Solution of Constraints in Theory of Relativistic String

0.18 MB·English

by  B. M. Barbashov

Solution of Constraints in Theory of Relativisti
String
1 1
B.M. Barbashov and V.N. Pervushin
1
Joint Institute for Nu
lear Resear
h,
141980, Dubna, Russia
6
0
0
2

n Abstra
t
a
J The Hamiltonian theory of a relativisti
string is
onsidered in a spe
ifi
referen
e frame in

7 terms the diffeo-invariant variables.
1 Theevolutionparameterandenergyinvariantwithrespe
ttothetime-
oordinatetrans-

formationsare
onstru
ted,sothatthedimensionofthekinemetri
groupofdiffeomorphisms
1

oin
ideswiththedimensionofasetofvariableswhosevelo
itiesareremovedbytheGauss-
v
5 type
onstraints in a

ordan
e with the se
ond No(cid:4)ther theorem. This
oin
iden
e allows us
1 to solve the energy
onstraint, and fulfil Dira
's Hamiltonian redu
tion.
1
1
0
6
0
/
h
t
- Invited Talk at INTAS Summer S
hool and Conferen
e
p
(cid:16)New trends in High-Energy Physi
s(cid:17)
e
h Yalta, Crimea, Ukraine, September 10 - 17, 2005
:
v
i
X
r
a
1
1. Introdu
tion
The main diffi
ulty of the theory of a relativisti
string is the group of diffeomorphisms, i.e.
general
oordinate transformations [1℄.
There is an essential differen
e between the frame group of the Lorentz (cid:21) Poin
ar(cid:1)e-type
leadingtoasetofinitialdataandthediffeomorphismgroupofgeneral
oordinatetransformations
restri
tingtheseinitialdataby
onstraints. Thisdifferen
ewasrevealedbytwoN(cid:4)othertheorems
1
[5℄ .
Starting position of the paper is the group of diffeomorphisms of this (cid:16)Hamiltonian frame(cid:17)
known as the kinemetri
one in general relativity [6℄. This kinemetri
group
ontains global
reparametrizations of the
oordinate time and lo
al transformations of a spatial
oordinate, and
it requires to distinguish a set of variables with the same dimension, velo
ities (or momenta) of
whi
h are removed by the Gauss-type
onstraints from the phase spa
e of physi
al variables in
a

ordan
e with the se
ond N(cid:4)other theorem.
A similar Hilbert-type [2, 3℄ geometro-dynami
formulation of spe
ial relativity (SR) [7℄ with
reparameterizations of the
oordinate evolution parameter showsthattheenergy
onstraintfixes
a velo
ity of one of the dynami
variables that be
omes a evolution parameter. In parti
ular,
in SR su
h a dynami
evolution parameter is well known, it is the fourth
omponent of the
Minkowskian spa
e-time ve
tor.
In order torealize a similar
onstru
tion of theDira
(cid:21)ADMHamiltonian GR [8℄, one should
point out in GR a homogeneous variable that
an be a diffeo-invariant evolution parameter in
the field spa
e of events [9℄ in a

ordan
e with the kinemetri
diffeomorphism group of GR in
the (cid:16)Hamiltonian frame(cid:17) [6℄.
Thus, inthestandard approa
h toarelativisti
string thishomogeneous evolution parameter
is not split, so that the dimension of the Hamiltonian
onstraints does not
oin
ide with the
dimension of the diffeomorphism group.
In the present paper, a evolution parameter is identified with the homogeneous
omponent
of the time-like variable [10℄ in a

ordan
e with the se
ond N(cid:4)other theorem.
2. Diffeo-Invariant Content of a Relativisti
String
To illustrate the invariant Hamiltonian redu
tion let us
onsider the a
tion for a relativisti
string [1℄ in the form whi
h has been done by L. Brink, P. Di Ve

