arXiv:gr-qc/0212117 v2 24 Jan 2003KANAZAWA-02-39

gr-qc/021

2117

Theexpectatio

nvalueofthemetric

operatorwith

respectto

Gaussia

nweavesta

tein

loopquantum

gravity

Tom

oyaTsushim

a�

Institu

teforTheoretica

lPhysics,

KanazawaUniversity

,Kanazawa920-1192,JAPAN

(Dated

:Jan

uary

27,2003)

Abstract

Westu

died

themetric

opera

torontheGaussian

weave

state,which

isacan

didate

ofdescrib

ing

thesem

iclassicalspace,

inloopquantum

gravity.

Inrecen

tyears,

thee�ect

ofquantum

gravity

isbein

gveri�

edbythecosm

ologica

lobservation

s.Iftherelativ

isticrelation

betw

eenenergy

and

mom

entumofcosm

icray

isdefo

rmed

bythequantumgrav

itye�ect

atPlan

ckscale,

wecan

explain

whytheultra

highenergy

cosm

icray

has

reached

theearth

,beyondtheupper

limitofenergy

ofthe

order

of10

11GeV

.How

ever,theparam

etersofdeform

ationterm

sare

sensitiv

eto

theobserva

tion.

Toderiv

etheparam

eters,itisimportan

tto

understan

dthesuitab

lequantum

statesandbehavior

ofthemetric

operato

r.In

thispaper,

wecalcu

latetheexpectation

valueofthemetric

operato

rand

evaluate

thesize

ofquantum

scaleofGaussian

weave

state,tow

ardunderstan

dingthesem

iclassical

approx

imatio

nof

thespace.

�Electro

nic

address:

tomoya@hep

.s.kanazawa-u.ac.jp1

I. INTRODUCTION

From 1980s to present, the canonical quantum gravity has progressed dramatically [1].

We call this theory the loop quantum gravity (LQG). The quantum state is characterized

by closed paths on three-dimensional space, and is called spin network state. Its norm is

positive de�nite. The spin network state is diagonalized with respect to three-dimensional

geometrical observables such as area and volume, and gives them the discrete eigenvalues

[2, 3, 4, 5, 6]. This result is derived independent of strength of the coupling constant, there-

fore, it is the non-perturbative e�ect. Using the quantum states and the area eigenvalues,

the black hole entropy is calculated by counting the paths of quantum states crossing the

(classical) event horizon [7]. Furthermore, the non-commutativity of space appears [8]. LQG

is not as complete as we call non-perturbative theory, because the Hamiltonian constraint

operator (HCO), which is strongly related to dynamics of the system, is not solved. The

matrix element of the HCO can be calculated with respect to the spin network state [9, 10].

By formal solution of the HCO, the picture similar to Feynman graph, as time evolution of

the spin network state, is obtained [11].

The ultra high energy cosmic ray (UHECR), which energy is larger than 1011 GeV, have

been detected by AGASA [12, 13]. Assume that the UHECR consists of protons. If its

energy is larger than 1011 GeV, it can not travel more than 102 Mpc because it interacts

the cosmic microwave background radiation. Since there is no sources near our galaxy, the

charged particle has to have a energy threshold, called GZK cuto� [14, 15]. However, it

con icts to the data from AGASA.

It can be solved by using the extra dimensions or the high-energy physics beyond the

standard model [16, 17, 18, 19, 20]. The semiclassical approximation of LQG also can solve

the problem. It deforms a relation between energy and momentum with Lorentz violated

terms E2 = ~p 2 +m2 +P�nj~p jn=Mn�2

Pl . The image of semiclassicalize method of LQG is

shown in FIG. 1. This deformation avoids the problem if the coeÆcients take suitable values

[21]. Moreover, these coeÆcients restricted by another observation [22]. Incidentally, the

method of Lorentz symmetry break down was �rst introduced by [23].

The Hilbert space of LQG is a space of holonomy, which describes spin networks. In the

semiclassical approximation of LQG, the HCO is regarded as just a Hamiltonian operator.

It acts at the vertices of the spin networks, so it is discrete operator. The method of

2

�

?

GeV

1019

...

1016

1015

1014...

Quantum gravity scale

{ regularized spacewhich consists of 3-dim. lattice {

discrete space

Semiclassical scale

{ deformed Lorentz invariance {continuous space

Classical scale

{ Lorentz invariance {

?

?

��������

FIG. 1: The picture of semiclassical approximation of LQG.

semiclassical approximation of LQG appears in [24, 25, 26]. It does not treats a holonomy

hI , but SU(2) connection Aia as the con�guration. In [25], the codreibein operator n�i

I ,

which is de�ned later, is expanded by the length of holonomy as

n�iI =

4

i�h�tr(� ihI V

nh�1I )

� 2

i�h�_saI [A

ia; V

n]� 1

i�h�_saI _s

bI [@aA

ib +

1

2�ijkAj

aAkb ; V

n] +O(j _saI j3); (1)

where V is a volume operator. The evaluation of the expectation values can not be calculated

because the SU(2) connection operator is not de�ned on the Hilbert space, and quantum

states is not determined. So, the expectation value is estimated as of the order of hAiai �

0 +O(1)� (`Pl=L)�=L. L is a scale that the space is regarded as continuum for �elds. � is

unknown parameter.

