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Transfer Matrix Formalism for Two-Dimensional

arXiv:hep-th/9302133v1 26 Feb 1993INS-969UT-633February 1993Transfer Matrix Formalism for Two-DimensionalQuantum Gravity and Fractal Structures of Space-timeH. KAWAI⋆, N. KAWAMOTO†, T. MOGAMI‡ and Y. WATABIKI†⋆Department of Physics, University of Tokyo,Bunkyo-ku, Tokyo 113, Japan† Institute for Nuclear Study, University of Tokyo,Tanashi, Tokyo 188, Japan‡ Department of Physics, Kyoto University,Kitashirakawa, Kyoto 606, JapanABSTRACTWe develop a transfer matrix formalism for two-dimensional pure gravity.

Bytaking the continuum limit, we obtain a “Hamiltonian formalism” in which thegeodesic distance plays the role of time. Applying this formalism, we obtain a uni-versal function which describes the fractal structures of two dimensional quantumgravity in the continuum limit.

Recent developments of two-dimensional gravity have provided us with an un-ambiguous definition of quantum gravity. This is based on the equivalence of thecontinuum formulation [1] and the dynamical triangulation.

[2] In two dimensions,we thus have a regularized quantum gravity which has a definite continuum limit.The remarkable success of the matrix models [3] further elucidated topological as-pects of the theory.However we still lack a general formulation for describingquantum fluctuations of space-time and for evaluating physical observables such asfractal dimensions. In this paper we propose a new formulation which is a kind ofHamiltonian formalism for quantum gravity.

We show that a geodesic distance de-fined on a dynamically triangulated surface can be regarded as the “time” variablefor defining the transfer matrix. We then obtain a “Hamiltonian” in the continuumlimit, and analyze the fractal structures of the space-time.Let us consider a cylinder with an entrance loop(c) and an exit loop(c′).

(SeeFig.1. )We introduce the following quantity which is formally defined in thecontinuum framework:N(L, L′; D; A) =ZDgV ol(Diff)δ(Zd2x√g −A)δ(Zcpgµνdxµdxν −L)δ(Zc′pgµνdxµdxν −L′)YP∈c′δ(d(P, c) −D).

(1)Here in the path integral the total area of the surface and the lengths of c and c′are constrained to A, L and L′, respectively. We further impose a constraint thatany point on c′ has a geodesic distance D from the entrance loop c. To be precise,the geodesic distance d(P, c) is defined as the minimum distance between the pointP and a point on the loop c. Note that the exit loop c′ is one of the loops which arecomposed of points having geodesic distance D from the entrance loop c. For laterconvenience we introduce a making point on the exit loop as is shown in Fig.1.2

An important property of N(L, L′; D; A) is the following composition law:N(L, L′; D; A) =∞Z0dA′∞Z0dL′′N(L, L′′; D′; A′)N(L′′, L′; D −D′; A −A′). (2)In other words, if we define N(L, L′; D) by N(L, L′; D) =R ∞0 dAN(L, L′; D; A)e−tA,where t is the cosmological constant, it can be regarded as the matrix element< L | e−D ˆH | L′ > for a “Hamiltonian” ˆH.

In this sense we call N(L, L′; D)the proper time evolution kernel. Roughly speaking, eq.

(2) asserts that a cylinderwith height D can be decomposed into two cylinders with height D′ and D −D′.More strictly, there could be several loops which have geodesic distance D′ fromthe entrance loop c. From only one of them, however, the exit loop c′ has geodesicdistance D −D′. (See Fig.2)We now give a constructive definition of N(L, L′; D; A) in terms of dynamicaltriangulation.The strategy to define its lattice counterpart ¯N(l, l′; d; n) is thefollowing.

We first introduce the notion of deforming a loop one step forward,whose precise definition is given in the next paragraph. As we will see there, aloop can split into several loops after a one-step deformation.

In that case eachloop is independently deformed in the next step. By repeating this procedure dtimes, we have a d-step deformation of a loop.

Then ¯N(l, l′; d; n) is defined as thenumber of possible triangulations of a cylinder satisfying the following conditionswhich correspond to the four δ-functions in eq. (1): (i) The number of triangles isn.