hia, P. Howe [11℄
γ
S = d2u√ ggαβ∂αxµ(u0,u1)∂βxµ(u0,u1), (u0,u1) = (τ,σ),
−2 − (1)
Z Z
xµ(τ,σ) (µ = 0,1,2,...,d 1)
where (cid:21) string
oordinates given in d-dimension spa
e-time − ,
gαβ(u0,u1)
(cid:21) is a se
ond-rank metri
tensor on the string surfa
e (two dimensional Riemannian
u0,u1
spa
e ).
Now let us
onsider the Hamiltonian s
heme whi
h is based on the Arnowitt(cid:21)Deser(cid:21)Misner
g
αβ
parametrization of metri
tensor [12℄
gαβ = Ω2(cid:18)N02N−1N12 −N11(cid:19), gαβ = Ω21N02 (cid:18)N11 N12N−1N02(cid:19), √−g = Ω2N0 (2)
with the
onformal invariant interval
ds2 =gαβduαduβ = Ω2[N02dτ2 (dσ+N1dτ)2],
− (3)
1
One of these theorems (the se
ond) was formulated by Hilbert in his famous paper [2℄ (see also its revised
version [3℄). We should like to thankV.V. Nesterenko who draw our attentionto this fa
t [4℄.
2
N0 N1(τ,σ)
where and are known in GR as the lapse fun
tion and (cid:16)shift ve
tor(cid:17), respe
tively.
Ω
Thea
tion (1) after thesubstitution(2)doesnot depend onthe
onformal fa
tor andtakes
the form S = γ τ2dτ ldσ (x˙µ−N1x′µ)2 N0x′2 ,
−2 Zτ1 Z0 " N0 − # (4)
x˙µ = ∂τxµ x′µ = ∂σxµ x˙µ N1x′µ = Dxµ
where , and − is the
ovariant derivative with respe
t to
Dx
µ
the metri
(3). The a
tion (4), the metri
(3) and the
ovariant derivative are invariant
τ τ˜= f1(τ), σ σ˜ = f2(τ,σ)
under the (cid:16)kinemetdriτ˜
(cid:17)=trfa˙1n(sτf)odrτmadtσ˜io=n f˙2→(τ,σ)dτ +f2′(τ,→σ)dσ . These transformations
of the differentials ,
orrespond to transformations of
the string
oordinates
xµ′(τ,σ) = x˜µ′(τ˜,σ˜)f2′(τ,σ),
x˙µ(τ,σ) = x˜˙µ(τ˜,σ˜)f˙1(τ)+x˜µ′(τ˜,σ˜)f˙2(τ,σ),
(5)
N0(τ,σ) = N˜0(τ˜,σ˜) f˙1(τ) , N1(τ,σ) = N˜1(τ˜,σ˜)f˙1 + f˙2,
f2′(τ,σ) f˜2′ f˜2′
Dτxµ(τ,σ) = f˙1(τ)Dτ˜x˜µ(τ˜,σ˜)
(6)
S N0 N1 N0,N1
The variation with respe
t to and leads to equation for determination
δS (Dx )2 (x˙x′)2 x˙2x′2
δN0 = N02µ +x′2 = 0 ⇒ N02 = (x′−2)2 , (7)
δS (x′ Dxµ) (x˙x′)
µ
δN1 = N0 ⇒ N1 = x′2 , (8)
Thesubstitutionoftheseequationina
tion(4)
onvertsitintothestandardNambu(cid:21)Gotoa
tion
of the relativisti
string [1℄
τ2 l
S = γ dτ dσ (x˙x′)2 x˙2x′2.
− Zτ1 Z0 − (9)
p
S
One
an
onstru
t the Hamiltonian form of . The
onjugate momenta are determined by
pµ = δδx˙Sµ = γx˙µ−NN01xµ′ ⇒ x˙µ = N0pγµ +N1xµ′ (10)
and density of Hamiltonian is obtained by the Legendre transformation
p2+γ2x′2
H = pµx˙µ L = N0 +N1(px′),
− 2γ (11)
then τ2 l
S = dτ dσ[p x˙µ H].
µ
−Zτ1 Z0 − (12)
S N0,N1
The se
ondary
lass
onstraints arises by varying with respe
t to
δS p2+γ2x′2 δS
= = 0, = (px′)= 0.
δN0 2γ δN1 (13)
3
The equations of motion take the form
δS ∂
δxµ = p˙µ− ∂σ(γN0x′µ+N1pµ) = 0,
(14)
δS p
δpµ = x˙µ−N0 γµ +N1x′µ = 0.
The standard gauge-fixing method is to fix the se
ond
lass
onstraints (orthonormal gauge)
N0 = 1,N1 = 0. xµ
In this
ase the equations of motion (14) redu
e to d'Alambert ones for
p
p˙ = γx ′′, x˙ = µ x¨ x ′′ = 0,
µ µ µ µ µ
γ ⇒ − (15)
the
