内容简介：
This book is intended as an introduction to gauge field theory for advanced
undergraduate and graduate students in high energy physics. The discussion
is restricted to the classical (nonquantum) theory. Furthermore, general relativity
is outside the limits of this book.
My initial plan was to review the self-interaction problem in classical gauge
theories with particular reference to the electrodynamics of point electrons
and Yang–Mills interactions of point quarks. The first impetus to summarize
current affairs in this problem came to me from the late Professor Asim Barut
at a conference on mathematical physics in Minsk, Belarus, during the summer
of 1994. He pointed out that developing a unified approach to self-interaction
in the classical context might help to illuminate the far more involved quantum
version of this problem. The idea of writing a review of this kind came up again
in my discussions with Professor Rudolf Haag during a physics conference at
the Garda Lake, in the autumn of 1998. He advised me to extend the initial
project to cover all attendant issues so that the review would meet the needs
of senior students.
Self-interaction is a real challenge. Traditionally, students become aware of
this problem in the course of quantum field theory. They encounter numerous
divergences of the S matrix, and recognize them as a stimulus for understanding
the procedure of renormalization. But even after expending the time and
effort to master this procedure one may still not understand the physics of
self-interaction. For example, it is difficult to elicit from textbooks whether
the orders of divergence are characteristic of the interaction or are an artifact
of the perturbative method used for calculations. On the other hand, the
structure of self-interaction is explicit in solvable models. We will see that
many classical gauge theory problems can be completely or partly integrated.
In contrast, quantum field theory has defied solvability, with the exception of
two- and three-dimensional models.
The self-interacting electron was of great concern to fundamental physics
during the 20th century. However, classical aspects of this problem are gradually
fading from the collective consciousness of theoretical physics. Textbooks which cover this topic in sufficient detail are rare. Among them, the best
known is the 1965 Rohrlich’s volume. This excellent review represents the
state of the art in the mid-1960s. Since then many penetrating insights into
this subject have been gained, in particular exact retarded solutions of the
Yang–Mills–Wong theory. It is therefore timely to elaborate a unified view of
the classical self-interaction problem. The present work is a contribution to
this task.
The book is, rather arbitrarily, divided into two parts. The first part, which
involves Chaps. 1–5 and 7, is a coherent survey of special relativity and field
theory, notably the Maxwell–Lorentz and Yang–Mills–Wong theories. In addition,
Mathematical Appendices cover the topics that are usually beyond
the standard knowledge of advanced undergraduates: Cartan’s differential
forms, Lie groups and Lie algebras, γ-matrices and Dirac spinors, the conformal
group, Grassmannian variables, and distributions. These appendices
are meant as pragmatic reviews for a quick introduction to the subject, so
that the reader will hopefully be able to read the main text without resorting
to other sources.
The second part of this book, stretching over Chaps. 6 and 8–10, focuses
on the self-interaction problem. The conceptual basis of this study is not
entirely conventional. The discussion relies heavily on three key notions: the
rearrangement of the initial degrees of freedom resulting in the occurrence of
dressed particles, and spontaneous symmetry deformation.
We now give an outline of this book.
Chapter 1 discusses special relativity. Following the famous approach of
Minkowski, we treat it as merely the geometry of four-dimensional pseudoeuclidean
spacetime. Section 1.1 offers the physical motivation of this point
of view, and introduces Minkowski space. Mathematical aspects of special
relativity are then detailed in Sects. 1.2 through 1.6.
