Reparametrization invariant operators in string field theory


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Reparametrization invariant operators in string field theory
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vi, 69 leaves : ill. ; 29 cm.
Kleppe, Gary, 1962-
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String models   ( lcsh )
Physics thesis Ph. D
Dissertations, Academic -- Physics -- UF
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Thesis (Ph. D.)--University of Florida, 1991.
Includes bibliographical references (leaves 66-68)
Statement of Responsibility:
by Gary Kleppe.
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University of Florida
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Full Text







To all of my teachers throughout the

years, especially my family, who had the

hardest job.


I owe many thanks to Professor Pierre Ramond, my thesis advisor, for

many enlightening discussions; for his encouragement, which led me to strive

to do better quality work; for his patience and support when my work became

especially difficult; and for his wit and humor, which made my work more

enjoyable. I am also grateful to Professor Richard Woodard for many long

and extremely informative discussions and for allowing me to collaborate with

him on another project. I also thank Doctor R. Raju Viswanathan for his

collaboration and friendship, both of which were invaluable during my stay

here, and Thomas McCarty for his long friendship and many useful discussions

including a preliminary reading of this thesis.

I would like to thank the University of Florida particle theory group for

useful discussions, especially those members who participated in the seminars

that I gave. In particular, I thank Professor Charles Thorn, Doctors Deok Ki

Hong, Diego Harari, and Phillial Oh, and future Doctors Bettina Kesthelyi,

Eric Piard, and Brian Wright, for comments and discussions which directly

influenced my work.

I wish to thank my family in Milwaukee for their support and encour-

agement. Finally, I also thank all those friends (whom I do not list, for fear

of accidentally omitting someone) who made my time in graduate school a

very enjoyable experience. In particular, I thank Zhu Chengjun for her warm

friendship and companionship during the past three years.


ACKNOWLEDGEMENTS . . . . . . . . . .iii

ABSTRACT . . . . . . . . . . . . . vi

INTRODUCTION . . . . . . . . . . . .. 1

Representations of the Reparametrization Algebra . . . . 8
Direct Product Rules . . . . . . . . . . .. 10
Functionals . . . . . . . . . . . . .. 11
Super-reparametrizations . . . . . . . . . .. 13
Representations of the Super-reparametrization Algebra . . .. .13
Direct Product Representations . . . . . . . .. .16

FREE FIELD THEORY . . . . . . . . . .. .18
Second Quantization . . . . . . . . . . .. .18
The Einbein . . . . . . . . . . . . .. 20
Mode Expansions . . . . . . . . . . .. .21
Left and Right Combinations . . . . . . . . .. .23
Anomaly Cancellation . . . . . . . . . .. .26
Construction of the Invariant Operator . . . . . . .. .28

CATALOG OF INVARIANTS . . . . . . . . .. .30
Operator Algebras . . . . . . . . . . .. .31
An Exam ple . . . . . . . . . . . . .. 36
Lorentz Symmetry Breaking . . . . . . . . .. .38

INTERACTIONS . . . . . . . . . . . .. .41
Second Quantization . . . . . . . . . . .. .41
Witten's Interaction . . . . . . . . . . .. .43
Manifestly Invariant Formulation . . . . . . . .. .45
Nonlocality . . . . . . . . . . . . .. 48
Ghost Insertion . . . . . . . . . . . .. 48

CONCLUSIONS . . . . . . . . . .





REFERENCES . . . . . . . . . .


. . . 50

. . . 53

. . . 58

. . . 61

. . . 64

. . . 66

. . . 69

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy




August 1991
Chairman: Pierre Ramond
Major Department: Physics

In this study, the role of the reparametrization group in the field theory of

strings is examined. Representations and direct product rules are constructed

for both the bosonic and supersymmetric reparametrization algebras. This

group theory is applied to free open bosonic string field theory. A catalog

is made of the various invariant operators which may be formed. These are

found to include a previously unknown symmetric tensor whose trace is the

BRST charge. It is found that only certain subsets of the projections of this

tensor are closed algebraically. The possible subgroups of the Lorentz group

which may be left unbroken by the choice of these subsets are classified. Finally,

reparametrization invariance is discussed in the context of the interacting open

string theory of Witten. It is argued that in this theory, reparametrization

invariance is hidden by a parametrization dependent choice of phases in a

set of basis wave functionals. An alternative, completely equivalent, version

of this theory is constructed, which does possess manifest reparametrization

invariance; however, the interaction in the new formulation has an apparently

nonlocal form.


The standard model of particle physics (see ref. 1 for reviews) is in agree-

ment with the results of all high energy physics experiments which have been

performed so far. Nevertheless, this model suffers from two shortcomings.

Firstly, many details of the model are not predicted by the theory and must be

put in "by hand." It is not just the numerical quantities like mass ratios and

couplings, but also the pattern of particle families, the gauge group structure,

even the fact that spacetime has four dimensions, which the theory makes no

attempt to predict. Secondly, the standard model does not include gravity.

This does not contradict experiment, because all experiments to date in ele-

mentary particle physics have probed physics at energy ranges far lower than

those at which the gravitational forces would become significant. However, it

does indicate that the theory is not fundamentally complete. It also means that

the theory cannot be applied to situations in which both quantum mechanics

and gravity are significant, a striking example of which is the early universe.

When one tries to define a quantum field theory containing gravity [2], one

quickly finds that the theory is non-renormalizable. This means that the theory

at high energies behaves badly enough that its predictive power is lost, at least

if one treats the interactions using the standard perturbative methods, which

are the only methods currently known which are generally useful in interacting

field theory. All of this seems to indicate that some new theory may be needed,

one which reduces to the old theory in the appropriate limit.

In contrast to local field theories, there are only a few different string the-

ories [3,4,5]. The simplest is the bosonic string [6,7], which has twenty-six


[8,9,10,11] spacetime coordinates xa(o), where / 0 = 0,1,...,25 labels the one

time and twenty-five space coordinates of the string, and a is a parameter

labeling the individual points along the string. The slightly more compli-

cated superstring [12] theories, of which there are only a few types, include

ten coordinates xP as above, and anticommuting coordinates which are the

supersymmetric partners of the xP. There is also a theory called the heterotic

string [13], which combines aspects of both the bosonic and supersymmetric

theories. This theory has a naturally arising gauge symmetry which contains

(among many other things) the gauge groups and structure of the standard

model; thus the heterotic string is the one which actually has the possibility of

being a workable theory of everything.

In addition to the different categorizations described above, strings have

the property of being either open or closed, closed strings being those whose

endpoints are constrained to coincide with one another: xa(0) = x/(27r). Be-

cause of the nature of string interactions, an interacting theory which contains

open strings must also contain closed strings. This is because two open strings

interact by joining at their endpoints to form one longer open string. Locally,

this interaction looks the same as the joining of the two ends of an open string

to form a closed string. The end of an open string cannot discriminate be-

tween its own other end and the end of another open string without violating

the locality of the interactions, thus ruining the quantum consistency of the

theory. Of the string theories mentioned above, the heterotic string and most

types of superstrings include only closed strings, the other type of superstrings

includes both, and the bosonic theory includes closed strings and may or may

not include open strings.

In short, there are only a few different string theories, and each has no

free parameters. Thus, given enough calculational facility, we could use string

theory to predict answers to all experimental questions. If any of these predic-

tions were not correct, string theory would be ruled out as the theory of the


On first examination, the phenomenology of the quantum theory of strings

would seem to bear little relation to the experimentally observed world. Het-

erotic string theory requires ten spacetime dimensions, and predicts a large

spectra of particles which are generally not observed. However, various schemes

have been proposed by which the ten dimensions of the heterotic string com-

pactify to yield a four-dimensional theory. There seem to be a huge number

of such schemes [14,15], each yielding different low-energy phenomenology. Al-

though the dynamics of the string should in principle determine which scheme

(if any) is favored, in practice it is not known how to make this determination.

One such scheme has been extensively investigated and found to yield realis-

tic results [16]; however, this is only one of thousands or millions of possible


The spectrum of the closed string is found [17,18] to automatically include

a massless spin-two particle, which may be identified as the quantum of the

gravitational field. This is of particular interest because string theories are

well-behaved in the high-energy regime, unlike other theories of gravity. Thus

string theory gives us at best a true theory of everything, and at worst a way of

including perturbative gravity into a quantum theory without losing predictive

power through nonrenormalizibility. Even if strings turn out not to explain the

entire universe, they will certainly show us new directions in which to search

for a quantum mechanical theory of gravity.

The standard approach to string theory is to first solve the quantum me-

chanics of the free string; then interactions are introduced in a way which

is manifestly perturbative [19,11,20,21]. Although this is sufficient for many

types of calculations, we would expect that a fully nonperturbative starting

point would be necessary before we could answer certain questions, such as

whether compactification is favored. Such a formulation of the theory is pro-

vided by the second-quantized theory of strings, known as string field theory


String field theory (for reviews see references 25 and 26) was developed

long ago [22] in the light cone gauge [11]. It was not until much later that

manifestly covariant field theories of strings were developed. The field theory

of open bosonic strings was developed by Witten [24]. This theory was shown

to reproduce the correct dual model amplitudes. It is based on a physically

appealing picture, as we shall discuss, in which three strings are created from

the vacuum at some instant of time. Energy conservation requires one or two

of the strings to be created moving back in time. Additionally, the strings

are required to overlap, so that half of one string (0 < a < 7r/2) is required to

overlap with the other half of another string (7r/2 < a < 7r). The generalization

of Witten's formalism to open superstrings [27] and closed strings [28] proved

to be fraught with difficulties [29,30]; in fact, the question of closed string field

theory seems not to be settled [31].

