Integrable string and hydrodynamical type models and nonlocal brackets

0.14 MB·English

by  V. D. Gershun

Published by Institute of Physics Publishing for SISSA/ISAS
Received: , 2005
Revised: , 2005
Accepted: , 2005
5
0
Integrable string and hydrodynamical type models and
0
J
2
nonlocal brackets
n H
a
J E

2 P
1
0

]
V. D. Gershun
I 0
S
O. I. Akhieser Institute for Theoretical Physics, NSC Kharkiv Institute of Physics and
n. (
Technology, Academy of Sciences of Ukraine E-mail: [email protected]
i
l 2
n
[ 0

Abstract: The closed string model in the background gravity field is considered as a bi-
2
0
v Hamiltonian system in assumption that string model is the integrable model for particular
1 kind of the background fields. The dual nonlocal Poisson brackets (PB), depending of 2
5
0 the background fields and of their derivatives, are obtained. The integrability condition )
1
is formulated as the compatibility of the bi-Hamiltonity condition and the Jacobi identity
1 0
4 of the dual Poisson bracket. It is shown that the dual brackets and dual Hamiltonians
0
0
can be obtained from the canonical PB and from the initial Hamiltonian by imposing the
/
n 0
second kind constraints on the initial dynamical system, on the closed string model in the
i
nl constant background fields, as example. The hydrodynamical type equation was obtained.
: Twotypesofthenonlocalbracketsareintroduced. Constantcurvatureandtime-dependent
v
i metrics are considered, as examples. It is shown, that the Jacobi identities for the nonlocal
X
brackets have particular solution for the space-time coordinates, as matrix representation
r
a
of the simple Lie group.
Keywords: bst, igt.
(cid:13)c SISSA/ISAS2008 http://jhep.sissa.it/JOURNAL/JHEP3.tar.gz
Contents
1. Introduction 1
2. Hydrodynamical type models 2
3. Closed string in the background fields. 4
J
4. Constant background fields (g = const.). 9
ab
H
5. Acknowledgments 11
E
P
0
0
1. Introduction
(
The bi-Hamiltonian approach to the integrable systems was initiated by Magri [1] for the 2
investigation of the integrability of the KdV equation and was generalized by Das, Okubo 0
[2]. Definition 1. A finite dimensional dynamical system with 2N degrees of freedom
0
xa,a = 1,...2N is integrable, if it is described by the set of the n integrals of motion
2
F ,...,F in involution under some Poisson bracket (PB)
1 n
)
F ,F = 0.
i k PB 0
{ }
0
The dynamical system is completely solvable, if n = N. Any of the integral of motion
(or any linear combination of them) can be considered as the Hamiltonian H = F . 0
k k
Definition 2. The bi-Hamiltonity condition [2] has following form:
dxa
x˙a = = xa,H = ... = xa,H . (1.1)
1 1 N N
dt { } { }
The hierarchy of new PB is arose in this connection:
, , , ,.. , .
1 2 N
{ } { } { }
The hierarchy of new dynamical systems arises under the new time coordinates t .
k
dxa
= xa,H = xa,H . (1.2)
n k+1 k n+1
dt { } { }
n+k
The new equations of motion describe the new dynamical systems, which are dual to the
original system, with the dual set of the integrals of motion.
There is another approach to the bi-Hamiltonian systems [1]. Two PB , and , are
1 2
{ } { }
called compatible if any linear combination of these PB c , +c , is PB. It is possible
1 1 2 2
{ } { }
to find two corresponding Hamiltonians H and H which are satisfy to bi-Hamiltonity
1 2
– 1 –
condition.
We used first approach to the closed string models as the bi-Hamiltonity systems. Second
approach was used to description of the hydrodynamical type models.