onformal interval (3) takes the usual form
2 2 2 2
ds = Ω [dτ dσ ],
−
(H = 0)
but the Hamiltonian (11) in view of the
onstraint (13) is equal to zero .
There is another way to introdu
e evolution parameter as the obje
t reparameterizations
in the theory being adequate to the initial (cid:16)kinemetri
(cid:17) invariant system and to
onstru
t the
non-zero Hamiltonian. We identify this evolution parameter with the time-like variable of the
(cid:16)
enter of mass(cid:17) (CM) of a string defined as the total
oordinate
1 l
X (τ) = x (τ,σ)dσ = x .
µ µ µ
l 0 h i (16)
Z
The redu
tion requires to separate the (cid:16)
enter of mass(cid:17) variables before variation of the a
tion
(4) whi
h after substitution
x = X (τ)+ξ (τ,σ), x′ (τ,σ) = ξ′(τ,σ),
µ µ µ µ µ
(17)
l
ξ (τ,σ) = 0
µ
0 (18)
Z
takes the form
S = γ τ2dτ X˙2(τ) l dσ +2X˙µ(τ) ldσ ξ˙µ−N1ξµ′ +
2 Zτ1 ( Z0 N0(τ,σ) Z0 N0 !
+ ldσ (ξ˙µ−N1ξµ′)2 N0ξ′2(τ,σ) .
0 " N0 − #) (19)
Z
The usual determination of the
onjugate momenta
P (τ) = δS = γX˙ (τ) l dσ +γ ldσ ξ˙µ(τ,σ)−N1ξµ′(τ,σ) ,
µ δX˙µ(τ) µ 0 N0(τ,σ) 0 N0(τ,σ) ! (20)
Z Z
π (τ,σ) = δS = γX˙ (τ) 1 +γξ˙µ(τ,σ)−N1ξµ′(τ,σ)
µ δξ˙µ(τ,σ) µ N0(τ,σ) N0(τ,σ) (21)
l
P (τ) π (τ,σ) π (τ,σ)dσ =
µ µ 0 µ
leadstothe
ontradi
tionbe
ause and arenotindependent,namely
P (τ)
µ
. R
4
P (τ)
µ
Thus for the
onsistent definition of the momentum of (cid:16)
enter-mass(cid:17) and momentum
π (τ,σ)
µ
of intrinsi
variables we have to put strong
onstraint in a
tion (19)
ξ˙µ(τ,σ) N1ξµ′(τ,σ)
dσ − = 0.
N0 ! (22)
Z
Then we obtain the following form of the redu
ed a
tion:
Sred = γ2 dτ (X˙2(τ)N(lτ) + 0ldσ"[ξ˙µ(τ,σ)N−0(Nτ,1σξ)µ′(τ,σ)]2 −N0(τ,σ)ξν′2(τ,σ)#), (23)
Z Z
N(τ)
where is the global lapse fun
tion
1 1 l dσ
= = N−1 .
0
N(τ) l 0 N0(τ,σ) h i (24)
Z
For global momenta we get
P (τ) = ∂Sred = γX˙ (τ) l
µ ∂X˙µ(τ) µ N(τ) (25)
and for the lo
al (intrinsi
) momenta
∂Sred ξ˙µ(τ,σ) N1(τ,σ)ξµ′(τ,σ)
π (τ,σ) = = γ −
µ ∂ξ˙µ(τ,σ) N0(τ,σ) (26)
with two strong
onstraints (18) and (22)
l l
ξ (τ,σ)dσ = 0, π (τ,σ)dσ = 0.
µ µ
0 0 (27)
Z Z
This separation
onserves the group of the (cid:16)kinemetri
(cid:17) transformation (5) and leads to Hamil-
tonian form of redu
ed a
tion, in view of
X˙µ(τ) = N(lτ)Pµγ(τ); ξ˙µ(τ,σ) = N0(γτ,σ)πµ(τ,σ)+N1(τ,σ)ξµ′(τ,σ), (28)
we get
H = PµX˙µ(τ)+πµξ˙µ L = 1 N0(τ,σ)P2(τ)+N0(τ,σ)[π2 +γ2ξ′2]+2γN1(πξ′) ,
− 2γ l (29)
(cid:26) (cid:27)
τ2 P2(τ) l π2+γ2ξ′2
S = dτ Pµ(τ)X˙µ(τ) N(τ) + dσ (πξ˙) N0 N1(πξ′) .
Zτ1 (cid:26) − 2γl Z0 (cid:20) − 2γ − (cid:21)(cid:27) (30)
The equation of motion is split into global one
δS δS P
= P˙ (τ) = 0, = X˙ (τ) N(τ) µ = 0
δX˙µ(τ) µ δPµ(τ) µ − γl
and lo
al one
δπµδ(Sτ,σ) = ξ˙µ(τ,σ)− Nγ0πµ−N1ξµ′ = 0, − δξµδ(Sτ,σ) = π˙µ− ∂∂σ(N1πµ+γN0ξµ′) = 0. (31)
5
N0(τ)
The variation of the a
tion (30) with respe
t to results in the
onstraint
δS N2(τ) P2(τ) π2(τ,σ)+γ2ξ′2(τ,σ)
= + = 0,
δN0(τ,σ) N02(τ,σ) 2γl2 2γ (32)
here it is ne
essary to take into a

