Introduction to String Theory

78 Pages·2003·0.501 MB·English

by  Mohaupt T.

Introduction to String Theory
Thomas Mohaupt
Friedrich-SchillerUniversita(cid:127)t Jena,Max-Wien-Platz 1, D-07743 Jena,Germany
2
0 Abstract. Wegiveapedagogicalintroductiontostringtheory,D-branesandp-brane
0 solutions.
2

l
u 1 Introductory remarks
J

7 These notes are based on lectures given at the 271-th WE-Haereus-Seminar
2
‘Aspects of Quantum Gravity’. Their aim is to give an introduction to string

theory for students and interested researches. No previous knowledge of string

1
theory is assumed.The focus is ongravitationalaspects and we explaininsome
v
detailhowgravityisdescribedinstringtheoryintermsofthegravitonexcitation

9 of the string and through background gravitational(cid:12)elds. We include Dirichlet
4
boundaryconditionsandD-branesfromthebeginninganddevoteonesection to
2
p-brane solutions and their relation to D-branes. In the (cid:12)nal section we brie(cid:13)y
7
0 indicate how string theory (cid:12)ts into the larger picture of M-theory and mention
2 some of the more recent developments, like brane world scenarios.
0 The WE-Haereus-Seminar ‘Aspects ofQuantumGravity’covered both main
/
h approaches to quantum gravity: string theory and canonical quantum gravity.
t Botharecomplementaryinmanyrespects.Whilethecanonicalapproachstresses
-
p background independence and provides a non-perturbative framework, the cor-
e nerstone of string theory still is perturbation theory in a (cid:12)xed background ge-
h
ometry. Another di(cid:11)erence is that in the canonical approach gravity and other
:
v interactions are independent fromeach other, while string theory automatically
i
X is a uni(cid:12)ed theory of gravity, other interactions and matter. There is a single
dimensionful constant and all couplings are functions of this constant and of
r
a vacuum expectation values of scalars. The matter content is uniquely (cid:12)xed by
the symmetries ofthe underlying stringtheory.Moreover, whenformulatingthe
theory in Minkowski space, the number of space-time dimensions is (cid:12)xed. As
we will see, there are only (cid:12)ve distinct supersymmetric string theories in ten-
dimensionalMinkowski space.
The most important feature of string perturbation theory is the absence of
UV divergencies. This allows one to compute quantum corrections to scatter-
ing amplitudes and to the e(cid:11)ective action, including gravitationale(cid:11)ects. More
recently, signi(cid:12)cant progress has been made in understanding non-perturbative
aspects ofthe theory,throughthe study ofsolitonsandinstantons,andthrough
stringdualitieswhichmapthe strongcouplingbehaviourofone stringtheory to
the weak coupling behaviour of a dual theory. Moreover, string dualities relate
2 Thomas Mohaupt
all(cid:12)ve supersymmetric string theories to one another andlead to the picture of
one single underlying theory, called M-theory. So far, only various limits of this
theory are known, while the problem of (cid:12)nding an intrinsic, non-perturbative
and background-independent de(cid:12)nition is unsolved. One expects that M-theory
has an underlying principle which uni(cid:12)es its various incarnations, presumably
a symmetry principle. One of the obstacles on the way to the (cid:12)nal theory is
that it is not clear which degrees of freedom are fundamental. Besides strings,
also higher-dimensional p-branes play an essential role. Moreover, there is an
eleven-dimensional limit,which cannot be described in terms of strings.