In any string theory one naively expects that observable quantities should

be independent of the parametrization of the string. This requirement plays a

central role in the first-quantized formalism [5]. However, Witten's string field

theory breaks reparametrization invariance by singling out the "middle point,"

a = 7r/2, the location of which clearly depends on a particular parametrization.


As Witten [24] points out, BRST [32,33] invariance is sufficient to guarantee

that the negative norm states decouple [34] from the physical states in the

theory, so reparametrization invariance is not necessary. Nevertheless, it seems

paradoxical that this theory, which completely reproduces the results of the

first-quantized theory, disregards a symmetry which was fundamental in the

original theory.

In the case of Yang-Mills field theory [35], BRST invariance arises as a

remnant of a gauge invariance after the gauge has been fixed. It has been con-

jectured that Witten's theory may be some gauge-fixed version of a more funda-

mental theory, one which includes reparametrization invariance, and probably

closed strings as well. There have been many attempts [36,37,38] to develop

a new version of Witten's interaction which does not lose reparametrization


This dissertation is organized as follows: Following this introduction, an

attempt is made to develop a systematic group theoretical technology of the

reparametrization group; for completeness, the supersymmetric extension of

the reparametrization group is also discussed. Then the role of the repara-

metrization group in the free string field theory is examined, in particular the

transition from first quantized strings to string fields is discussed. A catalog

of operators which are invariant under reparametrizations is presented; some

familiar ones are seen, along with a collection of new ones, whose utility is

not clear. Finally, we address the question of the apparent lack of repara-

metrization invariance in the interactions; this is explained as being due to a

parametrization-dependent choice of phases in the string field. It is shown how

to derive the interactions for any choice of phases, and the interactions are

derived for the special case in which the phase convention does not depend on


the parametrization. In this formulation the interactions are shown to be man-

ifestly parametrization invariant, and independent of any choice of a midpoint;

however, the interactions appear to be nonlocal in coordinate space.


We turn now to a discussion of the mathematical theory associated with the

reparametrization symmetry. Generally, the set of symmetry operations which

leave a physical system unchanged will necessarily satisfy the four axioms of

a group [39]. Namely, the successive action of two symmetry transformations

must itself be a symmetry transformation; this way of combining symmetries

is associative; there is an identity transformation, which is to do nothing; and

any transformation may be reversed so that the net result is no transformation.

Thus we may apply the powerful machinery of group theory to study any

system with symmetry.

Consider a set of quantities al, a2,..., aN, which under some symmetry op-

eration T transform to ai = D(T)ijaj, where D(T) is a matrix of numbers, and

summation over repeated indices is implied. The matrices D must obey the

same product law as the abstract group elements, i.e. D(T2T1) = D(Tl)D(T2),

for consistency. Then the set of matrices D is what is referred to as a repre-

sentation of the group.

The classification of all possible representations of a given group is a stan-

dard problem in group theory. Any group will have a trivial representation in

which every element is represented by the identity matrix (D(T)ij = 6ij for

all T), i.e., the quantities ai are invariant; the transformations have no effect

on them at all. One way of generating other representations is to start from a

set ai which are not invariant, and consider all possible products aiaj. These

products may themselves be arranged into representations of the group. Alter-

nately, we may use this procedure to search for some particular representation.


For example, we can use the product laws for representations to determine

all possible invariant products which may be formed starting from a given set

of quantities. These are necessary and useful in many ways; for example, in

constructing an action for a system which must be invariant under a given

symmetry group.

The reparametrization group is an example of a Lie group, which means

that it has an infinite number of elements, which are labeled by continuous pa-

rameters. In such a group, we may consider elements which are infinitesimally

close to the identity; these may be expanded as T(ej, e2,...) = 1 + eiMi, where

ei are infinitesimal parameters. The operators Mi are known as the generators

of the Lie algebra of the group. Since the product of two T's must give a

certain combination of T's, we find that the commutator of two M's must give

a certain combination of M's:

[Mi,Mj] = ifijkMk (1)

The numbers fijk are known as the structure constants of the algebra. Any

matrices which obey (1) provide a representation of the algebra.

Representations of the Reparametrization Algebra

We now apply the above general ideas to the special case of the reparamet-

rization group. Consider the open string whose coordinates are xl(a) where

the parameter a ranges from 0 to 7r. The parameter has no physical meaning,

it is merely a set of numbers which we associate with the points on the string

in order to facilitate a mathematical description of it. For a given string there

are many ways to attach such numbers, and we would expect the physics of

the string not to depend on the particular choice used. Thus under a repara-

metrization a -- F the dynamics should be unchanged. Under such a shift, the

value of the coordinates xp at some particular point will not change; however,
the functional form of xP(a) will look different because it is written in terms
of the new parameter. That is, xP'(ca) will transform to a new function TY(a)
such that 71(0(a)) = x/'(o) for all ar. For an infinitesimal reparametrization
a= + ef(a), where e is an infinitesimal parameter,

6Sf(e)XI,(a) = V(a) xa(a) = (ef()x' (7) (2)

where the primes denote derivatives with respect to a. For open strings, we
require f = 0 at the string's endpoints, as the location of the endpoints is
independent of the parametrization; they are fixed at 0 and 7r. However, for
mathematical purposes the reparametrizations can be formally extended to
include any arbitrary function f. For closed strings, f could be any periodic
function on the interval 0 to 27r.
From (2) we may calculate the algebra of reparametrizations [23]

[6fS(l), 6g(2)] = fg,-fg(ElE2) (3)

We are now interested in finding other representations of this algebra.
Let us concentrate specifically on representations using a single quantity q(a),
whose transformation law will not involve any other quantities. We can nar-

row down the possibilities by considering a uniform rescaling of the parameter,
a --* aa; this should clearly have no effect, therefore each term in the trans-
formation law should contain exactly one derivative with respect to ar. Thus
the most general possible transformation law is (suppressing the parameters 6)
bfq = -(vfq' + wf'q + zf') for some numbers v, w, z. Then we find that (3) is
satisfied if and only if v = 1:

6f q = -(fq' + wf'q + zf')

We will refer to such a quantity q as transforming according to the represen-

tation (w, z). For the special case z = 0, q is said to transform covariantly

with weight w. A quantity with w = 1 transforms into a total derivative,

so its integral over a is a reparametrization invariant, the contributions from

the endpoints vanishing due to the open string boundary condition f = 0 at

a = 0, 7r (in the case of closed string boundary conditions there are also no

such contributions because in this case f must be periodic). An example [23]

of a classical invariant which can be constructed from the string coordinates
xp(a) is the length of the string

Iid (5)

which contains a square root, a harbinger of trouble at the quantum level.

Direct Product Rules
We now examine the transformations of products of two quantities p(a) and

q(a). Following the usual group-theoretical treatment, we attempt to identify

combinations of the form

r(a) J f dadcr2K(a, aOl, '2)P(al)q(a2)

which transform into themselves. This turns out to only be possible in the

special case in which p and q transform covariantly. For this case we find two

covariant combinations [40,41]:

p(a)q(a), with weight Wp + Wq (6a)


wppq wqp'q, with weight Wp + Wq + 1



where Wp, Wq are the weights of p, q respectively. For Wp + Wq = 1, (6 a) will

yield an invariant quantity when integrated. However, for Wp + Wq = 0, (6 b)

is itself a total derivative, so its integral will be a trivial invariant (i.e. made

up entirely of endpoint contributions). It would be possible to get a nontrivial

invariant by (for example) combining the result of (6 b) with a third quantity

according to (6 a), such that the final result had weight one.

The above discussion does not take into account any ordering ambiguities

which may occur when the quantities used are operators in the Hilbert space of

the first-quantized string. The transformation properties of the product may

depend on the ordering of terms within the product, and this must be checked

in each case. Generally, the products of interest will be the normal-ordered

products. The implementation of the reparametrization group at the quantum

level hinges on the existence of a scheme according to which the product of two

or more covariant normal-ordered quantities yields covariant normal-ordered

results. We will discuss this point at length for the specific cases which occur

in string theory.


A functional 4[qi(a)] is a scheme by which a (generally complex) number

is associated with the quantities qi(o); that is, the values of all of the q's are

input (for all a), and a number is the output. Such functionals are used in the

Schr6dinger representation of string quantum mechanics, and as string fields.

Under reparametrizations, the functional will change due to the effect of the

change in the arguments qi; there may be additional terms in the transforma-

tion law for the field as well. If there are no such extra terms, the functional

is referred to as a scalar functional. This is analogous to the transformation

of local fields under the Lorentz group: There is a term in the transformation
law due to the change of the coordinates, and if the field is not a scalar then
there will be additional (spin) terms. Generally any extra terms are allowed
so long as the algebra is still satisfied.
For the special case of a scalar functional I, the change is given by

() = -ieMfS (7)

where the generators Mf have the form

Mf = -i j do(fq' + wif'qi + zif') (8)

where each qi transforms as (wi, zi). The M's satisfy the algebra

[Mf,Mg] = iMfg,_gf, (9)

An important consequence of this is that M must be built from an integrand
which transforms covariantly with weight two. To see this, write Mg in the
Mg= fdag(cr)M(o) (10)

i.e. we have integrated by parts to remove all derivatives of the function g.
Inserting this into (9) we find

Jdog(a) [Mf, M(o)] = i f da(fg' f'g)M(a)
J r (11)
= -i J dag(fM' + 2f'M)

showing that the integrand M transforms with weight two. Alternatively one
may see this from the explicit form

M(a) = i (wi 1)qI- + (wiqi + zi) ( (12)

For superstring field theory we seek a kinematical supersymmetry trans-
formation [41] Of which is the "square root" of the reparametrization 6f in the
sense that

[f (1)W, g(W2)]F(O') = -2Sfg( i2)F(a) (13)

for any field F(cr). The 's are anticommuting parameters. The commutation
relations of the O's with the reparametrizations 6 can be determined from the
Jacobi identity

[[Of, g], h] + [[6h,/f]1,g] + [[Og,6h],Uf] 0. (14)

We first note that the commutator of a reparametrization 6h with a super-
reparametrization f must be bilinear in f, h and their derivatives; this is
clear from (14) and (13). Furthermore, derivatives of order higher than one
are excluded due to the presence of the first term in (14) (since this identity
should hold for arbitrary functions). The commutator must therefore have the

[Ah(), Of()] = a hf,+3h'f,(E (15)

Using this relation and (13) in (14), we find that a = 1 and /3 -, i.e.

