Gauge Theory and
Variational Principles

GLOBAL ANALYSIS
Pure and Applied
Modem Methods for the Study of Nonlinear Phenomena in
Engineering, the Sciences, and Mathematics
In Two Series
— Advanced, graduate-level texts; monographs, and reference works
— Basic, advanced undergraduate - level texts
Ralph Abraham, Jerrold E. Marsden, Philip J. Holmes, Editors
NUMBER SERIES
1 David Bleecker, Gauge Theory and Advanced
Variational Principles. 1981 Graduate-level
Text
Other Numbers in preparation

Gauge Theory and
Variational Principles
DAVID BLEECKER
Department of Mathematics
University of Hawaii
▼▼
1981
ADDISON-WESLEY PUBLISHING COMPANY, INC.
Advanced Book Program/World Science Division
Reading, Massachusetts
London • Amsterdam • Don Mills, Ontario • Sydney • Tokyo

Library of Congress Cataloging in Publication Data
Bleecker, David.
Gauge theory and variational principles.
(Global analysis, pure and applied; no. 1)
Bibliography: p.
Includes index.
1. Gauge fields (Physics) 2. Variational principles.
I. Title. II. Series.
QC793.3.F5B55 530.1'43 81-17570
ISBN 0-201 -10096-7 AACR2
American Mathematical Society (MOS) Subject Classification Scheme A980)
53A55, 53C05, 55F10, 70G05, 81A30, 81A33, 81A36, 81A39, 81A54, 81A60
83E15
Copyright © 1981 by Addison-Wesley Publishing Company, Inc.
Published simultaneously in Canada.
All rights reserved. No part of this publication may be reproduced, stored in
a retrieval system, or transmitted, in any form or by any means, electronic,
mechanical, photocopying, recording, or otherwise, without the prior written
permission of the publisher, Addison-Wesley Publishing Company, Inc.,
Advanced Book Prog ram/World Science Division Reading, Massachusetts
01867, U.S.A.
Manufactured in the United States of America
ABCDEFGHIJ-HA-8987654321

Contents
Series Editors' Foreword ix
Preface xiii
Chapter 0. Preliminaries 1
0.1. Multilinear Algebra and Forms 2
0.2. Manifolds and Tensor Analysis 6
0.3. Lie Groups and Lie Algebras 18
Chapter 1. Principal Fiber Bundles and Connections ... .23
1.1. Principal Fiber Bundles 26
1.2. Connections 29
Chapter 2. Curvature and g-Valued Differential Forms . . .34
2.1. Graded Lie Algebra of §-Valued Forms 35
2.2. Curvature 37

vi CONTENTS
Chapter 3. Particle Fields, Lagrangians, Gauge
Invariance 42
3.1. Particle Fields 43
3.2. Gauge Transformations 46
3.3. Lagrangians and Gauge Invariance 50
Chapter 4. Lagrange's Equation for Particle Fields 54
4.1. The Principle of Least Action 55
4.2. Some Machinery 56
4.3. Lagrange's Equation 60
Chapter 5. The Inhomogeneous Field Equation 64
5.1. The Current 65
5.2. Inhomogeneous Field Equation 68
Chapter 6. Free Dirac Electron Fields 71
6.1. Covering the Lorentz Group 73
6.2. The Levi-Cevita Connection 77
6.3. Spin Structures and the Lagrangian 81
6.4. Dirac's Equation 84
Chapter 7. Interactions 89
7.1. Bundle Splicing 90
7.2. The (Nonfree) Dirac Electron Field 95
7.3. The Nucleon in a Yang-Mills Potential 99
Chapter 8. Calculus on Frame Bundle 106
8.1. Tensor Fields on L{M) 107
8.2. Pseudo-Riemannian Geometry 109
8.3. Metric Variations 116

CONTENTS vii
Chapter 9. Unification of Gauge Fields and
Gravitation 120
9.1. Gradients of Metric-Dependent Functionals .... 122
9.2. Conservation Laws from External Symmetry .... 127
9.3. The Einstein-Yang-Mills Action Principle 133
Chapter 10. Additional Topics 141
10.1. Geodesies and Forces on Classical Particles 141
10.2. Utiyama's Theorem 147
10.3. Spontaneous Symmetry Breaking 154
10.4. Characteristic Classes, Monopoles, and
Instantons 161
References 171
Selected Bibliography 173
Index of Notation 175
Index 177

Foreword
AIM AND SCOPE OF THE SERIES
What is Global Analysis?
From ancient times till Newton, mathematics meant geometry
and algebra. Then analysis (now called classical) was born, along
with the foundations of physics, engineering, and modern science.
Among the outstanding events of modern mathematics are the
syntheses of these fields, along common frontiers. The synthesis of
classical analysis and geometry is now called global analysis.
The History of Global Analysis and its Applications:
Important pioneers in the synthesis of global analysis were Henri
Poincare A880s), George Birkhoff A920s), Marston Morse A930s),
and Hassler Whitney A940s). The technical tools of differential
topology A950s) made the final synthesis possible A960s). Through
the efforts of Solomon Lefshetz A950s), the work of the Russian
school (Liapounov, Andronov, Pontriagin) on dynamics became
widely known in the west, and included in this synthesis. A veritable
explosion of new results and applications followed in the 1970s.
/x