Our presentation of string theory will be systematic rather than follow the
path of historical development. Nevertheless we feel that a short historical note
will be helpful, since many aspects which may seem somewhat ad hoc (such
as the de(cid:12)nition of interactions in section 3) become clearer in their historical
context.ThestorystartedwiththeVenezianoamplitude,whichwasproposedas
an amplitude for meson scattering in pre-QCD times. The amplitude (cid:12)tted the
known experimental data very well and had precisely the properties expected
of a good scattering amplitude on the basis of S-matrix theory, the bootstrap
program and Regge pole theory. In particular it had a very special soft UV
behaviour.Later work byY. Nambu,H.B. Nielsen and L.Susskind showed that
theVenezianoamplitude,andvariousgeneralizationthereofcouldbeinterpreted
as describing the scattering of relativistic strings. But improved experimental
data ruled out the Veneziano amplitude as a hadronic amplitude: it behaved
just to softly in order to describe the hard, partonic substructures of hadrons
seen in deep inelastic scattering. J. Scherk and J. Schwarz reinterpreted string
theory as a uni(cid:12)ed theory of gravity and all other fundamental interactions,
making use of the fact that the spectrum of a closed string always contains a
massless symmetric tensor state which couples like a graviton. This lead to the
development of perturbative string theory, as we willdescribe it in sections 2{4
of these lecture notes. More recently the perspective has changed again, after
the role of D-branes, p-branes and string dualities was recognized. This will be
discussed brie(cid:13)y in sections 5 and 6.
Fromthehistoricalperspective itappearsthatstringtheoryisatheorywhich
is ‘discovered’ rather than ‘invented’. Though it was clear from the start that
one was dealing with an interesting generalization of quantum (cid:12)eld theory and
generalrelativity,thesubjecthasgonethroughseveral‘phasetransitions’,andits
fundamentalprinciplesremaintobemadeexplicit.Thisisagaincomplementary
to canonical quantum gravity, where the approach is more axiomatic, starting
from a set of principles and proceeding to quantize Einstein gravity.
The numerous historicaltwists, our lackof(cid:12)nalknowledge about the funda-
mental principles and the resulting diversity of methods and approaches make
stringtheory asubject which isnoteasy to learn(or toteach). The 271-thWE-
Haereus-Seminar covered a broad variety of topics in quantum gravity, ‘From
Theory to Experimental Search’. The audience consisted of two groups: gradu-
ate students, mostly without prior knowledge of string theory, and researches,
working on various theoretical and experimental topics in gravity.The two lec-
Introduction to String Theory 3
tures on string theory were supposed to give a pedagogical introduction and to
prepareforlaterlecturesonbranesworlds,largeextradimensions,theAdS-CFT
correspondence and black holes. These lecture notes mostly followthe lectures,
but aim to extend them in two ways. The (cid:12)rst is to add more details to the
topicsIdiscussed inthe lectures. InparticularIwanttoexpandonpointswhich
seemed tobe either di(cid:14)cultorinteresting tothe audience.The second goalisto
includemore material,inorder to bringthe reader closer to the areas ofcurrent
active research. Both goals are somewhat contradictory, given that the result is
notmeanttobe abook,butlecture notesofdigestablelength.As acompromise
Ichoosetoexplainthosethingsindetailwhichseemed tobethemostimportant
ones for the participants of the seminar, hoping that they represent a reason-
able sample of potential readers. On the other side several other topics are also
covered, though in a more scetchy way. Besides summarizing advanced topics,
which cannot be fully explained here, I try to give an overview of (almost) all
the new developements of the last years and to indicate how they (cid:12)t into the
emerging overall picture of M-theory.
The outline of the lectures is as follows:sections 2{4 are devoted to pertur-
bativeaspects of bosonic andsupersymmetric stringtheories. They are the core
ofthe lectures. References aregivenatthe endofthesections. Stringtheoryhas
been a very active (cid:12)eld over several decades, and the vast amount of existing
literature is di(cid:14)cult to oversee even for people working in the (cid:12)eld. I will not
try to give a complete account of the literature, but only make suggestions for
further reading. The basic references are the books [1,2,3,4,5], which contain a
huge number of references to reviews and originalpapers. The reader interested
in the historical developement of the subject will (cid:12)nd information in the anno-
tated bibliography of [1]. Section 5 gives an introduction to non-perturbative
aspects by discussing a particularclass of solitons,the p-brane solutions oftype
II string theory. Section 6 gives an outlook on advanced topics: while sections
6.1{6.3 scetch how the (cid:12)ve supersymmetric string theories (cid:12)t into the larger
picture of M-theory, section 6.4 gives an overview of current areas of research,
together with references to lecture notes, reviews and some originalpapers.
