Electromagnetic dipole radiation of oscillating D-branes

8 Pages·0.16 MB·English

by  G. K. Savvidy

1
Electromagnetic dipole radiation of oscillating D-branes
G.K.Savvidya
aNational Research Center Demokritos,
Ag. Paraskevi,GR-15310 Athens,Greece
0
I emphasize analogy between Dp-branes in string theories and solitons in gauge theories comparing their
0
commonpropertiesandshowingdifferences. Instringtheorywedonothavethefullsetofequationswhichdefine
0
2 the theory in all orders of coupling constant as it was in gauge theories, nevertheless such solutions have been
foundassolutionsoflowenergysuperstringeffectiveactioncarryingtheRRcharges. Theexistenceofdynamical
n
RR charged extended objects in string theory has been deduced also by considering string theory with mixed
a
boundaryconditions, when typeIIclosed superstingtheoryisenriched byopen stringswith Neumannboundary
J
conditions on p+1 directions and Dirichlet conditions on theremaining 9-p transverse directions.
7
Wewillshowthatforcertainexcitationsofthestring/D3-branesystemNeumannboundaryconditionsemerge
1
from the Born-Infeld dynamics. Here the excitations which are coming down the string with a polarization

1 along a direction parallel to the brane are almost completely reflected just as in the case of all-normal Dirichlet
v excitations considered by Callan and Maldacena, but now the end of the string moves freely on the 3-brane
8 realizing Polchinski’s open string Neumannboundary condition dynamically.
9 Inthelowenergylimitω→0,i.e. forwavelengthsmuchlargerthanthestringscaleonlyasmallfraction∼ω4
0 oftheenergy escapes in theform ofdipole radiation. Thephysicalinterpretation isthatastring attachedtothe
1 3-brane manifests itself as an electric charge, and waves on the string cause the end point of the string to freely
0
oscillate and produce e.m. dipole radiation in the asymptotic outer region. The magnitude of emitted power is
0
in fact exactly equal to theone given by Thompson formula in electrodynamics.
0
/
h
t
- 1. INTRODUCTION ries we have mass hierarchy: the particle masses
p are of order O(e0) and the soliton masses are of
e ProbablythebestwaytointroduceD-branesis order O(1/e2). These states are invisible in per-
h
to recall that in non-Abelian gauge field theories
: turbation theory. One can probably think about
v there exist classical BPS solutions corresponding
these solutions as ” typhoons” of gauge bosons.
i to new particles, extended objects, vortices and
X The second important message which bring to
monopoles which are not present in the pertur-
r us these solutions is that extended objects in
a bative spectrum of the quantized theory [1–3].
gauge theories carry a new type of charge,in the
In these theories in addition to the point par-
givencaseitismagneticcharge[2],whichwasnot
ticle spectrum of perturbation theory consisting
present in perturbative spectrum, as well
of a massless photon, massive vector bosons and
Xiggs particles there also exist stable solutions g =2π/e. (2)
with masses inversely proportional to the square
ThemagneticchargesatisfiestheDiracquantiza-
of the coupling constant [1–3]
tion condition
(Mass)2BPS =4πMW/e2. (1) eg =2πn n Z, (3)
∈
where M is a vector bosonmass. In the case of which ties togetherthe units ofelectric andmag-
W
vorticesitisenergyperunitoflengthandshould netic charges[4,5]. (Generallyspeakingmagnetic
be associated with the string tension. These charge is realized as a genuine solution without
states cannot be seen in the weak coupling limit singularities when U(1) is embedded into non-
simplybecausethemassesofthesestatesbecome Abelian compact group [2].) With the existence
very large when e 0, therefore in gauge theo- of magnetic charges and also dyons – particles
→
2
carryingelectricandmagneticcharges[3]thethe- scales as
orymayexhibitfullelectromagneticduality[6–8]. 1
According to this conjecture, the strong coupling (Tension)RR (6)
≃ g
s
limit of the theory is equivalent to the weak cou-
pling limit with ordinary particles and solitons sothatbothtypesofRRcharges,electricp-brane
exchanged. and its dual magnetic 6-p brane, are solitons of
In string theory one can also expect solutions themassO(1/gs)whichissmallerthanthatofthe
which cannot be seen in the perturbation theory. NS-NSsolution(5). Thisbehaviour,intermediate
In string theory we do not have the full set of between fundamental string spectrum O(gs0) and
equations which define the theory to all orders solitonic brane O(1/gs2), was not known in field
of coupling constant as it was in gauge theories, theory (see Figure 3).