We consider the dynamical systems with constraints. In this case, first kind constraints
are generators of the gauge transformations and they are integrals of motion. First kind
constrains F (xa) 0, k = 1,2... form the algebra of constraints under some PB.
k
≈
F ,F = Cl F 0.
i k PB ik l
{ } ≈
The structure functions Cl may be functions of the phase space coordinates in general J
ik
case. The second kind constraints fk(xa) 0 are the representations of the first kind H
≈
constraints algebra. The second kind constraints is defined by the condition
E
P
f ,f = C = 0.
i k ik
{ } 6
0
The reversible matrix C is not constraint and also it is a function of phase space coor- 0
ik
dinates. The second kind constraints take part in deformation of the , PB to the Dirac (
{ }
bracket , . As rule, such deformation leads to nonlinear and to nonlocal brackets. The
{ }D 2
bi-Hamiltonity condition leadstothedualPBthatarenonlinearandnonlocalbrackets asa
0
rule. We suppose,that thedualbrackets can beobtained fromtheinitial canonical bracket
0
undertheimpositionofthesecondkindconstraints. Wehaveapplied[3,4,5,6], [7,8,9,10]
bi-Hamiltonian approach to the investigation of the integrability of the closed string model 2
in the arbitrary background gravity field and antisymmetric B-field. The bi-Hamiltonity
)
condition and the Jacobi identities for the dual brackets were considered as the integrabil-
0
ity condition for a closed string model. They led to some restrictions on the background
0
fields.
Nonlocal PB of a hydrodynamical type was obtained as the Dirac bracket by Ferapontov 0
[13] and by Maltsev [14].
The plan of the paper is the following. In the second section we briefly considered papers
about hydrodynamical type nonlocal brackets. In the third section we considered closed
string model in the arbitrary background gravity field. We suppose that this model is
an integrable model for some configurations of the background fields. The bi-Hamiltonity
condition and the Jacobi identities for the dual PB resulted in to the integrability con-
dition, which restrict the possible configurations of the background fields. As examples
we considered constant curvature space and time-dependent metric space. In the fourth
section we considered closed string model in the constant background gravity field. We
obtainedhydrodynamicaltypeequationforthestringmodelonthesecondkindconstraints
as configuration subspace embedded in a phase space.
2. Hydrodynamical type models
Mokhov and Ferapontov introduced the nonlocal PB [11]. The Ferapontov nonlocal PB
– 2 –
(or hydrodynamical type nonlocal PB ) [12] is:
L
∂
ui(x),uk(y) = gik(u) δ(x y) gijΓkulδ(x y)+ ω(s)i(u(x))ujν(x y)ω(s)k(u(y))ul,
{ } ∂x − − jl x − j x − l y
X
s=1
(2.1)
where ν(x y) = sgn(x y) = ( d )−1δ(x y), ui(x) are local coordinates, ui(x) =
− − dx − x
∂ ui(x),i = 1...N. The coefficients gik(x),Γk(x),ω(s)i(x) are smooth functions of local
x jl k
coordinates. This nonlocal PB is satisfy the Jacoby identity if and only if gik(u) is the
pseudo-Riemannian metric without torsion and also the coefficients satisfy the following
relations:
J
1. Γk(u) is the Levi-Civita connection,
jl
H
2. gik(u)ω(s)j(u) = gjk(u)ω(s)i(u),
k k
E
(s)i (s)i
3. ω (u) = ω , where is the covariant differential,
4. R∇ikj(ul) = L∇[lω(ks)iω(s)j ω∇(sk)jω(s)i], where Rij is Riemannian curvature tensor of the P
kl s=1 l k − l k kl
metric gik, P 0
(s)i (t)k (t)i (s)i
5. ω ω = ω ω . 0
k l k k
This nonlocal PB corresponds to an N-dimensional surface with flat normal bundle em-
(
bedded in a pseudo-Euclidean space EN+L [13]. There metric gik is the first fundamental
2
(s)i
form, ω is Weingarten operator of this embedded surface, which is define the second
k
0
fundamental form. The relations 2)-4) are the Gauss-Peterson-Codazzi equations. The
relations 5) are correspond to the Ricci equations for this embedded surface. 0
Dubrovin and Novikov have considered the local dual PB of the similar type [16] in the 2
application to the Hamiltonian hydrodynamical models. Dubrovin-Novikov PB ( or the
)
hydrodynamicaltypelocalPB)canbeobtainedfromthenonlocalPB(2.1)undercondition
0
(s)i
ω = 0. The Jacobi identity for this PB is satisfied if g is the Riemann metric without
k ik
0
torsion, the curvature tensor is equal to zero. The metric tensor is constant, locally.