2 Free bosonic strings
We start our study of string theories with the bosonic string. This theory is a
toy-model rather than a realistic theory of gravity and matter. As indicated by
its name it does not have fermionic states, and this disquali(cid:12)es it as a theory
of particle physics. Moreover, its groundstate in Minkowski space is a tachyon,
i.e., a state of negative mass squared. This signals that the theory is unstable.
Despite these shortcomings, the bosonic string has its virtues as a pedagogi-
cal toy-model: whereas we can postpone to deal with the additional techniques
needed to describe fermions, many features of the bosonic string carry over to
supersymmetric string theories, which have fermions but no tachyon.
4 Thomas Mohaupt
2.1 Classical bosonic strings
We start with a brief overview of classical aspects of bosonic strings.
Settingthestage. Letus(cid:12)rst(cid:12)xournotation.Weconsidera(cid:12)xedbackground
Pseudo-Riemannianspace-time ofdimensionD,withcoordinatesX =(X(cid:22)),
M
(cid:22)=0;:::;D 1.The metric is G (X) andwe takethe signature tobe ‘mostly
(cid:22)(cid:23)
(cid:0)
plus’,( )(+)D 1.
(cid:0)
(cid:0)
Themotionofarelativisticstringin isdescribed byitsgeneralizedworld-
M
line, a two-dimensional surface (cid:6), which is called the world-sheet. For a single
non-interacting string the world-sheet has the form of an in(cid:12)nite strip. We in-
troduce coordinates (cid:27) = ((cid:27)0;(cid:27)1) on the world-sheet. The embedding of the
world-sheet into space-time is given by maps
X : (cid:6) :(cid:27) X((cid:27)): (1)
(cid:0)!M (cid:0)!
The background metric induces a metric on the world-sheet:
@X(cid:22) @X(cid:23)
G = G ; (2)
(cid:11)(cid:12) @(cid:27)(cid:11) @(cid:27)(cid:12) (cid:22)(cid:23)
where (cid:11);(cid:12) = 0;1 are world-sheet indices. The induced metric is to be distin-
guished from an intrinsic metric h on (cid:6). As we will see below, an intrinsic
(cid:11)(cid:12)
metric is used as an auxiliary (cid:12)eld in the Polyakov formulation of the bosonic
string.
Auseful,butsometimesconfusingfactisthattheabovesettingcanbeviewed
from two perspectives. So far we have taken the space-time perspective, inter-
pretingthesystemasarelativisticstringmovinginspace-time .Alternatively
M
wemayview itasa two-dimensional(cid:12)eldtheory livingon the world-sheet, with
(cid:12)elds X which take values in the target-space . This is the world-sheet per-
M
spective,whichenablesustouseintuitionsandmethodsoftwo-dimensional(cid:12)eld
theory for the study of strings.
Actions. The natural action for a relativistic string is its area, measured with
the induced metric:
1
S = d2(cid:27) detG 1=2: (3)
NG (cid:11)(cid:12)
2(cid:25)(cid:11) j j
0 Z(cid:6)
This is the Nambu-Goto action, which is the direct generalization of the action
foramassiverelativisticparticle.Theprefactor(2(cid:25)(cid:11) ) 1istheenergyperlength
0 (cid:0)
ortensionofthe string,whichis thefundamentaldimensionfulparameter ofthe
theory. We have expressed the tension in terms of the so-called Regge slope (cid:11),
0
which has the dimension (length)2 in natural units, c = 1, ~ = 1. Most of the
time we will use string units, where in addition we set (cid:11) = 1.
0 2
The Nambu-Goto action has a direct geometric meaning, but is technically
inconvenient, due to the square root. Therefore one prefers to use the Polyakov
Introduction to String Theory 5
action, which is equivalent to the Nambu-Goto action, but is a standard two-
dimensional (cid:12)eld theory action. In this approach one introduces an intrinsic
metric on the world-sheet, h ((cid:27)), as additional datum. The action takes the
(cid:11)(cid:12)
form of a non-linear sigma-modelon the world-sheet,
1
S = d2(cid:27)phh(cid:11)(cid:12)@ X(cid:22)@ X(cid:23)G (X); (4)
P (cid:11) (cid:12) (cid:22)(cid:23)
4(cid:25)(cid:11)
0 Z(cid:6)
where h= deth .
