to a dependence of the results on the parameter f ofEq. (105). Nevertheless, one would expect, based largely onuniversality arguments, that critical exponents and scaling di-mensions (such as the ones obtained exactly in Ref. [2]) shouldbe universal, and therefore independent of the specific details ofthe ultraviolet cutoff, for which the introduction nevertheless isessential at some stage in order to regularize the inevitablequantum infinities.

6One can compare the above value for Gc obtained in theLorentzian 3þ 1 theory with the corresponding value in theEuclidean four-dimensional theory. There one finds Gc 0:6231 [29], which is within 10% of the above quoted value.The two Gc values are not expected to be the same in the twoformulations, due to the different nature of the UV cutoffs. Inparticular, in the lattice Hamiltonian formulation, the continuumlimit has already been taken in the time direction. Nevertheless,it is encouraging that they are quite comparable in magnitude.

7It is tempting to try to extract a critical exponent from theresult of Eq. (121). In analogy to the wave functional for a freescalar field with mass m, and thus correlation length � ¼ 1=m,one would obtain for the correlation length exponent � (with �defined by �� jg� gcj��) from the above wave function thesemiclassical estimate � ¼ 1

2 . In the 2þ perturbative expan-sion for pure gravity, one finds in the vicinity of the UV fixedpoint ��1 ¼ ðd� 2Þ þ 3

5 ðd� 2Þ2 þOððd� 2Þ3Þ [30–32]. Theabove lowest order lattice result would then agree only withthe leading, semiclassical term.


084012-16

obtain a partial differential equation for c ðRtot; VtotÞ. Theequation reads (in the following, we shall write Rtot as Rand Vtot as V to avoid unnecessary clutter)

@2c

@V2þ cV

@c

@Vþ cR

@c

@Rþ cVR

@2c

@V@Rþ cRR

@2c

@R2

þ c�c þ ccurvc ¼ 0: (124)

The coefficients in the above equation are given by

cV ¼ 11þ 9q

2q2� N3

V

¼ 11þ 9q02q20

� N3

Vþ 22þ 9q0

48ffiffiffi2

p31=3�q0

� N1=33 R

V4=3þOðR2Þ

cR ¼ � 2

9

R

V2þ 11þ 9q0

6q20� N3R

V2þOðR2Þ

cVR ¼ 2

3

R

VþOðR2Þ

cRR ¼ 2

9

R2

V2

c� ¼ 32�

q2G2¼ 32

G2q20þ 4

ffiffiffi2

p�

331=3�q0G� R

N2=33 V1=3

þOðR2Þ

ccurv ¼ � 16

G2q2� RV¼ � 16

G2q20� RVþOðR2Þ: (125)

Note that in the small curvature, large volume limit [this isthe limit in which, after all, Eq. (115) was derived], one cansafely set the coefficients cR and cRR to zero. It is then easyto check that the solution in Eq. (115) satisfies Eqs. (124)and (125), up to terms of order 1=V2. Also note that here,and in Eqs. (112)–(115), we take the large volume limitV ! 1, treating the number of tetrahedra N3 as a large,fixed parameter. A differential equation in the variable Vonly can be derived as well (with coefficients that arefunctions of R), but then one finds that the required coef-ficients are not real, which makes this approach lessappealing.8

In the limit R ! 0, Eq. (124) reduces to

@2c

@V2þ 11þ 9q0

2q20� N3

V� @c@V

þ 32�

G2q20c ¼ 0; (126)

which is essentially Eq. (84) in the same limit, withsolution given previously in Eq. (91).

VIII. NATURE OF THE WAVE FUNCTIONSOLUTION c

In this section, we discuss some basic physical proper-ties that can be extracted from the wave function solutionc ðV; RÞ. So far, we have not been able to find a generalsolution to the fundamental Eq. (124), but one mightsuspect that the solution is still close to a Bessel or hyper-geometric function, possibly with arguments ‘‘shifted’’according to Eqs. (122) and (123), as was the case in2þ 1 dimensions. As a consequence, some physicallymotivated approximations will be necessary in the follow-ing discussion. Let us discuss here in detail one possibleapproach. If one sets the troublesome coefficient cVR ¼ 0in Eq. (124), and keeps only the leading term in cV , thenthe relevant differential equation becomes