X FOREWORD
From the earliest work of Poincare and Liapounov onward, the
applications of geometry and analysis to astronomy, physics, and
engineering provided the explicit motivation for much of this work.
The current form of the theory reflects this pervasive influence in its
direct applicability to these fields. It has already created new and
powerful methods of applied mathematics, which complement exist-
existing tools such as pertubation methods, asymptotics, and numerical
techniques.
Far from being the exclusive preserve of pure mathematicians,
global analysis has its roots in physical problems, and can be
redirected to these problems once again, often with startling results.
Target of the Series: The Accessibility of Global Analysis
There is a great contrast between the potential importance of
global analysis and the great difficulty of learning about it. A
growing number of scientists of all disciplines have discovered that
the techniques of global analysis have important applications in their
own fields, and are looking seriously for keys to these techniques.
These Series will attempt to provide these keys.
Needed are books that introduce the basic concepts and their
applications, texts that develop the prerequisites for more serious
study accessibly and compactly, and advanced monographs which
make the research frontier available to a wide audience of scientists
and engineers who have acquired these prerequisites. To these ends,
these Series will deal with such subjects as:
Theory
Linear algebra and representation theory
Calculus on manifolds and bundles
Differential geometery and Lie theory
Manifolds of mappings and sections
Transversal approximations
Calculus of variations in the large
Dynamical systems theory and nonlinear oscillations
Nonlinear actions of Lie groups
Applications
Classical mechanics and field theory
Geometric quantization
Hydrodynamics
Elastomechanics

FOREWORD xi
Econometrics
Social theory
Morphogenesis
Network theory
and other topics of pure and applied global analysis.
AUDIENCES
Series A. The advanced texts will provide reports on theory or
applications from the research frontier in expository style for special-
specialists, or for nonspecialists who have the prerequisite mathematical
background. For example, graduate students of science or engineer-
engineering, as well as mathematics, will find them manageable.
Series B. The basic texts will provide a complete curriculum of
essential prerequisites, starting with advanced linear algebra and
calculus, for the advanced texts of Series A. These texts will be
suitable for advanced undergraduate courses in pure and applied
mathematics, or as reference works for research in engineering, the
sciences, or mathematics.
UNIQUE FEATURES
Through the basic texts (Series B) covering all the prerequisites in
a uniform style, and the advanced texts (Series A) building on this
foundation it will be possible for anyone to study the detailed
applications of global analysis to their own fields (as they appear in
the series), to form their independent evaluation of the new methods,
and to master the techniques for their own use if justified.
Series В starts from the post-calculus level, in textbook format
with worked examples, exercises with answers, and adequate illustra-
illustrations. The two Series will give a complete library of prerequisites,
together with new contributions to global analysis, and some out-
outstanding examples of its applications, illustrating the new methods in
applied mathematics. All the texts will be in English, and conform as
far as possible to a common notational scheme.
Ralph Abraham
Jerrold E. Marsden
Philip J. Holmes

Preface
By a very fortunate combination of events, I. M. Singer visited
Hawaii in the spring of 1978 and delivered two sterling lectures on
differential geometry and gauge theory. This was an inestimable
boon to my enthusiasm for this subject area. At his suggestion, I
conducted an interdepartmental seminar that was attended by a
handful of diverse mathematicians and a physicist. This book grew
out of the seminar.
Owing to the preliminary chapter, the text can be understood by
advanced undergraduate mathematics and/or physics majors. The
proofs are detailed enough so that nobody will sweat blood between
the lines. Because of this, I must ask for the patience of those
enthusiasts who are already familiar with the jargon and find the
details distracting. On a first reading, you should skip those details
that you find uninteresting.
To the uninitiated, it would seem that the use of fiber bundles
and connections to describe the basic forces of nature is a half-baked
scheme devised by some clique of mathematicians bent on producing
an application for their work. However, physicists themselves found
these notions forced upon them by their own perception of nature. In
xiii