nevertheless such solutions have been found as The existence of dynamical extended objects
solutions of low energy superstring effective ac- in string theory which carry RR charges can be
tion [9]. The lagrangian can be represented in deduced also by considering string theory with
the form: mixed boundary conditions [10]. Indeed one can
consider type II closed superstring theory which
1
S = d10x√ g has been completed by adding open strings with
16πG(10) Z − Neumann boundary conditions on p + 1 direc-
N
1 1 tions and Dirichlet conditions on the remaining
e 2ϕ[R+4(∂ϕ)2 H2] G2 (4)
{ − − 3 − 3 } 9-p transverse directions [10,11,13,14],
whereϕisthedilatonfield,H -theNS-NSgauge na∂aXµ =0, µ=0,...p, (7)
field and G - RR gauge field have essentially dif- Xµ =0, µ=p+1,...9. (8)
ferentorderindilatoncouplingforRRfields. The
corresponding equations have two types of solu- Here the open string endpoints live on the ex-
tions carrying NS-NS or RR charges [9]. tended object, D-brane, and in this indirect way
The NS-NS soliton is determined by the bal- defines a string soliton with p spatial and one
ancebetweenthefirstthreetermsanditstension timelike coordinates. The D-branes are surfaces
(tension mass/volume ), scales as the action of dimension p on which open string can end.
(4) ≡ This is a consistent string theory if p is even in
type IIA theory and is odd in type IIB theory
1 [10]. The identification with solitons can be seen
(Tension) (5)
NS−NS ≃ g2 if one compute the tension [10,11]
s
where e ϕ = g is the string coupling constant. 1
− s τ = (9)
In NS-NS sector the electrically charged object p gs(2π)p−21(2πα′)p+21
is the fundamental string of the mass O(g0) and
s Thisisthesamecouplingconstantdependenceas
is accompanied by the above solitonic D5-brane
for the branes carrying RR charges (6)! The RR
carring the dual magnetic charge of the mass
charge which carry D-brane is
O(1/g2). This is very similar to what we had
s
in gauge theories. 1
µ = . (10)
In RR sector the situation is different, all p (2π)−21(4π2α′)p−23
stringstatesareneutralwithrespecttoRRgauge
field and in perturbative string spectrum there is and satisfies the Dirac quantization condition
no fundamental objects carrying elementary RR
µ µ =2πn n Z (11)
p 6 p
charges [10,11]. In the RR sector the solution of − ∈
the equations is determined by the balance be- with n = 1. Thus RR gauge fields naturally
tween the first three and the fourth term in the couple to Dp-branes and identify Dp-branes as
effectiveaction(4)andthetensionofthesolution carrier of RR charge [10]. The general remark
3
is that the tension of the p-brane should have sions
dimension massp+1 and it is indeed so in (9), 1
L= d4x 1 E~2+(∂~x )2, (12)
but what is important is that it is a function of −(2π)3gs Z q − 9
thefundamentalstringtension1/2πα (aswellas
′
where g is the string coupling (α =1).
1/g ) and not just a new parameter of the the- s ′
s
The abovementionedtheorycontains6scalars
ory [15,16,18]. The D-branes are indeed the ”ty-
x ,....,x , which are essentially Kaluza-Klein
phoons” of strings. 4 9
remnantsfromthe10-dimensionalN =1electro-
dynamics afterdimensionalreductionto3+1di-
mensions. As we explain these extra components
2. THE D-BRANE EXCITATIONS
ofthee.m. fieldA ,...,A describethetransverse
4 9
deviations of the brane x ,...,x .