@2c

@V2þ cV

@c

@Vþ c�c þ ccurvc ¼ 0; (127)

with coefficients given in Eq. (125), except that from nowon only the leading term in cV and c� will be retained(otherwise, it seems again difficult to find an exact solu-tion). Note that the above equation still contains an explicitcurvature term proportional to R, from ccurv. Now a com-plete solution can be found in terms of the confluent hyper-geometric function of the first kind, 1F1ða; b; zÞ [33–35].Up to an overall wave function normalization constant, it is

c ðV;RÞ

’e�4i

ffiffiffi2�

pV

q0G ��ð11þ9q0ÞN3

4q20

þ iffiffi2

pR

q0Gffiffiffi�

p

��1�ð11þ9q0ÞN3

4q20

þ iffiffi2

pR

q0Gffiffiffi�

p

� 1F1

�ð11þ9q0ÞN3

4q20� i

ffiffiffi2

pR

q0ffiffiffiffi�

pG;ð11þ9q0ÞN3

2q20;8i


pV

q0G

�:

(128)

Here again q0 is just a number, given previously inEq. (51), and N3 is the total number of tetrahedra for agiven triangulation of the manifold. Note that this lastsolution still retains three key properies: it is a functionof geometric invariants ðV; RÞ only; it is regular at theorigin in the variable V (the irregular solution is discardeddue to the normalizability constraint); and, finally, itagrees, as it should, with the zero curvature solution ofEqs. (91) and (92) in the limit R ¼ 0.The above wave function exhibits some intriguing

similarities with the exact wave function solution foundin 2þ 1 dimensions; the difference is that the total

8It would of course be of some interest to derive a resultsimilar to Eq. (117) using an entirely different set of methods,such as the Wentzel–Kramers–Brillouin (WKB) approximation.Such an approximation was discussed, again on the lattice, inRef. [1], but the resulting equations there turned out to be toocomplicated to solve. In the context of a continuum WKBapproximation, one would expect the approximate results forthe wave function to contain some remnants of short distanceinfinities and therefore depend, at least in part, implicitly orexplicitly, on the specifics of the ultraviolet regularization pro-cedure. But, more generally, one would expect such a continuumexpansion to be poorly convergent in four dimensions, in view ofthe perturbative nonrenormalizability of ordinary gravity. Thelattice methods presented here are, on the other hand, genuinelynonperturbative in nature and therefore not immediately affectedby the escalating divergences encountered in the continuumtreatment in four dimensions.


084012-17

curvature R here plays the role of the Euler characteristic �there. Let us be more specific and discuss each argumentseparately. For the arguments of the confluent hypergeo-metric function of the first kind, 1F1ða; b; zÞ, one findsagain b ¼ 2a for R ¼ 0, with both a and b proportionalto the total number of lattice sites, as in 2þ 1 dimensions[2]. Specifically, here one has

ReðaÞ ¼ 11þ 9q04q20

N3 0:5464N3; (129)

whereas in 2þ 1 dimensions, the analogous result is

ReðaÞ ¼ 1

4N2: (130)

The curvature contribution in both cases then appears as anadditional contribution to the first argument (a) and ispurely imaginary. Here one has

ImðaÞ ¼ �ffiffiffi2

p

q0ffiffiffiffi�

pG

Zd3x

ffiffiffig

pR; (131)

whereas in 2þ 1 dimensions, the corresponding result is

ImðaÞ ¼ � 1


pG

Zd2x

ffiffiffig

pR: (132)

Finally, here again the third argument z is purely imaginaryand simply proportional to the total volume. From theabove solution,

z ¼ i8


pq0G

Zd3x

ffiffiffig

p; (133)

whereas in 2þ 1 dimensions,

z ¼ i2


pG

Zd2x

ffiffiffig

p: (134)

Nevertheless, we also find some significant differenceswhen compared to the exact 2þ 1-dimensional result,most notably the various gamma-function factors involv-ing the curvature R, which are entirely absent in the lowerdimensional case, as well as the fact that the critical(UV fixed) point is located at some finite Gc here [seeEq. (119)], whereas it is exactly at Gc ¼ 0 in 2þ 1dimensions [2].