(cid:11)(cid:12)
j j
The equation of motion for h is algebraic. Thus the intrinsic metric is
(cid:11)(cid:12)
non-dynamicalandcanbe eliminated,whichbringsus backtothe Nambu-Goto
action. Since
1 (cid:14)S 1
T := 2(cid:25)(cid:11)ph (cid:0) P =@ X(cid:22)@ X h @ X(cid:22)@(cid:13)X (5)
(cid:11)(cid:12) 0 (cid:14)h(cid:11)(cid:12) (cid:11) (cid:12) (cid:22)(cid:0) 2 (cid:11)(cid:12) (cid:13) (cid:22)
(cid:16) (cid:17)
is the energy momentum of the two-dimensional(cid:12)eld theory de(cid:12)ned by (4), we
caninterprettheequationofmotionofh asthetwo-dimensionalEinsteinequa-
(cid:11)(cid:12)
tion.Thetwo-dimensionalmetricisnon-dynamical,becausethetwo-dimensional
Einstein-Hilbertactionisatopologicalinvariant,proportionaltotheEulernum-
berof(cid:6).Thusitsvariationvanishes andthe Einsteinequationof(4)coupled to
two-dimensional gravity reduces to T = 0. Note that the energy-momentum
(cid:11)(cid:12)
tensor (5) is traceless, h(cid:11)(cid:12)T =0. This holds before imposingthe equations of
(cid:11)(cid:12)
motion(‘o(cid:11)shell’).Therefore T hasonlytwoindependent components, which
(cid:11)(cid:12)
vanishforsolutionstothe equationsofmotion(‘onshell’).Sincethe trace ofthe
energy-momentum tensor is the Noether current of scale transformations, this
shows thatthe two-dimensional(cid:12)eldtheory (4)is scale invariant.As we willsee
below, it is in fact a conformal(cid:12)eld theory.
The Polyakov action has three local symmetries. Two are shared by the
Nambu-Gotoaction, namely reparametrizations of the world-sheet:
(cid:27)(cid:11) (cid:27)~(cid:11)((cid:27)0;(cid:27)1): (6)
(cid:0)!
The third local symmetry is the multiplication of the metric h by a local,
(cid:11)(cid:12)
positive scale factor,
h ((cid:27)) e(cid:3)((cid:27))h ((cid:27)): (7)
(cid:11)(cid:12) (cid:11)(cid:12)
(cid:0)!
Thistransformationis calleda Weyltransformationbyphysicists, while mathe-
maticians usually use the term conformal transformation. The three local sym-
metries can be used to gauge-(cid:12)x the metric h . The standard choice is the
(cid:11)(cid:12)
conformalgauge,
!
h ((cid:27))=(cid:17) ; where ((cid:17) )=Diag( 1;1): (8)
(cid:11)(cid:12) (cid:11)(cid:12) (cid:11)(cid:12)
(cid:0)
While this gauge can be imposed globally on the in(cid:12)nite strip describing the
motionofasinglenon-interactingstring,itcanonlybe imposedlocallyonmore
general world-sheets, which describe string interactions. We will discuss global
aspects of gauge (cid:12)xing later.
6 Thomas Mohaupt
The conformal gauge does not provide a complete gauge (cid:12)xing, because (8)
isinvariantunder aresidualsymmetry.Onecanstillperformreparametrizations
underwhichthemetriconlychangesbyalocal,positivescalefactor,becausethis
factorcanbeabsorbedbyaWeyltransformation.Suchconformalreparametriza-
tions are usually called conformal transformations by physicists. Note that the
same term is used for Weyl transformations by mathematicians. A convenient
way to characterize conformal reparametrizations in terms of coordinates is to
introduce light cone coordinates,
(cid:27)(cid:6) =(cid:27)0 (cid:27)1 : (9)
(cid:6)
Then conformal reparametrization are precisely those reparametrizations which
do not mix the light cone coordinates:
(cid:27)+ (cid:27)~+((cid:27)+); (cid:27) (cid:27)~ ((cid:27) ): (10)
(cid:0) (cid:0) (cid:0)
(cid:0)! (cid:0)!