To obtain the coeÆcient restricted from the observation, beyond the order estimation,

we determine a quantum state, which satis�es semiclassical condition. We introduce a state,

called Gaussian weave state [27], which is superposition of in�nite number of states so that

it can describe semiclassical space. Then we calculate the codreibein operator before series

expansion. Furthermore, the metric operator, the squere of the codreibein operator, is

calculated in order to study the semiclassical spece. Then we evaluated the expectation

value of the metric operator with respect to the Gaussian weave state.

In next section, we explain a essence of LQG we need, and determine a notations. Section

III is main part, we introduce a Gaussian weave state, and calculate the expectation value

of the metric operator. The techniques for spin network calculation is in appendix.

3

II. LOOP QUANTUM GRAVITY

A. Real Ashtekar variable

In our starting point, we employ the Einstein-Hilbert action

S =1

16�G

Zdt d3x N

pq (R � (Ka

a)2 +KabK

ab) (2)

to describe the gravitational �eld. a; b; � � � are spatial indices and qab is a three-dimensional

metric and q = det(qab). R andKab are four-dimensional Ricci scalar and extrinsic curvature,

respectivly. G is Newton constant. The lapse function N corresponds to time-time compo-

nent of four-dimensional metric. Independent variables of this action are, naively, qab and

its canonical momentum. Let us introduce a codreibein �ia satis�es the relation qab = �ia�ib.

The new canonical variables are a dreibein of density weight one Eai = 1

2�abc�ijk�

jb�

kc and a

real Ashtekar variable

Aia = �ia +

pqEbiKab; (3)

where �ia is a spin-connection that is a function satis�es the torsionless condition @aEai +

�ijk�jaEak = 0 . The Immirzi parameter is a real number that cannot be determined in

the theoretical point of view. The indices i; j; � � � are degrees of freedom of a local internal

SO(3). This canonical pair forms a Poisson bracket

fAia(x); E

bj (y)gP = �ÆbaÆ

ijÆ

3(x; y); (4)

where � = 8� G, which reproduce the original one that made by metric and its conjugate.

The real Ashtekar variable Aia behaves as an SU(2) connection form. That is, we can

regard Aia and Ea

i as a vector potential and an electric �eld in the SU(2) gauge theory,

respectively. The constraints, which included by this system, are Hamiltonian constraint

H =

2�pq�ijkEa

i Ebj

�F kab � ( 2 + 1)�klmK l

aKmb

�; (5)

Gauss constraint Gi = 1�(@aEa

i + �ijkAjaE

ak ) and di�eomorphism constraint Da =

1�EbiF

iab �

AiaGi, where F i

ab = @aAib�@bAi

a+�ijkAj

aAkb is a curvature 2-form of Ai

a, and Kia =

1 (Ai

a��ia).

H, Gi and Da are caused by time reparametrization, SU(2) gauge transformation and spatial

di�eomorphism invariance, respectively. In addition, we treat SU(2) as gauge group instead

of SO(3).

4

B. Regularization

In (5), the weightpq locates a denominator, because the canonical momentum Ea

i is

density weight one and the integrand must be density weight one. As an example, the

electromagnetic Hamiltonian is

HEM [N ] =Zd3x

1

2Nqabpq(EaEb +BaBb): (6)

Since both electric and magnetic �eld Ea; Ba are density weight one, the integrand also has

an inverse weight. If we naively quantize this, a second order functional derivative emerges

at the same point and the gravitational sector diverges. Therefore this operator is ill-de�ned.

However, using volume variables and point splitting methods, we can solve these two types

of singularities [28, 29].

Consider a box, which satis�esRBox d

3y = �, centered at x, and f�(x; y) is unity if y in

the box and is zero otherwise. If we take a limit � ! 0, 1�f�(x; y) ! Æ3(x; y). The volume

variable in the box is

V�(x) =Zd3y f�(x; y)

qq(y)

=Zd3y f�(x; y)

s����16�abc�ijkEai E

bjE

ck

���� (y) (7)

This becomes 1�V�(x) !

qg(x) as � ! 0. Surprisingly, a Poisson bracket of Ai

a and V n

makes a codreibein of density weight (n� 1):

lim�!0

�1�n2

�nfAi

a(x); (V�(x))ngP =

�ia

(pq)1�n

!(x): (8)

In particular, the density weight becomes a negative if n < 1. Let this relation apply to (6).