[Ah(), Of()] = Ohf,'-hf/2() (16)

Henceforth, the parameters e and C will not be indicated explicitly.

Representations of the Super-reparametrization Algebra
Given fields transforming in a specified manner under reparametrizations,
we can deduce their possible transformation properties under super-reparametrizations.
First consider the case of a field a(a), either commuting or anticommuting,

transforming covariantly under reparametrizations with weight Wa, for which

we postulate the transformation law

fa = -fb (17)

where b is a field of opposite type (commuting or anticommuting) from a.
Equation (16) tells us that

-fSgb = ( f g Ogf,_gf /2)a (18)

and upon evaluating the right hand side of (18) we find

6gb = -gb'- (Wa + )g/b (19)

i.e. b transforms covariantly with weight wb = Wa + . Assuming that a and

b form a closed multiple involving no other fields,* the most general form for

the transformation of b under a super-reparametrization is

fb=An dna (20)
where the An's are functions of f and its derivatives. Using (13) with g = ,

we find

6b = -Sffb (21)

so that

Anfb = ffb' + 2wbff'b. (22)
Since b and its derivatives are all independent, we can equate coefficients on
either side to solve for the An's. We find that the only non-zero An's are

AO = -2Waf' and A1 = -f, i.e.,

fb= -(fa' + 2waf'a). (23)
In Appendix B we show that adding extra fields does not generate new
irreducible representations.

We have discovered one type of multiple on which the super-reparametrization
algebra is represented. The representation can be written in matrix form:

( a)= (f d I ftf a( ) (24)
ba 30 +fwa

whereas the transformation 8f is written as

a) f = do- f+(a+ )f)0 (a) (25)
=- + +(W. + )f

The representation is the same regardless of which field a or b is the com-
muting one. For this type of multiple, we will refer to the component a
transforming according to (17) as the light component, and to b which trans-
forms according to (23) as the heavy component. An important difference
between the two components is that if the integral of the heavy component is
reparametrization invariant (i.e. if it has weight one), then it is also super-
reparametrization invariant, as is evident from the transformation law (23).
The integral over the light component is never super-reparametrization invari-
An example of this type of representation is provided by the string coor-
dinates x1. These transform according to (17) into generalized Dirac matrices
[23] FP:
P =' = -fr (26)

F = -fxu' (27)

Because x1' has weight zero, the multiple (F', x'I') also transforms as (24),
with Fr/' as the light component. This multiple is of more direct use in string
field theory because it is translationally invariant.


Direct Product Representations

Given two doublets (a, b) and (c, d), we again wish to know all of the differ-

ent covariant super-reparametrization representations which can be built out

of products of these fields and their derivatives. One can form eight quantities

which transform covariantly:

weight w :
weight w + :

weight w + :

weight w + 2:

A1 = ac

A2 = ad and A3= bc

A4 = bd and A5 = wd C Waac

A6 = (wc + 1)ad Waadd' and A7 = wcb'c (Wa + I)b c
1 1
A8= (w + )bd (Wa + )bd'
2 2

In these equations, w = Wa + wc. Among these quantities, three combina-

tions may be identified as doublets:

(AI,A2 + A3),

(waA2 T wcA3, A5 + wA4),

(2A5 + A4, 2A7 + 2A6),

with weight (w,w + )

with weight (w + w + 1)
with weight (w + 1, w + -)

The upper (lower) sign of the 's in these equations is to be read in the

case where a is the commuting (anticommuting) member of its multiple. The

heavy component of both (29 b) and (29 c) reduce to total derivatives in the

cases in which their weight is one, so they yield only trivial invariants. The

remaining two quantities in (28) are members of a multiple containing non-

covariant quantities.

We have thus demonstrated the decomposition

2w 2v = 2v+w 6 2v+w+ E 2v+w+l E3 (non covariant)





The fact that the only covariant representation of the super-reparametrization

algebra found in the direct product of two doublets is again a doublet suggests

that no other covariant irreducible representations exist. This is true, and a

detailed proof is given in Appendix B.


In this section we apply the reparametrization group theory developed in
the previous section to the determination of the Schr6dinger equation for free

strings, and from that the equation of motion and action of free string field

theory. Although the results are very well known, the details of the construction

may not be.

Second Quantization
The free quantum mechanics of strings will be formulated in terms of the
wave functional T[x]. This functional obeys a Schr6dinger equation,

Q[=x] = 0 (31)

for some operator Q. Also of interest in quantum mechanics is the inner prod-

uct of two wave functionals, given by

<'l|x2 >= J DxP*[x]f2[x] (32)

The Schrodinger equation (31) will become the classical equation of motion for

the free string field. The string field generally resembles the wave functional

except that there may be a reality condition on the string field [42]. However,

this reality condition is irrelevant at the free level due to the linearity of the
equation of motion. We will return to this point later. The action functional

for the free field theory is then given by

S{} =< |IQl| >= J Dx4*[x]Q4[x] (33)

On physical grounds we would expect the quantum theory to be invariant

under the global reparametrization symmetry. We will take the string wave

functional to be a scalar, meaning that its change under reparametrizations is

solely that due to the change in the coordinates. Under a -- a + ef(oa), we


6T[x] = M4x'[X] (34)


MS-fx(/()') (35)

Reparametrization invariance of the quantum theory, and hence the free

field theory, requires two conditions: For any solution T of the Schr6dinger

equation, the reparametrized wave functional T + 5&P must also be a solution;

and the reparametrization operation must be hermitian with respect to the

inner product. The second condition is already met by the form (35); the first

will be met if and only if Mf commutes with Q for all functions f.

We might expect that the next step is to try to build Q using the coordi-

nates x and their functional derivatives. This was the approach of Marshall

and Ramond [23]. However, this approach is problematic. With great hind-

sight, we may understand the difficulty as follows: Physically, we would expect

a string's wave functional to associate a number with each distinct string con-

figuration. However, the functionals we use are functionals of parametrized

strings; strings which differ by reparametrization are treated as distinct. This

may not cause problems in the classical theory, but in the quantum theory we

will wish to perform path integrals over the space of all possible string func-

tionals; counting reparametrized strings as distinct will cause us to miscount in

the integral. We would need to find a measure which is reparametrization in-

variant, which seems to be impossible to do using the string coordinates alone.

Thus the functionals of parametrized strings should be used only for a fixed
More practically, the only nontrivial invariant operators which may be
formed using only the string coordinates x are made using X'(o)2, which is
not a well-behaved operator in the string Hilbert space. We must therefore seek
an alternative formulation of the theory if we wish to have reparametrization
invariant equations of motion.

The Einbein
Reasoning by analogy with gravity, many authors [43,44,45,46,47] have
realized that the dynamics of point particles can be reformulated by introducing
an einbein. Indeed they have shown that the free point particle action (a = d)

m j d7V2 (36)
can be replaced by
J dr [ i.~2 + m2e(r), (37)

where e(r) is an einbein transforming into a total derivative under reparamet-

rizations of the proper time parameter r:

5e(r) = --- [f(-)eQ7-)]

This action led to the same classical dynamics as the previous one, at the cost
of introducing a new quantity. We are interested in building reparametrization
invariant quantities; their construction shows how to generalize our expressions
while avoiding square roots. Thus consider [40] introducing a variable e(u)
transforming covariantly with some weight w

6e(a) = -e[f()e'(r) + wf'(a)e(O)]


which is the representation (w, 0) of the last section. However, the functional
derivative 6T, transforms with weight 1 w, so its integral is in general no
longer invariant. The more complicated operator e- = 6 does transform
be -6in e
with weight one, so its integral is invariant. So we make the change of variables
[48] to
(a) = lne(a) (39)

This transforms inhomogeneously, as (0, w):

() = -ef(a)d w d(40)

so its reparametrization generators are given by

MO icIda [f(a) d'+w df(a)l 641
f do 60(a)(4+

The quantity will be familiar as the bosonized ghost of string theory.

Mode Expansions
We can expand the various fields in Fourier modes. For the string coor-
dinates xP, the open string boundary conditions demand that x'P = 0 at the
endpoints a = 0, 7r. Thus xP has an expansion purely in terms of cosines:

xP (a) = xP + v/2- x^ cos na, (42)

whereas the functional derivative has the expansion
5 ~ 0 SC
L9Q 00 a
( o + V T cos. (43)
6XP (a) -x +vL X'Tona.
0 n=1

In terms of creation and annihilation operators (a background-dependent de-
a -i V2 aL-9+nxp) (44)


a = -iV( a nx (45)

Xn V2-n~ _n

A_ [ + a-,] (46)

the expansion of x1 and of the functional derivatives become, respectively,

XP P + i 00 1[/t-a
(a) X E n [cn -n] cosncr (47)
& 00
-- iza + i Z [a" + ap ] cos no. (48)
O ~ n-----
*X, n=1 n
The a's satisfy

[a', a'] =mgPm+n,o (49)

[x, a'] = igl".