The Dp-brane is a dynamical extended object 4 9
The solution, which satisfies the BPS condi-
which can fluctuate in shape and position and
tions, is necessarily also a solution of the linear
these fluctuations are described by the attached
theory, the N =4 super-Electrodynamics [20],
open strings. The boundary condition (8) al-
lows open string ends to be at any place in- E~ = c e~ , ∂~x = c e~ , x = c , (13)
side the p-brane with coordinates x0,...,xp and r2 r 9 r2 r 9 −r
xp+1 = ... = x9 = 0. In this way the open string where c = 2π2g is the unit charge, and can be
s
describes the excitations of the Dp-brane, and as thoughtofas setting the distance scale. Here the
itiswellknowntheopenstringmasslessstatesare scalar field represents the geometrical spike, and
vectorandspinorfieldsoften-dimensionalN =1 the electric field insures that the string carries
supersymmetricU(1)gaugetheory inten dimen- uniform NS charge along it. The RR charge of
sions. The massless field Aµ(xν),µ,ν = 0,...,p the 3-brane [19]
propagatesasgaugebosonsonthep-braneworld-
ΩMNL =ǫabc∂ XM∂ XN∂ XL (14)
volume,whiletheothercomponentsofthevector a b c
potentialA4(xν),...,A9(xν)canbeinterpretedas cancels out on the string, or rather the tube be-
oscillation in the position of the p-brane. Thus havesasakindofRRdipolewhosemagnitudecan
the theory on the p-brane worldvolume is a ten- be ignored when the tube becomes thin. It was
dimensional supersymmetric U(1) gauge theory demonstratedin[20]thattheinfiniteelectrostatic
dimensionally reduced to p+1 dimensions. The energy of the point charge can be reinterpreted
low energy effective action for these fields is the asbeingduetotheinfinitelengthoftheattached
Dirac-Born-Infeldlagrangian[12,31]. Thekeydif- string. The energy per unit length comes from
ference with previously known p-branes and su- the electric field and corresponds exactly to the
permembranes[17] is the inclusion of a worldvol- fundamental string tension.
ume electromagnetic field. Polchinski,whenheintroducedD-branesasob-
Callan and Maldacena [20] showed that the jects on which strings can end, required that the
Dirac-Born-Infeld action, can be used to build string has Dirichlet (fixed) boundary conditions
a configuration with a semi-infinite fundamental forcoordinatesnormaltothebrane(8),andNeu-
string ending on a 3-brane1, whereby the string mann (free) boundary conditions for coordinate
is actually made out of the brane wrapped on directions parallel to the brane (7) [10–12].
S2 (see also [21]). The relevant action can be It was shown in [20] that small fluctuations
obtained by computing a simple Born-Infeld de- whicharenormaltoboththestringandthebrane
terminant,dimensionallyreducedfrom10dimen- aremostlyreflectedbackwithaphaseshift π
→
which indeed corresponds to Dirichlet boundary
condition (8). See also [32] and [30] for a super-
1We review the favorite case of the 3-brane first because gravity treatment of this problem.
of its non-singular behaviour in SUGRA, and secondly
In the paper [23] it was demonstrated that
it is suggestive of our own world which is after all 3-
dimensional. P-wave excitations which are coming down the
4
string with a polarization along a direction par- (∂ φ)2(1+B~2)+(∂~φ)2+(B~ ∂~φ)2
0
− ·
allel to the brane are almost completely reflected (E~ ∂~φ)2+2∂ φ(B~[∂~φ E~])
0
just as in the case of all-normal excitations, but − × ×
the endofthe stringmovesfreelyonthe 3-brane, We will proceed by adding a fluctuation to the
thus realizing Polchinski’s open string Neumann background values (13) :
boundary condition (7) dynamically. As we will
see a superpositionof excitations ofthe scalarx9 E~ =E~0+δE~, B~ =δB~, φ=φ0+η .
and of the electromagnetic field reproduces the
Then keeping only terms in the Det which are
required behaviour,e.g. reflectionof the geomet-
linearandquadraticinthefluctuationwewillget
rical fluctuation with a phase shift 0 (Neu-
→
mann boundary condition). δDet=δB~2 δE~2 (E~ δB~)2 (∂ η)2(16)
0 0
In[23]itwasalsoobservedtheelectromagnetic − − −
+(∂~η)2+(δB~∂~φ)2 (E~ ∂~η)2 (δE~ ∂~φ)2
dipoleradiationwhichescapestoinfinityfromthe − 0× − ×
place wherethe stringisattachedtothe 3-brane. 2(E~0 ∂~η)(δE~ ∂~φ) 2(E~0δE~)+2(∂~φ∂~η)
− × × −
It was shown that in the low energy limit ω 0,
→ Note that one should keep the last two linear
i.e. for wavelengths much larger than the string
terms because they produce additional quadratic
scale a small fraction ω4 of the energy escapes
∼ terms after taking the square root. These terms
to infinity in the form of electromagnetic dipole
involve the longitudinal polarization of the e.m.
radiation. The physical interpretation is that a
field and cancel out at quadratic order. The re-
string attached to the 3-brane manifests itself as
sulting quadratic Lagrangianis [23]
an electric charge, and waves on the string cause
the end point of the string to freely oscillate and 2L =δE~2(1+(∂~φ)2) δB~2+(∂ η)2 (17)
q 0
produce electromagnetic dipole radiation in the −
asymptotic outer region of the 3-brane. This re- (∂~η)2(1 E~02)+E~02(∂~η δE~) .
