From W. Siegel,
*Introduction
to String Field Theory*
(World Scientific, Singapore, 1988)
# 1. INTRODUCTION

## 1.1. Motivation

The experiments which gave us quantum theory and
general relativity are now quite old, but a satisfactory theory which
is consistent with both of them has yet to be found. Although the
importance of such a theory is undeniable, the urgency of finding it
may not be so obvious, since the quantum effects of gravity are not
yet accessible to experiment.
However, recent progress in the problem has indicated that the
restrictions imposed by quantum mechanics on a field theory of
gravitation are so stringent as to *require* that it also be a
unified theory of all interactions, and thus quantum gravity would
lead to predictions for other interactions which can be subjected to
present-day experiment. Such indications were given by supergravity
theories [1.1], where finiteness was found at some higher-order loops as a
consequence of supersymmetry, which requires the presence of matter
fields whose quantum effects cancel the ultraviolet divergences of the
graviton field. Thus, quantum consistency led to higher symmetry
which in turn led to unification. However, even this symmetry was
found insufficient to guarantee finiteness at all loops [1.2] (unless
perhaps the graviton were found to be a bound-state of a truly finite
theory). Interest then returned to theories which had already
presented the possibility of consistent quantum gravity theories as a
consequence of even larger (hidden) symmetries: theories of
relativistic strings [1.3-5]. Strings thus offer a possibility of
consistently describing all of nature. However, even if strings eventually
turn out to disagree with nature, or to be too intractable to be useful
for phenomenological applications, they are still the only consistent
toy models of quantum gravity (especially for the theory of the graviton
as a bound state), so their study will still be useful for discovering
new properties of quantum gravity.
## 1.2. Known models (interacting)

Although many string theories have been invented which are consistent
at the tree level, most have problems at the one-loop level. (There
are also theories which are already so complicated at the free level
that the interacting theories have been too difficult to formulate to
test at the one-loop level, and these will not be discussed here.)
These one-loop problems generally show up as anomalies. It turns out
that the anomaly-free theories are exactly the ones which are finite.
Generally, topological arguments based on reparametrization
invariance (the ``stretchiness'' of the string world sheet) show that
any multiloop string graph can be represented as a tree graph with
many one-loop insertions [1.10], so all divergences should be representable as
just one-loop divergences. The fact that one-loop divergences
should generate overlapping divergences then implies that one-loop
divergences cause anomalies in reparametrization invariance, since the
resultant multi-loop divergences are in conflict with the
one-loop-insertion structure implied by the invariance. Therefore,
finiteness should be a necessary requirement for string theories (even
purely bosonic ones) in order to avoid anomalies in reparametrization
invariance. Furthermore, the absence of anomalies in such global
transformations determines the dimension of spacetime, which in all
known nonanomalous theories is D=10. (This is also known as the
``critical,'' or maximum, dimension, since some of the dimensions can be
compactified or otherwise made unobservable, although the number of
degrees of freedom is unchanged.)
## 1.3. Aspects

There are several aspects of, or approaches to, string theory which can
best be classified by the spacetime dimension in which they work: D = 2,
4, 6, 10. The 2D approach is the method of first-quantization in
the two-dimensional world sheet swept out by the string as it propagates,
and is applicable solely to (second-quantized) perturbation theory, for
which it is the only tractable method of calculation. Since it
discusses only the properties of individual graphs, it can't discuss
properties which involve an unfixed number of string fields: gauge
transformations, spontaneous symmetry breaking, semiclassical solutions
to the string field equations, etc. Also, it can describe only the
gauge-fixed theory, and only in a limited set of gauges. (However, by
introducing external particle fields, a limited amount of information on
the gauge-invariant theory can be obtained.) Recently most of the
effort in this area has been concentrated on applying this approach to
higher loops. However, in particle field theory, particularly for
Yang-Mills, gravity, and supersymmetric theories (all of which are
contained in various string theories), significant (and sometimes
indispensable) improvements in higher-loop calculations have required
techniques using the gauge-invariant field theory action. Since such
techniques, whose string versions have not yet been derived, could
drastically affect the S-matrix techniques of the 2D approach, we do not
give the most recent details of the 2D approach here, but some of the
basic ideas, and the
ones we suspect most likely to survive future reformulations, will be
described in chapters 6-9.
## 1.4. Outline