We can similarly expand the field 0,

n o"'lr (50)
-i= ?7 On cos no
60 n=-oo


[/1m,1n] = ??m6m,-n (51)

[o0, /3] = iSm,0 (52)

and q = 1. The ground state <0 of the representation is defined to satisfy

amno = /3PmO = 0 for all m > 0. The mode operators 03m create states

of positive or negative norm depending on the choice for 77. Expanding the

generators M in modes shows the need to redefine M when the full quantum

nature of the string Hilbert space is taken into account. Since x'(a) does not
commute with y-,, their product has a possible ordering ambiguity. In such
a case, we need a convention to fix the ordering. The normal ordered product
is defined to be the ordering in which all positive modes an are on the right,
and all negative modes a-n on the left. The new normal ordered M differs by
a finite correction:

Mf =: Mflasal : (53)

These generators are related to a subgroup of the Virasoro algebra:

M(nna=- Ln --L-n) (54)

M-lassical (Ln -- L-n + Cn) (55)
v'2v sin n o, 7

where Ln are the well-known Virasoro operators,

Ln = L + L (56)

L = a n-l (57)
1 .wn
Lt- 2Z OlIn-i + i2n (58)


Cn for n even, (59)
S0 for n odd.
Note that the redefinition of Mf to remove the c-number anomaly term Cn
affects neither the algebra (9) nor the free invariance condition [Q, Mf] = 0.

Left and Right Combinations
The generators (35) can be rewritten in terms of the commuting combina-

tions 6xZ-)



dof ( X/+ )2 (60)

R +1 [do, _ 2
M f X'- i1 (61)
This split-up is familiar from first-quantized string theory as the splitting of the
string coordinate into left and right movers; we identify x1 = x' + i7r as the

derivative of the left-moving part of the string coordinate, and x x' i5 r
as that of the right-moving part. This identification is confirmed by the mode
expansions in terms of exponentials

x?(a) = -Oa, + i (62)

P ( = X(-0) (63)

which agree with the conventions of the first-quantized theory. [5,3,4]. Note

that this split-up would not be possible in the presence of a curved space-time

metric gpv.

The Mf generators for the field 0 split in a similar manner, with

.=+7 (64)


OR 0- i(65)

M (f 2 + wf'(l q0) (66)

Because of this split-up, we expect that it should be possible to build covari-

ant quantities purely out of left or right moving fields alone. For example, the

normal-ordered exponential : eaL : can be seen [40] to transform covariantly
with weight a(w 77a/2).
The left and right combinations commute with each other; the relation (9)
then requires that

[Mfr M 1 + [MN M] f iM .fg-f'g + Mfg,_f'g). (67)

Since ML contains only left pieces, and MR only right ones, and left movers
commute with right movers, (67) can be separated into left and right pieces,
which must each be zero up to a c-number:

[Mf,Mg ^= iMfgfg + cfg
M Mg fg Cfg(68)

To evaluate the c-number we must expand in Fourier modes and carefully take
into account ordering effects; we shall do so in the next subsection. However,
the form of the c-number may be deduced from algebraic properties alone.
Applying the Jacobi identity for the commutator of three ML generators yields
a relation that the c-number term must satisfy:

Cf,gh,-hgl + Cg,hf,-fh, + Ch,fgl-gfl = 0. (69)

Additionally, by the antisymmetry of the commutator,

Cf,g + Cg,f = 0. (70)

Since the generators Mf are linear in f and its derivatives, the most general
form for the c-number Cf,g is

d o 0 Cng(a) f( a) (71)
Cfg = 7r cng1 (o)

where we have integrated by parts to remove all derivatives of f. The constraint
(70) is satisfied if and only if cn = 0 for all even n. The constraint (69) (after
integration by parts) requires

Scn [(-1)n '(gh'- hg')- 2h'dng
7r E don don

dn+lg ,, dnh dn+-h 1
-h + 2g dnh + g d---f f =0 (72)
dofl+l dcT d0.f+l I
which is satisfied if and only if n = 0,1, or 3. Thus the most general form
which satisfies both constraints is [49]

Cfg = (cl fg + c3f'g). (73)

An advantage of the left-right split-up is that the operator

Cf = Mf M (74)

is automatically a covariant dynamical operator of weight two, provided that
the anomaly Cf,g is zero. With the fields x1 and this operator has the form

1 f= lJ : (r2 2 x2 _r2 + 275)
2 7r 6X 0

This operator contains second-order time derivatives, hence it is dynamical,
unlike the purely kinematical reparametrization operator Mf; because of this,
it is referred to as a generalized Klein-Gordon operator. The split-up of M
into ML and MR is of great physical significance as ML + MR contains no time
derivatives and also does not depend on the metric of the background space.

Anomaly Cancellation
We now evaluate the c-number anomaly term in (68). This may be found
directly by expanding the generators in modes, or by evaluating the product

ML(Ol)ML (a2), where al -, 0o2, as a series of normal-ordered operators; the
anomaly c3 will be given by the c-number term in the series. We will show this
approach to illustrate the technique. Recall

ML(a) = aL2 + 7'2 + 2wro" (76)

Using Wick's theorem, the c-number term is given by
c = 2 < x2r(71), v(o72) >< x ( 'J), v(2) >
+2< 0'(61),O(a2) >2 +4w2 <(7 (al),L(a2)>
where the correlation functions are defined as, e.g.,
< ( =< 0|1L((l) a2)|0 >
L(0(a1) L(2)- : (1ai), 0'(92):
These may be evaluated directly:

< (91), O (02) >=< o011nC ^me )o >
n m
=< 00 > +77 neif(1-2)-f (79)
where E > 0 has been included for convergence. We then sum the series and
evaluate it for a1 a2. The result is

<2.), (>i_-)2 (80)
< LclOL(0'2) >= 77 62(80)

Then the total anomaly is

c= (2(D + 1)+ 4w2 67) 6) (81)
\ ~ (0\-1 0'2 + iE)

It is clear that we must have 7 = -1, otherwise the anomaly can only cancel if
D + 1 < 0, not physically reasonable. Then the anomaly is found to cancel for

D = 12w2 -1 (82)


Construction of the Invariant Operator

Finally, we are now ready to construct an invariant free field equation.

The quantity ML(o) (the integrand of Mf) has the right dynamical structure
(e.g. it contains -2), but it is of weight two; therefore we must multiply by a

measure of weight -1, which is provided by the exponential : ea1L :.

When r = -1, the invariant operator is given by

Q f d : eaL(C)ML() : +(L - R) (83)

where a = 1, 2 so that the exponential has weight -1. The operator Q is

invariant under reparametrizations. It acts in a Fock space with a spectrum

bounded from below provided that it can be written in a normal ordered form.

Recall that in terms of modes, we have

Mn = Ln L-n,

1 00
Ln = Z- (-n-k ak + 7/7n-kk) : + riwn/n. (84)
We need to be check that overall normal ordering does not affect the invariance

of this operator. We find that

[LnQ n(n+ 1) d --ino(, d a2 aw 2aw ea +(L R).
[LQ] 2 J0 7 d(z 2 3 3): :+(L R).

The integrand is a total derivative iff. 2aw = -3 and a2 = 1, the contribution

at the lower limit cancelling between left and right pieces. The only solutions
are a = 1 corresponding to w = :F respectively. These solutions are of

course equivalent since redefining -- - would change from w = +7 to
w = -3; we will make the latter choice so that a = +1. This gives us a

unique invariant operator whose construction, by (82), is only possible in 26

dimensions. This of course is the usual BRST charge, and one can check that

Q is in fact nilpotent. Also, if w = --, while : eL() : has weight -1, the

other combination : e-OL() : has weight two; it anticommutes with : eOL() :

to give a delta function, and thus can be taken to be the conjugate antighost

field b++ corresponding to c+ which is : e () : [5].

Due to the nilpotency of Q, if b is, a solution of the free field equation

Qr = 0, then D+QA will also be a solution, where A is an arbitrary functional.

This is a gauge invariance of the string field theory which insures that the

negative norm states of the first-quantized theory decouple from the physical

states [34].


In this section we digress to list and discuss the various reparametrization-
invariant operators present in the bosonic theory. These are found by the sim-
ple group- theoretical procedure described previously, i.e. to combine covariant
quantities using the direct product rules for representations, and to check that
normal ordering does not affect the invariance. We find many familiar results,
along with some surprises. The list is:
(1) The momentum vector
PP- -if' da (86
Pt ^/O (^). (86)

(2) The ghost number

NG =-i dafo. (87)

(3) The Lorentz generators

MV i 7rda gPPx- 9gVPxP- ) (88)

(4) The symmetric space-time tensor
Q1V =/ -- d- + 3O)L : +(1 --1 R). (89)

This invariant tensor depends on the space-time geometry. The BRST operator
is obtained by taking its its trace:

Q = g9,Qt'. (90)

(5) An additional symmetric tensor

B" = j -- L -xL xL) : +(L -+ R). (91)



We note that although the dilatation operator D = f du : x* 6 : has

the right weight to be a classical invariant, it transforms anomalously due to

ordering effects. Thus the largest space-time symmetry seems to be that of the

Poincar6 group. We remark that there does not exist an invariant 26-vector

which serves as the string position in space-time. This is not too surprising

since the theory is not (space-time) conformally invariant. On the other hand,

by specializing the Poincare generators to the relevant space-like surfaces, we

can define a physical position for the string in 25 (at equal time) or 24 (light

cone) space dimensions.

Operator Algebras

All of the operators we have described are normal ordered and invariant

under the complex reparametrization algebra. We proceed to study the algebra

obeyed by these special quantities [50]. P1 and Myv form the algebra of the

Poincare group, and commute with ghost number, while Q/I has ghost number

1 and BP' ghost number 2:

[N, QI] = QO ; [NV, Buy] = 2B". (92)

Furthermore, the translationally invariant QlI and B v transform as second

rank tensors under the Lorentz group.

