STRING
ˇ
Cestm´ır Burd´ık
Department of Mathematics, Czech Technical University Faculty of Nuclear Sciences and Physical Engineering
Trojanova 13, 120 00 Prague 2, Czech Republic e-mail: [email protected]
Ondˇrej Navr´atil
Department of Mathematics, Czech Technical University Faculty of Transportation Sciences
Na Florenci 25, 110 00 Prague, Czech Republic e-mail: [email protected]
(Received 5 January 2007; accepted 5 July 2007) Abstract
Dirac formulation of open relativistic strings as systems with constraints is made explicitly. Classical theory is given in the standard light-cone and covariant center-of-mass gauges. It is mentioned that the well-known resultD = 26 is af-fected by using the standard quantization of the mutually in-dependent nonphysical boson creation and annihilation opera-tors. It is shown that in the Dirac formulation these operators are not independent in both the gauges.
1
Hamilton description of classical open string
We will study the Nambu–Goto [1] free open string in dimensionD. We assume the sign convention gµν = diag(−1,1, . . . ,1), where
µ, ν = 0,1, . . . , D−1. The string is described by the functions
Xµ(τ, σ), whereτ∈Randσ∈ h0, πi. The classical string is described
by Lagrangian
L(X) =−ω
Z π
0 Ldσ,
whereω >0 is a constant and the Lagrangian densityL(τ, σ) is
L= q
˙
XX02
− X˙2
X02
.
A dot means partial derivation with respect toτ, a dash with respect toσ, and
XY =gµνXµYν=XµYν.
The boundary conditions areXµ0(τ,0) =Xµ0(τ, π) = 0. In the Hamiltonian formulation we define momenta
Pµ(τ, σ) = δL
δX˙µ(σ) =ω ˙
Xµ(X0X0)−Xµ0( ˙XX0) q
˙
XX02
− X˙2
X02
. (1)
From (1) we obtain the relations
Φ1= 12
P2+ω2 X02
= 0, Φ2=P X0= 0 (2)
called constraints. For the Poisson brackets of two functionalsF(X, P) andG(X, P) we have
F, G = Z π
0
δF δXµ(σ)
δG δPµ(σ)−
δF δPµ(σ)
δG δXµ(σ)
dσ . (3)
In particular, the relation
Xµ(σ), Pν(σ0) =gµνδ(σ−σ0) is valid. The Hamiltonian of the system with constraints (2) is
H = Z π
where the Hamiltonian density is given by
H = PµX˙µ− L+λ1(σ)Φ1+λ2(σ)Φ2=λ1(σ)Φ1+λ2(σ)Φ2= = 1
2λ1(σ)
P2+ω2 X02
+λ2(σ)P X0.
The equations of motion of the string are
˙
Xµ = Xµ, H =λ1Pµ+λ2(Xµ)0 ,
˙
Pµ =
Pµ, H = d dσ
λ1ω2Xµ0 +λ2Pµ
, (4)
Φ1 = Φ2= 0.
Starting from the boundary conditions Xµ0(τ,0) =Xµ0(τ, π), it is possible to extend the functions Xµ0(τ, σ) continuously in variables
σ into odd periodic functions with the period 2π. Therefore, the functions Xµ(τ, σ) and Pµ(τ, σ) are even with the period 2π. So we have
Xµ(τ, σ) = √1 2πX
µ 0(τ) +
1 √
π
∞ P n=1
Xnµ(τ) cosnσ ,
Pµ(τ, σ) = √1 2πP
µ 0(τ) +
1 √
π
∞ P n=1
Pnµ(τ) cosnσ ,
(Xµ)0(τ, σ) = −√1
π
∞ P n=1
nXnµ(τ) sinnσ ,
where
Xnµ(τ) = √1
π
Z π
−π
Xµ(τ, σ) cosnσdσ ,
X0µ(τ) = √1 2π
Z π
−π
Xµ(τ, σ) dσ ,
Pnµ(τ) = √1
π
Z π
−π
Pµ(τ, σ) cosnσdσ ,
P0µ(τ) = √1 2π
Z π
−π
Pµ(τ, σ) dσ .