Thus we are left with an in(cid:12)nite-dimensional group of symmetries, which in
particular includes scale transformations.
Equations of motion, closed and open strings, and D-branes. In order
to proceed we now spezialize to the case of a (cid:13)at space-time, G =(cid:17) , where
(cid:22)(cid:23) (cid:22)(cid:23)
(cid:17) =Diag( 1;+1;:::;+1).In the conformalgauge the equation of motion for
(cid:22)(cid:23)
(cid:0)
X reduces to a free two-dimensionalwave equation,
@2X(cid:22) =@(cid:11)@ X(cid:22) =0: (11)
(cid:11)
Note that when imposing the conformal gauge on the Polyakov action (4), the
equationofmotionforh ,i.e.,T =0,becomes a constraint, whichhas to be
(cid:11)(cid:12) (cid:11)(cid:12)
imposed on the solutions of (11).
Thegeneralsolutionof(11)isasuperpositionofleft-andright-movingwaves,
X(cid:22)((cid:27))=X(cid:22)((cid:27)+)+X(cid:22)((cid:27) ): (12)
L R (cid:0)
However, we also have to specify boundary conditions at the ends of the string.
One possible choice are periodic boundary conditions,
X(cid:22)((cid:27)0;(cid:27)1+(cid:25))=X(cid:22)((cid:27)0;(cid:27)1): (13)
They correspond to closed strings. A convient parametrization of the solution
is:
X(cid:22)((cid:27))=x(cid:22)+2(cid:11)0p(cid:22)(cid:27)0+ip2(cid:11)0 (cid:11)n(cid:22)ne(cid:0)2in(cid:27)+ +ip2(cid:11)0 (cid:11)~n(cid:22)ne(cid:0)2in(cid:27)(cid:0) : (14)
n=0 n=0
X6 X6
Reality of X(cid:22) implies: (x(cid:22))? = x(cid:22) and (p(cid:22))? = p(cid:22) and ((cid:11)(cid:22))? = (cid:11)(cid:22) and
m m
((cid:11)~(cid:22))? =(cid:11)~(cid:22) . Here ? denotes complex conjugation.While x(cid:22) is the pos(cid:0)ition of
m m
the center (cid:0)ofmass of the string at time (cid:27)0,p(cid:22) is its totalmomentum.Thus, the
centerofmassmovesonastraightlineinMinkowskispace,likeafreerelativistic
Introduction to String Theory 7
particle.Theadditionaldegrees offreedomare decoupled left-and right-moving
waves on the string, with Fourier components (cid:11)(cid:22) and (cid:11)~(cid:22).
m m
Whennotchoosingperiodicboundaryconditions,theworld-sheethasbound-
aries and we have open strings. The variationof the world-sheet action yields a
boundaryterm,(cid:14)S d(cid:27)0@ X(cid:22)(cid:14)X .Thenaturalchoicetomakethebound-
’ @(cid:6) 1 (cid:22)
ary term vanish are Neumann boundary conditions,
R
@ X(cid:22) =0; @ X(cid:22) =0: (15)
1 j(cid:27)1=0 1 j(cid:27)1=(cid:25)
With these boundary conditions, momentum is conserved at the ends of the
string. Left- and right-movingwaves are re(cid:13)ected at the ends and combine into
standing waves. The solution takes the form
X(cid:22)((cid:27))=x(cid:22)+(2(cid:11)0)p(cid:22)(cid:27)0+ip2(cid:11)0 (cid:11)n(cid:22)ne(cid:0)in(cid:27)0cos(n(cid:27)1): (16)
n=0
X6
There is, however, a second possible choice of boundary conditions for open
strings, namely Dirichlet boundary conditions. Here the ends of the string are
kept (cid:12)xed:
X(cid:22) =x(cid:22) ; X(cid:22) =x(cid:22) : (17)
j(cid:27)1=0 (1) j(cid:27)1=(cid:25) (2)
With these boundary conditions the solution takes the form
X(cid:22)((cid:27))=x(cid:22)(1)+(x(cid:22)(2)(cid:0)x(cid:22)(1))(cid:27)(cid:25)1 +ip2(cid:11)0 (cid:11)n(cid:22)ne(cid:0)in(cid:27)0sin(n(cid:27)1): (18)
n=0
X6
More generally we can impose Neumann boundary conditions in the time and
in p space directions and Dirichlet boundary conditions in the other directions.