On the other hand, the QPV obey a more complicated anticommuting al-


{Q", QP'} = i(gPBO" + gPB10' + guOBvP + gvoB1P)

+ 2[(gIBP + gP(Bl) (g"gVP + gIPgV -g"VgPa)C1
13 13


C=j de2XI x + (L -* R); (94)
J0 7r L L
it is an invariant tensor, but it is not normal ordered. Hence the algebra of
the Q/I does not close on normal-ordered invariant operators, which is hardly
surprising. The algebra is completed by noting that Bfl commutes with itself
and with QA'
[BV, QPU] = [B ,, Bpa] = 0. (95)

In order to determine the largest non-anomalous algebra among our oper-
ators, let us introduce a set of 26 x 26 matrices alc such that the projections

QI = aUQ1V = Tr(&1Q) (96)

satisfy a non-anomalous algebra. It is easy to see that

f{Q, QJ} -2i{ a, I} p + -Z{BJTr(ce)) + B1Tr(&J)}
13 (97)
{QIQ -} i~I J 1uB
-{Tr(o~aj) 16Tr(a1)Tr(CJ)}C,(
13 26-
where B1 = al- Blv. Thus the algebra will not be anomalous as long as we


Tr a)= -Tr(I)Tr(&J) (98)

where the trace operation is meant to act between covariant and contravariant

indices, i.e.

Tr a, = (aI)p (99)

This condition is obeyed by the metric itself
Tr(g g) = -Trg Trg. (100)
26In fact this gives the only Lorentz-invariant projection,
In fact this gives the only Lorentz-invariant projection,

Q = g1Q0U;


this is the nilpotent BRST charge:

{Q,Q}=0. (102)

Similarly, we see that Bly is the BRST transform of QII

{Q, Q1,} = 2iB1, (103)

so that it commutes with Q.
We can redefine the matrices a' by subtracting out their traces:

I I --a MTral (104)

Thus it is enough to concentrate on the redefined set of 26 x 26 matrices which
now satisfy the following conditions

(a1), =(a1)p ; ()P = 0 ;(0
(a)P(aJ)_ = 0.

The traceless projections of Q obey the algebra

{QI, QJ} = 2i{a1, aj})vB, (106)

These matrices a do not exist in Euclidean space. In D-dimensional
Minkowski space, K, the number of linearly independent matrices obeying
these properties, can be written as the number of symmetric traceless matrices
less the number of constraints, i.e.

K-- D(D + 1) K(K1) (107)
2 2 (107)

which is satisfied when K = D 1. Thus, including the Lorentz invariant
BRST charge, there are at most 26 linearly independent Q1 which satisfy a
non-anomalous algebra.

None of these, except the trace, commutes with the full Lorentz group.
These projections can be transformed by Lorentz transformations into other
operators which are not in the set {Q/} since the Lorentz algebra has been
broken by the very choice of the a1 matrices. Hence, it is only in the context
of a broken Lorentz algebra that the operators Q1 need to be considered.
Manifest Lorentz invariance yields an algebra of one extra operator, the BRST
operator of the usual open string field theory. This suggests that we try to
arrange the inequivalent sets of a matrices in terms of the unbroken sub-Lorentz
algebras. If there were a set in which the Lorentz algebra were broken to a
3+1 dimensional algebra, it would possibly be of physical relevance, since the
world of experience indicates only Lorentz invariance in four dimensions, not

in 26 dimensions.
In general, the set of a matrices is closed under linear combinations; its
structure is that of a linear vector space [51]. An arbitrary element of this
space can be written in the form

a V (-TrA a) (108)

where a is a 25-vector and A is a symmetric 25 x 25 matrix. The constraints

(105) imply
Tr(A1AJ) + TrA'TrAJ -= 2a' a- (109)

Thus the quantity

(a', a) 1 (Tr(AIAJ) + TrAITrAJ) = a a (110)

is an inner product on the space of a's. We can choose a basis in this space
which is orthonormal with respect to this inner product. Since only D 1


orthonormal vectors a may be found, we see again that the maximum number

of a matrices is D 1.

If a particular a' is nilpotent so is the corresponding operator Q1, and

we can show that there is at most one nilpotent matrix for each set of the

matrices. Nilpotency of a given a matrix implies the equations

A2=a aT
Aa = (TrA)a.

These equations can be solved by introducing the null 26-dimensional vector

ap with components (TrA, a), in terms of which we have

afiv = apav ; aa" = 0. (112)

Let a^v = bYbv be another nilpotent matrix (bpbu = 0). We see that Tr(aa') =

0 implies that a,,bP = 0, so that a, and bp must be proportional to one an-

other: there is only one nilpotent per set, besides the Lorentz invariant BRST

nilpotent. (One can also show that there is no nilpotent linear combination of

Q and QI; otherwise it would imply

ca(aI + 1) = 0, (113)

meaning that a, has only 0 or -1 as eigenvalues. Since it is to be traceless,

it can only have zero eigenvalues, and thus is similar to a triangular matrix,

which contradicts its pseudosymmetry.)
SIt is not very hard, albeit tedious, to produce sets of singular a matrices

by inductive procedures. For instance, if we know a set of such matrices in

D 1 dimensions, we can insert them in a set of D x D matrices with the Dth

column and row being zeros, and the remaining a matrix will have zeros in its

first D 1 rows and columns, and a null vector in its Dth row and column.

The eigenvalues of the a matrices are in general complex numbers forming
conjugate pairs. In D dimensions, they are determined by 2D 5 parameters
and one overall normalization. There can be at most 7 commuting elements
in the non-singular set in 26 dimensions, although this number will increase
whenever singular elements are present.

An Example
We now give a specific example. It is very simple and serves to illustrate
our procedure, which up to now has been of a general nature. Let ak and bn
be two null vectors with the properties

apap = 0 ; blbl = 0 ; abp = 1 (114)

and introduce a set of 24 transverse orthonormal vectors c( which satisfy

CLau = c4'1 = 0 ; cc =- (115)

Then it is easy to see that a set of a matrices can be formed out of these
according to

a() = apav (116a)

Wi (i) Wi
v pa c ) + avc i-- 1,... ,24. (116b)

This is a rather degenerate example, since all these matrices are singular. The
ensuing projections Q and Qi satisfy the superalgebra

{QO,QO} = 0,
{QO,Qi} =0, (117)

{QiQ} = -2iiJBo .

The algebra can be enlarged to include the BRST charge, Q, with the results

{Q, Q} = 2iBo,
{Q,Qi} = 2iB (.

The relevant projections of the Lorentz generators which map the set into itself

are obtained through the antisymmetric light-cone matrices

+rnp = apbV aib ,

mi = c(i) ci)

3ij _(0C) (_ e)C(i)
mtiV -P a cv p V

It is easy to see that the only subalgebra which keeps the set within itself
is that generated by the projections M'1 = m"i A/I/ and M+- = m+-MW,
,UV pIL'
which together generate the subgroup SO(24) x SO(1, 1), with the transfor-

mation properties

[M+-,Qi] =Qi ; [M+-,QO] = QO, (120)


[M1, Q] = 0 ; [M3, Qk] = isikQ ikQ (121)

In this very simple example, we see that the algebra preserves Lorentz invari-

ance in 1 + 1 dimensions, together with an internal SO(24) algebra, and two

nilpotent operators. Note that the set {Q1} forms a reducible representation of

the unbroken algebra made up of a singlet Q and the Qi which form a vector

representation under the transverse subgroup.

Having the nilpotent operator Qo, we may ask what nontrivial cohomology

is possessed by the operator. For simplicity we will look at the ghost-free


sector. States |^ > satisfying Q0 1 >= 0 in this sector are linear combinations

of states of the form
> I>= I(ay a P0> (122)
where the an are the Fourier modes of X'(oa) (not to be confused with the

matrices apuv), and the ground state 10 > is annihilated by an and cn for n > 0

and bn for n > 0. These states have zero norm, so they are trivial in the BRST

sense, i.e. they can be written as Q acting on some other state. However not

all of them can be written as Q0 on other states. Specifically, states which can

be so written are of the form
1b >triv= i a'- &-nLOmIO > (123)
for any m > 0, where

LO a. -apa -am-p. (124)
The set of states which are not of this form gives the cohomology of the

nilpotent operator Q0. Since these states are BRST trivial, they do not con-

tribute to the dynamics of the usual BRST-based string theory. They would

represent the physical states in some hypothetical theory based on Qo.

Lorentz Symmetry Breaking

We have seen that only a subalgebra of the Lorentz algebra will transform

the set {Q'} into itself; thus the set of QI's forms a representation (in general

reducible) of the unbroken subalgebra. Put differently, under the relevant

decomposition of the SO(25, 1) Lorentz algebra, Qpv itself decomposes into

a sum of representations of the subalgebra, in terms of which the Q1 must

be expressed. This is a powerful requirement which can be used to rule out

some imbeddings. For instance, consider the imbedding of F4 into the Lorentz


algebra. It does not work because the symmetric traceless tensor splits up into

two F4 representations, the smallest having 26 dimensions while there are only

25 Q"s.

For any matrix a the Lorentz-transformed quantity 65o will automati-

cally satisfy TrSa- = 0 and Tra l6aS = 0, because these quantities are the

Lorentz transformations of the invariant traces Tr cI and Tr(al)2 respectively.

We can think of starting with some matrix a and building a set by applying

the Lorentz transformations of some subgroup until a set closed under the sub-

group is obtained. However at each step the condition Tr(6al)2 = 0 must be

checked for all 8 in the unbroken subalgebra.