Let us now continue here with a discussion of the mainproperties of the wave function in Eq. (128). First, let usintroduce some additional notational simplification. Byusing the coupling g [see Sec. IV and Eq. (36)], one canmake the above expression for c slightly more transparent,

c ðV; RÞ

’ e�4iV

q0g ��ð11þ9q0ÞN3

4q20

þ 2iRq0g

3

��1� ð11þ9q0ÞN3

4q20

þ 2iRq0g

3

� 1F1

�ð11þ 9q0ÞN3

4q20� 2iR

q0g3;ð11þ 9q0ÞN3

2q20;8iV

q0g

�:

(135)

We remind the reader that, by virtue of Eq. (51), in allthe above expressions, q0 is just a numerical constant, q0 �2�=cos�1ð13Þ ¼ 5:1043. Note that for weak coupling the

curvature terms become more important due to the 1=g3

coefficient. The resulting probability distribution jc ðV; RÞj2is shown, for some illustrative cases, in Figs. 3–5.One important proviso should be stated here first. We

recall that having obtained an (exact or approximate)expression for the wave function does not lead immedi-ately to a complete solution of the problem. This should beevident, for example, from the general expression for theaverage of a generic quantum operator OðgÞ,

hOðgÞi � h�jOj�ih�j�i ¼

Rd�½g� �OðgijÞ � j�½gij�j2R

d�½g� � j�½gij�j2;

(136)

where d�½g� is the appropriate (DeWitt) functional measureover the three-metric gij. Because of the general coordinate

invariance of the state functional, the inner products shownabove clearly contain an infinite redundancy due to the

0

5

10

V

1.0

0.5

0.0

0.5

1.0

R

0

1

2

3

2

FIG. 3 (color online). Wave function of Eq. (135) squared,jc ðV;RÞj2, plotted as a function of the total volume V and thetotal curvature R, for coupling g ¼ ffiffiffiffi

Gp ¼ 1 and N3 ¼ 10. One

notes that for strong enough coupling g the distribution incurvatures is fairly flat around R ¼ 0, giving rise to largefluctuations in the curvature. These become more pronouncedas one approaches the critical point at gc.


084012-18

geometrical indistinguishability of three-metrics which dif-fer only by a coordinate transformation [7]. Nevertheless,this divergence is of no essence here, since it cancels outbetween the numerator and the denominator.

On the lattice, the above average translates into

hOðl2Þi � h�jOj�ih�j�i ¼

Rd�½l2� �Oðl2Þ � j�½l2�j2R

d�½l2� � j�½l2�j2 ; (137)

where d�½l2� is the Regge–Wheeler lattice transcription ofthe DeWitt functional measure [7] in terms of edge lengthvariables, here denoted collectively by l2. The latterincludes an integration over all squared edge lengths, con-strained by the triangle inequalities and their higherdimensional analogs [27]. Again, because of the continu-ous local diffeomorphism invariance of the lattice theory,the individual inner products shown above will contain aninfinite redundancy due to the geometrical indistinguish-ability of three-metrics, which differ only by a latticecoordinate transformation. And, again, this divergencewill be of no essence here, as it is expected to cancelbetween numerator and denominator [19].

It seems clear then that, in general, the full functionalmeasure cannot be decomposed into a simple product ofintegrations over V and R. It follows that the averageslisted above are, in general, still highly nontrivial to evalu-ate. In fact, quantum averages can be written again quite

generally in terms of an effective (Euclidean) three-dimensional action,

h�j ~OðgÞj�i ¼ NZ

d�½g� ~OðgijÞ exp f�Seff½g�g; (138)

with Seff½g� � � ln j�½gij�j2 and N a normalization con-

stant. The operator ~OðgÞ itself can be local, or nonlocal asin the case of correlations such as the gravitational Wilsonloop [36]. Note that the statistical weights have zeroscorresponding to the nodes of the wave function �, sothat Seff is infinite there.

9

Nevertheless, it will make sense here to consider asemiclassical expansion for the 3þ 1-dimensional theory,where one simply focuses on the clearly identifiable sta-tionary points (maxima) of the probability distribution jc j,

0

5

10

V

1.0

0.5

0.0

0.5

1.0

R

0

1

2

3

2

FIG. 4 (color online). Same wave function of Eq. (135)squared, jc ðV;RÞj2, plotted as a function of the total volumeV and the total curvature R, but now for weaker coupling g ¼ffiffiffiffiG

p ¼ 0:5 and still N3 ¼ 10. For weak enough coupling g, thedistribution in curvature is such that values around R ¼ 0 arealmost completely excluded, as these are associated with a verysmall probability. Note that, unless the total volume V is verysmall, the probability distribution is markedly larger towardpositive curvatures.