− − ·
sultisalsoinagoodagreementwiththeinterpre-
Let us introduce the gauge potential for the fluc-
tation of the open string ending on D-brane (7), tuation part of the e.m. field as (A ,A~) and sub-
0
accordingtowhichtheopenstringstatesdescribe
stitute the values of the background fields from
the excitations of the D-brane. Indeed as we will
(13)
see open string massless vector boson can propa-
gate inside the D-brane and in our case this ex- 2L =(∂ A~ ∂~A )2(1+ 1 ) (~ A~)2(18)
citation is identified with electromagnetic dipole q 0 − 0 r4 − ∇×
radiation. Thus not only in the static case, but +(∂ η)2 (∂~η)2(1 1 )+ 1 (∂ A~ ∂~A ) ∂~η .
also in a more general dynamical situation the 0 − − r4 r4 0 − 0 ·
above interpretation remains valid. This result The equations that follow from this lagrangian
provides additional support to the idea that the contain dynamical equations for the vector po-
electron(quark)maybeunderstoodastheendof tential and for the scalar field, and a separate
a fundamental string ending on a D-brane. equation which represents a constraint. These
equations in the Lorenz gauge ∂~ A~ =∂ A are
0 0
3. THE LAGRANGIAN AND THE E-
1 1
QUATIONS −∂02A~(1+ r4)+∆A~+ r4∂~∂0(A0+η)=0(19)
1 1
LetuswriteoutthefullLagrangianwhichcon- ∂2A +∆A +∂~ ∂~(A +η) ∂~ ∂ A~ =0(20)
tains both electric and magnetic fields, plus the − 0 0 0 r4 0 − r4 0
1 1
scalar x9 ≡φ [23] −∂02η +∆η −∂~r4∂~(A0+η)+∂~r4∂0A~ =0(21)
L= d4x√Det (15) Equation (20) is a constraint: the time deriva-
−Z tive of the lhs is zero, as can be shown using the
where Det=1+B~2 E~2 (E~ B~)2 equation of motion (19).
− − ·
5
Let us choose A = η. This condition can be
0
−
viewed as (an attempt to) preserve the BPS re-
lation which holds for the background: E~ = ∂~φ. δz δz
η
Another point of view is that this fixes the gen-
eral coordinate invariance which is inherent in x9xy x9xy
the Born-Infeld lagrangian in such a way as to z z
a b
make the given perturbation to be normal to the
surface. Of course transversality is insured auto-
matically but this choice makes it explicite. The
Figure1. InorderthatallpointsontheS section
generaltreatment of this subject can be found in 2
of the tube ( which is schematically shown here
[31].
as a circle) move all in the same direction zˆ by
Withthisconditiontheequations(20)and(21)
an equal distance δz, the field η has to take on
become the same, and the first equation is also
different values at, say,opposite points of the S .
simplified [23]: 2
In effect, η = δ 1 = 1 zδz. If, on the other
1 −r r2r
∂2A~(1+ )+ ∆A~ =0 , (22) hand,itistakentobeanS-wave(asinthepaper
− 0 r4 [30]) that would correspond to Fig 1b, which at
1
∂2η+∆η+∂~ ∂ A~ =0 . (23) best is a problematic ‘internal’degree of freedom
− 0 r4 0 of the tube.
Thisshouldbe understoodto implythatoncewe
obtainasolution,A isdeterminedfromη,butin
0
additionwe are now obligedto respect the gauge follows from the first by differentiation. This is
condition which goes over to ∂~A~ = ∂ η. because the former coincides with the constraint
0
−
in our ansatz.
4. NEUMANN BOUNDARY CONDITI- Thereforetheproblemisreducedtofindingthe
ONSFROMBORN-INFELDDYNAM- solutionofasinglescalarequation,anddetermin-
ICS [23] ingtheotherfieldsthroughsubsidiaryconditions.