Carrying out this analysis (see appendix C), we find that the possibilities


1) The space of a's is 26-dimensional, but has no nilpotent element. There

is no unbroken subgroup of the Lorentz group.

2) The space is 26-dimensional and contains a nilpotent element; the un-

broken subgroup is SO(1, 1).

3) The space is d-dimensional, where 0 < d < 26, with no nilpotent; the

unbroken subgroup if SO(26 d).

4) The space has dimension d, where 0 < d < 26, and has a nilpotent

element; the unbroken subgroup is SO(1, 1) SO(26 d).

5) The space is zero dimensional, thus containing only the BRST charge

which was subtracted out as the trace; the full Lorentz group is unbroken.

The lack of a scheme in which the noncompact part of the unbroken sub-

group is SO(3, 1) seems particularly unfortunate for any phenomenological

applications of this procedure.


In conclusion, we have drawn attention to a curious algebra of normal-

ordered, reparametrization invariant operators. The algebra closes only in a

very few cases, one of which gives only the BRST charge.We have learned

from this exercise that the requirement of reparametrization invariance in the

oscillator basis can lead to the standard BRST results. We have arrived at this

conclusion by demanding algebraic consistency. It is clear that in order to go

further in this approach, we would have to find a different basis, i.e. a new

ground state.


We now study the important question of interactions in string theory. We

will begin with a general discussion of the transition from quantum mechan-

ics to field theory, in order to make several relevant points. We will then

discuss reparametrization invariance in Witten's string field theory, the most

well-studied example of an interacting string field theory. We will show that

the apparent lack of reparametrization invariance is due to a parametrization-

dependent choice of phases in a set of basis functions. We will construct an

equivalent theory with an invariant choice of phases. This will result in an

equivalent interaction which is explicitly reparametrization invariant, but not

explicitly local in x(a).

Second Quantization

Let again consider the passage from the first quantized theory of open

strings to the second quantized one. The Schr6dinger wave functions in the first

quantized theory are built as a linear combination of a set of basis functions:*

0= ZCn O[x(0)9 (125)
The basis functions are in turn built by acting on the ground state with the

creation operators Jn, where n > 0:

S[X = ( )1(2)n2... 4[X] (126)
The string field of course is also a functional of the Faddeev-Popov ghosts.
The treatment of the ghosts is in most cases identical to that of the coordinates;
therefore we will not explicitly indicate such ghost dependence except in cases
where the ghosts require special handling.


The ground state is annihilated by the destruction operators, which are a/ for

all positive n. The dynamics of the system are enforced by the Schr6dinger

equation, which in this case is

QP = 0 (127)

where Q is the BRST operator.

In passage to the second-quantized theory, we define the (classical) string

field -1[x]:

41[x(a)] = an Onn[X(0-)] (128)
This looks similar to (125). Indeed the string field (128) satisfies (127) as its

classical equation of motion. However, there is an important distinction be-

tween (125) and (128): In (125) the coefficients Cn are complex numbers with

arbitrary phases. Although the relative phases of two states will be important

in a superposition such as (125), an individual phase by itself has no conse-

quence. In the field theory defined by (128), the phases may be handled in two


1) Complex Field Theory: The phases are allowed to take any value. The

real and imaginary parts of the coefficients an in (128) are independent, and

upon quantization they will become associated with two different types of

quanta. In this type of field theory there will be "strings" and "anti-strings".

These ideas do not agree with the well- known picture of string interactions in

which there is only one type of string.

2) Real Field Theory: The phases are fixed to some specified values. Which

values are used is of no physical consequence. Changing the phase convention

would not change the theory in any fundamental way. Indeed, the free equation

of motion and the inner product are independent of the phase choices; however,

if an interaction is added its form might appear to be quite different. This type

of field theory has only one type of quanta; the strings are their own "anti-

In Witten's theory of interacting open strings, the phase condition is that

:*[x(o)] = $[x(7r o)], (129)

which is clearly parametrization dependent. In terms of the basis functions
(126), this condition means that the coefficient a in (128) should be real for
states built from an odd number of odd modes, and pure imaginary otherwise.
In other words, in Witten's theory, in+lan are "real" operators, in the sense
that acting with them preserves the phase condition.

Witten's Interaction

Witten's interaction [24] is constructed as a generalization of the interac-
tions of point particles in local field theory. We may think of point particle
interactions as the creation of several particles from the vacuum at some in-
stant of space-time, in such a way that energy-momentum is conserved. Thus
one or more particles will have negative energy, and thus may be interpreted as
incoming positive energy particles. The result is that the interaction is formed

as a simple product of fields at a point.

Witten's generalization of this involves the definition of a product operation
which maps two string fields D, T into a single field 4 T. It is of the form

( r)[x] = J DyDz K[x, y, z] I)[y] T[z] (130)

where the kernel K[x, y, z] is given by

7r/2 7r/2 ir
K[x,y,z] = f 6[x(oa)-y(u)] 11 [y(7r-a)-z(a)] 6[x(oa)-z(a)] (131)
0=0 0=0 a=7r/2

A crucial consistency check is that ) *4) must satisfy the phase condition (129)
for any 4 which does; this is easily seen to be true in this case. The interaction
vertex which is then found using the inner product (32) is then guaranteed to
be real for 4 satisfying (129).

V{ } = I Dx V*[x] (4 1)[ x]
J (132)
= /DxDyDz K'[x,y,z] )[x] b[y] D[z]

7r/2 7r/2 7Ir/2
K[x, y, z] = [X(7r ) y(o)] 1 6y(7r -0) fi 6[Z7 -~0)-~x(o)]
a=O o'=O 0'=0
The redefined kernel K is cyclically symmetric in x, y, z. It satisfies the follow-
ing overlap equations for all a < r/2:

(x(a) z(7r )) K[x, Y,z =0 (134a)

( 5) + ( )) ki[xyz] = 0 (134b)
\o <) +bz(v ])

and all cyclic permutations in (x,y,z). Note that the solution is possible
because the operators in (134 a) and (134 b) commute. We may interpret these
equations as the generalization of momentum conservation, i.e. the density of
momentum is conserved on the overlapping strings. These overlap equations
are an alternate starting point leading to the interaction form (133).
We are now in a position to understand what becomes of reparametrization
invariance in this theory. When we act with a reparametrization it will in gen-
eral not preserve the phase condition (129). Therefore the reparametrization
must be accompanied by an appropriate transformation of the phases of the
coefficients of the basis functionals. The generators of reparametrizations are
made up of Virasoro operators Ln. In this theory, in+o1 n are real operators,

hence Ln is real for even n and imaginary for odd n. Thus we need to shift the
phases by using the real operators inLn to generate reparametrizations. The
full generators of reparametrizations become inLn i-nL-n = i'Kn which are
known [27] to be an exact kinematical symmetry of Witten's theory.
We therefore see that this theory does in fact possess reparametrization in-
variance, where the reparametrizations must be accompanied by corresponding
phase shifts in the coefficients of the basis functionals. We could have chosen
the basis functionals in a way such that the Ln operators are all real, in other
words so that reparametrizations do not affect the phase choice. This will be
the subject of the next section.

Manifestly Invariant Formulation
We may restore manifest parametrization invariance by choosing an invari-
ant phase condition, namely

4*[x()] = 4[x(a)] (135)

This condition makes all an pure imaginary operators, therefore all Ln will
be real. We can translate the form of the overlap equations (134) to the new
phase convention by the substitution

an --+ n an (136)

It is convenient to work in terms of the left and right combinations. Under
(136) we find, using (62),

{ XLa+t)+tQ0, fora<7<;
R xL( - ) + 0 for a <
x L (0) --7 -> 37r- (137a)
xn( --a or aO, for ur >

(7- 09)+^ ot fora < (137b)
xRv (09 XR(a 7) + -o0, for a > 7.

resulting in the new overlap equations, valid for 0 < o- < r:

(xL()+ ZRa)z =)-x,,] 0 (138)

and all cyclic permutations in (x, y, z). Note that these equations are repara-
metrization invariant, since left and right combinations transform indepen-
dently, and oz0 (the string's total momentum) is an invariant. The equations
are consistent, since the commutator of XL with itself cancels with that of zR
with itself. The solution to these equations (see Appendix D) does not involve
delta function overlaps; rather it is*
K[x, y, z] = 6[x(7r) + x(0) z(7r) z(0)J 8[y(7r) + Y(0) z(7r) z(0)]

exp -- xy + yz + zx yx zy xz) (139)
[ 2 J TT 7r +
Several remarks about (139) need to be made. First, the overlap equations
(138) do not hold at the endpoints a = 0, 7r due to surface terms in the integral
over ao in (139). This is to be expected because the overlap equations at the
endpoints came from those at the midpoint in the original Witten formulation,
and x'(acr) was discontinuous at the midpoint. Transforming the phases we see
that at the endpoints only the following equations need be satisfied:

x(7r) + x(0) = y(7r) + y(0) = z(r) + z(0) (140a)

__- 6 ) [ ]=0 (140b)
Sx(0) + 6x(- ) + Sy(O) + 6y(--) Sz(0) S z(7r) xyz 0 (140b)
Hence the delta function overlaps at the endpoints. Note that the equations
do not require e.g. x(0) = z(7r) as in the previous formulation.
Secondly, the solution gives a manifestly reparametrization invariant in-
teraction vertex (up to anomalies, which will be discussed shortly). It is easy
* Because of (135), there is no distinction between K and k as in the last
section. Note also that no normal ordering is implied in the exponential; the
K's are functionals, not operators which act on functionals.

to see this directly using the generator (35). Applying this to the interaction
(132) we find, after integration by parts, the condition for an invariant vertex

(x'(U) ()+y'(CF ) + z'(O) z(-))- K[x,Vy,z] = 0 (141)

which can be easily shown to hold (disregarding for now the anomalous effect
of the normal ordering) for (139). Alternatively, we note that (for example)
x(cr) and y'(u) transform covariantly with weights 0 and 1 respectively. Since
there is no ordering ambiguity in the product xy', it has weight 0+1=1, thus
its integral is an invariant. Note that the endpoint terms in (139) do not affect
reparametrization invariance since the endpoints of the string are unaffected
by reparametrizations (they are always fixed at 0, 7r).
Thirdly, this vertex is guaranteed to possess all of the appealing properties
of Witten's vertex, such as unitarity, correct modular properties, and BRST
gauge invariance which persist despite ordering difficulties. Since our interac-
tion is just Witten's with changes in the phase conventions, these properties
follow automatically.
Finally, we need to check that (139) gives a vertex which gives a real number
when evaluated for a function satisfying (135). This is indeed seen to be true
for the particular form (139). Using (135),

({J DxDyDz K*[x,y,z]
but from (139),
K*[x,y,z] =K[y,x,z] (143)

so, relabelling x +-+ y, we see that (142) is equal to (132).