0.5

1.0

1.5

2.0

g

1.0

0.5

0.0

0.5

1.0

R

0

1

2

3

Ψ 2

FIG. 5 (color online). Curvature distribution in R as a functionof the coupling g ¼ ffiffiffiffi

Gp

. The strong coupling relationship be-tween the average volume and the coupling g [Eq. (64)] allowsone to plot the wave function of Eq. (135) squared as a functionof the coupling g and the total curvature R only (we use againhere N3 ¼ 10 for illustrative purposes). Then, for strong enoughcoupling g ¼ ffiffiffiffi

Gp

, the probability distribution jc j2 is againfairly flat around R ¼ 0, giving rise to large fluctuations in thecurvature. The latter are interpreted here as signaling the pres-ence of a massless particle. On the other hand, for weak enoughcoupling g, one notices that curvatures close to R ¼ 0 haveessentially vanishing probability. The distribution shown herepoints therefore toward a pathological ground state for weakenough coupling g < gc [given in Eq. (119)], with no sensiblecontinuum limit.

9In practical terms, the averages in Eqs. (136) and (137) aredifficult to evaluate analytically, even once the complete wavefunction is known explicitly, due to the nontrivial nature of thegravitational functional measure; in the most general case, theseaverages will have to be evaluated numerically. The presence ofinfinitely many zeros in the statistical weights complicates thisissue considerably, again from a numerical point of view.


084012-19

obtained by squaring the solution in Eq. (128) or (135).In the following, we will therefore focus entirely on theproperties of the probability distribution jc ðV; RÞj2obtained from Eq. (128) or (135). For illustrative purpose,the reader is referred to Figs. 3–5 below.

As discussed previously, the asymptotic expansion forthe wave function at large volumes is suggestive of a phasetransition at someG ¼ Gc [see for example Eqs. (118) and(119)]. In addition, the explicit solution in Eq. (135) allowsa more precise non-perturbative characterization of the twophases. In view of the non-trivial and generally complexarguments of both the gamma function and the confluenthypergeometric function, the analytic properties of thewave function, and therefore of the probability distribu-tion, are quite rich in features, at least for the more generaland physically relevant case of non-zero curvature.

One first notes that for strong enough coupling g thedistribution in curvature is fairly flat around R ¼ 0, givingrise to large fluctuation in the latter (see Figure 3). On theother hand, for weak enough coupling g the probabilitydistribution in curvature is such that values around R ¼ 0are almost excluded, since they are associated with a verysmall probability. Furthermore, unless the volumeV is verysmall, the probability distribution is also generally mark-edly larger towards positive curvatures (see Figure 4).

In order to explore specifically the curvature (R) depen-dence of the probability distribution, it would be desirableto factor out or remove the dependence of the wave func-tion c ðV; RÞ on the total volume V. To achieve this, onecan employ a mean-field-type prescription, and replace thetotal volume V by its average hVi. After all, the probabilitydistribution in the volume is well behaved at large G [seeSec. VI], and does not exhibit any marked change inbehavior for intermediate G [as can be inferred, forexample, from the asymptotic form of the wave functionin Eq. (115)]. Consequently we will now make the replace-ment in c ðV; RÞ

V ! hVi � N3hV�i ¼ 0:2643Gffiffiffiffi�

p ¼ 0:3738g; (139)

obtained by inserting the result of Eq. (64). This replace-ment then makes it possible to plot the wave function ofEq. (135) squared as a function of the coupling g and thetotal curvature R only (in the following we use againN3 ¼ 10 for illustrative purposes); see Figure 5. One thennotes that for strong enough gravitational coupling

g ¼ ffiffiffiffiG

pthe probability distribution is again fairly flat

around R ¼ 0, giving rise to large fluctuations in thecurvature. On the other hand, for weak enough couplingg one observes that curvatures close to zero have nearvanishing probability. The distributions shown suggesttherefore what seems a pathological ground state forweak enough coupling g<gc [or G<Gc, see Eq. (119)],with no sensible four-dimensional continuum limit.