Chapter 2 covers the relativistic mechanics of point particles. Newton’s
second law is embedded in the four-dimensional geometry of Minkowski space
to yield the dynamical law of relativistic particles. We define the electromagnetic
field through the Lorentz force law, and the Yang–Mills field through
the Wong force law. Electromagnetic field configurations are classified according
to their algebraic properties. We develop a regular method of solving
the equation of motion for a charged particle driven by a constant and uniform
electromagnetic field. Section 2.5 reviews the Lagrangian formalism of
relativistic mechanical systems. Reparametrization invariance is studied in
Sect. 2.6. It is shown that a consistent dynamics is possible not only for massive,
but also for massless particles. Section 2.7 explores the behavior of free
spinning particles. Since the rigorous two-particle problem in electrodynamics
is a formidable task, we pose a more tractable approximate problem, the
so-called relativistic Kepler problem. We then analyze a binary system composed
of a heavy magnetic monopole and a light charged particle. Collisions
and decays of relativistic particles are briefly discussed in the final section.
Chapter 3 gives a derivation of the equation of motion for the electromagnetic
field, Maxwell’s equations. We show that some of the structure of
Maxwell’s equations is dictated by the geometrical features of our universe,
in particular the fact that there are three space dimensions. The residual information
translates into four assumptions: locality, linearity, the extended
action-reaction principle, and lack of magnetic monopoles.
Chapter 4 covers solutions to Maxwell’s equations. It begins by considering
static electric and constant magnetic fields. Some general properties of solutions
to Maxwell’s equations are summarized in Sect. 4.2. Free electromagnetic
fields are then examined in Sect. 4.3. We use the Green’s function technique to
solve the inhomogeneous wave equation in Sect. 4.4. The Li´enard–Wiechert
field appears as the retarded solution to Maxwell’s equations with a point
source moving along an arbitrary timelike smooth world line. A method of
solving Maxwell’s equations without resort to Green’s functions is studied in
Sect. 4.7. This method will prove useful later in solving the Yang–Mills equations.
We show that the retarded electromagnetic field F generated by a single
arbitrarily moving charge is invariant under local SL(2,R) transformations.
This is the same as saying there is a frame in which the Li´enard–Wiechert
field F appears as a pure Coulomb field at each observation point. The chapter
concludes with a discussion of the electromagnetic field due to a magnetic
monopole.
Chapter 5 covers the Lagrangian formalism of general field theories, with
emphasis on systems of charged particles interacting with the electromagnetic
field. Much attention is given to symmetries and their associated conservation
laws in electrodynamics. These symmetries are of utmost importance in the
theory of fundamental interactions. The reader may wish to familiarize himself
or herself with these concepts early in the study of field theory; the Maxwell–
Lorentz theory seems to be a good testing ground. An overview of strings and
branes completes this chapter. This material may be useful in its own right,
and as an application of the calculus of variations to systems that combine
mechanical and field-theoretic features.
Chapter 6 treats self-interaction in electrodynamics. We begin with the
Goldstone and Higgs models to illustrate the mechanism of rearrangement
whereby the original degrees of freedom appearing in the Lagrangian are rearranged
to give new, stable modes. We then introduce the basic concept
of radiation, and derive energy-momentum balance showing that mechanical
and electromagnetic degrees of freedom are rearranged into dressed particles
and radiation. The Lorentz–Dirac equation governing a dressed particle is discussed
in Sect. 6.4. Two alternative ways of deriving this equation are given
in Sect. 6.5.
The essentials of classical gauge theories are examined in Chap. 7. Section
7.1 introduces the Yang–Mills–Wong theory of point particles interacting
with gauge fields, in close analogy with the Maxwell–Lorentz theory. We
briefly review a Lagrangian framework for the standard model describing the three fundamental forces mediated by gauge fields: electromagnetic, weak, and
strong. Section 7.3 outlines gauge field dynamics on spacetime lattices.
Exact solutions to the Yang–Mills equations are the theme of Chap. 8.
It seems impossible to cover all known solutions. Many of them are omitted,
partly because these solutions are of doubtful value in accounting for
the subnuclear realm and partly because they are covered elsewhere. The emphasis
is on exact retarded solutions to the Yang–Mills equations with the
source composed of several colored point particles (quarks) moving along arbitrary
timelike world lines. The existence of two classes of exact solutions
distinguished by symmetry groups is interpreted as a feature of the Yang–
Mills–Wong theory pertinent to the description of two phases of subnuclear
matter.