This is in the case of n = 1=2, and there is a factor �1=2�1=2 = � on a numerator. Then we

should insert a point splittingRd3y 1

�f�(x; y) to eliminate the �. Thus,

Zd3x

qabpqXaY b = lim

�!0

1

�

Zd3x

Zd3y f�(x; y)

�iaX

a

(pq)1=2

!(x) �

�ibY

b

(pq)1=2

!(y)

=16

�2lim�!0

Zd3x

Zd3y f�(x; y)

��fAi

a;qV�gPXa

�(x)

�fAi

b;qV�gPY b

�(y) (9)

This corresponds to the regularized electromagnetic Hamiltonian. Similarly, The gravita-

tional Hamiltonian (constraint) can be explained by volume variables and point splitting

5

method. The �rst term of (5) is

lim�!0

HE� = lim

�!0

�2�abcF i

abfAic; V�gP =

2�pq�ijkEa

i EbjF

kab: (10)

The superscript E means Euclidean. If space-time is Euclidean metric and = 1, the

gravitational Hamiltonian vanishes except for this term. For the remainder term, we use

K ia =

1� 2fAi

a;K�gP with K� = fHE� ; V�gP = 2K i

aEai , so

H� lim�!0

HE� = �

2�pq�ijkEa

i Ebj � ( 2 + 1)�klmK l

aKmb

= � lim�!0

2 + 1

�4 3�abc�ijkfAi

a;K�gPfAjb;K�gPfAk

c ; V�gP : (11)

So the volume variable plays a very important role for regularization of the Hamiltonian

operator.

We divide a spatial integration into regions speci�ed by �. We make the region at most

includes one vertex of quantum states, which de�ned in next subsection. By this division,

if we quantize a regularized variable, that becomes well-de�ned operator independently �.

Thus, the limit will be eliminated.

C. Quantum states

The SU(2) holonomy is a path ordered integral of the connection form along the smooth

path e,

h(p)e (A) = P exp��Z 1

0ds _ea(s) Ai

a(e(s)) �i(p)

�(12)

on the three dimensional space. We call the path e an edge, and parametrize the orbit ea of

the edge by s 2 [0; 1]. _ea(s) is a tangent vector of the edge. The beginning point ea(0) and

the endpoint ea(1) are called vertices. The space of holonomies constructs the con�guration

space of connections modulo SU(2) gauge transformations. � i(p) is an anti-hermite generator

of su(2) (p + 1)-dimensional representation, or, equivalently, spin-p=2 representation. p

called a color of the edge. The holonomy changes by gauge transformation from he to

g(e(1))heg�1(e(0)) for g(e(1)); g(e(0)) 2 SU(2).

Quantum state is characterized by closed graph constructed by edges. Edges meet at a

vertex, but they don't have to connect smoothly. A vertex that connects n pieces of edges is

6

�����

PPPP

PBBBBB

�����r

P0

P1

P2

P3

P4

- ����PP

PP BBBB

����

r rr

��AAA

P0

P1

P2

P3

P4

i2

i3

FIG. 2: An example of pentavelent vertex. Five pieces of edges with color P0; � � � ; P4 have connected

at the vertex. The pentavalent vertex constructed by three virtual trivalent vertices with two virtual

edges of color i2; i3.

called n-valent vertex. In trivalent vertex, if colors of edges are a; b and c, respectively and

if a and b are given, c can only take ja� bj; ja� bj+2; � � � ; a+ b� 2; a+ b , because of SU(2)

invariance of quantum states. Using (n � 3) pieces of `virtual' edges with gauge invariant

set of colors ~i = fi2; i3 � � � ; in�2g, we can compose the n-valent vertex by (n � 2) `virtual'

trivalent vertices. The virtual edges have many degrees of freedom with respect to its colors,

we regard as basis of the vertex. The way of connection between virtual edges is not unique,

but if we �x the basis, the other way of connection can be described by linear combination of

the basis we chose. This basis, the way of connection between edges, is called the intertwiner

(or intertwining tensor.) In other words, the intertwiner is a map from tensor product of

incoming edgesN

eI(1)\v(p(eI) + 1) to tensor product of outgoing edgesN

eI(0)\v(p(eI) + 1).

FIG. 2 shows pentavalent vertex, for example.

The spin network S = f ; ~p; ~�g that is a set of a graph , colors ~p = fp(e1); � � � ; p(en)g ofedges and intertwiner ~� = f�~i1(v1); � � � ; �~im(vm)g of vertices ~v = fv1; � � � ; vmg. It describesa quantum state called a spin network state. The wave function is explained by

S(A) = (he1 � � � hen) � (�~i1(v1) � � � �~im(vm)) (13)

for the spin network S. Now we de�ne a norm of states of S1 = f 1; ~p1; ~�1g and S2 =

f 2; ~p2; ~�2g. There is a larger graph = [nI=1eI � 1[ 2, then for S1 and S2 on the graph

, we can explain the norm

hS1jS2i =Z(SU(2))n

d�(he1) � � � d�(hen) fS(he1; � � � ; hen) qS(he1 ; � � � ; hen)