It need to consider the linear combination of the local and the nonlocal Poisson brackets to 0
obtain the hydrodynamical type equations [17]. There we consider Mokhov, Ferapontov,
nonlocal PB [11] for the metric space of constant Riemannian curvature K, as example:
d ∂hk ∂hi d
ui(x),uk(y) = c ηik δ(x y)+c ( + Kuiuk) δ(x y)
{ } 1 dx − 2 ∂ui ∂uk − dx −
∂2hk
+( Kδiuk)ulδ(x y)+Kuiν(x y)uk. (2.2)
∂ui∂ul − l x − x − y
The canonical form of PB (2.2) was first presented by Pavlov [15]. The Jacobi identity is
satisfied on the following relations:
∂2hi ∂2hj ∂2hj ∂2hi
= ,
∂uk∂un∂un∂ul ∂uk∂un ∂un∂ul
∂hn ∂hi ∂2hk
( + Kuiun) = i j .
∂ui ∂un − ∂uj∂un { ←→ }
First of this equations is the WDVV [18, 19] consistence local condition. The system of
hydrodynamical type is a bi-Hamiltonian system with the PB , and , if:
FM ND
{ } { }
u˙i(x) = ui(x),H = ui(x),H .
1 FM 2 ND
{ } { }
– 3 –
Where Hamiltonians H and H are following:
1 2
1 K
H = ui(x)ui(x)dx, H = [hi(u(x))ui(x) uiuiukuk]dx.
1 2 Z 2 Z − 8
3. Closed string in the background fields.
Thestringmodelinthebackgroundgravityfieldisdescribedbythesystemoftheequations:
x¨a x′′a+Γa (x)(x˙bx˙c x′bx′c) = 0, g (x)(x˙ax˙b+x′ax′b) =0, g (x)x˙ax′b = 0,
bc ab ab
− −
where x˙a = dxa, x′a = dxa. We will consider the Hamiltonian formalism. The closed string J
dτ dσ
H
in the background gravity field is described by first kind constraints in the Hamiltonian
formalism: E
h = 1gab(x)p p + 1g (x)x′ax′b 0, h = p x′a 0, (3.1)
1 a b ab 2 a P
2 2 ≈ ≈
wherea,b= 0,1,...D 1, xa(τ,σ),p (τ,σ)aretheperiodicalfunctionsonσ withtheperiod 0
a
−
on π. The original PB are the canonical PB: 0
(
xa(σ),p (σ′) = δaδ(σ σ′), xa(σ),xb(σ′ = p (σ),p (σ′) = 0.
b 1 b 1 a b 1
{ } − { } { }
2
The Hamiltonian equations of motion of the closed string, in the arbitrary background
0
π
gravity field under the Hamiltonian H1 = h1dσ and PB , 1, are 0
{ }
R0
2
x˙a =gabp , p˙ = g x′′b 1∂gbcp p 1∂gbcx′bx′c+ ∂gacx′bx′c. )
b a ab − 2 ∂xa b c− 2 ∂xa ∂xb
0
The dual PB are obtained from the bi-Hamiltonity condition
0
π π
0
x˙a = xa, h (σ′)dσ′ = xa, h (σ′)dσ′ , (3.2)
{ Z 1 }1 { Z 2 }2
0 0
π π
p˙ = p , h (σ′)dσ′ = p , h (σ′)dσ′ .
a { a Z 1 }1 { a Z 2 }2
0 0
They have the following form:
Proposition 1.