ount that the variation over the global lapse fun
tion (24)
leads to δN(τ) N2(τ)
= .
2
δN0(τ,σ) N0(τ,σ) (33)
N1(τ,σ)
The variation of the (30) with respe
t to results in the
onstraint for lo
al variables
δS
= (πξ′(τ,σ)) = 0.
δN1(τ,σ) (34)
Now we introdu
e the Hamiltonian density for lo
al ex
itations
π2(τ,σ)+γ2ξ′2(τ,σ)
(τ,σ) =
H − 2γ (35)
and rewrite (32) in the form
N(τ)
P2(τ) = 2γ (τ,σ).
N0(τ,σ)l µ H (36)
q q
σ
One integrates (36) over and taking into a

ount the normalization equality (24)
1 l N(τ)
dσ = 1,
l 0 N(τ,σ) (37)
Z
it leads to global
onstraint
l
Mst = Pµ2(τ) = 2γ (τ,σ)dσ = l 2γ ,
0 H h Hi (38)
q Z p p
Mst
where is mass of the string. The lo
al part of the
onstraint (36)
an be obtained by
substitution (38) into (36)
N(τ) (τ,σ)
= H .
N0(τ,σ) √ (39)
p
h Hi
Finally, after substitution (382), (39) into a
tion (30) we
an derive the
onstraint-shell a
tion,
N(τ)P (τ)/2γl = N0(τ,σ) (τ,σ)dσ
whereas the equation H
τ2 l R
Sconst−shell = dτ PµX˙µ(τ)+ dσ πµ(τ,σ)ξ˙µ(τ,σ) N1(τ,σ)πµ(τ,σ)ξµ′(τ,σ) .
Zτ1 (cid:26) Z0 h − i(cid:27) (40)
N1(τ,σ) (πµ(τ,σ)ξ′µ(τ,σ)) = 0.
Againthevariationwithrespe
tto resultsinsubsidiary
ondition
l
Pi(τ) = 0, P0(τ) = 0dσ 2γ (τ,σ)
Now in the (cid:16)
enter-mass(cid:17)
oordinate system H the a
tion
(40) takes the form R p
τ2 l
Sconst−shell = dτ dσ 2γ (τ,σ)X˙0(τ)+πµ(τ,σ)ξ˙µ(τ,σ) ,
Zτ1 Z0 np H o (41)
6
whi
h des
ribes the dynami
s of a lo
al (intrinsi
)
anoni
al variables of a string with non-zero
Hamiltonian be
ause (41)
an be rewritten in the form
x2 l ∂ξµ(X0,σ)
Sconst−shell = dX0 dσ πµ(X0,σ) + 2γ (X0,σ) ,
Zx1 Z0 (cid:26) ∂X0 H (cid:27) (42)
p
where 2γ (X0,σ) = [π2(X0,σ)+γ2ξ′2(X0,σ)]
H −
dX0 = X˙0(τ)dτ dτ˜ = f1dτ.
and time is invariant with respe
t to
N0(τ,σ) = 1,
Inthegauge-fixingmethod,byusingthekinemetri
transformation,wehavetoput
N1(τ,σ) = 0
(this requirement does not
ontradi
t to Eq. (37) in view of Eqs. (38), (39)). Then
a