XIV PREFACE
the balance of this Preface, I hope to suggest how this could happen,
and to provide a philosophical matrix into which you can pour the
olla podrida of theorems found in the main text.
Let M be the space-time continuum. In semiclassical (first-
quantized) physics, a particle is described in terms of a particle field
(or wave function) \p: M -»V where V is some vector space (typically
over the complex numbers). Some fixed basis of V is chosen with the
basis corresponding to certain states of the particle. Implicit in the
determination of ip(x) is the choice of a reference frame at x. By
"reference frame" we do not necessarily mean a choice of space-time
axes, but we could mean a choice of zero-phase angle or a choice of
axes in isospin space. Let Px denote the space of all possible reference
frames at x. Any two reference frames are uniquely related by an
element of some group G of transformations (e.g., rotations). If
p G Px and gtEG, thenpg denotes the transformed frame. Now g also
produces a transformation of V (say w\-+g-w, for we V). If \p(p) is
the value of \p relative topE. Px, then \p(pg) — g~l">P(P) *s tne value
relative to pg. A smooth concatenation P of the various Px as x
ranges over M is called a principal fiber bundle with group G; Px is
the fiber above x. If pE Px is chosen, then g^pg gives a topological
equivalence of G with Px, but in general P is not equivalent to
M X G, since P may be twisted.
If U is a subregion of M, then a function au: £/-» P, such that
au(y) E Pv for all >>e U, is called a gauge (i.e., a continuous choice of
reference frame). A wave function ip should be regarded as a function
on P, \p: P-^V such that \p(pg) = g~]-\p(p). However, given the
gauge au: U -»P, we can pull \p down to U С М to obtain a local
wave function \pu: U-^V given by $u(y)= 4/(au(y)) f°r У^ U. If ow:
W-^V is another gauge, then we can write ow(y)= ou(y)gUH.(y)
where guw: UDW-^G. Consequently, we have >pw(y)= ^{ow{y)) =
'P(°u(y)gUK(y))=gu*(y)~l-'Pu(y)> which shows how the local wave
functions change under a change of gauge.
Presumably, physically meaningful quantities should be indepen-
independent of the choice of gauge (unlike the local wave functions). Hence
the physicist attempts to construct real-valued functions on M that
depend on the wave function (and its differential), so that the result
is independent of the choice of gauge. Such a function is called an
Action density. (We use a capital "A" to distinguish this "Action"
from the mathematical group "action.") In Theorem 3.3.5 we show

PREFACE XV
that it is not possible to construct an Action density (solely out of
the wave function) that depends nontrivially on the differential of
the wave function. Indeed, what is needed is an additional object
(called a gauge potential or connection) that lives on P and trans-
transforms in a way so that when it is incorporated into a hypothetical
Action density, it leads to a truly gauge-invariant Action density.
Theorem 3.3.6 shows precisely how connections solve the gauge-
invariance problems of defective Action densities.
There is a standard way to compute the field strength (or curva-
curvature) of a gauge potential (or connection). In the case where G = U(l)
(or equivalently, the group of rotations in the plane), the gauge
potential is essentially the 4-vector potential of electromagnetism and
the field strength is the electromagnetic field. Although these terms
go back (at least a century) to the days of Maxwell, it appears to
have been H. Weyl [1918] who introduced the concepts of gauge
transformation and gauge invariance. With the possible exception of
general relativity, the gauge concept in physics was mostly limited to
the study of electromagnetic interactions, until 1954 with the paper
of C. N. Yang and R. L. Mills [1954]. (Actually, O. Klein [1939] had
considered a non-Abelian gauge theory 15 years before.) Yang and
Mills introduced gauges prescribing a point-dependent choice of
isotopic spin axes. In this case, the group is 5GB) (or equivalently,
the unit quaternions). There was no immediate application for the
associated gauge potentials and field strengths, because these fields
were found to be massless, whereas nuclear forces were known to be
mediated by massive particles (e.g., 77 mesons). Nevertheless, the
Yang-Mills model was a precursor to the apparently successful
model of Weinberg and Salam [1967] for weak interactions. Indeed,
we shall see in Section 10.3 that the mechanism of spontaneous sym-
symmetry breaking (developed by Higgs [1966]) allows gauge fields to
acquire mass (consider, e.g., the massive "intermediate vector bosons"
in the Weinberg-Salam model). In spite of these refinements, the
basic fact remains that the existence of gauge fields is a consequence
of the existence of gauge-invariant Action densities for particle fields.
In the beginning there was a gauge-invariant Action density, and
then there was radiation (photons, pions, intermediate vector bosons,
gluons, etc.).
The Action density is a measure of the superfluous manifesta-
manifestations of the fields involved. Nature obeys the principle of least