∂A ∂B ∂
A(σ),B(σ′) = [[ωab(σ)+ωab(σ′)]ν(σ′ σ)+[Φab(σ)+Φab(σ′)] δ(σ′ σ)
{ }2 ∂xa∂xb − ∂σ′ −
∂A ∂B
+[Ωab(σ)+Ωab(σ′)]δ(σ′ σ)]+ [[ω (σ)+ω (σ′)]ν(σ′ σ)+
ab ab
− ∂p ∂p −
a b
∂
+[Φ (σ)+Φ (σ′)] δ(σ′ σ)+[Ω (σ)+Ω (σ′)]δ(σ′ σ)]+
ab ab ∂σ′ − ab ab −
∂A ∂B ∂A ∂B ∂
+[ + ][[ωa(σ)+ωa(σ′)]ν(σ′ σ)+[Φa(σ)+Φa(σ′)] δ(σ′ σ)]
∂xa∂p ∂p ∂xa b b − b b ∂σ′ −
b b
– 4 –
∂A ∂B ∂A ∂B
+[ ][Ωa(σ)+Ωa(σ′)]δ(σ′ σ)
∂xa∂p − ∂p ∂xa b b −
b b
The arbitrary functions A,B,ω,Φ,Ω are the functions of the xa(σ),p (σ). The functions
a
ωab,ω , Φab,Φ are the symmetric functions on a,b and Ωab,Ω are the antisymmetric
ab ab ab
functions to satisfy the condition A,B = B,A . The equations of motion under the
2 2
π { } −{ }
Hamiltonian H = h (σ′)dσ′ and PB , are
2 2 2
{ }
R0
x˙a = ωaxb+2ωabp +2Φabp′′ 2Φax′′b+2Ωax′b 2Ωabp′+
b b b b b b
− − −
π J
+Z dσ′[ωbax′a+ ddωσa′bpb]ν(σ′−σ)+ ddΦσabp′b− ddΦσabx′b, H
0 E
p˙a = ωabxb 2Φabx′′b+2Ωabx′b+2ωabpb+2Φbap′b′ +2Ωbap′b+ P
− −
π 0
+ dσ′[ω x′b+ dωabp ]ν(σ′ σ) dΦabx′b+ dΦbap′.
Z ab dσ′ b − − dσ dσ b 0
0
(
The bi-Hamiltonity condition (3.2) is led to the two constraints
2
ωaxb+2ωabp +2Φabp′′ 2Φax′′b+2Ωax′b 2Ωabp′+ 0
− b b b − b b − b
0
π
dσ′[ωax′a+ dωabp ]ν(σ′ σ)+ dΦabp′ dΦabx′b = gabp , 2
Z b dσ′ b − dσ b− dσ b
)
0
0
ω xb 2Φ x′′b+2Ω x′b+2ωbp +2Φbp′′ +2Ωbp′+ 0
ab ab ab a b a b a b
− −
0
π
+ dσ′[ω x′b+ dωabp ]ν(σ′ σ) dΦabx′b+ dΦbap′ =
Z ab dσ′ b − − dσ dσ b
0
+g x′′b 1∂gbcp p 1∂gbcx′bx′c+ ∂gacx′bx′c.
ab − 2 ∂xa b c− 2 ∂xa ∂xb
In really, there is the list of the constraints depending on the possible choice of the
unknown functions ω, Ω, Φ. In the general case, there are both the first kind constraints
and the second kind constraints. Also it is possible to solve the constraints equations as
the equations for thedefinition of the functionsω,Φ,Ω. We considered the latter possibility
and we obtained the following consistent solution of the bi-Hamiltonity condition:
∂ωab
Φab = 0, Ωab = 0, Φa = 0, Ωa = 0, xc+2ωab = gab,
b b ∂xc
1 ∂2ωcd ∂ωac
ω = p p , ωa = p ,
ab 2∂xa∂xb c d b − ∂xb c
Φ = 1g ,Ω = 1(∂Φbc ∂Φac)x′c, ∂ωab = 0.