At this point some preliminary conclusions, based on thebehavior of the wave function discussed previously inSec. VII and the shape shown in Figs. 3–5, are as follows.For large enough G>Gc, but nevertheless close to thecritical point, the flatness in the curvature probabilitydistribution implies that different curvature scales are allequally important. The corresponding gravitational corre-lation length is finite in this region as long as G>Gc andexpected to diverge at the critical point, thus presumablysignaling the presence of a massless excitation at Gc [seethe argument after Eq. (121)]. On the other hand, for weakenough coupling, G<Gc, we observe that the probabilitydistribution appears to change dramatically. The mainevidence for this is the shape of the approximate wavefunction given in Eq. (128), which points to a vanishingrelative probability for metric field configurations forwhich the curvature is small R 0. This would suggestthat the weak coupling phase, for which G<Gc, has nocontinuum limit in terms of manifolds that appear smooth,at least at large scales. The geometric character of themanifold is then inevitably dominated by nonuniversalshort-distance lattice artifacts; no sensible scaling limitexists in this phase.If this is indeed the case, then the results obtained in the

present, Lorentzian, 3þ 1 theory generally agree withwhat is found in the Euclidean case, where the weakcoupling phase was found to be pathological as well[17,18] (it bears more resemblance to a branched polymerand has thus no sensible interpretation in terms of smoothfour-dimensional manifolds). In either case, the onlyphysically acceptable phase, leading to smooth manifoldsat large distances, seems to be the one with G>Gc. It is asimple consequence of renormalization group argumentsthat in this phase the gravitational coupling at large dis-tances can only flow to larger values, implying thereforegravitational antiscreening as the only physically possibleoutcome. Nevertheless, it needs to be emphasized hereagain that these conclusions have been obtained from adetermination of the wave functional at small curvatures; itshould be clear that when the curvature cannot be regardedas small, higher order terms in the curvature expansion ofEqs. (107) and (128) need to be retained, which leads usbeyond the scope of the present work.

IX. SUMMARYAND CONCLUSIONS

In this work, we have discussed the nature of gravita-tional wave functions that were found to be solutions of thelattice Wheeler–DeWitt equation for finite simplicial latti-ces. The main results here were given in Eqs. (124), (128),and (135). While there are many aspects of this problemthat still remain open and unexplored, we have neverthe-less shown that the very structure of the wave function issuch that it allows one to draw a number of useful andperhaps physically relevant conclusions about ground stateproperties of pure quantum gravity in 3þ 1 dimensions.


084012-20

These include the observation that the theory exhibits aphase transition at some critical value of Newton’s constantGc [given in Eq. (119)].

The structure of the wave function further suggests thatthe weak coupling phase, for which the coupling G<Gc,is pathological and cannot be interpreted in terms ofsmooth manifolds at any distance scale. In view of theseresults, it is therefore not entirely surprising that calcula-tions that rely on the weak field, semiclassical, or small Gexpansion run into serious trouble and uncontrollable di-vergences very early on. Such an expansion does not seemto exist if the nonperturbative lattice results presented hereare taken seriously. The correct physical vacuum appar-ently cannot in any way be obtained as a small perturbationof flat, or near-flat, spacetime. On the other hand, the strongcoupling phase does not exhibit any such pathology and istherefore a good candidate for a physically acceptableground state for pure quantum gravity. It is then a simpleconsequence of standard renormalization group argumentsthat in this phase Newton’s constant grows with distance,so that this phase is expected to exhibit gravitational anti-screening. Still, to make the problem tractable, most of theresults presented in this work have been obtained inthe limit of small curvatures. This is clearly a limitationof the present approach. A more general treatment of theproblem, where a variety of curvature squared terms areretained in the expansion of the wave functional, should befeasible by the methods presented here but is for nowclearly beyond the scope of the present work.

Let us mention here that in the Euclidean latticetheory of gravity in four dimensions it was also found earlyon [17,18] (see Ref. [29] for more recent numerical inves-tigations of four-dimensional lattice gravity, includingthe determination of the critical point and scaling expo-nents) that the weak coupling (or gravitational screening)phase is pathological with no sensible continuum limit,corresponding to a degenerate lower dimensional branchedpolymer. The calculations presented here can be regarded,therefore, as consistent with the conclusions reached ear-lier from the Euclidean lattice framework. No new sur-prises have arisen so far when considering the Lorentzian3þ 1 theory, using what can be regarded as an entirelydifferent set of tools.