XVi PREFACE
Action. This means that the only fields allowed by nature are those
that leave the integral of the Action density fixed (to first order) with
respect to all suitable variations of the field. This condition is
equivalent to requiring that the field satisfy a certain differential
equation, known as Lagrange's equation. In Chapter 4, we derive this
equation and find that it is most natural to regard it as an equation
on the bundle space P. Physicists are used to pulling the equation
down to M (by a choice of gauge), typically at the expense of
elegance. We derive in great detail the following special cases of
Lagrange's equation: (A) the Klein-Gordon equation for a charged
spin-0 particle (e.g., a 77" meson); (B) the Dirac equation for the
relativistic spin-5 electron; (C) the equation for the spin-5 nucleon
(proton-neutron doublet) subject to a Yang-Mills gauge potential.
Physicists are as familiar with these equations as differential geome-
geometers are with connections and principal fiber bundles. However, some
residual novelty is gained by considering these equations as based on
a principal fiber bundle over a curved space-time. Also, it is clearly
demonstrated that Dirac's equation follows from Lagrange's equa-
equation for the usual Action density provided that the Levi-Cevita
connection for space-time is used.
The gauge field strengths obey two differential equations. The
first equation, known as the homogeneous field equation or Bianchi
identity, is a consequence of the way in which the field strength is
defined. The second field equation is generally inhomogeneous,
because of a term that is the current of the particle field. The
inhomogeneous equation results from insisting that the gauge poten-
potential obey the principle of least Action for the sum of Action density
of the particle field and the so-called self-Action of the gauge
potential itself. The current is then the first variation (of the first
term in this sum) with respect to a change of potential. In the case
where G = (/(l), the two field equations together are equivalent to
Maxwell's four equations. In general, one can prove that the current
is conserved either by using the inhomogeneous equation or by using
the gauge invariance of the Action density. We do it both ways.
Although no explicit reference to Noether's theorem is made, the
idea that conservation laws follow from symmetries (in this case,
gauge transformations) is largely her own (Noether [1918]).
So far, we have considered only transformations that change the
reference frames at the various points xE.M, but leave the points

PREFACE xvii
themselves fixed. For this reason, the gauge theories and transforma-
transformations discussed above are called internal. The Einstein field equation
of general relativity is based on a principle of least Action that is
invariant under transformations of space-time itself. In this sense,
general relativity is an external gauge theory of sorts. Here the gauge
transformations can be regarded as coordinate changes, which leave
the geometry of space-time unchanged. Thus the integral of the
geometrically expressed Action density (scalar curvature) of general
relativity is invariant under these transformations. The conservation
law arising from this external symmetry is interpreted (via the
Einstein field equation) as conservation of the nongravitational en-
energy and momentum that shape space-time.
There is an interesting and unexpected harmony that arises from
a certain blending of the internal and external gauge theories through
the natural imposition of a geometry on the bundle space P over
space-time. Given a geometry (or metric) on M and a gauge potential
on P, there is a natural way to define a bundle metric on P. The
Action density is defined to be the scalar curvature of this bundle
metric. By demanding that the integral of the Action density be
stationary under variations of the metric on M, we obtain the
Einstein field equation, with the energy-momentum source arising
from the field strength of the gauge potential. Similarly, variation of
the gauge potential leads to the second field equation (or Yang-Mills
equation) for the field strength of the potential. Actually, the idea
here (which we study in Chapter 9) has its roots in the classical
Kaluza-Klein unified field theory (Klein [1926]). Although they
consider only the case in which G = U(l), the situation for an
arbitrary Lie group is only technically more difficult. There are many
accounts of this generalized Kaluza-Klein theory (e.g., Cho [1975]),
Herman [1978], Tabensky [1976]), but I am not aware of any that
bother to go through the complete calculation of Chapter 9. In
Section 10.1 it is proved that the geodesies (i.e., straightest possible
paths) relative to the bundle metric on P project to the (generally
nongeodesic) paths of "charged" particles under the force of the
gauge field strength; the charge is essentially the fiber component of
the geodesic in P. Kaluza and/or Klein proved this in the case where
A number of interesting physical consequences arise when it
happens that P is twisted. The mathematical consequences (e.g.,

XViii PREFACE
nonvanishing characteristic classes) were well known by 1950, but
the physical interpretations (e.g., magnetic monopoles, solitons, in-
stantons) have been explored only recently in this context (Belavin,
Polyakov, Schwarz, and Tyupkin [1976], Jackiw and Rebbi [1976],
't Hooft [1974], etc.)- A brief encounter with such notions is found in
Section 10.4. It is perhaps regrettable that the relatively recent work
of Atiyah, Hitchin, Singer, Ward, and many others in this area could
not be given adequate coverage. Actually, these works speak well for
themselves, and are best received by those with some background in
algebraic geometry or with the analytical tools needed to understand
the Atiyah-Singer index theorem (see Booss [1977], Mayer [1981],
Palais [1965]).
It must be conceded that this book does not by any means bring
you up to the level of many of the current research areas in gauge
theory. For example, there is nothing in the text about the second
quantization of gauge potentials or particle fields. Without this, very
little of real physical significance (scattering cross sections, particle
lifetimes, etc.) can be properly discussed or computed. However, the
mathematical foundations of interacting second-quantized field the-
theories are plagued by multifarious divergent expressions. You may
wish to consult the book of Faddeev and Slavnov [1980] and the
references therein. Should you wish to broaden a general background
in mathematical physics, you will find the books by Abraham and
Marsden [1978], Arnol'd [1978], Thirring [1978], and Sachs and Wu
[1977] particularly helpful.
There are many people who have helped bring this book to its
present form. For helpful comments, encouragement, and willing
ears, I thank Chris Allday, Joe Gerver, Dave Johnson, Bob Little,
Ernest Ma, Jerry Marsden, M. E. Mayer, Dick Palais, I. M. Singer,
Joel Weiner, and Les Wilson. I owe much to the typists, Kathleen
Kikuta and Joanne Nakamura, for their patience, accuracy, and
punctuality in the face of unabated notational monstrosities. I apolo-
apologize and give thanks to those who have indirectly influenced this
volume or who figure prominently in the field and yet do not appear
among the references or in the selected bibliography (which scarcely
represents the vast and growing literature).
David Bleecker