From (3) we obtain the Poisson brackets
Xmµ, Xnν =
Pmµ, Pnν = 0,
By using the variables Xµ
n(τ) and Pnµ(τ) it is possible to formulate the problem as the system with the Poisson brackets
f, g =gµν
∞ P n=0
∂f
∂Xnµ
∂g ∂Pν n − ∂f ∂Pν n ∂g ∂Xnµ
, (5)
with the Hamiltonian
H =λ1,0Φ1,0+ P k∈N
λ1,kΦ1,k+λ2,kΦ2,k
,
whereλ1,k andλ2,kare constants and
Φ1,0 = 12gµν
P0µPν 0 +
∞ P n=1
Pµ
nPnν+ω2n2XnµXnν
,
Φ1,k = 12gµν h√
2P0µPν k +
∞ P n=1
Pnµ+kPν
n+ω2n(n+k)X µ n+kX
ν n
+
+12 k−1
P n=1
Pkµ−nPnν−ω2n(k−n)X µ k−nX
ν n
i
,
Φ2,k = −
gµν 2
h√ 2kPν
0X µ k + ∞ P n=1
(k+n)Pν nX
µ
k+n−nP ν k+nX
µ n + + k−1 P n=1
nPkν−nXnµ i
and with the constraints
Φ1,0= Φ1,k= Φ2,k= 0.
The equations of motion of the system are
˙
Xnµ=
Xnµ, H , P˙nµ=
Pnµ, H .
Since
Φ1,k,Φ1,l = 12ω2 (k−l)Φ2,k+l+ (k+l)Φ2,k−l
, k > l ,
Φ1,k,Φ2,l = 12 (l−k)Φ1,k+l+ (l+k)Φ1,|k−l|
,
Φ2,k,Φ2,l = 12 (l−k)Φ2,k+l+ (l+k)Φ2,k−l
, k > l ,
(6) we have the system with the first class constraints.
As a next step it is useful to define for anyn≥1 complex variables
aµ±n= √1 2 P
µ
and the functions
L0= Φ1,0, Ln= Φ1,n−iωΨ2,n, L−n= Φ1,n+ iωΨ2,n. It is easy to see thataµn=aµ−n andLn =L−n hold.
In these variables the Poisson brackets (5) have the form (Z0 = ±1,±2, . . .)
f, g =gµν
∂f
∂X0µ ∂g ∂Pν
0 − ∂f
∂P0µ ∂g ∂Xν
0
+ iω P n∈Z0
n ∂f ∂aµn
∂g ∂aν −n
(7)
and for the constraints we obtain
L0 = 12(P0)2+ ∞ P n=1
ana−n = 0,
Lk = P0ak+ ∞ P n=1
ak+na−n+12 k−1
P n=1
ak−nan= 0, k≥1,
L−k = P0a−k+ ∞ P n=1
a−k−nan+12 k−1
P n=1
an−ka−n= 0, k≥1. (8) Now we can write the Hamiltonian in the form
H = P n∈Z
κnLn, (9)
whereκn=κ−n is valid for constantsκn, and equations (6) give
Lk, Ln = iω(n−k)Ln+k.
2
Gauge conditions, Dirac brackets and
quantiza-tion
First we will briefly recall the main idea of the Dirac formulation of the description of the Hamiltonian systems with the first class constraints [2, 3].
of the constantsκn, i.e. the functions, whose Poisson brackets with constraints vanish on the set defined byLn= 0.
To determine the constantsκn, we choose the system of additional conditions Rn = 0, the so-called gauge conditions. We require the regularity of the matrix
C=
L, L
L, R
R, L
R, R
!
on the set M, where Ln =Rn = 0 is valid. The constants κn are obtained from the equations
dRn dτ =
Rn, H +
∂Rn
∂τ = 0, (10)
which give the time conservation of the calibration conditions. In the Dirac terminology the system of equations Ln =Rn = 0 forms the constraints of second class. For these systems the Dirac brackets defined by
f, g
Dir=
f, g +
f, L, ,
f, R C−1
g, L
g, R
(11)
play a very important role. The Dirac brackets have on the set M similar properties as the Poisson brackets, but in addition for any constraintsLn and gauge conditionsRn we have
f, Ln Dir=f, Rn Dir= 0.
In consequence, for calculation of the Dirac brackets, it makes no difference, contrary to the Poisson brackets, if we use the equations of constraints before or after calculation.