(ii) The entrance loop (c) consists of l links. (iii) The exit loop (c′) consits ofl′ links.

(iv) The exit loop is one of the connected components obtained after a dstep deformation of the entrance loop.In order to give a precise definition of the deformation, let us consider a loop con a triangulated surface. To deform c one step forward means to remove trianglesattached to c in the forward direction and two-fold links on c. A typical exampleis illustrated in Fig.3, where c is represented by a solid line, and the forward3

direction is assumed to be inward. In this example, c has three two-fold links on itas indicated by α, β and γ, but we still regard it as a single loop.

After a one-stepdeformation, this c will split into three loops which are indicated by the dottedlines.The notion of two-fold links might seem an idle complexity.As we willsee, however, it plays an important role in the explicit evaluation of the transfermatrix. Note that the deformation of a loop considered here is closely related tothe geodesic distance on the dual lattice.

Actually, if a loop c′ is obtained fromloop c after a d-step deformation, any point on c′ has geodesic distance d from c.This is why we impose the condition (iv) in order to express the last δ-function ineq. (1).Since ¯N(l, l′; d; n) is the lattice counterpart of N(L, L′; D; A), it satisfies thesame composition law as eq.(2).

Therefore, if we define the d-step evolution kernel¯N(l, l′; d) by ¯N(l, l′; d) = P∞n=0 ¯N(l, l′; d; n)Kn and regard it as the l, l′ elementof a matrix ˆN(d), it can be decomposed into a product of single step evolutionkernels:¯N(l, l′; d) ≡( ˆN(d))l,l′ = ( ˆN(1)d)l,l′,(3)which means that ˆN(1) plays the role of a transfer matrix.Next we show that the transfer matrix ˆN(1) can be evaluated by a combina-torial analysis. To proceed, we first need to evaluate the disk amplitude˜F(y; K) =∞Xl,n=0ylKnF(l; n),(4)where F(l; n) is the number of possible triangulations of a disk which has a bound-ary of length l and consists of n triangles.

The disk amplitude ˜F(y; K) can beevaluated by the large-N φ3 matrix model [4] as˜F(y; K) = 1yn12(1y −Ky2) + K2 (1y −c)r(1y −a)(1y −b)o,(5)4

where a, b and c are functions of K determined bya = 1K −c +r2c( 1K −c),b = 1K −c −r2c( 1K −c),1K = c( 1K −c)(c −12K ). ( b < 0 < a < c )(6)As is easily seen from (5) and (6), ˜F(y; K) is analytic in both K and y withfinite convergence radii.

At the critical point where ˜F(y; K) becomes singular asa function of K, the values of a and c coincide and the convergence radius withrespect to y is equal to 1/a. These critical values can be easily obtained from eq.

(6):K2c = 1/12√3, yc = (31/4 −3−1/4)/2, ac = cc = 1/yc, bc = (1 −√3)/2Kc.Near this critical point we take the continuum limit by parametrizing K andy asK = Kce−ǫ2t,y = yce−ǫζ. (7)Then a, b and c are calculated from eq.

(6) up to order ǫ asa = ac −431/4ǫ√t,b = bc,c = cc +231/4 ǫ√t. (8)We point out that the parametrization of eq.

(7) is a natural one in the followingsense. If we replace Kn and yl in eq.

(4) with Kn = Knc e−nǫ2t and yl = ylce−lǫζ,respectively, it is clear that the continuum limit ǫ →0 corresponds to taking then →∞and l →∞limit with the physical area A = nǫ2 and the physical lengthL = lǫ kept finite. Thus t and ζ are the conjugate variables to the area and theboundary length, respectively.5

Substituting eqs. (7) into eq.

(5), we obtain the following disk amplitude nearthe continuum limit:˜F(ζ; t) = 1 +√32(1 −√3ǫζ) + (1 +√3)5/24f(ζ, τ)ǫ32 + O(ǫ2),(9)wheref(ζ, τ) = (2ζ −√τ)qζ + √τ,√τ =43 +√3√t. (10)Here f(ζ, τ) is a universal function in the sense that it does not depend on thedetails of the triangulation prescription.We now use combinatorics to evaluate the generating function of the matrixelement of the transfer matrix ˆN(1):˜N(y, y′; K) ≡∞Xl,l′yly′l′ ˆN(1)l,l′ ≡∞Xl,l,n=0yly′l′Kn ¯N(l, l′; 1; n).