Gauge Theory and
Variational Principles

CHAPTER
О
Preliminaries
Roughly speaking, you should be familiar with the basic concepts
and notation of elementary set theory (e.g., unions, intersections,
Cartesian products, and functions); you must know some linear
algebra (abstract vector spaces, bases, linear transformations, dual
spaces, diagonalization of symmetric matrices, etc.); a little group
theory (homomorphisms, kernels, direct products, etc.); and the
basics of analysis on R" (open sets, continuous functions, the funda-
fundamental theorem of calculus, partial derivatives, integration of smooth
functions of several variables, etc.). You should be familiar with the
use of indices, and realize that sums are taken over repeated indices.
If you lack motivation, you should read the introduction to Chapter
1, as well as the Preface.
David D. Bleecker, Gauge Theory and Variational Principles ISBN 0-201-10096-7
Copyright © 1981 by Addison-Wesley Publishing Company, Inc., Advanced Book
Program/World Science Division. All rights reserved. No part of this publication
maybe reproduced, stored in a retrieval system, or transmitted, in any form or by
any means, electronic, mechanical, photocopying, recording, or otherwise, without
the prior permission of the publisher.
1

2 GAUGE THEORY AND VARIATIONAL PRINCIPLES
Nearly everything in this chapter plays some role in mathematical
physics and will be used later in the book. In Section 0.1, we
introduce concepts of multilinear algebra such as tensors, metrics,
orientations, and star operators. In Section 0.2, we define manifolds
and extend the notions of 0.1 to objects on manifolds. Exterior
differentiation and codifferentiation of forms on manifolds are de-
defined, as well as the integration of «-forms on «-manifolds. Stokes'
theorem is proved for compactly supported forms on manifolds
without boundary. Maxwell's equations are explicitly written in
terms of differential forms in 0.2.22. In Section 0.3, we define Lie
groups, Lie algebras, and related notions (e.g., exponential maps and
structure constants). The case of the special unitary group SU(n),
which is important in physical gauge theories, is discussed in 0.3.12.
0.1 MULTILINEAR ALGEBRA AND FORMS
0.1.1 Notation. Let E and F be vector spaces over U. Let E be the
dual space of E, and let T°'°(E,F)=F. For p,q>0,Tp'q(E,F) is
the space of multilinear functions (F-valued tensors) f: Ё X • — X
EXEX---XE^F. We denote Tp-q(E,U) by Tp-\E). If и,,...,
up(EE and v\...,vq(EE, then ux® ■ ■ ■ ®up®tf® • • • ®vq<ETp-q{E)
is defined by (м, <8> • • • ®vq){x\ ..., xp, yx,..., yq)= х\щ) ■ ■ ■
xp(up)v\yx) ■ ■ ■ vq(yc/). Let ex,...,en be a basis for E, with e\...,e"
the dual basis. Any/E Tp~ q{E) can be uniquely written as
f=
for components fj*.:'? (EM. Let Ak(E,F) be the subspace of T°-k(E,F)
consisting off such that f(uu...,uk) is antisymmetric in uu...,uk^E;
A°(E,F) = F and Ak(E) = Ak(E,R). For k
where w, ..., E.U is antisymmetric in the indices il,...,ik.

0 PRELIMINARIES 3
0.1.2 Definition. For абЛ'(£) and j8eAJ(£), we define аЛ/?е
= 7777 2 (-1)а«(МаA)."-."а(/))^(«а(,Ч-1).•••.«»(,•+;))
/Ле ямт м сюег /Ле set of all permutations о of {1,...,/ + j), and
(— 1 )a = ± 1 is the sign of a. The components of (а Л /?) are given by
where
For a£A°(£), we je/
0.1.3 Definition. Л /mtfnc o« E is a g^T°'2(E) such that g is
symmetric and nondegenerate (i.e., if g(u,v) — 0 for all vE.E, then
м = 0). An orthonormal (o.n.) basis of E relative to g is a basis
ex,...,en such that g(ej,ej)= ±8tj where S:J is the Kronecker delta
(Sjj = 1 if i=j; 5,7 = 0 if i4=j). A volume element of E relative to g is a
juGA"(£') given by ё] Л • • • Аё" for an o.n. basis ex,...,en; note that
ё' ЕЛ'(£) and 1х(е^,...,еп)=\. Actually ju depends on the choice of
o.n. basis, but only by a factor of ±1. A choice of the two possible
volume elements is.called an orientation of E. If ц is an orientation for
E, then a basis vx,...,vn of E is positively oriented if [x(vi,...,vn)>0.
0.1.4 Definition. If g is a metric on E, then there is an induced
metric g^T°-2(Ak(E)) defined as follows. Let vx,...,vnbe any basis of
E, and let gtj —g(vf, vs). Take g'J to be the (i, j)th entry of the inverse
of the matrix (g,7). For a, fi^Ak(E), we define g(a, /3) in terms of
components (relative to vx,...,vn) by
p(a B)= У ?'l>lg'272 . . . o'kJ'kn R
8\aP) ZS 5 5 (xiiP-