ab −2 ab ab 2 ∂xa − ∂xb ∂p
c
– 5 –
Remark 1. In distinct from the PB of the hydrodynamical type, we need to introduce
the separate PB for the coordinates of the Minkowski space and for the momenta because,
the gravity field is not depend of the momenta. Although, this difference is vanished under
the such constraint as f(xa,p ) 0.
a
≈
Consequently, the dual PB for the phase space coordinates are
xa(σ),xb(σ′) = [ωab(σ)+ωab(σ′)]ν(σ′ σ),
2
{ } −
∂2ω (σ) ∂2ω (σ′)
p (σ),p (σ′) = [ cd p p + cd p p ]ν(σ′ σ) J
{ a b }2 ∂xa∂xb c d ∂xa∂xb c d − −
H
1[g (σ)+g (σ′)] ∂ δ(σ′ σ)+[∂gac ∂gbc]x′c(σ)δ(σ′ σ)
−2 ab ab ∂σ′ − ∂xb − ∂xa − E
∂ωac(σ) ∂ωac(σ′) P
xa(σ),p (σ′) = [ p + p ]ν(σ′ σ),
{ b }2 − ∂xb c ∂xb c − 0
∂ωbc(σ) ∂ωbc(σ′) 0
p (σ),xb(σ′) = [ p + p ]ν(σ′ σ). (3.3)
{ a }2 − ∂xa c ∂xc c − (
The function ωab(x) is satisfied on the equation: 2
0
∂ωab
xc+2ωab = gab. (3.4) 0
∂xc
2
The Jacobi identities for the PB , are led to the nonlocal consistence conditions on the
2
{ } )
unknown function ωab(σ). We can calculate unknown metric tensor gab(σ) by substitution
of the solution of the consistence condition for ωab to the equation (3.4). The Jacobi 0
identity 0
xa(σ),xb(σ′) xc(σ′′) J (3.5) 0
{ } } ≡
xa(σ),xb(σ′) xc(σ′′) + xc(σ′′),xa(σ) xb(σ′) + xb(σ′),xc(σ′′) xa(σ ) = 0
{ } } { } } { } }
is led to the following nonlocal analogy of the WDVV [18, 19] consistence condition:
∂ωab(σ) ∂ωac(σ)
[ [ωdc(σ)+ωdc(σ′′)] [ωdb(σ)+ωdb(σ′)]]ν(σ′ σ)ν(σ′′ σ)+
∂xd − ∂xd − −
∂ωcb(σ′) ∂ωab(σ′)
[ [ωda(σ′)+ωda(σ)] [ωdc(σ′)+ωdc(σ′′)]]ν(σ σ′)ν(σ′′ σ′)+
∂xd − ∂xd − −
∂ωac(σ′′) ∂ωcb(σ′′)
[ [ωdb(σ′′)+ωdb(σ′)] [ωda(σ′′)+ωda(σ)]]ν(σ σ′′)ν(σ′ σ′′)= 0. (3.6)
∂xd − ∂xd − −
This equation has the particular solution of the following form:
∂ωab(σ) ∂ωac(σ)
[ωdc(σ)+ωdc(σ′′)] [ωdb(σ)+ωdb(σ′)] =
∂xd − ∂xd
[Tb,Tc]Ta]f(σ,σ′,σ′′)ν(σ′′ σ)ν(σ′ σ),
− −
– 6 –
where Ta,a = 0,1,...D 1 is the matrix representation of the simple Lie algebra and
−
f(σ,σ′,σ′′) is arbitrary function. The Jacobi identity is satisfied on the Jacobi identity of
the simple Lie algebra in this case:
([Ta,Tb]Tc]+[Tc,Ta]Tb]+[Tb,Tc]Ta])f(σ,σ′,σ′′) = 0
and we used the relation ν2(σ′ σ) = 1. The local solution of the Jacobi identities leads
−
to the constant metric tensor. The rest Jacobi identities are cumbrous and we do not
reduce this expressions here. The symmetric factor of σ,σ′ of the antisymmetric functions
ν(σ′ σ), ∂ δ(σ σ′) in the right side of the PB can be both sum of the functions of σ and