Let us denote the Neumann directions by (Xm) = (X0;X1;:::;Xp) and the
Dirichlet directions by (Xa)=(Xp+1;:::;XD 1).
(cid:0)
The most simple choice of Dirichlet boundary conditions is then to require
that all open strings begin and end on a p-dimensional plane located at an
arbitrarypositionXa =xa alongtheDirichletdirections.Suchaplaneiscalled
(1)
ap-dimensionalDirichlet-membrane,orD-p-brane,orsimplyD-brane forshort.
While the ends of the strings are (cid:12)xed in the Dirichlet directions, they still can
move freely alongthe Neumann directions. The world-volumeof a D-p-brane is
(p+1)-dimensional.TheNeumanndirections arecalledthe world-volumeorthe
paralleldirections,whilethe Dirichletdirectionsarecalledtransverse directions.
An obvious generalization is to introduce N > 1 such D-p-branes, located
at positions xa , where i = 1;:::;N, and to allow strings to begin and end on
(i)
anyof these. In this setting the mode expansion for astring starting on the i-th
D-brane and ending on the j-th is:
Xm((cid:27)) = xm+(2(cid:11)0)pm(cid:27)0+ip2(cid:11)0 (cid:11)nmn e(cid:0)in(cid:27)0cos(n(cid:27)1);
n=0
X6
Xa((cid:27)) = xa(i)+(xa(j)(cid:0)xa(i))(cid:27)(cid:25)1 +ip2(cid:11)0 (cid:11)nane(cid:0)in(cid:27)0sin(n(cid:27)1): (19)
n=0
X6
8 Thomas Mohaupt
(One might also wonder about Dirichlet boundary conditions in the time direc-
tion.This makessense, atleast forEuclidean space-time signature,and leads to
instantons, called D-instantons, which we willnot discuss in these lectures.)
Dirichletboundaryconditionshavebeenneglected forseveralyears.Therea-
sonisthatmomentumisnotconserved atthe ends ofthe strings,re(cid:13)ecting that
translation invariance is broken along the Dirichlet directions. Therefore, in a
complete fundamentaltheory the D-branes mustbe new dynamicalobjects, dif-
ferent from strings. The relevance of such objects was only appreciated when it
became apparent that string theory already includes solitonic space-time back-
grounds, so called (’RR-charged’) p-Branes, which correspond to D-branes. We
willreturn to this point later.
Promotingthe D-branes to dynamicalobjects impliesthat they willinteract
through the exchange ofstrings. This means that in general they willrepulse or
attract, and therefore their positions become dynamical. But there exist many
static con(cid:12)gurations of D-branes (mainly in supersymmetric string theories),
where the attractive and repulsive forces cancel for arbitrary distances of the
branes.
2.2 Quantized bosonic strings
The de(cid:12)nition of a quantum theory of bosonic strings proceeds by using stan-
dard recipies of quantization. The two most simple ways to proceed are called
‘old covariant quantization’ and ‘light cone quantization’. As mentioned above
imposingtheconformalgaugeleavesuswitharesidualgaugeinvariance.Inlight
cone quantization one (cid:12)xes this residual invariance by imposing the additional
condition
X+ =! x++p+(cid:27)+ ; i.e.; (cid:11)+ =! 0; (20)
m
where X = 1 (X0 XD 1) are light cone coordinates in space-time. Then
(cid:6) p2 (cid:6) (cid:0)
the constraints T = 0 are solved in the classical theory. This yields (non-
(cid:11)(cid:12)
linear) expressions for the oscillators (cid:11) in terms of the transverse oscillators
(cid:0)n
(cid:11)i, i= 1;:::D 2. In light cone coordinates the world-sheet is embedded into
n (cid:0)
space-time alongthe X0;XD 1 directions. The independent degrees of freedom
(cid:0)
are the oscillations transverse to the world sheet, which are parametrized by
the (cid:11)i. One proceeds to quantize these degrees of freedom. In this approach
n
unitarity of the theory is manifest,but Lorentz invariance is not.