4 GAUGE THEORY AND VARIATIONAL PRINCIPLES
We can verify that g(a, ft) is independent of the choice of basis. For
a,P<EA°(E), we let g(a,P) = aP.
0.1.5 Definition. Suppose that we are given a metric g on E and h
on F. Then there is a metric (gh) on Ak(E, F) defined as follows. Let
/i,-..,/„, be a basis for F and write a^Ak(E, F) as a = 1a"fa,a" E
Ak(E,R). Then, for a,/3(EAk(E, F), let (gh)(a,/3) = lhahg(a«,ph)
where hah=h(fa, fh). Again, (gh)(a, ft) is independent of the basis.
0.1.6 Theorem. Let g be a metric on E, and suppose that \x is a
volume element of E relative to g. For n = dim E, there is a unique
linear isomorphism *: Ak(E)-^>A"~k(E), such that aA*fi=g(a,fi)[x
for alia,
Proof. For уеЛ""^), define (py: Ak(E)^R by <py(a)ix=aAy.
We can prove that if <py(a) = 0 for all обЛк(£), then y = 0. Thus,
yi-^<p defines a one-to-one linear map А"~к(Е)-^Ак(Е)л. Since
this map is an isomorphism. Thus, for each fiE:Ak(E), there is a
unique y<EA"~k(E) such that q>y(a)-g(a,/S) for all a<EAk(E). We
take */? to be this y. Then аА*@=аАу = (ру(а)ц=£(а,>8)/х. From
this equation, it is clear that * is one-to-one (and hence an isomor-
isomorphism). ■
0.1.7 Notation. Let g be a metric on E, and let g'1 be as in 0.1.4
{relative to the arbitrary basis vx,...,vn of E). For io^Ak(E) with
components w, ...,■, we define
,,J\ ■■•/*= V ohj\ . . . aidk,.,_
We write \g\ = \det(gIJ)\. If eu...,en is an o.n. basis for E, let tj,7 =
g(ei,ej)=±8ij, and set (-l)g^=det(r]IJ). If «y-2al7<?,-, then gtj =
g(vi,vJ) = '2akiamj7}km or G—ATr\A in terms of matrices. Thus, \det A\
= |g|1/2, and it follows that ц=ё^ A ■■■ Ae" = |g|1/2t31 Л • • • Av" if

0 PRELIMINARIES 5
v),..., vn is positively oriented relative to jjl. Finally, we let e, =
8j2h::"j, and note that n = leit... teh® ■ ■ ■ <8>e'". '' '"
0.1.8 Theorem. Let g be a metric on E with orientation iueA"(£'),
and let vx,...,vnbe a positively oriented basis of E. For w = 2 w, . t5'>
Л • • • Ли'* еЛ'(£), we have (in the notation of 0.1.6 and 0.1.7)
Proof. Let *w denote the right-hand side of A). By 0.1.6, we need
only verify that aA*/S = g(a,/S)^ for all a,/S^Ak(E). Now
k\{n-k)\ k\
a Bjl '"Jke
k\
where we have used the antisymmetry of a and /? in the upper and
lower indices. ■
0.1.9 Theorem. For weA*(£), we have
Proof. In terms of components relative to some positively oriented

6 GAUGE THEORY AND VARIATIONAL PRINCIPLES
basis, we have
1 \\g\1<»mi""Hkemi...m?r"J-kt
(n-k)l k\
1 1
(n-k)\ k\
g g
| ■■■mkji---j^kej, ■■■л-t'l ■■■'*
' em\ ■ ■ ■ m'kj, ■ ■ /„-A, ■ ■ ■/„_*/, ■■■!'*
(n-k)l k!
■ ■ ■ Jn-km'\ ■ ■ ■ m'k£jl ■ ■ -Jn-kh ■■'*
0.1.10 Remark. There is a star operator *: Ak(E,F)-^A" k(E,F)
defined componentwise relative to a basis of F using * on Ak(E).
0.2 MANIFOLDS AND TENSOR ANALYSIS
0.2.1 Definition (C°° л-Manifold). Let M be a set, and suppose M
is the union of a number of subsets Ut where i ranges over some
(possibly infinite) index set I (i.e., M= U;e/(/;). In Figure 1, R" —
{(x\...,x")\x' ED?}, and let yt: U^U" be a one-to-one function such
that <Pj(Ut) is open. Assume that for all i,jE.I, we have q^o^r1;
(py((/, П Ц-)-»ф,-Щ П Uj) is C°° (i.e., has continuous partial derivatives
of all orders). A subset VdM is open г/(р;((/; П V) is open for all /E/.
The collection of open sets is called the topology of M relative to