(11)What we are going to do is to sum up all possible triangulations of a cylinder suchthat the exit loop c′ is one of the connected components of the one-step deformationof the entrance loop c. In order to define a natural matrix multiplication of thetransfer matrix, we introduce a marking point on c′ but not on c.(See Fig.4.) Letus first consider the triangle which is attached to the marked link.

There are onlythree possible types for this marked triangle; 1), 2), and 3) in Fig.5a, where F in3) denotes an arbitrary triangulation of a disk which is connected to the markedtriangle through a vertex. After a one-step deformation, c proceeds to the markedlink on c′ and the disk part will be disconnected.

As we can see in eq. (11), thepower of y, y′, and K counts the number of links on c, links on c′, and trianglesrespectively.

Then the contribution to ˜N(y, y′; K) from the marked triangles iseasily evaluated as follows: 1) yy′2K, 2) yy′2K, 3) y2y′K ˜F(y; K).6

We then examine the structure of the rest of the triangulation by going alongthe exit loop c′. There are four different types of basic structures 1),2),3) and4) as depicted in Fig.5b.Their weights are easily evaluated as 1) yy′2K, 2)y2y′K ˜F(y; K), 3) y2 ˜F(y; K), and 4) yK( ˜F(y; K) −1).

Starting with the markedtriangle, we can attach any of these four structures one by one repeatedly, and thencome back to the original marked triangle. Therefore the contribution of these fourstructures to ˜N(y, y′; K) can be expressed by a geometric series.

We thus obtainthe following form for the generating function of the transfer matrix elements:˜N(y, y′; K) = {2yy′2K + y2y′K ˜F(y; K)}×∞Xn=0{yy′2K + y2y′K ˜F(y; K) + y2 ˜F(y; K) + yK( ˜F(y; K) −1)}n=2yy′2K + y2y′K ˜F(y; K)1 −yy′2K −y2y′K ˜F(y; K) −y2 ˜F(y; K) −yK( ˜F(y; K) −1). (12)We next consider the continuum limit of this transfer matrix.

In order to seehow the continuum limit should be taken, let us express the composition law interms of the generating functions:12πiI dzz˜N(y, z; d1; K) ˜N(1z, y′; d2; K) =˜N(y, y′; d1 + d2; K),(13)where ˜N(y, y′; d; K) = P∞l=0P∞l′=0P∞n=0 yly′l′Kn ¯N(l, l′; d; n). From eq.

(13) it isclear that the continuum limit for y and y′ should be taken around different valueswhich are inverse to each other. Furthermore, since the structure of the entranceloop is similar to the boundary loop of the disk amplitude, it is natural to expectthat the continuum limit can be taken by settingy = yce−ǫζ,y′ = y−1c e−ǫζ′,K = Kce−ǫ2t.

(14)7

If this is true, the composition law (13) can be re-expressed asǫ2πii∞Z−i∞dξ ˜N(ζ, ξ; d1; t) ˜N(−ξ, ζ′; d2; t) =˜N(ζ, ζ′; d1 + d2; t),(15)where ˜N(ζ, ζ′; d; t) stands for ˜N(y, y′; d; K) for the values of y, y′ and K given by(14).The validity of this continuum limit is explicitly checked by substituting (14)into eq.(12). We obtain˜N(y, y′; K) ≡˜N(ζ, ζ′; t) = 1ǫ1ζ′ + ζ −αǫ12f(ζ, τ)+ O(ǫ0),(16)where α =q2/(9√3 −1) and f(ζ, τ) is given by eq.(10).

In this eqation we havethe following miraculous cancellations, which convince us that the continuum limitwe consider here is in fact the right one. First of all O(ǫ0) terms in the denominatorof ˜N(y, y′; K) cancel out.

Secondly, the coefficients of ζ and ζ′ are 1. Thirdly, theresidue of ˜N(y, y′; K) with respect to the ζ +ζ′ term is 1/ǫ, which, as we see below,exactly cancels the factor ǫ in (15).