In old covariant quantization one imposes the constraints at the quantum
level. Lorentz covariance is manifest, but unitarity is not: one has to show that
there is a positive de(cid:12)nite space of states and an unitary S-matrix. This is the
approach we will describe in more detail below.
One might also wonder about ‘new covariant quantization’, which is BRST
quantization. This approach is more involved but also more powerful then old
covariantquantization.When dealingwith advanced technical problems,forex-
ampletheconstructionofscatteringamplitudesinvolvingfermionsinsuperstring
theories, BRST techniques become mandatory. But this is beyond the scope of
these lectures.
Introduction to String Theory 9
The Fock space. The (cid:12)rst step is to impose canonical commutation relations
on X(cid:22)((cid:27)) and its canonical momentum (cid:5)(cid:22)((cid:27)) = @ X(cid:22)((cid:27)). In terms of modes
0
one gets
[x(cid:22);p(cid:23)]=i(cid:17)(cid:22)(cid:23) ; [(cid:11)(cid:22);(cid:11)(cid:23)]=m(cid:17)(cid:22)(cid:23)(cid:14) : (21)
m n m+n;0
Forclosed strings there are analogousrelationsfor(cid:11)~(cid:22).Therealityconditionsof
m
the classical theory translate into hermiticity relations:
(x(cid:22))+ =x(cid:22) ; (p(cid:22))+ =p(cid:22) ; ((cid:11)(cid:22))+ =(cid:11)(cid:22) : (22)
m m
(cid:0)
While the commutation relations for x(cid:22);p(cid:23) are those of a relativistic particle,
the (cid:11)(cid:22) satisfy the relations of creation and annihilation operators of harmonic
m
oscillators,though with an unconventional normalization.
To proceed, one constructs a Fock space on which the commutation rela-
F
tions (21) are repesented. First one chooses momentum eigenstates k , which
j i
are annihiliatedby half of the oscillators:
p(cid:22) k =k(cid:22) k ; (cid:11)(cid:22) k =0=(cid:11)~(cid:22) k ; m>0: (23)
j i j i mj i mj i
Then a basis of is obtained by acting with creation operators:
B F
= (cid:11)(cid:22)1 (cid:11)~(cid:23)1 k m ;n >0 : (24)
B f (cid:0)m1(cid:1)(cid:1)(cid:1) (cid:0)n1(cid:1)(cid:1)(cid:1)j ij l l g
A bilinear form on , which is compatible with the hermiticity properties (22),
F
cannot be positive de(cid:12)nite. Consider for example the norm squared ofthe state
(cid:11)(cid:22) k :
(cid:0)mj i k ((cid:11)(cid:22) )+(cid:11)(cid:22) k (cid:17)(cid:22)(cid:22) = 1: (25)
h j (cid:0)m (cid:0)mj i(cid:24) (cid:6)
However,theFockspace isnotthespace ofphysicalstates, because westillhave
toimpose the constraints. The real question iswhether the subspace ofphysical
states contains states of negative norm.
The Virasoro Algebra. Constraints arise when the canonical momenta of a
system are not independent. This is quite generic for relativistic theories. The
mostsimpleexampleisthe relativisticparticle,where the constraintis themass
shell condition, p2+m2 =0. When quantizing the relativistic particle, physical
states are those annihilated by the constraint, i.e., states satisfying the mass
shell condition:
(p2+m2)(cid:8) =0: (26)
j i
When evaluating this in a basis of formal eigenstates of the operator x(cid:22), one
obtains the Klein-Gordon equation, (@2+m2)(cid:8)(x) = 0, where (cid:8)(x) = x(cid:8) is
h j i
interpreted as the state vector in the x-basis. This is a clumsy way to approach
thequantumtheoryofrelativisticparticles,andoneusuallyprefers touse quan-
tum (cid:12)eld theory (‘second quantization’) rather than quantum mechanics (‘(cid:12)rst
quantization’).Butinstringtheoryitturns outthatthe(cid:12)rst quantizedformula-
tionworksnicelyforstudyingthespectrum andcomputingamplitudes,whereas
string (cid:12)eld theory is very complicated.