σ′, a−nd pr∂oσducti−on of them. Last possibility can be used in the vielbein formalism. J
Proposition 2. The bi-Hamiltonity condition can be solved in the terms PB , , which H
2
{ }
have the following form: E
xa(σ),xb(σ′) = ea(σ)eb(σ′)ν(σ′ σ), P
2 µ µ
{ } −
0
∂ec(σ′)
xa(σ),p (σ′ = ea(σ) µ p (σ′)ν(σ′ σ), 0
{ b }2 − µ ∂xb c −
(
∂ec(σ) ∂ed(σ′) ∂
{pa(σ),pb(σ′)}2 = ∂µxa pc(σ) ∂µxb pd(σ′)ν(σ′−σ)−eµa(σ)eµb(σ′)∂σ′δ(σ′ −σ)+ 2
+[∂eµaeµ ∂eµb eµ ∂eµc eµ+ ∂eµc eµ]x′c(σ)δ(σ′ σ), (3.7) 0
∂xc b − ∂xc a − ∂xa b ∂xb a − 0
where veilbein ea is satisfied on the additional conditions:
µ 2
gab = ηµνeaeb, g = η eµeν )
µ ν ab µν a b
0
and ηµν is the metric tensor of the flat space.
0
The particular solution of the Jacobi identity is
0
∂ea(σ) ∂ea(σ)
µ eb(σ′)ed(σ)ec(σ′′) µ ec(σ′′)ed(σ)eb(σ′) =
∂xd µ ν ν − ∂xd µ ν ν
[Tb,Tc]Ta]f(σ,σ′,σ′′)ν(σ′′ σ)ν(σ′ σ).
− −
As example let me consider the the constant curvature space.
Example 1. Theconstant curvaturespace is described by the metric tensor g (x(σ)) and
ab
by it inverse tensor g−1:
ab
kx x
g =η + a b , gab g−1 = η kx x .
ab ab 1 kx2 ≡ ab ab− a b
−
Proposition 3. Dual (PB) , are:
2
{ }
x (σ),x (σ′) = [η kx (σ)x (σ′)]ν(σ′ σ),
a b ab a b
{ } − −
x (σ),p (σ′) = kx (σ)p (σ′)ν(σ′ σ),
a b a b
{ } −
p (σ),p (σ′) = kp (σ)p (σ′)ν(σ′ σ)
a b a b
{ } − −
– 7 –
1 kx x kx x ∂ x x′ x x′
[2η + a b (σ)+ a b (σ′)] δ(σ′ σ)+ a b− b aδ(σ′ σ). (3.8)
− 2 ab 1 kx2 1 kx2 ∂σ′ − 2(1 kx2) −
− − −
The Jacobi identity (3.5) is led to the equation
[η x (σ′′) η x (σ′)]ν(σ′ σ)ν(σ σ′′)+[η x (σ) η x (σ′′)]ν(σ σ′)ν(σ′ σ′′)+
ab c ac b bc a ba c
− − − − − −
[η x (σ′) η x (σ)]ν(σ′ σ′′)ν(σ′′ σ) = 0.
ca b cb a
− − −
The particular solution of this equation is:
J
η x (σ′′) η x (σ′)= [T ,T ]T ]f(σ,σ′,σ′′)ν(σ′′ σ)ν(σ′ σ). (3.9)
ab c ac b b c a
− − − H
Consequently, the space-time coordinate x (σ) is the matrix representation of the simple E
a
Lie algebra. The Jacobi identity x (σ),x (σ′) p (σ′′) is led to the equation P
a b c J
{ } }
0
kη p (σ′′)ν(σ′ σ)[ν(σ′′ σ)+ν(σ′′ σ′)] =0. (3.10)
ab c
− − − 0
These results can be obtained from the veilbein formalism under the following ansatz for (
the veilbein of the constant curvature space: 2
0
ea(s) = n (m(s)na+√ km(s)xa),eµ(s) = nµg (m(s)nb+√ km(s)xb),
µ µ 1 − 2 a ab 1 − 2 0
where n2 = 1, m(s)m(s) = 1, m(s)m(s) = 1, m(s)m(s) = 0, nanb = δab and (s) is number 2
µ 1 1 2 2 1 2
of the solution of the equations )
0
eaeb = gab, eµeµ = g , eaeµ = δa.