Quantization is understood as assignment to any function f of the operatorf, so that
f, g Dir=h −→
f,g
= ih. (12)
The constraint L= 0 and gauge conditionR = 0 give us the condi-tions between the operators, or we can use these for definition of the physical states|ψiphys by the equations
3
Light cone gauge
This gauge is often used in the string theory (see e.g. [4]) but it is not Lorentz covariant. For explicit description of this gauge we define new variables
P0±= √1 2 P
0 0 ±P
D−1 0
, X0±= √1 2 X
0 0±X
D−1 0
,
a±k =√1 2 a
0 k±a
D−1 k
.
We have
XY =−X+Y−−X−Y++ D−2
P β=1
XβYβ
and the Poisson brackets (7) take the form
f, g = ∞ P n=0
− ∂f
∂Xn+
∂g ∂Pn−
− ∂f
∂Xn−
∂g ∂Pn+
+ ∂f
∂Pn+
∂g ∂Xn−
+ ∂f
∂Pn−
∂g ∂Xn+
+ D−2
P β=1
∂f
∂Xnβ
∂g
∂Pnβ − ∂f
∂Pnβ
∂g
∂Xnβ
. (13)
We define the gauge conditions by the equations
R0=
X0+
P0+ −τ= 0, Rk = a+k
iωkP0+ = 0. (14)
From the Poisson brackets (13) we obtain on the set M
Ln, Lk =
Rn, Rk = 0 and
Ln, Rk =−δn,−k. (15) Equations (9) and (10) give in this gauge
κn= 0 forn6= 0 and κ0= 1, (16) and equations of motion are
˙
X0µ=P0µ, P˙0µ= 0, a˙µn= iωnaµn. (17)
With respect to (15) the Dirac bracket (11) of the functionsf and
g is
f, g Dir=
f, g + P n∈Z
f, Ln g, R−n −
f, Rn g, L−n
If we define new variables by (β = 1, . . . , D−2)
P+=P+
0 , Xb− =
X0−P0+−X0+P0− P0+ =X
− 0 −τ P
− 0 ,
Pβ=Pβ
0 , Xbβ =
X0βP0+−X0+P0β P0+ =X
β 0 −τ P
β 0 ,
b
X+=X+ 0 −τ P
+
0 , a
β k, a
+
k , k∈Z0, b
L0=P0+P −
0 , Lbk =P0+a − k ,
(18) we obtain that the only nonzero Dirac brackets are as follows
P+, b
X− Dir= 1,
Pβ, b
Xγ
Dir=−δ βγ,
aβk, aγ
n Dir= iωkδ βγδ
k,−n,
b
Xβ, b
L0 Dir=Pβ, Xbβ,Lbk Dir=aβk,
aβ
n,Lbk Dir= iωnaβn+k, n6=−k , aβk,Lb−k Dir= iωkPβ,
b
Ln,Lbk Dir= iω(n−k)Lbn+k.
Equations (14) and (8) have in the new variables the following form:
X+ = a+k = 0,
b
L0 = 12 D−2
P β=1
Pβ2 + ∞ P n=1 D−2 P β=1 aβ na β −n= 0,
b
Ln = D−2
P β=1
P0βaβn+ ∞ P k=1
aβn+kaβ−k+12 n−1
P k=1
aβn−kaβk, n >0
b
L−n = D−2
P β=1
P0βaβ−n+ ∞ P k=1
aβ−n−kaβk +12 n−1
P k=1
aβk−naβ−k n >0
and for the equations of motion (17) we obtain
˙
f =
f, HDir Dir, (19)
where
HDir= ∞ P n=1 D−2 P β=1
ana−n=Lb0−12 D−2
P β=1
Pβ2=−1 2PµP
µ= 1 2M
It results from the quantization principle (12) that we have to assign the operators to functions in such a way that
P+,Xb−
= i, Pβ,Xbγ
=−iδβγ,
aβk,aγn
=−ωkδβγδk,−n,
b
Xβ, b
L0
= iPβ,
aβk,Lb0
=−ωkaβk,
b
Xβ, b
Lk
= iaβk,
aβ n,Lbk
=−ωnaβn+k, n6=−k ,
aβk,Lb−k
=−ωkPβ,
b
Ln,Lbk
=ω(k−n)bLn+k
(20) hold.