10 Thomas Mohaupt
Proceeding parallel to the case of a relativistic particle one (cid:12)nds that the
canonical momentum is (cid:5)(cid:22) =@ X(cid:22). The constraints are
0
(cid:5)(cid:22)@ X =0; (cid:5)(cid:22)(cid:5) +@ X(cid:22)@ X =0: (27)
1 (cid:22) (cid:22) 1 1 (cid:22)
In the Polyakov formulation they are equivalent to T = 0. It is convenient
(cid:11)(cid:12)
to express the constraints through the Fourier components of T . Passing to
(cid:11)(cid:12)
light cone coordinates, the tracelessness of T , which holds without using the
(cid:11)(cid:12)
equation of motion or imposingthe constraints, implies
T =0=T : (28)
+ +
(cid:0) (cid:0)
Thusweareleftwithtwoindependent components,T andT ,whereT
++
@ X(cid:22)@ X .Forclosed strings, where @ X(cid:22) areperiodic in(cid:27)1(cid:0),(cid:0)we expand(cid:6)T(cid:6) ’
(cid:22)
in(cid:6)a Fo(cid:6)urier series and obtain Fourier(cid:6)coe(cid:14)cients L ;L~ , m Z. For op(cid:6)e(cid:6)n
m m
2
strings, observe that (cid:27)1 (cid:27)1 exchanges @ X(cid:22) and @ X(cid:22). Both (cid:12)elds can
+
! (cid:0) (cid:0)
be combined into a single (cid:12)eld, which is periodic on a formally doubled world-
sheet with (cid:25) (cid:27)1 (cid:25). In the same way one can combine T with T .
++
(cid:0) (cid:20) (cid:20) (cid:0)(cid:0)
By Fourier expansion on the doubled world-sheet one then obtains one set of
Fourier modes for the energy-momentum tensor, denoted L . This re(cid:13)ects that
m
left- and right-movingwaves couple through the boundaries.
The explicit form for the L in terms of oscillators is
m
1
L = 1 (cid:11) (cid:11) ; (29)
m 2 m(cid:0)n(cid:1) n
n=
X(cid:0)1
with an analogous formula for L~ for closed strings. We have denoted the con-
m
traction of Lorentz indices by ‘’ and de(cid:12)ned (cid:11)(cid:22) = 1p(cid:22) = (cid:11)~(cid:22) for closed strings
and(cid:11)(cid:22) =p(cid:22) foropenstrings. In(cid:1) terms ofthe Fo0urier2modes,0the constraints are
0
L =0, and, for closed string, L~ = 0. Translations in (cid:27)0 are generated by L
m m 0
for open and by L +L~ for closed strings. These functions are the world-sheet
0 0
Hamiltonians.The L satisfy the Witt algebra algebra,
m
L ;L =i(m n)L ; (30)
m n P.B. m+n
f g (cid:0)
where ; is the Poisson bracket. For closed strings we have two copies
P.B.
f(cid:1) (cid:1)g
of this algebra. The Witt algebra is the Lie algebra of in(cid:12)nitesimal conformal
transformations.Thusthe constraintsre(cid:13)ect thatwehavearesidualgaugesym-
metry corresponding to conformal transformations. Since the constraints form
a closed algebra with the Hamiltonian, they are preserved in time. Such con-
straints are called (cid:12)rst class, and they can be imposed on the quantum theory
without further modi(cid:12)cations (such as Dirac brackets).
In the quantum theory the L are taken to be normal ordered, i.e., annihi-
m
lationoperators are moved to the right. This is unambigous, except for L . We
0
will deal with this ordering ambiguity below. The hermiticiy properties of the
L are:
m
L+ =L : (31)
m m
(cid:0)