ording to [10℄ 1 l Mst
(τ,σ) = dσ (τ,σ) =
H l 0 H √2γ l (43)
Z
p p
2γ (τ,σ)
means that the Hamiltonian H is
onstant. In this
ase the equations for the lo
al
variables obtained by varyingpthe a
tion (42) take the form
δS ∂ξµ(X0,σ) πµ(X0,σ)
= = 0,
δπµ(X0,σ) ∂X0 − (π2+γ2ξ′2) (44)
−
p
δS ∂πµ(X0,σ) 2 ∂ ξµ′(X0,σ)
= +γ = 0.
δξµ(X0,σ) − ∂X0 ∂σ " (π2+γ2ξ′2)# (45)
−
(π2+γ2ξ′2) = γp, (Mst = γl)
If we put a

ording to Eq. (43) − , then it leads again to
ξµ(X0,σ)
d'Alambert equation for p
2 2
∂ ξµ(X0,σ) ∂ ξµ(X0,σ)
= .
∂X02 ∂σ2 (46)
The general solution of these equations in
lass of fun
tions (27) with boundary
onditions for
ξ′µ(X0,0) = ξ′µ(X0,l) = 0
the open string is given as usually by the Fourier series
1
ξµ(X0,σ) = [Ψµ(X0 +σ)+Ψµ(X0 σ)],
2√γ −
1
ξ′µ(X0,σ) = [Ψ′µ(X0+σ) Ψ′µ(X0 σ)],
2√γ − − (47)
πµ(X0,σ) =γ∂ξµ(X0,σ) = √γ[Ψ′µ(X0+σ)+Ψ′µ(X0 σ)],
∂X0 2 −
where Ψµ(z) = i αnµe−iπlnz, Ψ′µ(z) = π αnµe−iπlnz
n l (48)
n6=0 n6=0
X X
(n = 0)
it is very important to stress that (48) does not
ontain zero harmoni
s 6 ). The substi-
ξ π
µ µ
tution of and in this form into (35) and taken into
onsideration (43) leads to density of
Hamiltonian 1 M2
H = −4 Ψ′2µ(X0 +σ)+Ψ′2µ(X0−σ) = 2γslt2. (49)
(πξ′) =(cid:2)0 Ψ′ (cid:3)
µ
From the
onstraint (34) in terms of the ve
tor (47) we obtain
1
πµ(X0,σ)ξ′µ(X0,σ) = 4 Ψ′2µ(X0 +σ)−Ψ′2µ(X0−σ) = 0, (50)
(cid:0) (cid:1) (cid:2) (cid:3)
7
then from (49) and (50) finally we get
2
M
Ψ′µ2(X0+σ) = Ψ′µ2(X0−σ) = −l2γst. (51)
Ψ′ (z)
µ
It means that is the modulo-
onstant spa
e-like ve
tor and in terms of its representation
(48) the equalities (51)
an be rewritten
2 ∞ 2
−Ψ′µ2(z) = πl2 Lke−iπlkz = Ml2γst, (52)
k=−∞
X
where L = α αµ , L∗ = L .
k − nµ k−n k −k
n6=k,0
X
Now we
an see that the zero harmoni
of this
onstraint determines the mass of a string
Ms2t = π2γL0 = −πγ αnµαµ−k, αµ−k = α∗kµ (53)
n6=0
X
and
oin
ides with standard definition in string theory [1℄, however the non-zero harmoni
s of