= Æ 1; 2

0@Ye2

Æp1(e);p2(e)�p1(e)

1A �0@Yv2

(�1(v); �2(v))

1A (14)

7

with inner product

(�1(v); �2(v)) =

Y~e

Æp1(~e);p2(~e)�p1(~e)

! Y~v

�(p1(e1~v); p1(e2~v); p1(e3~v))

!; (15)

where d� is SU(2) Haar measure, which normalizedRSU(2) d� = 1. ~e; ~v mean a virtual edge

and a virtual trivalent vertex at vertex v, respectively. e1~v; e2~v; e3~v are edges (or virtual ones)

that connected with the virtual vertex ~v. The symmetrizer �a and the �-net �(a; b; c) are

de�ned by (A9,A10).

For simplicity, we sometimes denote a spin network state as only the graph , omitting

the colors of edges and intertwiners. The holonomy hs along to the segment s, is just a

product operator that operates to adding the edge to the graph, i.e., hs = s[ .

The operator corresponds to derivative operator is so-called the left (right) invariant vector

(he� i)AB@=@(he)AB ((� ihe)AB@=@(he)AB.) This operation corresponds to connect � i with

the vertex of the edge in the graph.

The volume operator includes three derivative operators as in (A18), they act at vertex.

Therefore, we aim at one vertex, and investigate the intertwiner on it. The normalized

intertwiner ~�~i of the n-valent vertex can be written graphically as

~�i2;���;in�2(P0; � � � ; Pn�1) = N~i(P0; � � � ; Pn�1) � �r r r� � �

P0 P1 P2 Pn�2 Pn�1

i2 in�2

(e0) (e1) (e2) (en�2) (en�1)

(16)

with the normalization factor

N~i(P0; � � � ; Pn�1) =

vuut Qn�2x=2�ixQn�2

x=1 �(ix; Px; ix+1): (17)

where i1 = P0; in�1 = Pn�1. e0; � � � ; en�1 are edges connecting at the vertex, and P0; � � � ; Pn�1are its colors respectively. The inner product (15) of ~�~i are given graphically,

(~�~i ;~�~k) = N~iN~k

��

��r

rrr

� � �P0 P1 Pn�2 Pn�1

k2

i2

kn�2

in�2

=n�2Yx=2

Ækxix : (18)

If an operator X act to it, we obtain a matrix element X~i~k = (~�~k; X

~�~i).

8

1 2

�����

BBBBB

�����

BBBBB

r rp

p

p

p

0

1 1

2 2

FIG. 3: The graph constructed by 1; 2 in the left. The path of the graph is a1(0) ! a1(1) =

a2(0)! a2(1) = a1(0) The vertual vertex can take the color-0; 2; � � � ; 2p, but we treat the color-0

for simply.

III. THE EXPECTATION VALUE OF THE METRIC OPERATOR

A. Gaussian weave state

In LQG, the space is constructed by \excitation" of a graph. Actually, if the graph is not

include the multivalent vertex that valence is higher than three, the volume of the space is

zero. Thus, the \ground state" is not a at space. In order to obtain a at space, the graph

must include in�nite number of vertices. Therefore, the weave state W =Qv2Rwv in �nite

region R in three-dimensional space is de�ned as following [27, 30]. Let us consider the two

closed edges 1; 2 crossing at the vertex v. The wave fuction of color-p is

�p = tr(h(p) 1h(p) 2

) = (h(p) 1)AB (h(p) 2

)CD (�0(p; p; p; p))ABCD: (19)

constructed by the edges, as FIG. 3. The inner product of �p is normalized. We take a

weave state for each vertex asP1

p=0 cp�p, and determine the coeÆcient as following:

wv = N exp(��2(�1 � 2)2); (20)

where N , � are normalization factor and any real paremeter, respectivly. Using the formula

of SU(2) tensor products [30]:

(�1)n =

Xp

(p+ 1) � n!(n�p2 )!(n+p2 + 1)!

�p ;n� p

2= 0; 1; 2; � � � (21)

and generating function of Hermite polynomials, (20) is

wv = Ne�4�21Xn=0

�nHn(2�)

n!(�1)

n

9

= Ne�4�21Xp=0

1Xn=0

p+ 1

n!(n+ p+ 1)!�2n+pH2n+p(2�) ��p: (22)

Then we obtain the coeÆcient as

cp(�) = Ne�4�21Xn=0

(p+ 1)�2n+pH2n+p(2�)

n!(n+ p+ 1)!: (23)

Although the series diverges depending on the value of �, it is at most exp(4�2). However,

in numerical point of view, it is preferable to converge it. Moreover, in order to avoid a

contribution of higher color, we employ � = 3=4 same as [27]. The norm for each vertex as

hwvjwvi = P1p=0(cp(

34))2.