We note that the interactions in this formulation of the theory do not

possess manifest spacetime locality; the interactions do not consist of delta

function overlaps, except for the constraint that the position midway between

the two endpoints must be the same for all three strings. In exposing the

hidden reparametrization invariance of the theory we seem to have hidden the

locality in the same way.

We should remember, however, that Witten's theory has a different kind of

nonlocality, namely, nonlocality in terms of components. Whereas the action

of a local theory contain only a finite number of derivatives, the vertices of any

string field theory when written in components as (42,43) contain factors of

exp (7-0)2, which are responsible for the ultraviolet convergence of the theory.

This can be traced to the fact that strings with different values of the zero

mode coordinate xO may interact. Thus when the field theory is written in

terms of components so that x0 is the coordinate and the other modes Xn are

internal degrees of freedom, the interactions are nonlocal. This nonlocality

does not show up in perturbation theory, however it poses a great problem in

any nonperturbative formulation of the theory [52] which has still not been

solved. A discussion of this problem would be beyond the scope of this thesis;

we simply remark that in terms of components, the new formulation is no less

(and no more) local than in Witten's original formulation.

Ghost Insertion
Up to now we have not discussed the ghost part of the vertex. The same

analysis applies in this case, and the result is almost identical. However, we

know that in Witten's formulation there is an extra ghost insertion exp -(-),

which is necessary so that the interaction term in the action has the same
ghost number as the kinetic term. Carrying out our substitution, we find
that the required ghost insertion is exp : ((0) + q(7r)). This appears to
be parametrization invariant because it is made from endpoint contributions;
however, following the analysis of Gross and Jevicki [53], we find that this term
gives an anomalous contribution to the effect of the reparametrization which
exactly cancels the anomaly due to the normal ordering of the generators.
Finally, we give the complete form of the vertex, including the ghosts:

K[x, y, z; , x] =6[x(7r) + x(O) z(7r) z(0)] S[y(7r) + y(0) z(7r) z(0)

6[(7r) + 0(0) x(r) x(O)] 6[(7r) + (O) X(7 X(0)]

exp [ -- xy + yz + zx yx zy xz
2 J7r
+ 9' + uX' + X0' ' X6' (144)


We have arrived, through our algebraic and group-theoretical approach, at

our final result, which is an understanding of the apparent lack of reparamet-

rization invariance in Witten's interacting string field theory. We have shown

that this is due to the phase convention used in building the string field. We

have shown one example of the same theory written using a different phase

convention, choosing the particular choice which maintains reparametrization

invariance. It is clear that one could, if one wished, construct the theory with

any arbitrary phase conventions.

To conclude, we identify directions for future research in this subject mat-


(1) The field theory of closed strings must be developed. Although there

has been progress in this area, it is clear that closed string field theory is not

well understood. The interaction term between open and closed string fields

must also be found.

(2) The issue of the nonlocality of string field theory must be dealt with.

Some new formulation of the theory may be necessary, which is not manifestly

perturbative, but one in which the nonlocality does not render the theory

unsuitable. At present it is hard to imagine what this formulation might be.

(3) Some method for calculating in the low-energy limit of string physics

must be developed and applied. Since exactly soluble models are rare, it is

likely that some systematic approximation method will be appropriate. After

such a procedure is developed, we will be able to properly examine string theory

to decide whether it is or is not able to describe the universe that we live in.

Finally let us briefly mention what work is already under progress to ad-

dress these issues:

(1) Some work still continues into the development of string field theories

with closed strings. However the prevailing belief among string theorists is that

string field theory is not the correct formalism through which to study string


(2) Many ideas have been proposed which purport to be new nonperturba-

tive formulations of string theory. So-called "matrix models" were speculated

to be one such candidate recently. It is possible to do a few nonperturbative

calculations in this formalism. However, as these models require the dimen-

sion of spacetime to be one or less, it seems unlikely that these models will be

related in any way to reality. Related models such as topological field theories

and W-gravity theories have been considered; while these models are of math-

ematical interest, they do not seem to describe the world in which we live. Yet

another string-inspired approach is to consider models [54] which are nonlocal

but which do not include the higher modes of the string. Such theories may

evade the nonperturbative problems associated with nonlocality by the fact

that they are local at the classical level. This approach can produce a finite

theory of gravity, however since this form of nonlocalization is apparently pos-

sible for any field theory, and in fact there are many ways to do it for each

theory, it does not reduce the arbitrariness of the standard model (in fact, it

makes the problem worse).

(3) So far no reasonable justification has been suggested for the belief that

strings will naturally compactify to give the standard model (or that they

will compactify at all). In the formalism of string field theory it seems evident

that no compactification will be stable, because of the nonlocality problem [52].


Other formalisms currently known depend on perturbation theory about a par-

ticular spacetime background so it is not possible to even ask such questions

in these formalisms. The difficulty of these questions has led many researchers

to abandon searching for realistic models and instead to study unrealistic the-

ories which are solvable, in the hope that some insight may be gained. Others

now believe that any attempt at studying particle physics at the inaccessibly

high scales at which gravitational effects are significant is futile due to the lack

of experimental evidence, and instead confine their efforts to physics at lower

energy scales. In the absence of any great new ideas, it seems difficult to find

a path between these two extreme points of view.



Metric: goo = -1, gii = 1

For open strings, 0 < a < -r.

Reparametrization representations



For a quantity q(7) transforming according to (w, z),

6fq = -(fq' + wf'q + zf')



String coordinates xP(r) transform as (0, 0).
Bosonized ghosts 0(o-) transform as (0, -3/2).


[Sf, 6g] = 6fg,_-f,g

Product rules:

For p, q transforming as (wp, 0) and (wq, 0),
pq transforms as (Wp + Wq, 0);

wppqI Wqp' q transforms as (wp + wq + 1,0);


Commutation Relations

1 6 X'(02)]

= 9 ("61 02)

6l (2)]= 6(01 02)

[I pXv-n] = 9v^m,n

On]= 6m,n

[XL'(1), X ('2)] = -2irg" d 6(Oi --02)
[ L (01),al

[X'(0o1),X (2)] = -


[O'(O"l), O'(0.2)]=

2i7rgP 6(01 0a2)

+2i7r--06(0'1 0-2)
dcirl6o 2

Mode Expansions

X (O) = ao + v2 E Xn cos nc"

= o+ V2 T. On COS nu

-- COS no-

--n COS )




x pR(01) -XI'(0T) ~ 6
1 6
OL, R(O") 0 '(09) T= V -









6 1
6x (0~) 7r

6 1
60(g) 7r







X T ao0 +i E 1-ne


,= -iv2 a n4x)
\O (x-np n)

+n = +iV2


for n > 0, and

^ = ia

a= 0


For any functional 4,

6f'= -iEmfC


mClassical Ei if daf(a) \x'(C)
f in I


+0(a) + 6 3 ( )I

[Mf,Mg] =iMfg,-fg

Mf =: Mlassical :


= (L L_)

mvivClassical -7=(L -i
vin2 sin(Ln Ln + cn)

C= n(D+l) or n even,
S0 for n odd.

LR(U) =

OL,R(O) =

















Lx Ta a^^ n-l
2 1
Ol c~

(n- + i On )

x 12 of -301/


Invariant Operators

NG -i -do6),

Mp~v = i 7rdo, XU x xv-
Jo ( 6v 6f1

Q = gUQ^

B 1 7r : e22' gI ) + (L R)

String Field Dynamics

Equation of Motion:

S{I} < IJQ,4 > +v{}

< |T >= IDxl*[x]T[x]

( *)[x] = J DyDz K[x,y,z] D[y] T[z]

n =- 72

ML-l jda f(0.)
f 4 o 7





7f(a) ( i2 -

Q 7r

6e L L L I










Mf =+ 1f i>r

9 V (01
L + 301/) ii\
26 LL+ I

V{}=f Dx -*[x] (D*4)[x]

= DxDyDz K[x,y,z] 1[x] -[y] 4[z]

Witten Formulation:

*[()] = [X(7r o)]

7r/2 7r/2 7r
1 [X(=)-Y(O')] H [y(7r-7)-Z(a)] 1 [X()-z()]
C'=0 O'=0 o-=7r/2

7r/2 7r/2 7r/2
<[Xy,z] = i H X(7r o)-y(o)] 1 6[y(7r a) z()] 1 8[z(7r a) x()]
07=0 o=0 =u=O
New Formulation:

*[x] = "[]


K[x,y,z] = K[x,y,z]

= 6[x(r) + x(O) z(7r) z(O)] 6[y(r) + y(O) z(7r) z(O)]
exp -if d xy/ + v2 + zxT yX ~ / ~xz


K[x,y,z] =






In this appendix we shall prove that the doublet representation given

by (17) and (23) is the only irreducible linear representation of the super-

reparametrization algebra whose basis elements are a finite number of fields

which transform covariantly under reparametrizations. We will show that

given a set of covariant fields which transform into each other under super-

reparametrizations, the representation can be reduced into a series of dou-

blets. We will use the notation aw,i to denote the ith field of weight w in the

collection, where i = 1 to Nw for each value of w.

