Introduction to String Field Theory - W. Siegel.pdf

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INTRODUCTION to
STRING FIELD THEORY
Warren Siegel
University of Maryland
College Park, Maryland
Present address: State University of New York, Stony Brook
19689270.001.png
CONTENTS
Preface
1. Introduction
1.1. Motivation 1
1.2. Known models (interacting) 3
1.3. Aspects
6. Classicalmechanics
6.1. Gauge covariant
120
6.2. Conformal gauge
122
6.3. Light cone
125
4
Exercises
127
1.4. Outline
6
7. Light-cone quantum mechanics
7.1. Bosonic
2. General light cone
2.1. Actions 8
2.2. Conformal algebra 10
2.3. Poincar´ealgebra 13
2.4. Interactions 16
2.5. Graphs 19
2.6. Covariantized light cone 20
Exercises
128
7.2. Spinning
134
7.3. Supersymmetric
137
Exercises
145
8. BRST quantum mechanics
8.1. IGL(1)
146
8.2. OSp(1,1
|
2)
157
23
8.3. Lorentz gauge
160
3. General BRST
3.1. Gauge invariance and
constraints
Exercises
170
25
9. Graphs
9.1. External fields
171
3.2. IGL(1)
29
9.2. Trees
177
3.3. OSp(1,1
|
2)
35
9.3. Loops
190
3.4. From the light cone
38
Exercises
196
3.5. Fermions
45
10. Light-cone field theory
197
3.6. More dimensions
46
Exercises
203
Exercises
51
11. BRST field theory
11.1. Closed strings
4. General gauge theories
4.1. OSp(1,1
204
|
2)
52
11.2. Components
207
4.2. IGL(1)
62
Exercises
214
4.3. Extra modes
67
12. Gauge-invariant interactions
12.1. Introduction
4.4. Gauge fixing
68
215
4.5. Fermions
75
12.2. Midpoint interaction
217
Exercises
79
Exercises
228
5. Particle
5.1. Bosonic
References
230
81
Index
241
5.2. BRST
84
5.3. Spinning
86
5.4. Supersymmetric
95
5.5. SuperBRST
110
Exercises
118
PREFACE
First, I’d like to explain the title of this book. I always hated books whose titles
began “Introduction to...” In particular, when I was a grad student, books titled
“Introduction to Quantum Field Theory” were the most di cult and advanced text-
books available, and I always feared what a quantum field theory book which was
not introductory would look like. There is now a standard reference on relativistic
string theory by Green, Schwarz, and Witten, Superstring Theory [0.1], which con-
sists of two volumes, is over 1,000 pages long, and yet admits to having some major
omissions. Now that I see, from an author’s point of view, how much effort is nec-
essary to produce a non-introductory text, the words “Introduction to” take a more
tranquilizing character. (I have worked onaone-volume, non-introductory text on
another topic, but that was in association with three coauthors.) Furthermore, these
words leave me the option of omitting topics which I don’t understand, or at least
being more heuristic in the areas which I haven’t studied in detail yet.
The rest of the title is “String Field Theory.” This is the newest approach
to string theory, although the older approaches are continuously developing new
twists and improvements. The main alternative approach is the quantum mechanical
(/analog-model/path-integral/interacting-string-picture/Polyakov/conformal- “field-
theory”) one, which necessarily treats a fixed number of fields, corresponding to
homogeneous equations in the field theory. (For example, there is no analog in the
mechanics approach of even the nonabelian gauge transformation of the field theory,
which includes such fundamental concepts as general coordinate invariance.) It is also
an S-matrix approach, and can thus calculate only quantities which are gauge-fixed
(although limited background-field techniques allow the calculation of 1-loop effective
actions with only some coecients gauge-dependent). In the old S-matrix approach
to field theory, the basic idea was to startwiththeS-matrix, and then analytically
continue to obtain quantities which are off-shell (and perhaps in more general gauges).
However, in the long run, it turned out to be more practical to work directly with
field theory Lagrangians, even for semiclassical results such as spontaneous symmetry
breaking and instantons, which change the meaning of “on-shell” by redefining the
vacuum to be a state which is not as obvious from looking at the unphysical-vacuum
S-matrix. Of course, S-matrix methods are always valuable for perturbation theory,
but even in perturbation theory it is far more convenient to start with the field theory
in order to determine which vacuum to perturb about, which gauges to use, and what
power-counting rules can be used to determine divergence structure without specific
S-matrix calculations. (More details on this comparison are in the Introduction.)
Unfortunately, string field theory is in a rather primitive state right now, and not
even close to being as well understood as ordinary (particle) field theory. Of course,
this is exactly the reason why the present is the best time to do research in this area.
(Anyone who can honestly say, “I’ll learn itwhen it’s better understood,” should mark
adateonhiscalendar for returning to graduate school.) It is therefore simultaneously
the best time for someone to read a book on the topic and the worst time for someone
to write one. I have tried to compensate forthisproblem somewhat by expanding on
the more introductory parts of the topic. Several of the early chapters are actually
on the topic of general (particle/string) field theory, but explained from a new point
of view resulting from insights gained from string field theory. (A more standard
course on quantum field theory is assumed as a prerequisite.) This includes the use
of a universal method for treating free fieldtheories,which allows the derivation of
asingle,simple, free, local, Poincare-invariant, gauge-invariant action that can be
applied directly to any field. (Previously, only some special cases had been treated,
and each in a different way.) As a result, even though the fact that I have tried to
make this book self-contained with regard tostring theory in general means that there
is significant overlap with other treatments, within this overlap the approaches are
sometimes quite different, and perhaps in some ways complementary. (The treatments
of ref. [0.2] are also quite different, but for quite different reasons.)
Exercises are given at the end of each chapter (except the introduction) to guide
the reader to examples which illustrate the ideas in the chapter, and to encourage
him to perform calculations which have been omitted to avoid making the length of
this book diverge.
This work was done at the University of Maryland, with partial support from
the National Science Foundation. It is partly based on courses I gave in the falls of
1985 and 1986. I received valuable comments from Aleksandar Mikovic, Christian
Preitschopf, Anton van de Ven, and Harold Mark Weiser. I especially thank Barton
Zwiebach, who collaborated with me on most of the work on which this book was
based.
June 16, 1988
Warren Siegel
Originally published 1988 by World Scientific Publishing Co Pte Ltd.
ISBN 9971-50-731-5, 9971-50-731-3 (pbk)
July 11, 2001: liberated, corrected, bookmarks added (to pdf)
1.1. Motivation
1
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
yetaccessible 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 insucient to guarantee finiteness at allloops[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.
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 orclosed)1-dimensional object in space,
which sweeps out a 2-dimensional world-sheet as it propagates through spacetime:

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