B. Matrix elements of metric operators

(8) is rewritten by the holonomies:

n�iI(v) = � 4

�ntr(� ihIfh�1I ; (Vv)

ngP ); (24)

where hI = h(1)sI, sI is a segment that endpoint is at v. Let us quantize (24) by usual

procedure f�; �gP ! 1i�h [�; �], then

n�iI(v) = � 4

i�h�ntr(� ihI [h

�1I ; V n

v ]) =4

i�h�ntr(� ihI V

nv h

�1I ): (25)

We operate it to wv. First, h�1I acts to a trivalent vertex ~�k(p; p; p; p):

h�10~�k(p; p; p; p) = Nk(p; p; p; p) � � �r r

p p p p

k�1

= Nk(p; p; p; p)Xq=�1

�q(p) � �r rp p p p

krr1

1

p+ q

p

; (26)

where �+1(p) = 1; ��1 = � pp+1. It can be regarded as a pentavalent vertex for volume

operator. Denote

~�ij(�1; q; p; p; p) = Nij(�1; q; p; p; p) � �r r r�1 q p p p

i j

(e0) (e1) (e2) (e3)

; (27)

10

where the check �1 means that the volume operator does not operate to it. By (A33), matrix

element of volume operator is found, V n ~�ij =P

s;t Vnijst ~�st, then

V nh�10~�k =

Xq=�1

Xs;t

V npk

st(�1; p+ q; p; p; p)Nst(�1; p + q; p; p; p)Nk(p; p; p; p)

Npk(�1; p + q; p; p; p)

� � �r rp p p p

trr1

1

p+ q

s

(28)

Therefore, the action of codreibein operator with density weight (n� 1) is obtained as

n�i0~�k = � 4

i�h�n

Xq=�1

p

2+1X

s2=j p

2�1j

pXt2=0

Nk(p; p; p; p)

NstV

npkst

Npk

!(�1; p + q; p; p; p)

�Tet

264 s p p + q

1 1 2

375�(s; p; 2)

� ii���r 2 r� �r r

p p p p

t

s : (29)

In right hand side of (29), su(2) generator connects the intertwiner. It is not construct a

quantum state because the spin network state is a function of only holonomies. Thus, we

consider the metric, which operates codreibein two times,

nq(sI ; sJ )(v; v0) = n�i

I(v)n�i

J(v0): (30)

By spin network calculation, its expectation value is vanishes when v 6= v0. Then we treat

the metric operator on the same point v. Let the graphical part of right hand side of (29)

denote as

f ist = ii���r 2 r� �r r

p p p p

t

s

(e0) (e1) (e2) (e3)

: (31)

The codreibein operator acts it. There is two cases, I = J and I 6= J , in the operation. In

the case of I = J = 0, the action of h�10 is

h�10 f ist = ii���r 2 r� �r r

p p p p

t

s�1 =Xv=�1

�v(s)ii���r 2 r� �r r

p p p p

t

js + vj

s

s

1

1

rr

: (32)

11

The volume operator regards it as ~�st(�1; js + vj; p; p; p). Therefore, the matrix element of

the metric operator of same direction is

nq(s0; s0)(v; v) ~�k = � 8

(�h�n)2Xq=�1

p

2+1X

s2=j p

2�1j

pXt2=0

Xv=�1

pXm2=0

��q(p)�v(s)�Nk

Nm

�(p; p; p; p)

� NstV

npkst

Npk

!(�1; p+ q; p; p; p)

�NpmV

nstpm

Nst

�(�1; js+ vj; p; p; p)

�Tet

264 s p p+ q

1 1 2

375Tet264 p s js+ vj1 1 2

375�p�(s; p; 2)

~�m

=Xm

nq(s0; s0)(p)km ~�m: (33)

Similarly, in the case of I = 1; J = 0,

h�11 f ist = ii���r 2 r� �r r

p p p p

t

s 1 =Xu=�1

�u(p)ii���r 2 r� �r r

p p p p

t

s

p + u

p

1

1

rr

=Xu=�1

Xa

�u(p)

8><>: 1 p+ u a

t s p

9>=>; ii���r 2 r� �r r

p p p p

t

s p + u

a

11r

r : (34)

The volume operator regards it as ~�at(s; �1; u; p; p). Then,

nq(s1; s0)(v; v) ~�k = � 8

(�h�n)2Xq=�1

p

2+1X

s2=j p

2�1j

pXt2=0

Xu=�1

Xa

Xm;r

��q(p)�u(p)8><>: 1 p + u a

t s p

9>=>;�Nk

Nr

�(p; p; p; p)

� NstV

npkst

Npk

!(�1; p+ q; p; p; p)

�NmrV

natmr

Nat

�(s; �1; p + u; p; p)

�Tet

264 s p m

1 1 2

375�(m; 1; p)�(p; p; r)

~�r

=Xr

nq(s1; s0)(p)kr ~�r: (35)