0 PRELIMINARIES 7
{ф(.|/Е/}. We assume that the topology of M is Hausdorff (i.e., for
x, y^M with хфу, there are disjoint open sets Vx and Vy with xEF
and y^Vy). We add the technical assumption that M=DkUi where
ik E/, A: = 1,2,3,... . Under these assumptions, {ф(.|/Е/} is called an
atlas of M. Two atlases are equivalent if their union is an atlas. An
equivalence class of atlases is called a differentiable structure on M,
and M together with a differentiable structure is called a (Cx) n-
manifold, n being the dimension of M. Any tpf. Ц^>М" in some
representative atlas is called a chart or coordinate system.
0.2.2 Definition. // M and N are manifolds, then a function f:
M->N is called a (C°°) map, provided that (for all charts (p;: Ul■ -»IR"
onMandxPy. V^R1" onN)\Pj°f°<p7l: «p.((/. n/"'(^.))-»R" is C00.
If f has an inverse that is a map, then f is called a diffeomorphism. IfR
is given its usual differentiable structure, then the collection of maps f:
M->R is denoted by CX(M).
M
Figure 1

8 GAUGE THEORY AND VARIATIONAL PRINCIPLES
0.2.3 Definition. A curve through a point x EM is a map y: (a, £>)-»
M (a<0<b) such that y@) = x. Curves у, and y2 through x are called
equivalent if (y°Yi)'@) = (<p°Уг)'(О) for some (and hence any) chart
(p: £/-»R" with xEU. An equivalence class of curves through x is
called a tangent vector at x; the set of all tangent vectors at x is denoted
by TXM. We write y\Q) or
(=0
for the vector in TXM determined by y. Note that TXM has a natural
vector space structure. If YXETXM (say Yx=y'@)) and /EC°°(M),
then (/o y)'(O) EIR is called the derivative off along Yx, and is denoted
by Yx[f].
0.2.4 Definition. Let TM=UxeMTxM. A vector field on M is a
function Y:M-^TM such that YXETXM and (for all /E CX(M)) the
function x\->Yx[f] is in CX(M); we denote this function by Y[f]. The
set of all vector fields on M is denoted by T(TM). If Y, ZET(TM),
then [Y,Z] is that vector field such that [Y,Z]x[f]=Yx[Z[f]]-
Zx[Y[f]\. We omit the proof of the existence and uniqueness of[Y, Z]
(see Kobayashi and Nomizu [1963]). Observe that [Y,Z]=-[Z,Y],
and [Y,[Z,W]] + [W,[Y,Z]] + [Z,[W,Y]] = 0. The latter relation is
called the Jacobi identity.
0.2.5 Definition. ///: M-^ N is a map andxEM, then Дх: TXM->
Tf(x)N is the linear function (differential of f at x) f*x(y'(O)) =
(/° Y)'@) where у is a curve through x. If YE T(TM) and_ YE T(TN)
are such that f^x(Yx)= Yf(x), then we write Y1.Y. If YUY and ZL,Z,
then [Y, Z]1,[Y, Z] can be proved. In the event that f is onto, we write
this conclusion as fJ,Y, Z] — [fifY, f^Z].
0.2.6 Definition. Let <p: U-^U" be a chart. The coordinate vector
fields 3,,..., Зи on UCM are defined by
dt
(=0

0 PRELIMINARIES 9
where et is the ith standard unit vector in U". Any YET(TM), when
restricted to U, can be expressed as_Y=1a' 3, where a' E CX(U). If y:
U-+ R" is another chart with UD U¥=4>, then (for xE Un U) you may
verify that
<p(x)
where У—2а7Эу and у — (у',..., q>") on U. By an abuse of notation,
we can write
, dxj
Эх'
Vector fields can be regarded systems of functions (a1,..., a"), defined
on coordinate domains, that transform according to the rule above
under a change of coordinates.
0.2.7 Definition. Let YET(TM) be such that (for each xEM)
there is a curve yx: R->M through x with y^(t)= У (() for all t£R.
Such a Y is called complete. For tGU, define yt: M-^M by yt(x) =
yx{t). We can prove that q>, is a diffeomorphism, and that <ps°<p, = <ps+,
for all s,tGU. The set {<pt\tEM} is called the one-parameter group
generated by Y. If ZET(TM), then the Lie derivative of Z along Y is
the vector field LYZ defined by
(=0
{I.e.,
TXM).
( = 0
We can prove that LYZ = [Y, Z]. For a chart (p: £/->R" with Y=la'di
and Z=2bjdj (see 0.2.6) we have [Y, Z] = 2(aldi[bJ]-b'dl[aJ])dJ =
LYZ on U.

