Intro to String Theory - G. terHooft, Angielskie techniczne
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INTRODUCTIONTOSTRINGTHEORY
¤
version14-05-04
Gerard’tHooft
InstituteforTheoreticalPhysics
UtrechtUniversity,Leuvenlaan4
3584CCUtrecht,theNetherlands
and
SpinozaInstitute
Postbox80.195
3508TDUtrecht,theNetherlands
e-mail:
g.thooft@phys.uu.nl
Contents
1StringsinQCD. 4
1.1Thelineartrajectories.............................. 4
1.2TheVenezianoformula.............................. 5
2Theclassicalstring. 7
3Openandclosedstrings. 11
3.1TheOpenstring.................................11
3.2Theclosedstring.................................12
3.3Solutions.....................................12
3.3.1Theopenstring. ............................12
3.3.2Theclosedstring.............................13
3.4Thelight-conegauge...............................14
3.5Constraints....................................15
3.5.1foropenstrings:.............................16
¤
Lecturenotes2003and2004
1
3.5.2forclosedstrings:............................16
3.6Energy,momentum,angularmomentum....................17
4Quantization. 18
4.1Commutationrules................................18
4.2Theconstraintsinthequantumtheory.....................19
4.3TheVirasoroAlgebra..............................20
4.4Quantizationoftheclosedstring.......................23
4.5Theclosedstringspectrum...........................24
5Lorentzinvariance. 25
6Interactionsandvertexoperators. 27
7BRSTquantization. 31
8ThePolyakovpathintegral.Interactionswithclosedstrings. 34
8.1Theenergy-momentumtensorfortheghostfields...............36
9
T
-Duality. 38
9.1Compactifyingclosedstringtheoryonacircle.................39
9.2
T
-dualityofclosedstrings............................40
9.3
T
-dualityforopenstrings............................41
9.4Multiplebranes..................................42
9.5Phasefactorsandnon-coinciding
D
-branes. .................42
10Complexcoordinates. 43
11Fermionsinstrings. 45
11.1Spinningpointparticles.............................45
11.2ThefermionicLagrangian............................46
11.3Boundaryconditions...............................49
11.4Anticommutationrules.............................51
11.5Spin........................................52
11.6Supersymmetry..................................53
11.7Thesupercurrent. ...............................54
2
11.8Thelight-conegaugeforfermions.......................56
12TheGSOProjection. 58
12.1Theopenstring. ................................58
12.2Computingthespectrumofstates. ......................61
12.3Stringtypes....................................63
13Zeromodes 65
13.1Fieldtheoriesassociatedtothezeromodes. .................68
13.2Tensorfieldsand
D
-branes. ..........................71
13.3
S
-duality.....................................73
14MiscelaneousandOutlook. 75
14.1Stringdiagrams.................................75
14.2Zeroslopelimit.................................76
14.2.1TypeIItheories.............................76
14.2.2TypeItheory..............................77
14.2.3Theheterotictheories.........................77
14.3Stringsonbackgrounds.............................77
14.4Coordinateson
D
-branes.Matrixtheory....................78
14.5Orbifolds.....................................78
14.6Dualities.....................................79
14.7Blackholes...................................79
14.8Outlook.....................................79
3
1.StringsinQCD.
1.1.Thelineartrajectories.
Inthe’50’s,mesonsandbaryonswerefoundtohavemanyexcitedstates,calledres-
onances,andinthe’60’s,theirscatteringamplitudeswerefoundtoberelatedtothe
so-calledReggetrajectories:
J
=
®
(
s
),where
J
istheangularmomentumand
s
=
M
2
,
thesquareoftheenergyinthecenterofmassframe.Aresonanceoccursatthose
s
values
where
®
(
s
)isanonnegativeinteger(mesons)oranonnegativeintegerplus
1
2
(baryons).
Thelargest
J
valuesatgiven
s
formedtheso-called‘leadingtrajectory’.Experimentally,
itwasdiscoveredthattheleadingtrajectorieswerealmostlinearin
s
:
®
(
s
)=
®
(0)+
®
0
s:
(1.1)
Furthermore,therewere‘daughtertrajectories’:
®
(
s
)=
®
(0)
¡n
+
®
0
s:
(1.2)
where
n
appearedtobeaninteger.