The equation (24) itself surprisingly turned out
We will seek a solution with definite energy
tobethe onefamiliarfrom[20]forthetransverse
(frequency w) in the following form: A~ should
fluctuations. Thereitwassolvedbygoingoverto
have only one component A , and η be an l = 1
z a coordinate ξ which measures the distance radi-
spherical P-wave
ally along the surface of the brane
z
A =ζ(r) e iωt , η = ψ(r) e iωt r
z − r − c2
ξ(r)=ω du 1+ ,
Thegeometricalmeaningofsucha choiceforη is Z r u4
√c
explainedinFigure1,andtheparticularchoiceof
z dependence corresponds to the polarization of and a new wavefunction
theoscillationsalongthez directionofthebrane. ζ˜=ζ (c2+r4)1/4 .
With this ansatz the equations become
This coordinate behaves as ξ r in the outer
c2 c2 ∼
1+ ω2ζ+ ∂ (r2∂ ζ)=0 , (24) region(r )andξ 1/r onthe string(r
(cid:18) r4(cid:19) r2 r r →∞ ∼− →
0). The exact symmetry of the equation r
z z 1 z 2 ↔
ω2ψ+ ∂ (r2∂ ψ)+ ψ c/r goes over to ξ ξ. The equation, when
r r r2 r r r r2 − written in this coor↔din−ate becomes just the free
c2 waveequation,plusanarrowsymmetricpotential
iω∂ ζ =0 , (25)
z(cid:18)r4 (cid:19) at ξ 0
∼
with the gauge condition becoming ∂ ζ = iωψ. d2 5c2/ω2
r ζ˜+ ζ˜=ζ˜. (26)
It can be seen again, that the second equation −dξ2 (r2+c2/r2)3
6
The asymptotic wave functions can be con- Secondly, the i that enters causes the superposi-
structed as plain waves in ξ, tion of the incoming and reflected waves to be-
come a cosine from a sine, as is the case for ζ
ζ(r)=(c2+r4)−1/4e±iξ(r) , waves. This corresponds to a 0 phase shift for the
η wave(Figure2)andimplies thereforeNeumann
or in the various limits:
boundary condition , that is it allows open string
1
r 0 ζ e±iξ(r) , ends to be at any place inside the p-brane .
→ ∼ √c
1
r ζ e±iξ(r) . 5. ELECTROMAGNETIC DIPOLE RA-
→∞ ∼ r DIATION OF OSCILLATING D-
Theseformulaegiveustheasymptoticwavefunc- BRANE [23]
tion in the regions ξ , while around ξ =
→ ±∞ Because of the ω factor in the gauge condition
0 (r =1)thereis asymmetric repulsivepotential
whichdropsveryfast 1/ξ6 oneithersideofthe we need to be careful about normalizations,thus
∼ we shall chooseto fix the amplitude of the δz (or
origin. The scattering is describedby a single di-
η) wave to be independent of frequency ω. Then
mensionless parameter ω√c, and in the limit of
small ω and/or coupling c = 2π2g the potential the magnitude of the electromagnetic field in the
s
inner region becomes independent of ω as well,
becomes narrow and high, and can be replaced
by a δ-function with an equivalent area ∼ ω√1gs thus the amplitude of ζ wave at r →0 is
under the curve. Therefore the scattering matrix Az =ζ0e−iω(t+c/r)
becomes almostindependent ofthe exactformof
the potential. We refer the reader to pp 18-20 of
Combined with the transmission factor
the thesis [22] for the more detailed calculation
of transmission amplitude (29). The end result T = iω√c+O(ω2), (29)
−
is that most of the ζ amplitude is reflected back
this gives the following form of the dipole radia-
with a phase shift close to π.
tion field at infinity2
In order to obtain ψ (and η) we need to differ-
entiate ζ with respect to r:
T ζ iω 2π2g
1 4r3 Az = r· 0e−iωt = − pr sζ0e−iωt
iωψ= −4 (c2+r4)5/4e±iξ(r) iωeζ d˙
iω c2 1/2 = − r 0e−iωt = r, (30)
1+ e iξ(r) . (27)
±
±(c2+r4)1/4 (cid:18) r4(cid:19) where
d=eζ e iωt.
