arxiv:gr-qc/0212117 v2 24 jan 2003 · arxiv:gr-qc/0212117 v2 24 jan 2003 kanaza w a-02-39...
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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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