These are highly nontrivial results.By substituting (16) into (15) for d1 = 1 and d2 = d, we obtain˜N(ζ, ζ′; d + 1; t) =12πii∞Z−i∞dξ1ζ + ξ −αǫ12f(ζ, τ)˜N(−ξ, ζ′; d; t) ,=˜N(ζ −αǫ12f(ζ, τ), ζ′; d; t),(17)which leads to˜N(ζ, ζ′; d + 1; t) −˜N(ζ, ζ′; d; t) = −α√ǫf(ζ, τ) ∂∂ζ˜N(ζ, ζ′; d; t). (18)We then take the continuum limit of eq.

(18) by introducing D ≡α√ǫd and takingǫ →0. We thus obtain the following continuum differential equation for ψ(ζ; D) =8

˜N(ζ, ζ′; D; t):∂∂Dψ(ζ, D) = −f(ζ, τ) ∂∂ζ ψ(ζ, D). (19)In other words, the proper time evolution kernel can be identified with the followingmatrix element:˜N(ζ, ζ′; D; t) =< ζ | e−DH | ζ′ >, where H is a “Hamiltonian”given byH = f(ζ, τ) ∂∂ζ .

(20)It should be noted that the miraculous cancelations that occurred in eq. (16) arecrucial in the derivation of the continuum differential equation (19).The initial value problem for the differential equation (19) can be easily solvedby the use of the characteristic curve method as ψ(ζ; D) = ψ(ζ′; 0), where ζ′ isdefined byR ζζ′ dζ′′/f(ζ′′, τ) = D as a function of ζ and D. We can now evaluatethe proper time evolution kernel N(L, L′; D) which is given by the inverse Laplacetransformation of ˜N(ζ, ζ′; D; τ) for ζ and ζ′.⋆By carrying out an inverse Laplacetransformation for the solution of (19) with the initial condition ψ(ζ; 0) = e−L′ζ,we obtainN(L, L′; D) =12πii∞Z−i∞dζeLζe−L′ζ′.

(21)Note that eq. (21) indeed reproduces the corresponding initial condition N(L, L′; 0) =δ(L −L′), because we have ζ′ = ζ for D = 0.For later use we give N(L, L′; D) for small values of L and τ:N(L, L′; D) =L′√πLn 2D3 −τ 310(D + L′D ) + τ32 17(D3 + 12L′D) + O(τ2)oe−L′/D2+ O(L0).

(22)⋆N(L, L′; D) is related to its discrete version ¯N(l, l′; d) by N(L, L′; D) = ǫ−1ycl−l′ ¯N(l, l′; d).9

We also calculate the inverse Laplace transformation of the disk amplitude as adouble series expansion with respect to ǫ and t:F(L) =12πii∞Z−i∞dζ eLζ ˜F(ζ; t)ǫ−3/2= 1 +√32δ(L −√3ǫ)ǫ−3/2+ 3(1 +√3)5/28√πL−5/2 −τ2L−1/2 + τ3/23 L1/2+ O(τ2),(23)where ˜F(ζ; t) is given by eq. (9).We now apply the formalism developed here to the analysis of the fractalstructures of the space-time.

[5, 6, 7] We consider a large enough space-time withspherical topology for two-dimensional pure gravity. We take an arbitrary pointP and consider the set of points S(P; D) whose geodesic distances from P are lessthan or equal to D. The boundary of S(P; D) usually consists of many loops withvarious lengths.

Let ρ(L; D)dL be the number of loops belonging to the boundaryof S(P; D) whose lengths lie between L and L + dL. As we explain below, thequantity ρ(L; D) can be evaluated asρ(L; D) =limL0→0,τ→0∂2∂τ 2N(L0, L; D) 1LF(L)∂2∂τ 2 1L0F(L0),=1D2G(x) +4 ǫ−3/27(1 +√3)3/2D3(1 −x2)δ(L −√3ǫ),(24)whereG(x) ≡37√π(x−5/2 + 12x−3/2 + 143 x1/2)e−x,(25)with x = L/D2 as a scaling parameter.