Again it is easy to obtain the simplified limiting 0 −
form: In order to make a comparison with Thompson
r3 iω√c formula
r →0 iωψ ∼(cid:18)−c5/2 ± r2 (cid:19)e±iξ(r) I = 4ω4e2ζ 2
0
3
1 iω
r iωψ − e±iξ(r) for the total power emitted by an oscillating
→∞ ∼(cid:18)r2 ± r (cid:19)
dipole, we note that the agreementis guaranteed
This brings about several consequences for ψ. if the exact coefficient behind iω√c in (29) is
Firstly,itcausesψtogrowas 1/r2whenr 0. equal to one. As it was shown−in [22], the ap-
∼ →
This is the correct behaviour because when con- proximate computation of the transmission am-
vertedtodisplacementinthezdirection,itmeans plitude T (29) gives T i.92ω√c. The later
∼ −
constant amplitude [23]:
2Theunitofelectricchargeinournotationis2π2gs=e2.
z c z c This is because of the scaling of the fields needed to get
η = rψ =δ−r = rr2δz ⇒ δz ∼const .(28) theU(1)actionwith1/e2 infrontofit.
7
V( ξ ) V( ξ )
ζin ηin
ζout ξ ηout ξ
a b
Figure 2. The figure 2a depicts the scattering
of the ζ wave. Note the discontinuity in the
derivative which is proportional to 1/ω. Figure
2b shows the scattering of the η wave. Being
Figure3. InNS-NSsectortheelectricallycharged
the derivative of ζ, it results in a discontinuity objectisthefundamentalstringofthemassO(g0)
s
of the function itself, making it into a cosine,
and is accompanied by the solitonic D5-brane
whichmeansfree(Neumann)boundarycondition
carring the dual magnetic charge of the mass
at ξ =0. O(1/g2). In the RR sector both types of RR
s
charges, electric p-brane and its dual magnetic
6 p brane, are solitons of the mass O(1/g )
s
−
computation of this coefficient using the numeri- whichissmallerthanthatoftheNS-NSsolution.
cal solution of the equation (24) shows that it is Thisbehaviour,intermediatebetweenfundamen-
indeed equal to one. To solve properly the equa- tal string spectrum O(g0) and solitonic brane
s
tion on a computer it is necessary to introduce O(1/g2), was not known in field theory.
s
new variables [24]
c y+ y2+4c
y =r , or r(y)= , (31)
− r p2 contribute to the radiation at spatial infinity, as
canbeshownfromtheintegraloftheenergyden-
so that the equation (24) take the form
sity (∂ η)2d3r ω4/r2 4πr2dr ω4. This is
r
ω2(c2+r(y)4)ζ+(c+r(y)2)∂y(c+r(y)2)∂yζ =0(32) notaRltogethersu∼rpRrising,a·sweared∼ealingwitha
supersymmetric theory where the different fields
and boundary condition at r =ωc/2πn, n
0 →∞ are tied together. In fact, it is easy to show that
the form
the fields still do asymptotically satisfy the BPS
1 (2πn)2 conditions at infinity, and 1/4 of the SUSY is
ζ(y )= , ζ (y )= i ,
0 √c ′ 0 − ωc3/2 preserved. Thus the observer at spatial infinity
ωc 2πn will see both an electromagnetic dipole radiation
at y0 = . field and a scalar wave. Interestingly, the direc-
2πn − ω →−∞
tiondependencesofthetwoconspiretoproducea
Measuring the amplitudes yζ(y) (y+ π )ζ(y+
± 2ω spherically symmetric distribution of the energy
π )aty (r )onecangetthatthecoeffi-
2ω →∞ →∞ radiated.
cientbehind iω√cin(29)isequaltoone. Thus
I wish to thank the organizers of the confer-
−
themagnitude ofemittedpoweris infactexactly
ence QG99 in Villasimius and especially Vittorio
equal to the one given by Thompson formula in
de Alfaro for the invitation and for arranging in-
electrodynamics3.
teresting and stimulating meeting. I also thank
In conclusion,we need to analyze the outgoing
Konstantin Savvidy for the enjoyable collabora-
scalar wave. This wave has both real and imag-
tion on the work reported here. The work was
inary parts, the imaginary part is 1/r2 and
supported in partby the EECGrantno. HPRN-
∼
drops faster than radiation. The real part does
CT-1999-00161 and ESF Grant ”Geometry and
3seealso[25–29] Disorder: From Membranes to Quantum Grav-
8
ity”. Jul. 1987, 10pp; T.A. Arakelian and G.K.
Savvidy, Phys. Lett. B214 (1988) 350.