To realize the operators we put
b
Ln= D−2
P β=1
Lβn, Lβ0 =12(Pβ)2+E0β, Lβk =Pβaβk+Eβk, k6= 0,
(21) where
P+,Eβn =
Pγ,Eβn =
X−,Eβn =
Xγ,Eβn = 0.
From (20) we obtain foraβk ,Eβk the following relations:
aβk,aγ n
=−kωδβγδ k,−n,
aβk,Eγ n
=−kωaβk+nδβγ,
Eβk,Eγ n
=ω(n−k)Eβk+nδβγ, aβ 0 = 0.
These operators Eβk and aβk form an algebra which will be denoted byA.
The gauge conditions X+ = a+
k = 0 give X
+ = a+
k = 0 and the representations of the operators P+,
b
X−, Pβ and b
Xβ (where
β= 1, . . . , D−2) can be obtained in a standard way. In the following, we concentrate on the representation of the algebra A.
3.1 Standard quantization of string
By the standard quantization (see [4]) the main emphasis is placed on the relations
aβk,aγn
It is well known that these operators are possible to realize on the Fock space V which is generated by the action of the operators aβk,
k >0 on the vacuum state|0iwhich is defined by the conditions
aβk|0i= 0 for k <0.
The operatorsEβk are then defined by
Eβ0 = ∞ P n=1
aβ na
β −n,
Eβ n=
∞ P k=1
aβn+kaβ−k+12 n−1
P k=1
aβn−kaβk,
Eβ−n= ∞ P k=1
aβkaβ−n−k+1 2
n−1 P k=1
aβk−naβ−k.
(22)
The normal ordering of the operators guarantees the convergence of operators (22), but if we put
Ek= D−2
P β=1
Eβk,
we obtain
En,Ek
=ω(k−n)En+k−
ω2
12(D−2)n(n 2
−1)δn,−k, (23)
which is commonly used in the string theory .
We see that this choice of the representation of the operators Ek is in contradiction with the relation
En,Ek=ω(k−n)En+k. (24) which follows from the standard classical theory.
Jµν are given by
J01 = 12 P+Xb−+Xb−P+
=P+Xb−−12i,
J+β = −P+Xbβ,
Jµν = PνXbµ−PµXbν− i
ω
∞ P n=1
aµ
naν−n−aνna µ −n
n ,
J−β = 1
P+
PβJ01+P−J+β−
−i
ω
D−2 P α=1
∞ P n=1
Pαa
α na
β −n−a
β na
α −n
n +
∞ P n=1
Enaβ−n−a β nE−n
n
,
whereP−= P+−11 2
D−2 P β=1
PβPβ+E 0
. The relation
J−β,J−γ
= 0 (25)
is violated since from the commutation relation we obtain
J−β,J−γ= ∞ P n=1
ω(D−26)n
12 a β na
γ
−n−aγna β −n
−
−2
n
P+P−−1 2
D−2 P α=1
PαPα−E 0−
ω(D−2) 24
aβ na
γ
−n−aγna β −n
.
Therefore to obtain relation (25) we have to putD= 26 and change the constraint, which definesP−, to
P+P−= 12 D−2
P β=1
PβPβ+E0+ω .
3.2 Dirac quantization
IfM+,M− > n, the direct calculation leads to the formula
M+
P k=1
aβn+kaβ−k+1 2
n−1 P k=1
aβn−kaβk,
M− P k=1
aβkaβ−n−k+1 2
n−1 P k=1
aβk−naβ−k
=
=ω
M+n P k=M+1
(k+n)aβkaβ−k−2ωn
M+n P k=n
aβkaβ−k−ω
n−1 P k=1
(n+k)aka−k−
−ω 2
n−1 P k=1
(n−k) aβkaβ−k+aβ−kaβk,
(26) where M = min M+, M−. The statement (26) can be rewritten in a different way. For the normal ordering aβ
na β
−n, which we used in 3.1, we have
ω
M+n P k=M+1
(k+n)aβkaβ−k−2ωn
M+n P k=1
aβkaβ−k−ω 2
12n n 2−1
(27)
and in the symmetric ordering we obtain
1 2ω
M+n P k=M+1
(k+n) aβka−βk+aβ−kaβk −ωn
M+n P k=1
aβkaβ−k+aβ−kaβk
. (28)
Formula (27) was used in 3.1. We see that the first term is zero on the Fock space but the second changes commutation relation (24) to (23).