12

This operator, which we de�ned as (30), is not hermite because of the codreibein operator

is non-commutative: [�iI(v); �

iJ(v)] 6= 0. Actually, in the case of I = 0; J = 1,

nq(s0; s1)(v; v) ~�k = � 8

(�h�n)2Xq=�1

Xl=�1

Xs;t

Xu

Xv=�1

Xr

��q(p)�v(u)�2pu

8><>: 1 p+ q p+ l

k p p

9>=>;8><>: 2 1 u

s p 1

9>=>;��Nk

Nr

�(p; p; p; p)

� NstV

np+l;k

st

Np+l;k

!(p; �1; p+ q; p; p)

�NprV

nstpr

Nut

�(�1; ju+ vj; p; p; p)

�Tet

264 1 p u

t s p + q

375Tet264 1 u 2

p 1 ju+ vj

375�p�(t; p; u)

~�r

=Xr

nq(s0; s1)(p)kr ~�r; (36)

that is, nq(s1; s0) 6= nq(s0; s1). Since the metric operator acting on only at one vertex remains,

we can re-de�ne it hermitian 12(

nq(sI ; sJ) + nq(sJ ; sI))(v).

Concretely, we show the expectation value of nq(s0; s0). In (23), let the norm be normalizePp(cp)

2(�) = 1, and we set a parameter � = 3=4, then (c0)2 = 0:414892; (c1)2 = 0:482013

and (c2)2 = 0:0972374. Since the coeÆcients of higher color more than p = 3 are degligible

such as (c3)2 = 1:28919�10�6 , it is suÆcient to evaluate the contribution of p = 1; 2. Then,

nq(s0; s0)(1)00 =1

(�h�n)2( 3(V n

1010(1; 0; 1; 1; 1))2

�6V n10

10(1; 0; 1; 1; 1)V n10

10(1; 2; 1; 1; 1)

+3(V n10

10(1; 2; 1; 1; 1))2

+3V n10

12(1; 0; 1; 1; 1)V n12

10(1; 0; 1; 1; 1)

�3V n10

12(1; 2; 1; 1; 1)V n12

10(1; 0; 1; 1; 1)

�3V n10

12(1; 0; 1; 1; 1)V n12

10(1; 2; 1; 1; 1)

+3V n10

12(1; 2; 1; 1; 1)V n12

10(1; 2; 1; 1; 1)

+12V n10

32(1; 2; 1; 1; 1)V n32

10(1; 2; 1; 1; 1) )

nq(s0; s0)(2)ij =1

(�h�n)2(16

3V n

0220(1; 1; 2; 2; 2)V n

2002(1; 1; 2; 2; 2)

+32

9(V n

2020(1; 1; 2; 2; 2))2

13

�64

9V n

2020(1; 1; 2; 2; 2)V n

2020(1; 3; 2; 2; 2)

+32

9(V n

2020(1; 3; 2; 2; 2))2

+32

9V n

2022(1; 1; 2; 2; 2)V n

2220(1; 1; 2; 2; 2)

�32

9V n

2022(1; 3; 2; 2; 2)V n

2220(1; 1; 2; 2; 2)

�32

9V n

2022(1; 1; 2; 2; 2)V n

2220(1; 3; 2; 2; 2)

+32

9V n

2022(1; 3; 2; 2; 2)V n

2220(1; 3; 2; 2; 2)

+32

9V n

2024(1; 1; 2; 2; 2)V n

2420(1; 1; 2; 2; 2)

�32

9V n

2024(1; 3; 2; 2; 2)V n

2420(1; 1; 2; 2; 2)

�32

9V n

2024(1; 1; 2; 2; 2)V n

2420(1; 3; 2; 2; 2)

+32

9V n

2024(1; 3; 2; 2; 2)V n

2420(1; 3; 2; 2; 2)

+32

3V n

2042(1; 3; 2; 2; 2)V n

4220(1; 3; 2; 2; 2)

+32

3V n

2044(1; 3; 2; 2; 2)V n

4420(1; 3; 2; 2; 2) ) (37)

Since n is not essential in numerical evaluation, we treat the case of n = 1, i.e., density

weight zero. From (A33),

1q(s0; s0)(1)00 =3

16

s37

2+ 10

p6 �

q786 + 370

p6 �h� = 0:255522�h� > 0

1q(s0; s0)(2)00 = 0:254895�h� > 0 (38)

Therefore, we obtain the expectation value of the metric operator

h1q(s0; s0)i = hwvj 1q(s0; s0) wvi=

1Xp=0

(cp(3

4))2 � 1q(s0; s0)(p)00 = 0:14795�h� (39)

IV. CONCLUSION

We calculated that the expectation value of the metric operator with respect to Gaussian

weave state. The diagonal element of metric is surely positive de�nite. It is claimed that

hwvjnq(sI; sI)wvi = Pi jjn�i

Iwvjj2 � 0. This is the non-trivial result because

nq(sI ; sI) =4

(�h�n)2

�2tr(hI V

2nh�1I )��tr(hI V

nh�1I )�2�

: (40)

14

Now, can we acquire the method beyond the order estimation of classical approximation

by this result? The operator Aia is not de�ned. We can not expand a holonomy, and

can not derive the form of (1). Let (1) be semiclassical approximation as ansatz, with

Aia _s

aI = 2tr(� i ln(hsI )) and a derivative of connection is displaced as the di�erential between

two vertices. Then it becomes well-de�ned operator for quantum states. Since we have

already known the background metric, h1q(s0; s0)i � qab _sa0 _sb0, the scale of _s

a0 can be evaluated.