17. M.J.Duff, Supermembranes: The firstfifteen
REFERENCES
weeks, Class. Quant. Grav. 5 (1988) 189; Su-
1. H.B. Nielsen and P. Olesen, Nucl. Phys. B61 permembranes, hep-th/9611203.
(1973) 45. 18. G.K. Savvidy, Symplectic and large-N gauge
2. G.´t Hooft Nucl. Phys. B79 (1974) 276; A. theories, in NATO ASI Series v. 255 (1991)
Polyakov,JETP Lett. 20 (1974) 194. 415.
3. B. Julia and A. Zee, Phys. Rev. D11 (1975) 19. B. Biran, E. Floratos and G. Savvidy, Phys.
2227. Lett. B198 (1987) 329.
4. P.A.M. Dirac, Proc. Roy. Soc. A133 (1934) 20. C.G. Callan and J. Maldacena, Nucl. Phys.
60. B513 (1998) 198, hep-th/9708147.
5. J. Schwinger, Phys. Rev. 144 (1966) 1087. 21. G. Gibbons, Nucl. Phys. B514 (1998) 603,
6. C. Montonen and D. Olive, Phys. Lett. 72B hep-th/9709027.
(1977) 117. 22. K.Savvidy,Born-InfeldActionin StringThe-
7. E. Witten and D. Olive, Phys. Lett. 78B ory, PhD Thesis (Princeton University, June
(1978) 97. 1999) hep-th/9906075.
8. N. Seiberg and E. Witten, Nucl. Phys. B426 23. K.SavvidyandG.Savvidy,Neumannbound-
(1994) 19. ary conditions from Born-Infeld dynamics,
9. G.T. Horowitz and A. Strominger, Nucl. hep-th/9902023, Nucl. Phys. B561 (1999)
Phys. B360 (1991) 197; A. Dabholkar, G. 117.
Gibbons, J. Harvey and F. Ruiz Ruiz, Nucl. 24. K. Savvidy, October 1999, unpublished.
Phys. B340 (1990) 33; S.M. Hull and P.K. 25. C.G. Callan and A. Guijosa, Undulating
Townsend, Nucl. Phys. B438 (1995) 109; E. strings and gauge fields, hep-th/9906153.
Witten, Nucl. Phys. B443 (1995) 85; A. 26. M. Abou-Zeid and M. Costa, Radiation from
Strominger, Nucl. Phys. B451 (1995) 85; K. accelarated branes, hep-th/9909148.
Stelle, Lectures on Supergravity p-branes, 27. C. Bachas and B. Pioline. High energy scat-
hep-th/9710046; M.J. Duff, R.R. Khuri and tering on distant branes, hep-th/9909171.
J.X. Lu, String solitons, Phys. Rep. 259 28. R.JanikandR.Peschanski,Highenergyscat-
(1995) 213-326. teringandtheAdS/CFTcorrepondance,hep-
10. J.Polchinski,Phys.Rev.Lett.75(1995)4724, th/9907177.
hep-th/9510017; J. Polchinski, Tasi lectures 29. M. Rho, S-J. Sin and I. Zahed, Elastic
on D-branes, hep-th/9611050. Parton-Parton scattering from AdS/CFT,
11. J.Dai,R.G.Leigh,J.Polchinski,Mod.Phys. hep-th/9907126.
Lett. A4 (1989) 2073. 30. S.-J. Rey, J.-T. Lee, hep-th/9803001.
12. R.G.Leigh,Mod.Phys.Lett.A4(1989)2767. 31. M. Aganagic, C. Popescu and J.H. Schwarz,
13. C.Bachas,(Half)alectureonD-branes,hep- Nucl. Phys.B495(1997)99, hep-th/9612080.
th/9701019. 32. S.Lee,A.PeetandL.Thorlacius,Nucl.Phys.
14. E. Kiritsis, Supersymmetry and Duality 514 (1998) 161, hep-th/9710097.
in Field Theory and String Theory, hep- 33. C.G. Callan and L. Thorlacius, Nucl. Phys.
ph/9911525. B534 (1998) 121, hep-th/9803097.
15. G. Baseyan, S. Matinyan and G. Savvidy,
Pisma Zh. Eksp.Teor. Fiz. 29 (1979)641; G.
Savvidy, Nucl. Phys. B246 (1984) 302.
16. M.E. Laziev and G.K. Savvidy, Phys. Lett.
B198 (1987) 451; Sov. J. Nucl. Phys. 47
(1988)304;G.K.Savvidy,Sov.J.Nucl.Phys.
47 (1988) 748; Preprint-CERN-TH-4779/87,