In the Dirac quantization we have to use formula (28). However, in this case, we cannot use the representation of operatorsaβn on the Fock space because the series
1 2
D−2 P β=1
∞ P n=1
aβna β −n+a
β −naβn
|0i
is divergent.
So we must find other representations of aβk for which
lim m→∞m
D−2 P β=1
n P r=1
aβk+m+raβ−m−r+aβ−m−raβk+m+r= 0
In the limits the first term of (28) will be zero and for definition of the operatorsEk we can use
Eβ0 = 12 lim N→∞
N P n=1
aβnaβ−n+aβ−naβn,
Eβk = lim N→∞
N P k=1
aβn+kaβ−k+12 n−1
P k=1
aβn−kaβk,
Eβ−k = lim N→∞
N P k=1
aβkaβ−n−k+12 n−1
P k=1
aβk−naβ−k.
The second, alternative, possibility is to start from any represen-tation of the algebraA and try to define the physical states as the vectors|ψiphys, for which
lim N→∞
D−2 P β=1
Eβ0−1 2
N P n=1
aβnaβ−n+aβ−naβn
|ψiphys= 0,
lim N→∞
D−2 P β=1
Eβn− N P k=1
aβn+kaβ−k−1 2
n−1 P k=1
aβn−kaβk|ψiphys= 0,
lim N→∞
D−2 P β=1
Eβ−n− N P k=1
aβkaβ−n−k−1 2
n−1 P k=1
aβk−naβ−k|ψiphys= 0.
In both cases the commutator (24) and the Lorentz covariance will no longer be broken, but the explicit realization of this programme is an open question.
4
Lorentz covariant gauge
The gauge conditions, which we used in Section 3, are not Lorentz covariant. In this part, we will study the Lorentz covariant case. This gauge was partially used by Rohrlich in [5]. Specifically, we will take gauge conditions of the form
R0=
P0X0
P2 0
−τ = 0, Rk =
P0ak iωkP2
0
= 0, k∈Z0, (29)
where
XY =gµνXµYν and P02=P0P0=gµνP0µP ν
For the Poisson brackets (7) of these functions and constraints (8) on the setM(where the equations are valid (8) and (29)) we obtain
Ln, Lk = 0, Ln, Rk =−δn,−k, Rn, Rk = iδn,−k
ωnP2 0
.
For the constant from equations (10)κnwe have again the conditions (16) and the equations of motion (17).
In this case, the Dirac brackets (11) are
f, g Dir = f, g + P n∈Z
f, Ln g, R−n −
f, Rn g, L−n − − i ωP2 0 P k∈Z0
f, Lk g, L−k
k .
Particularly, for the Dirac brackets between the variables we have
P0µ, Pν
0 Dir= 0,
P0µ, Xν
0 Dir=−gµν+
P0µPν 0
P2 0
,
X0µ, X0ν Dir= 1
P2 0
P0µX0ν−P0νX0µ− i
ω ∞ P n=1 aν na µ
−n−aµnaν−n
n
=
=−J µν
P2 0
,
P0µ, aν
k Dir= iωk
P0µaν k
P2 0
,
aµk, X0ν Dir= 1
P2 0
P0µaνk+ iωk X0ν−2τ P0νaµk,
aµk, aν n Dir=
iωkn P2
0 P s6=0,−k,n
aµk+saνn−s
s , n+k6= 0,
aµk, aν
−k Dir= iωk
gµν−P µ 0P0ν
P2 0
+iωk
2
P2 0
P n∈Zk
aµk−naν n−k
n ,
in which in agreement with the constraints we putτ= P0X0
P2 0
.
The equations of motion can be written in the form (19), where
HDir=−1 2gµνP
µ 0P ν 0 = 1 2M 2.
If we put
Pµ=P µ 0
M , Q
µ=M Xµ 0 −τ P
µ 0
we obtain the Dirac brackets in the form
Pµ,Pν
Dir = 0,
Qµ,Pν Dir=g
µν+PµPν,
Qµ,Qν
Dir = Q
µPν− QνPµ− i
ω
∞ P n=1
aµ
naν−n−aνna µ −n
n =J
µν,
Pµ, aνk
Dir = 0,
Qµ, aνk
Dir=P νaµ
k,
aµk, aνn Dir = − iωkn
M2 P s∈Z+0
aµk+saν n−s
s , n+k6= 0, (30)
aµk, aν−k Dir = iωk gµν+PµPν −iωk
2
M2 P n∈Z0
aµk−naνn−k
n ,
M2,Pµ
Dir =
M2,Qµ
Dir= 0,
M2, aµk Dir=−2iωkaµk.