The concrete calculation in consideration of what is described above is under execution. It

will appear in next paper.

Acknowledgments

I would like to thank Ken-Ichi Aoki, for helpful comments and useful advice.

APPENDIX A: SPIN NETWORK CALCULATION

1. Spin network and recoupling theory

In this subsection, we refered to [5]. The SU(2) invariant tensor, a matrix XAB and its

trace are graphically written as

ÆAB =A

B ; ÆADÆBC = � �

�AAA B

C D ; i�AB = ��A B

i�AB =

��A B ; XA

B = X

A

B; trX = � X��

��: (A1)

Since we add the minus signature to the crossing line, a symmetric tensor becomes anti-

symmetric line. We derive the binor identity

+ ��AA +

���� = 0: (A2)

Thus, the lines behave as knots satisfying the Reidemeister moves O, I, II and III:

O :���� = ;

I : �@

��=

��;

II :���� =

���� ;III : �

�@

@� ��� = ��@

@��� � (A3)

15

in the knot theory if they are �xed the indices upper or lower position. A line of color-a is

a pieces of lines that anti-symmetrized:

a = � � �� � ���� �a

=1

a!

�� � � � �

��DDD � � � +� � � �

�; (A4)

where the white box means anti-symmetrizing of lines. A trivalent vertex de�ned as

ra

b c = ��a

b ci

jk ;

8>>>><>>>>:i = 1

2(�a+ b+ c)

j = 12(a� b+ c)

k = 12(a+ b� c)

: (A5)

The generator of su(2) anti-hermite 2-dimensional representation is

� i = ii = ii���r 2 r : (A6)

This generator acts to a line of color-a as

ii���r 2 a = a � ii

���r 2 ra

a

: (A7)

Summation of product of generators

3Xi=1ii���r 2ii���r 2

=1

2��Æ2

;Xi;j;k

�ijk

ik���r 2ij���r 2ii���r 2

= �1

2���r

2

2

2: (A8)

The symmetrizer

�a =����a = (�1)a(a+ 1): (A9)

The �-net

�(a; b; c) =��

��r r

c

ba

=(�1)m+n+p(m+ n+ p + 1)!m!n!p!

a!b!c!; (A10)

where m = 12(�a+ b+ c); n = 1

2(a� b+ c) and p = 12(a+ b� c). The tetrahedral net

Tet

264 a b e

c d f

375 =��

@@

@@

��

r rr

r �

�a

b c

def =

IE

Xm�S�M

(�1)S(S + 1)!Qi(S � ai)! �Qi(bi � S)!

; (A11)

a1 = (a+ d+ e)=2; b1 = (b+ d+ e+ f)=2;

16

a2 = (b+ c+ e)=2 ; b2 = (a+ c+ e+ f)=2;

a3 = (a+ b+ f)=2; b3 = (a+ b+ c+ d)=2 ;

a4 = (c+ d + f)=2;

m = maxfaig ; M = minfbigE = a!b!c!d!e!f ! ; I =

Qij(bj � ai)!:

The 9-j symbol 8>>>><>>>>:a b c

d e f

g h i

9>>>>=>>>>; =

��

��

� �� � �r

rrr

rr

a b c

d e f

g h i

: (A12)

The exchanging of lines in a trivalent vertex

��ra b

c= �abc �@ra b

c; �abc = (�1) 12 (a(a+3)+b(b+3)�c(c+3)): (A13)

The recoupling theorem

@�

�@r r

a

b c

df =

Xe

8><>: a b e

c d f

9>=>; @�

�@rra

b c

d

e : (A14)

The relation between a 6-j symbol and a tetrahedral net8><>: a b e

c d f

9>=>; =�e

�(a; d; e)�(b; c; e)Tet

264 a b e

c d f

375 : (A15)

The reduction formulas

����rr

a

b c

d

= Æad�(a; b; c)

�a

a ; (A16)

��rrr a

bc

de f =

Tet

264 a b e

c d f

375�(a; f; b)

��r

a

f

b: (A17)

2. Matrix elements of volume operator

The volume operator for the vertex v is

V =(�h�)3=2

4

s XI<J<K

���iW[IJK]

��� (A18)

17

where jXj =qXyX. The meaning of the square root is that its diagonalized matrix equiv-

alent to square root of XyX . The summation I; J;K is the label of edges eI connected with

the vertex v. The operator W[IJK] adding the three su(2) generators:

W[IJK] =

� ����r

(eI) (eJ ) (eK)

2 2 2(A19)

(times minus two) to the edges eI ; eJ; eK (see (A8)). Let us compute this operation to the

n-valent vertex. Its matrix element is

fW[IJK]~i

~k = (~�~k; W[IJK]~�~i)

= N~iN~kPIPJPK �

��

��r

rr

rrr

rrr

P0PI

PI

PJ

PJ

PK

PKPn�1

iI

kI

iI+1

kI+1

iJ

kJ

iJ+1

kJ+1

iK

kK

iK+1

kK+1

� � �� �r

2 22

: (A20)

We can calculate the graphical part of (A20). Now we separate it four part (i) e0-eI , (ii)

eI -eJ , (iii) eJ -eK and (iv) eK-en�1.

(i)

�� r

rr

P0PI

PI

iI

kI

iI+1

kI+1

��2

= �PI2PI���

��rrr

P0PI

PI

iI

kI

iI+1

kI+1

2

= ��iI+12kI+1(

rYx=2

Ækxix )�(P0; P1; i2)

�i2

(r�1Yx=2

�(ix; Px; ix+1)

�ix+1

)

�Tet

264 kI+1 iI+1 iI

PI PI 2

375�(kI+1; iI+1; 2)

����Ær

iI+1

kI+1

2

(A21)

= �i ����Ær

iI+1

kI+1

2

(A22)

18

(iv)

��r

rr

Pn�1

PK

PK

iK

2

kK

iK+1

kK+1

= (n�2Yx=t+1

Ækxix )(n�3Yx=t+1

�(ix; Px; ix+1)

�ix

)�(in�2; Pn�2; Pn�1)

�in�2

�Tet

264 iK kK iK+1

PK PK 2

375�(kK; iK; 2)

���r

iK

2

kK

(A23)

= �iv ���r

iK

2

kK

(A24)

Then the graph of (A20) is

��

��r

rr

rrr

rrr

P0PI

PI

PJ

PJ

PK

PKPn�1

iI

kI

iI+1

kI+1

iJ

kJ

iJ+1

kJ+1

iK

kK

iK+1

kK+1

� � �� �r

2 22

= �i�iv �

��

��

rrrr

rPJ

PJ

iI+1

kI+1

iJ

kJ

iJ+1

kJ+1

iK

kK� � �� �r

2 22

: (A25)

Next, we calculate the remaining parts (ii) and (iii).

(ii)

��r

rr

2

kI+1

iI+1

kJ�1

iJ�1

kJ

iJ

Ps�1 =

s�1Y

x=r+1

Tet

264 kx kx+1 2

ix+1 ix Px

375�(kx+1; ix+1; 2)

! ��r2

kJ

iJ

(A26)

= �ii ���r2

kJ

iJ

(A27)

(iii)

��� r

rr

2

kJ+1

iJ+1

kJ+2

iJ+2

kK

iK

Ps+1= �iK2

kK�

��� r

rr

2

kJ+1

iJ+1

kJ+2

iJ+2

kK

iKPs+1

19

=�iK2kK

�iJ+12kJ+1

t�1Y

x=s+1

Tet

264 kx kx+1 2

ix+1 ix Px

375�(kx; ix; 2)

! ��� r

2

kJ+1

iJ+1

(A28)

= �iii �

��� r

2

kJ+1

iJ+1

(A29)

Then we obtain

fW[IJK]~i

~k = N~iN~kPIPJPK �

��

��r

rr

rrr

rrr

P0PI

PI

PJ

PJ

PK

PKPn�1

iI

kI

iI+1

kI+1

iJ

kJ

iJ+1

kJ+1

iK

kK

iK+1

kK+1

� � �� �r

2 22

= N~iN~kPIPJPK � �i�ii�iii�iv �

8>>>><>>>>:kJ PJ kJ+1

iJ PJ iJ+1

2 2 2

9>>>>=>>>>; : (A30)

ifW[IJK] is a pure imaginally anti-symmetric matrix, there is a unitary matrix such that

diagonalize it.

XI<J<K

jiW[IJK]j ~�~i =X~j

(X

I<J<K

Uy[IJK]jifW diag

[IJK]jU[IJK] )~i~j~�~j =

X~j

fW~i~j~�~j: (A31)

Since summation of hermite matrix is also hermite, (A18) can be diagonalized:

V 2 ~�~i =(�h�)3

16

X~j

fW~i~j~�~j =

(�h�)3

16

X~j

(RyfW diagR)~i~j~�~j

=X~j

(RyV diagRRyV diagR)~i~j~�~j (A32)

Thus, the matrix element of V n is

V n = Ry(V diag)nR

=(�h�)3n=2

4nRy [ R (

XI<J<K

U[IJK]

� j U[IJK] ( ifW[IJK] ) Uy[IJK] j U[IJK] ) R

y ]n=2 R: (A33)

20

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