Chapter 9 deals with selected issues concerning self-interaction in gauge
theories. The initial degrees of freedom in the Yang–Mills–Wong theory are
shown to rearrange to give dressed quarks and Yang–Mills radiation. We address
the question of whether the renormalization procedure used for treating
the self-interaction problem is self-consistent. A plausible explanation for the
paradoxes of self-interaction in the Maxwell–Lorentz theory is suggested in
Sect. 9.3.
To comprehend electrodynamics as a whole, one should view this theory
from different perspectives in a wider context. For this purpose Chap.
10 generalizes the principles underlying mechanics and electrodynamics. The
discussion begins with a conceivable extension of Newtonian particles (governed
by the second order equation of motion) to systems whose equations
of motion contain higher derivatives, so-called ‘rigid’ particles. Most if not
all of such systems exhibit unstable behavior when coupled to a continuum
force field. Electrodynamics in various dimensions is another line of generalizations.
Two specific examples, D + 1 = 2 and D + 1 = 6, are examined
in some detail. If Maxwell’s equations are preserved, then a consistent description
for D + 1 = 6 is attained through the use of rigid particle dynamics
with acceleration-dependent Lagrangians. With these observations, we revise
Ehrenfest’s famous question: ‘In what way does it become manifest from
the fundamental laws of physics that space has three dimensions?’ Nonlinear
versions of electrodynamics, such as the Born–Infeld theory, are analyzed in
Sect. 10.4. We modify the Maxwell–Lorentz theory by introducing a nonlocal
form factor in the interaction term. The final section outlines the direct-action
approach in which the interactions of particles are such that they simulate the
electromagnetic field between them.
With rare exceptions, each section has problems to be solved. Some problems
explore equations that appear in the main text without derivation, while
other problems introduce additional ideas or techniques. The problems are
an integral part of the book. Many of them are essential for the subsequent
discussion. The reader is invited to read every problem and look for its solution.
When running into difficulties with a particular problem, the reader
may consult the answer or hint. References to problems are made by writing the number of the section in front of the number of the problem, for example,
Problem 10.4.2 for the second problem of Sect. 10.4.
Each chapter ends with Notes where some remarks and references for further
reading can be found. The reader should be warned that these Notes do
not pretend to provide a complete guide to the history of the subject. The selection
of the literature sources is a matter of the author’s personal taste and
abilities. Preference is given to the most frequently cited books, representative
reviews, classical original articles, and papers that are useful in some sense.
References are listed by the name of the author(s) and the year of publication.
I am indebted to many people with whom I have discussed the issues
addressed in this book. I am especially thankful to Professors Irina
Aref’eva, Vladislav Bagrov, Asim Barut, Iosif Buchbinder, Gariˇı Efimov,
Dmitriˇı Gal’tsov, Iosif Khriplovich, Vladimir Nesterenko, Lev Okun, Valeriˇı
Rubakov, and Georgiˇı Savvidi for their illuminating remarks. E-mail correspondences
with Professors Terry Goldman, Matej Pavˇsiˇc, Martin Rivas, and
Fritz Rohrlich were of great benefit to this project.
I thank Professor Rudolf Haag for his kind encouragement, interesting
comments, and letter of support to the International Science and Technology
Center (ISTC) with his recommendation to allot funds for writing this book.
The financial support from ISTC, under the project # 1560, during the
time the first version of this book was written, is gratefully acknowledged.
Finally, I am grateful for the assistance of Professor Richard Woodard. He
was my foreign collaborator in the ISTC project. Although the role of foreign
collaborators in project implementation is sometimes a formality, Richard
voluntarily shouldered proofreading of this rather long manuscript. He read
carefully all parts of the text, made numerous corrections of my wording,
and pointed to several mathematical mistakes. His comments, criticism, and
suggestions concerning the most difficult physical issues are of inestimable
help. In spite of his best efforts, errors minor and major, and obscurities are
solely my responsibility.
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