The constraints (8) and the gauge conditions (29) have in these vari-ables the form
PQ=Pak =PP+ 1 = 0, P
n∈Z
ana−n =M2,
P n∈Z
anak−n= 0, for k6= 0.
By the quantization of the first five equations in (30), we obtain
Pµ,Pν = 0,
Qµ,Pν= i gµν+PµPν,
Qµ,Qν = iPνQµ−PµQν− i
ω
∞ P n=1
aµ
naν−n−aνna µ −n
n
= iJµν,
Pµ,aνk
= 0,
Qµ,aνk
= iPνaµk.
Directly from these commutation relations it follows that
h
Pβ,
Qµ,Qµi
= igβµPν−igβνPµ, h
Qρ,Qµ,Qµi=gµρQν−gνρQµ
and by using the Jacobi identities we have
Jµν,Jρσ = −h
Qµ,Qν
,
Qρ,Qσi=
= hQρ,
Qσ,[Qµ,Qν]i
−hQσ,
Qρ,[Qµ,Qν]i =
which are the quantum relations for the operators of angular mo-mentaJµν.
Again, a big problem is how to find explicit realization of the operatorsaβk.
5
Conclusion
We have studied the Dirac theory of a free open string. We gave the explicit formulae for the Dirac brackets in the light-cone and Lorentz calibrations. We should like to emphasize that standard quantization, which is used now in modern string theory, changes a classical system by quantization.
Physically, the main difference between these two approaches is that in the standard method of quantization we suppose mutual in-dependence of the oscillators corresponding to the operatorsaβk and in the Dirac method these oscillators are strongly dependent.
We gave more attention to the ideas of Dirac quantization and formulated some new problem for quantization of the classical open string theory. The change of the classical system by quantization will not be needed and solution will be possible in any dimensionD.
The principal matter for solution of the problems of Dirac quan-tization of the string is to study the representations of the operators
aβk which are not of Fock type.
Partially supported by GACR 201/05/0857 and by research project MSM 6840770039.
References
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Comment on
DIRAC FORMULATION OF FREE OPEN
STRING
Andrzej Horzela
H. Niewodnicza´nski Institute of Nuclear Physics
Polish Academy of Sciences
Krak´ow, Poland
this by investigating the free open string in dimension D. They de-velop step by step classical canonical formalism for such a model and arrive at sets of Dirac brackets for two choices of gauge conditions: the so-called light cone gauge being Lorentz noncovariant and Lorentz co-variant gauge generalizing the condition proposed by Rohrlich about thirty years ago. Obviously, to get classical Dirac brackets is only a primary step if one wants to investigate a quantum system. The real challenge is to find a representation of an operator algebra obtained by formal replacement of Dirac brackets by commutators. Trying to solve this problem Burdik and Navratil are partially successful - for the light cone gauge they find a Fock-type representation which for
D = 2 and D = 26 is consistent with results following from stan-dard classical theory. Unfortunately, they do not give an explanation whether these specific values ofD have any physical meaning or are purely incidental. Investigating the Lorentz covariant gauge the au-thors arrive at results which for me seem more intriguing. For this case the quantization of Dirac brackets leads to the algebra in which canonical commutation rules between coordinateQµ and momentum
Pν operators are modified by terms proportional toPµPν, momenta commute and commutators between coordinates are proportional to the components of angular momentum. Such an algebra coincides with noncanonical algebras found in investigations rooted in quan-tum gravity and leading to theories with fundamental lenght [1] and for the so-called Wigner quantum systems [2], [3]. More detailed in-vestigation of such a coincidence seems to be an interesting problem as well as it would be extremely useful to find representations of this algebra. This, I believe, will be the subject of Burdik and Navratil’s further research.
References
[1] A.KempfJ. Phys.A:Math.Gen.30(1997) 2093
[2] R.C.King, T.D.Palev, N.I.Stoilova and J.Van der Jeught J. Phys.A: Math. Gen.36(2003) 4337,ibid. 36(2003) 11999