String theory can be considered a particular kind of particle theory, in
that its modes of excitation correspond to different particles.
All these particles, which differ in spin and other quantum numbers, are
related by a symmetry which reflects the properties of the string. As
discussed above, quantum field theory is the most complete framework
within which to study the properties of particles. Not only is this
framework not yet well understood for strings, but the study of string
field theory has brought attention to aspects which are not well
understood even for general types of particles. (This is another
respect in which the study of strings resembles the study of
supersymmetry.) We therefore devote chapts. 2-4 to a general study of
field theory. Rather than trying to describe strings in the language of
old quantum field theory, we recast the formalism of field theory in a
mold prescribed by techniques learned from the study of strings. This
language clarifies the relationship between physical states and gauge
degrees of freedom, as well as giving a general and straightforward
method for writing free actions for arbitrary theories.
## REFERENCES

### Preface

### Chapter 1

The fundamental difference between a particle and a string is that a particle is a 0-dimensional object in space, with a 1-dimensional world-line describing its trajectory in spacetime, while a string is a (finite, open or closed) 1-dimensional object in space, which sweeps out a 2-dimensional world-sheet as it propagates through spacetime:

The nontrivial topology of the coordinates describes interactions. A string can be either open or closed, depending on whether it has 2 free ends (its boundary) or is a continuous ring (no boundary), respectively. The corresponding spacetime figure is then either a sheet or a tube (and their combinations, and topologically more complicated structures, when they interact).

Strings were originally intended to describe hadrons directly, since the observed spectrum and high-energy behavior of hadrons (linearly rising Regge trajectories, which in a perturbative framework implies the property of hadronic duality) seems realizable only in a string framework. After a quark structure for hadrons became generally accepted, it was shown that confinement would naturally lead to a string formulation of hadrons, since the topological expansion which follows from using 1/N_color as a perturbation parameter (the only dimensionless one in massless QCD, besides 1/N_flavor), after summation in the other parameter (the gluon coupling, which becomes the hadronic mass scale after dimensional transmutation), is the same perturbation expansion as occurs in theories of fundamental strings [1.6]. Certain string theories can thus be considered alternative and equivalent formulations of QCD, just as general field theories can be equivalently formulated either in terms of ``fundamental'' particles or in terms of the particles which arise as bound states. However, in practice certain criteria, in particular renormalizability, can be simply formulated only in one formalism: For example, QCD is easier to use than a theory where gluons are treated as bound states of self-interacting quarks, the latter being a nonrenormalizable theory which needs an unwieldy criterion (``asymptotic safety'' [1.7]) to restrict the available infinite number of couplings to a finite subset. On the other hand, atomic physics is easier to use as a theory of electrons, nuclei, and photons than a formulation in terms of fields describing self-interacting atoms whose excitations lie on Regge trajectories (particularly since QED is not confining). Thus, the choice of formulation is dependent on the dynamics of the particular theory, and perhaps even on the region in momentum space for that particular application: perhaps quarks for large transverse momenta and strings for small. In particular, the running of the gluon coupling may lead to nonrenormalizability problems for small transverse momenta [1.8] (where an infinite number of arbitrary couplings may show up as nonperturbative vacuum values of operators of arbitrarily high dimension), and thus QCD may be best considered as an effective theory at large transverse momenta (in the same way as a perturbatively nonrenormalizable theory at low energies, like the Fermi theory of weak interactions, unless asymptotic safety is applied). Hence, a string formulation, where mesons are the fundamental fields (and baryons appear as skyrmeon-type solitons [1.9]) may be unavoidable. Thus, strings may be important for hadronic physics as well as for gravity and unified theories; however, the presently known string models seem to apply only to the latter, since they contain massless particles and have (maximum) spacetime dimension D = 10 (whereas confinement in QCD occurs for D ≤ 4).