µ µ a b ab µ b b 0
0
The following example is time-dependent metric space.
Example 2. The time-dependent metric in the light-cone variables has form:
ds2 = g (x+)dxidxk +g (x+)dx+dx+ +2g dx+dx−. (3.11)
ik ++ +−
We are used Poisson brackets (3.3) for the space coordinates xa = xi,x+,x− , i =
{ }
1,2...D 2. We introduced the light-cone gauge as two first kind constraints:
−
F (σ) = x′+ 0, F (σ) = p′ 0,
1 2 −
≈ ≈
and we imposed them on the equations of motion and on the Jacobi identities. The Jacobi
identities are reduced to the simple equation
∂ωab ∂ωac
ω+c ω+b =0.
∂x+ − ∂x+
We obtained following result from this equation and additional condition (3.4): there is
constant background gravity field only for the non-degenerate metric.
– 8 –
4. Constant background fields (g = const.).
ab
In this section we aresupplemented thebi-Hamiltonity condition (3.2) by the mirrortrans-
formations of the integrals of motion.
π π
x˙a = xa, h dσ′ = xa, h dσ′ .
{ Z 1 }1 { Z ± 2 }±2
0 0
The dual PB are
J
xa(σ),xb(σ′) = gabν(σ′ σ), xa(σ),p (σ′) = 0,
±2 b ±2
{ } ± − { }
H
∂
p (σ),p (σ′) = g δ(σ′ σ). E
{ a b }±2 ∓ ab∂σ′ −
P
The dual dynamical system
0
x˙a = xa, H = xa,H .
2 1 1 ±2 0
{ ± } { }
(
is the left(right) chiral string
x˙a = x′a, p˙ = p′. 2
a a
± ±
0
Another way to obtain the dual brackets is the imposition of the second kind constraints
on the initial dynamical system, by such manner, that F = F for i = k,i,k = 1,2,... on 0
i k
6
the constraints surface f(xa,pa)= 0. 2
Example 3 The constraints f(−)(x,p) = p g x′b 0 or f(+) = p +g x′b 0 (do not
a a ab a a ab )
− ≈ ≈
simultaneously) are the second kind constraints.
0
f(±)(σ),f(±)(σ′) = C(±)(σ σ′)= 2g ∂ δ(σ′ σ). 0
{ a b }1 ab − ± ab∂σ′ −
0
The inverse matrix (C(±))−1 has following form C(±)ab(σ σ′) = 1gabν(σ′ σ). There is
− ±2 −
only one set of the constraints, because consistency condition
f(±)(σ),H = f′(±)(σ) 0, ... , f(±)(n)(σ),H =f(±)(n+1)(σ) 0.
1 1 1 1
{ } ≈ { } ≈
is not produce the new sets of constraints. By using the standard definition of the Dirac
bracket, we are obtained following Dirac brackets for the phase space coordinates.
1 1 ∂
xa(σ),xb(σ′) = gabν(σ′ σ), p (σ),p (σ′) = g δ(σ′ σ),
{ }D ±2 − { a b }D ∓2 abσ′ −
1
xa(σ),p (σ′) = δaδ(σ′ σ).
{ b }D 2 b −
equation The equations of motion under the Hamiltonians H = h ,H = h and Dirac
1 1 2 2
bracket
x˙a = xa,H = xa,H = gabp = x′a,
1 D 2 D b
{ } { } ±
p˙ = p ,H = p ,H = g x′b = p′.
a a 1 D a 2 D ab a
{ } { } ±
– 9 –