®
(0)dependsonthequantumnumberssuchas
strangenessandbaryonnumber,but
®
0
appearedtobeuniversal,approximately1GeV
¡
2
.
Ittooksometimebeforethesimplequestionwasasked:supposeamesonconsistsof
twoquarksrotatingaroundacenterofmass.Whatforcelawcouldreproducethesimple
behaviorofEq.(1.1)?Assumethatthequarksmovehighlyrelativistically(whichis
reasonable,becausemostoftheresonancesaremuchheavierthanthelightest,thepion).
Letthedistancebetweenthequarksbe
r
.Eachhasatransversemomentum
p
.Then,if
weallowourselvestoignoretheenergyoftheforcefieldsthemselves(andput
c
=1),
s
=
M
2
=(2
p
)
2
:
(1.3)
Theangularmomentumis
J
=2
p
r
2
=
pr:
(1.4)
Thecentripetalforcemustbe
F
=
pc
r=
2
=
2
p
r
:
(1.5)
Fortheleadingtrajectory,atlarge
s
(sothat
®
(0)canbeignored),wefind:
r
=
2
J
p
s
=2
®
0
p
s
;
F
=
s
2
J
=
1
2
®
0
;
(1.6)
or:theforceisaconstant,andthepotentialbetweentwoquarksisalinearlyrisingone.
Butitisnotquitecorrecttoignoretheenergyoftheforcefield,and,furthermore,the
aboveargumentdoesnotexplainthedaughtertrajectories.Amoresatisfactorymodelof
themesonsisthe
vortexmodel
:anarrowtubeoffieldlinesconnectsthetwoquarks.This
4
linelikestructurecarriesalltheenergy.Itindeedgeneratesaforcethatisofauniversal,
constantstrength:
F
=d
E=
d
r
.Althoughthequarksmoverelativistically,wenowignore
theircontributiontotheenergy(asmall,negativevaluefor
®
(0)willlaterbeattributed
tothequarks).Astationaryvortexcarriesanenergy
T
perunitoflength,andwetake
thisquantityasaconstantofNature.Assumethisvortex,withthequarksatitsend
points,torotatesuchthattheendpointsmovepracticallywiththespeedoflight,
c
.At
apoint
x
between
¡r=
2and
r=
2,theangularvelocityis
v
(
x
)=
cx=
(
r=
2).Thetotal
energyisthen(putting
c
=1):
E
=
Z
r=
2
p
1
¡v
2
=
Tr
T
d
x
Z
1
0
(1
¡x
2
)
¡
1
=
2
d
x
=
1
2
¼Tr;
(1.7)
¡r=
2
whiletheangularmomentumis
J
=
Z
r=
2
p
1
¡v
2
=
1
2
Tr
2
Z
1
p
1
¡x
2
=
Tr
2
¼
x
2
d
x
8
:
(1.8)
¡r=
2
0
Thus,inthismodelalso,
E
2
=
1
J
2
¼T
=
®
0
;
®
(0)=0
;
(1.9)
buttheforce,or
stringtension
,
T
,isafactor
¼
smallerthaninEq.(1.6).
1.2.TheVenezianoformula.
1
4
2
3
Considerelasticscatteringoftwomesons,(1)and(2),formingtwoothermesons(3)
and(4).Elasticheremeansthatnootherparticlesareformedintheprocess.Theingoing
4-momentaare
p
(1)
¹
and
p
(2)
¹
.Theoutgoing4-momentaare
p
(3)
¹
and
p
(4)
¹
.Thec.m.energy
squaredis
s
=
¡
(
p
(1)
¹
+
p
(2)
¹
)
2
:
(1.10)
Anindependentkinematicalvariableis
t
=
¡
(
p
(1)
¹
¡p
(4)
¹
)
2
:
(1.11)
Similarly,onedefines
u
=
¡
(
p
(1)
¹
¡p
(3)
¹
)
2
;
(1.12)
5
Tvx
d
x
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