It seems also worthwhile at this point to compare withother attempts at determining the phase structure of quan-tum gravity in four dimensions. Besides the Regge latticeapproach, there have been other attempts at searching for anontrivial renormalization group UV fixed point in fourdimensions using continuum methods. In one popular fieldtheoretic approach, one develops a perturbative diagram-matic 2þ continuum expansion using the backgroundfield method to two-loop order [30–32]. This then leads toa nontrivial UV fixed point Gc ¼ OðÞ close to twodimensions. Two phases emerge, one implying againgravitational screening and the other antiscreening. In the

truncated renormalization group calculations for gravity infour dimensions [37,38], where one retains the cosmologi-cal and Einstein–Hilbert terms and possibly later somehigher derivative terms, one also finds evidence for a non-trivial UV fixed point scenario. As in the case of gaugetheories, both of these methods are ultimately based onrenormalization group flows and the weak field expansionand are therefore unable to characterize the nonperturba-tive features of either one of the two ground states. Indeed,within the framework of the weak field expansion inherentin these methods, only the weak coupling phase has achance to start with. It is nevertheless encouraging thatsuch widely different methods tend to point in the samedirection, namely, a nontrivial phase structure for gravity infour dimensions.Let us add here a few more comments, aimed at placing

the present work in a wider context. Over the years, anumber of attempts has been made to obtain results forthe gravitational wave functional �½g� in the absence ofsources. Often these have relied on the weak field expan-sion in the continuum; see, for example, Refs. [9,10].In 3þ 1 dimensions, one then finds

�½hTT� ¼ N exp

�� 1

4

Zd3k khTTik ðkÞhTT�ik ðkÞ

�; (140)

where hTTik ðkÞ is the Fourier amplitude of transverse-

traceless modes for the linearized gravitational field. It isclear that the above wave functional describes a collectionof harmonic oscillator contributions, one for each of thephysical modes of the linearized gravitational field. It is notnecessary to use Fourier modes, and, as in the case of theelectromagnetic field, one can write equivalently theground state wave functional in terms of first derivativesof the field potentials,

�½hTT� ¼ N exp

�� 1

8�2

Zd3x

Zd3y

hTTik;lðxÞhTT�ik;l ðyÞjx� yj2

�:

(141)

Nevertheless, it is generally understood that the aboveexpressions represent only the leading term in an expan-sion involving infinitely many terms in the metric fluctua-tion hij (in an expansion about flat space, the cosmological

constant contribution does not appear). Since Eq. (140) isjust the leading term in the weak field expansion, no issueof perturbative renormalizability appears to this order.Nevertheless, higher orders are expected to bring in ultra-violet divergences which cannot be reabsorbed into a sim-ple redefinition of the fundamental couplings G and �.Then the results presented in this paper [namely,Eqs. (124), (128), and (135)] can be viewed therefore asa first attempt in extending nonperturbatively the resultof Eq. (140), beyond the inherent limitations of the weakfield limit.


084012-21

We see a number of additional avenues by which thepresent work could be extended. One issue, which couldbe rather laborious to work out in 3þ 1 dimensions, isthe systematic determination of the relevant lattice wavefunctionals for the regular triangulations of the sphereto higher order in the weak field expansion, as wasdone, for example, in 2þ 1 dimensions [2]. It shouldalso be possible to obtain the lattice wave functionalnumerically in cases where the triangulation itself is notregular but is described instead by an average coordi-nation number hqi, as described earlier in the text.Another interesting problem would be the derivationof the general form of the lattice wave functional bymethods which differ from the Frobenius power seriesmethod presented here, such as the WKB approximation[7] or the Raleigh–Schrodinger approach. Finally, it

would also be of some interest to rederive the formof the lattice wave functional for other discrete triangu-lations, such as the case of the three-torus T3 (theKasner model) [39].10

ACKNOWLEDGMENTS

The work of H.W.H. was supported in part by the MaxPlanck Gesellschaft zur Forderung der Wissenschaften. Hewishes to thank Hermann Nicolai and the Max PlanckInstitut fur Gravitationsphysik (Albert-Einstein-Institut)in Potsdam for warm hospitality. The work of R.M.W.was supported in part by the UK Science and TechnologyFacilities Council. The work of R. T. was supported in partby a DED GAANN Student Fellowship.

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