Consider the fields awo,i, where w0 is the lowest weight in the set. Since the

super-reparametrizations increase the weight by these fields must transform

into weight w0 + fields. We can choose the basis for these fields so that

fawo,i = -faw+1., i 1 to Nwo (B.1)

Applying a second super-reparametrization operator, the covariance of awo,i


Sfawo+i = -(fa'0wo,i + 2wof'awo,i) i = 1 to Nw, (B.2)

i.e. the combinations (awo,i, awo+1-i) form Nwo independent doublets. We now

show that with an appropriate choice of basis, the elements of these doublets do

not appear elsewhere in the representation. First consider the other elements,

i.e. Nwo
(16) for these elements is
fawo+,i = S Aij(fawoj + 2wofawo,j) fawo+l,i (B.3)
with an appropriate choice of basis for the weight w + 1 elements. By changing

the basis for the weight wo + elements we can obtain elements which do not

transform into the weight wo elements. Redefining
awo+,i --+ %a+i E A 0jawo+j' (B.4)
we obtain

awo+,i = -fawo+l,i (B.5)

We now show that the elements in the doublets (aw0, awo+i) do not appear
elsewhere in the algebra. (Here the subscripts i are left as implicitly under-

stood). Let aw be the first (i.e. lowest weight) element whose transformation

law involves one of these elements. Then there are two cases to be considered:

1) w w0 = n is an integer, and the transformation of aw involves aw+.

aw could possibly have the transformation law
faw =EAmd- d m + Xf (B.6)
where Ai are coefficients and Xf is some quantity which does not involve the

elements in the doublet. Xf is found to transform to

2 t + d mf dn-m
fXf = f2' + 2wff'aw + Admfrnmfa/ + 2wOf'aw). (B.7)
aEAm dam don-M fa 10 2ow)
The transformation of Xf involves awo; since we assumed that no field of

lower weight than w has this property, Xf must have higher weight; the only

possibility is

Xf = fa.+I.


Since Xf has no derivatives of f, the only possible Ai's which could be nonzero
are those which are multiplied by f, which in this case is only A0. Then we

d- =AO+ (-f 1)' + 2wf(-faW 0p) + 2wo-a
+ f(faw)' + 2w5faw
It is easy to see by substituting from (B.6) that this cannot be satisfied unless
2) w w0 is half integral (w = w0 + n -), and the transformation of aw
involves awo. The details of this case are similar to case 1. We find
nA dmf dn-m
faw= E m dam danmawo + faw+ (B.10)
f dmf dn-m ( )
fdZ dAm (fam wo+i) + f2a + 2wff'aw (B.11)
2W+ dam dan-m )
requiring Am = 0 except for m = 0 and m = n; then
aW+1 =Aod (f 2 a' 2wOff'awo) + nf (-f 0o 2wof'awo)
df 2 o d on

A d" f dn'f
du (Aodn (fawo+) + An 'iawo+)
do do"

+ 2wf' (Aon (fawo+) + An daw,+)
which again cannot be satisfied unless A0 and An are zero.
We have shown that the lowest weight fields are part of doublets which
decouple from all other fields under super-reparametrizations. One may apply
the same procedure to what remains, again and again until the whole repre-
sentation is reduced to doublets. So any arbitrary representation in terms of

covariant quantities may be reduced to doublets.



In this appendix we derive the results of chapter 4, namely the different

possible subgroups of the Lorentz group which may be unbroken by the set of

matrices a&v.

Under a Lorentz transformation 6a, the matrix auv transforms as

8apatlv "= gapapv g/?pav + gavcicY ggvace (C.1)

We see that the Lorentz-transformed quantity o8a will automatically satisfy

Tr a = 0 and TraSa = 0, because these quantities are the Lorentz transfor-

mations of the invariant traces Tr a and Tr(a2) respectively. We can think of

starting with some matrix a and building a set by applying the Lorentz trans-

formations of some subgroup until a set closed under the subgroup is obtained.

However at each step the condition Tr(ba)2 = 0 must be checked for all 6 in

the unbroken subalgebra.

We have already shown an example which preserves S0(1, 1) symmetry.

Let us postulate an SO(2, 1) unbroken symmetry, and show that a contradic-

tion results. By a change of basis we can arrange things so that the hypothet-

ical unbroken symmetry acts on the first three coordinates, so the unbroken

generators are 601,602, and 612-

We can take a set of a's which transforms irreducibly under the unbroken

symmetry, i.e. one which is zero except in the first 3 by 3 block. To prove

that this is always possible, we notice that acting on any general a with any

of the unbroken generators produces another a (which, by the assumption of



unbroken symmetry, must still be in the set) which has zeros in the last 23

by 23 block. Further application of these transformation will not yield any

nonzero contribution to this block. Therefore there exists a subset of the a

matrices, with the same unbroken symmetry, whose last 23 by 23 blocks are

zero. Now we can obtain c matrices whose 3 by 23 off-diagonal blocks are zero,

by acting with the casmir operator 2(61 + 6 +812)- 1, which will leave the

first 3 by 3 block unchanged, but annihilate the off-diagonal blocks.

Thus without loss of generality we may concentrate on a set of matrices of

the form

a, = B D E (C.2)
where we have shown only the first three rows and columns. The trace condi-

tions on this matrix require
A=D+F (C.3a)


D2 + E2 + F2 + DF = B2+ C2 (C.3b)

Computing the transformed matrices,
/ 2B A+D E)
60a1=2 A+D 2B C (C.4a)
E C 0
2C E A+F)
602a = 2 E 0 B (C.4b)
(A+F B 2C
/0 C -B\
oia = 2 C 2E F D (C.4c)
(-B F-D -2E
which (as advertised) are automatically traceless; however, we must impose

the condition that the trace of the squared matrix is zero. This gives three


4C2 +B2 = E2 + (A+ D)2


B2 + 4C2 = E2 + (A + F)2 (C.5b)

4E2 + (F- D)2 = C2+ B2 (C.5c)

Solving (C.3) and (C.5), we find the general solution
( D+F VrD2 + DF fFTT~DF)
= /D +- -DF D V/D-F (C.6)
S/F2 + DDF F
which has the form apav, where ap = (v/-D + F, v/D, v/F). This is the same
form as the nilpotent matrix in the example of chapter 4. However, we have
shown that there can be only one nilpotent matrix in the set. Therefore the
transformed matrices (C.4) cannot have this form, and indeed by examining
them it is easy to see that they do not; therefore transforming a second time
will necessarily take us out of the set. Thus we have shown that it is impossible
to have an unbroken SO(2, 1) subgroup except in the trivial case when only
the BRST charge is included. Higher subgroups SO(N, 1) are of course also
excluded, since they contain SO(2, 1).
It is possible to have an unbroken SO(n) subgroup, by using a smaller
set of matrices. For example, if we use 23 matrices, which are all zero in the
last three rows and columns, then we have an unbroken SO(3) group. Such
a construction does not work for the noncompact subgroups because taking
the first row (the timelike one) and column to be zero makes it impossible to
satisfy the trace conditions.



We present the details of the derivation of the solution (139) to the overlap

equations (138). First we act with -d and reexpress the left and right combi-

nations in terms of coordinates and functional derivatives; then the equations


x' + i7r+ zI + i- K[x, y, z] = 0

( y' + 3rx X + i'r 8 I[x, y, z]=0

[Z' + Z'6 y + z7r -) K[x, y, z] = 0
b z 6y)
We substitute K = eA, and change variables to the

which transform irreducibly under permutations of x, y, z:

u = x +y + z

v = x + ey + e*z

v = x + e*y + ez

where e exp 2i. The equations (D.1) become

6iir-A = 0

(1 e)v' = i7r(1 + e)--A
(I e*)v*' = ir(l + e*) 6 A





U, V) V*







Then using

- = -z itan- = -iv3
1+e 3


we find the solution

A d0(v*v' v*'v) (D.5)
2 V3-7r

where we have integrated by parts to write the solution in the form which
will give a real vertex for fields satisfying the reality condition. Note that
integration by parts will create surface terms which may affect the overlap
equations at the endpoints, which will need to be checked subsequently. In
terms of our original variables,

A = -(xy xy + y'z yz + xz x'z) (D.6)

which is the solution (139). The endpoint overlaps require x(0) + x(7r) to be
the same for all three strings. It is not hard to see that (D.6) is consistent with
this, due to the total antisymmetry of A under permutations of x, y, z.


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Nucl. Phys. B.


Gary Kleppe was born in Milwaukee, Wisconsin, on October 28, 1962. Af-

ter graduating from the Milwaukee Public School system, he enrolled at Case

Western Reserve University in Cleveland, Ohio; it was there that he was intro-

duced to theoretical high energy physics by Professor Robert W. Brown. After

graduating from CWRU, he went immediately to graduate school in physics

at the University of Florida, and subsequently began doing research under the

supervision of Professor Pierre Ramond. His research interests currently in-

clude field theoretic approaches to unified theories and quantum gravity, and

the study and uses of nonlocal field theories.

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August 1991 _________
Dean, Graduate school