onstraint (52)
µ
Lk6=0 = − αnkαk−n = 0 (54)
n6=0,k
X
strongly differ from the standard theory be
ause they do not depend on the global motion(do
P P Ψµ′
µ µ
not depend on ) and do not
ontain the interferen
e term , be
ause of our
onstraint
(51) we
an rewrite l2γΨ′ 2(z)+P2 = 0
µ µ
(55)
instead standard one [1℄ (P +l√γΨ′ (z))2 = 0.
µ µ
(56)
Therefore diffeo-iPnvξaµria=nt0,appProaπ
µh=
oi0n
idePs wαiµth=th0e, Ro(cid:4)nhr=li
h0 one [13℄, whi
h is based on the
µ µ µ n
gauge
ondition ⇒ 6 . One use of that
ondition for
ξ0,π0 (Pi = 0)
eliminating thetime
omponents being
onstru
ted inthe frame of referen
e leads
Ψ′0 αn0
to formula (47)(cid:21)(54), where and are equal to zero.
3. Con
lusions
Thestandardapproa
htoarelativisti
stringisnotinvariantwithrespe
ttothetime
oordinate
transformations. Itistheproblemas,indiffeo-invarianttheories,allobservablequantitiesshould
be the diffeo-variant ones.
We propose here to solve this problem as in [7℄, where the referen
e frame is redefined by
pointing out diffeo-invariant homogeneous (cid:16)time-like variable(cid:17) in a

ordan
e with the dimension
of the diffeomorphism group.
Just this
hoi
e of an evolution parameter simplifies the Hamiltonian equations and leads
to exa
t resolution of the energy
onstraint with respe
t to the
anoni
al momentum of the
evolution parameter. We have shown that this diffeo-invariant approa
h to a relativisti
string

orresponds to the Ro(cid:4)hrli
h spe
trum [13℄.
A
knowledgement
The authorsaregrateful toA.V.Efremov, E.A. Kuraev, V.V.Nesterenko, for interesting and

riti
al dis
ussions.
8
Referen
es
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tion in the Relativisti
String Theory (World
S
ientifi
, 1990).
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hri
hten von der K(cid:4)on. Ges. der Wissens
haften zu Go(cid:4)ttingen, Math.-Phys.
Kl., Heft 3, 395 (1915).
[3℄ D. Hilbert, Math. Annalen. 92, 1 (1924).
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Dubna, 1986.
[5℄ E. Noether, Go(cid:4)ttinger Na
hri
hten, Math.-Phys. Kl., 2, 235 (1918).
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(1973); Yu.S. Vladimirov, Referen
e Frames in Theory of Gravitation, Mos
ow, Energoizdat,
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al and Mathemati
al
Physi
s, 132, 1045 (2002).
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ontres: 1967, Le
tures in Mathemati
s and Physi
s, edited by
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(1967).
[10℄ B.M.BarbashovandN.A.Chernikov,Classi
al Dynami
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