In eq. (24), the limit L0 →0 correspondsto shrinking the entrance loop to a point P, and the limit τ →0 is taken to have10

the thermodynamic ilmit. The second derivatives with respect to τ in eq.

(24) areintroduced to avoid small area dominance in the τ →0 limit. Any higher derivativeworks for this purpose without changing the final answer.

The numerator on theRHS corresponds to an operation of gluing a disk with a boundary of length Lto the exit loop of the cylinder. The factors 1/L0 and 1/L in front of F’s areintroduced to convert marked boundaries to unmarked ones.

As we have alreadyexplained, there are several loops which have geodesic distance D from the entranceloop. Recognizing that the exit loop is one of these loops, we easily realize thatthe ratio in eq.

(24) counts the number of loops which have the boundary length L.One of the surprising properties of the function ρ(L; D) is that it is essentiallya universal scaling function of the scaling parameter x = L/D2. For small valuesof L, however, ρ(L; D) includes non-universal parts which have negative powerdependence on the lattice constant ǫ.

In order to examine the scaling property, itis convenient to introduce the following quantities:< Ln > =∞Z0dL Lnρ(L; D). (26)From eq.

(24) it is easy to show that< L0 > ∝const × D3ǫ−3/2 + const × Dǫ−1/2 + const × D0,< L1 > ∝const × D3ǫ−1/2 + const × D2,< Ln > ∝const × D2n(n ≥2). (27)Since D2ρ(L; D) is essentially the universal function G(x) of the dimensionlessparameter x, it is an ideal quantity to measure in computer simulations.

In fact anumerical study for pure gravity shows a clear universal scaling behavior accordingto G(x). [8] As we can see in eq.

(27), < L0 > and < L1 > include ǫ dependentnon-universal part as a dominant contribution while < Ln > for n ≥2 includes11

only universal part and thus is expected to show clear fractal behaviors. The con-troversial numerical results for the fractal dimension by Agishtein and Migdal [6]can be understood by the non-universal behavior of the corresponding quantities.Although we have only treated pure gravity (c = 0) in this paper, it is im-portant to investigate quantum gravity coupled to matter fields.In the limitc →−∞it is known that the two-dimensional gravity system approaches tothe classical limit.We thus expect that the wildly branching space-time sur-face approaches to a smooth surface.On the smooth surface there is only asingle boundary at a geodesic distance D from a given point.We then haveρ(L; D) = δ(L−2πD) = 1/Dδ(L/D −2π), which again suggests a scaling functionwith respect to a scaling parameter x = L/D.

It is thus very natural to expect thatthe gravity, for example, with c = −2 matter shows the similar scaling behavior.In fact our recent numerical investigation shows a clear scaling behavior of thefunction ρ(L; D) which has a similar scaling parameter x = L/D1.9±0.2 but has aslightly different power behavior from (25). [9] It would be very interesting if wecan find an analytic formulation for deriving ρ(L; D) in a model with c ̸= 0.In this paper we have given an explicit form of “Hamiltonian” by eq.

(20) whichcould be viewed as a time evolution generator of closed string, where time is iden-tified with the geodesic distance. This formulation may provide a new formulationof closed string field theory.Acknowledgements: We thank N. Tsuda and T. Yukawa for informing us of the re-sults of their numerical simulations prior to publication.

We also thank Y. Okamotofor valuable comments.12

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FIGURE CAPTIONSFig. 1Cylinder with entrance loop c and exit loop c′.Fig.

2Decomposition of a cylinder with height D into two cylinders with heightD′ and D −D′.Fig. 3A Typical example of the deformation of a loop.The dotted loops areobtained after a one-step deformation of the solid loop.Fig.

4A triangulation contributing to ˜N(y, y′; K). The dotted loop (c′) is one ofthe components of the one-step deformation of the solid line (c).Fig.

5a) Three types for the triangle attached to the marked link. b)Four basicstructures which may appear along c. The solid and dotted lines stand for apart of the entrance(c) and the exit(c′) loops, respectively.14

DAcc'Fig.1

D-D'D'cc'D'D'Fig.2

Fig.3αβγ

Fig.4c'c

FFFF-1Fig5 a)2)3)1)2)3)4)b)1)

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