In fact, there are only four such theories:

I: N=1 supersymmetry, SO(32) gauge group, open [1.11] IIA,B: N=2 nonchiral or chiral supersymmetry [1.12] heterotic: N=1 supersymmetry, SO(32) or E(8)E(8) [1.13] orbroken N=1 supersymmetry, SO(16)SO(16) [1.14]

All except the first describe only closed strings; the first describes open strings, which produce closed strings as bound states. (There are also many cases of each of these theories due to the various possibilities for compactification of the extra dimensions onto tori or other manifolds, including some which have tachyons.) However, for simplicity we will first consider certain inconsistent theories: the bosonic string, which has global reparametrization anomalies unless D=26 (and for which the local anomalies described above even for D=26 have not yet been explicitly derived), and the spinning string, which is nonanomalous only when it is truncated to the above strings. The heterotic strings are actually closed strings for which modes propagating in the clockwise direction are nonsupersymmetric and 26-dimensional, while the counterclockwise ones are N=1 (perhaps-broken) supersymmetric and 10-dimensional, or vice versa.

The 4D approach is concerned with the phenomenological applications of the low-energy effective theories obtained from the string theory. Since these theories are still very tentative (and still too ambiguous for many applications), they will not be discussed here. (See [1.15,0.1].)

The 6D approach describes the compactifications (or equivalent
eliminations) of the 6 additional dimensions which must shrink from
sight in order to obtain the observed dimensionality of the macroscopic
world. Unfortunately, this approach has several problems which inhibit
a useful treatment in a book: (1) So far, no justification has been
given as to why the compactification occurs to the desired models, or to
4 dimensions, or at all; (2) the style of compactification
(Kauza-Klein,
Calabi-Yau, toroidal, orbifold, fermionization, etc.)
deemed most promising changes from year to year; and (3) the string
model chosen to compactify (see previous section) also changes every few
years. Therefore, the 6D approach won't be discussed here, either
(see [1.16,0.1]).
~~l~~

What is discussed here is
primarily the 10D approach, or second quantization, which seeks to
obtain a more systematic understanding of string theory that would allow
treatment of nonperturbative as well as perturbative aspects, and
describe the enlarged hidden gauge symmetries which give string theories
their finiteness and other unusual properties. In particular, it would
be desirable to have a formalism in which all the symmetries (gauge,
Lorentz, spacetime supersymmetry) are manifest, finiteness follows from simple
power-counting rules, and all possible models (including possible 4D
models whose existence is implied by the 1/N expansion of QCD and
hadronic duality) can be straightforwardly classified. In ordinary
(particle) supersymmetric field theories [1.17], such a formalism (*
superfields* or *superspace*) has resulted in much simpler rules for
constructing general actions, calculating quantum corrections (*
supergraphs*), and explaining all finiteness properties (independent
from, but verified by, explicit supergraph calculations). The
finiteness results make use of the background field gauge, which can be
defined only in a field theory formulation where all symmetries are
manifest, and in this gauge divergence cancellations are automatic,
requiring no explicit evaluation of integrals.

In chapts. 5-6 we discuss the mechanics of the particle and string. As mentioned above, this approach is a useful calculational tool for evaluating graphs in perturbation theory, including the interaction vertices themselves. The quantum mechanics of the string is developed in chapts. 7-8, but it is primarily discussed directly as an operator algebra for the field theory, although it follows from quantization of the classical mechanics of the previous chapter, and vice versa. In general, the procedure of first-quantization of a relativistic system serves only to identify its constraint algebra, which directly corresponds to both the field equations and gauge transformations of the free field theory. However, as described in chapts. 2-4, such a first-quantization procedure does not exist for general particle theories, but the constraint system can be derived by other means. The free gauge-covariant theory then follows in a straightforward way. String perturbation theory is discussed in chapt. 9.

Finally, the methods of chapts. 2-4 are applied to strings in chapts. 10-12, where string field theory is discussed. These chapters are still rather introductory, since many problems still remain in formulating interacting string field theory, even in the light-cone formalism. However, a more complete understanding of the extension of the methods of chapts. 2-4 to just particle field theory should help in the understanding of strings.

Chapts. 2-5 can be considered almost as an independent book, an attempt at a general approach to all of field theory. For those few high energy physicists who are not intensely interested in strings (or do not have high enough energy to study them), it can be read as a new introduction to ordinary field theory, although familiarity with quantum field theory as it is usually taught is assumed. Strings can then be left for later as an example. On the other hand, for those who want just a brief introduction to strings, a straightforward, though less elegant, treatment can be found via the light cone in chapts. 6,7,9,10 (with perhaps some help from sects. 2.1 and 2.5). These chapters overlap with most other treatments of string theory. The remainder of the book (chapts. 8,11,12) is basically the synthesis of these two topics.

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