10 GAUGE THEORY AND VARIATIONAL PRINCIPLES
0.2.8 Definition. Let TP'C>(M) = UxeMTp-c>(TxM). A 1-form on M
is a function a: M->T°-\M) such that axET0A(TxM) and (for any
YET(TM)) the function a(Y) given by a(Y)(x) = ax(Yx) is in CX(M).
A tensor field of type (p, q) on M is a function S: A/-» Tp-q(M) such
that Sx ETp-q(TxM) and (for any l-forms a,,..., ap and vector fields
YX,...,Y on M) the function S(ax,...,ap,Yx,...,Yq) given by
S(ax,...,ap,Yx,...,Y4)(x) = S(aXx,...,apx,YXx,...,Yqx) is in C"(M).
The space of all tensor fields of type (p,q) on M is denoted by
0.2.9 Definition. A k-form on M is a tensor field w E 5" од( М) such
that wx E.Ak(TxM). The space of k-forms on M is denoted by Ak(M).
For aeA'(M) and /3eAJ(M), we define aA/3eAi+J(M) by (aA/3)x
= ax/\fix. Ify: U-^R" is a chart <p = (x\...,x") (x' ECX(U)), then
dx\...,dx" are defined to be those l-forms on V with dx'(dJ) = 8j. Any
w&Ak(M) can be written on U as
"=
'л '■• Adx'k
where uir..ik = u(dli,...,dik)eC'x'(U).
0.2.10 Definition. If feCx(M), then dfEA\M) is defined by
df(Y) = Y[f]for arbitrary' YET(TM). For wEAk(M), we define dec
to be the (k+\)-form that when restricted to U (in the notation of
0.2.9) is given by
We can prove that dw, as defined, is independent of the choice of
coordinates. In fact, die can be defined (without reference to coordi-
coordinates) as that (k+X)-form such that for any Xx,..., Xk+X ET(TM) we

0 PRELIMINARIES 11
have
A +1
die(X^,..., Xk + ]) = 2j v 1
1 = 1
К i < у < и
where the circumflex means that the symbol beneath it is to be omitted.
The operation d: Ak(M)-^Ak+\M) is called exterior differentiation.
If aEA'(M) and /3&AJ(M), then (from the coordinate definition) we
easily obtain d(aAfi) = daAfi + (— \)'af\df5; and d2 =d° d=0.
0.2.11 Definition. ///: M-^N is a map and ic&Ak(N), then
the pull-back /*uEA'(M) is defined by (/*ш)А.(У,,..., Yk) =
A)* f
C°°(M). It can be proved that df*w=f*dw, /*(аЛ/?)=/*аЛ/*(/3),
and (/og)*w = g*/*w.
0.2.12 Definition. In order to integrate forms, we introduce some
topological notions. A subset WCM is closed if its complement Wc =
{xE.M\x£ W) is open. The closure A of an arbitrary subset ACM is
the intersection of all closed subsets of M that contain A. Note that A is
the smallest closed set containing A. An open covering of A CM is a
collection % of open subsets whose union contains A. A subcover %' of
% is a subcollection (i.e., %' C%) such that %' is an open covering. A
subset К of M is compact if every open cover of К has a finite subcover.
The Hausdorff property of M ensures that a compact subset is closed. If
S is a tensor field on M, then the support of S, denoted supp S, is the
closure of {x<EM\S(x)J=0}.
0.2.13 Definition (Integration). A nowhere zero n-form v on an
n-manifold M is called an orientation for M. The pair (M, v) is called
an oriented manifold. Let a be an n-form on the oriented manifold
(M, v) such that K=supp a is compact. The compactness of К ensures
that there is a finite number of charts (p,: Ut->M,i=l,...,N, such that
\ U ■ ■ • U UN, and (p,((/,)ClR" is bounded, and (see Kobayashi

12 GAUGE THEORY AND VARIATIONAL PRINCIPLES
and Nomizu [1963]) there exist functions p, EC°°(M) such that supp p,
С(/„ 0<р,-<1, and 2fp,-(x)=l /or all xEK. If /3 is an n-form
defined on a bounded open subset D of R" such that supp/SCZD is a
closed subset ofR", then we define
?= / b
where b is the real-valued function defined by /3 = bdxx Л • • ■ Adx".
By an interchange of the components of <pt (if necessary), we can
assume that (on Ц) <p*(dxl Л ■ ■ • Adx") is a positive multiple of the
orientation v for all i — 1,..., N. Then, we define
N
fa=2f <РГ*(Р,«).
JM i=\J<p(U,)
We omit the proof that JMa is independent of the choice of coordinate
covering, and so on (see Spivak [1971]). However, the underlying
reason is that the component of an n-form (in local coordinates)
changes under a change of coordinates by a factor equal to the Jacobian
of the change. Thus, the change of variable formula for integrals
applies.
0.2.14 Theorem (Stokes' Theorem). Let M be an oriented n-
manifold and suppose that а£Л"~'(М) has compact support. Then we
have
da = 0.
M
Proof. Let iC=suppa and let (p,: Ц-+Ш" and p,EC°°(M) (i =
1,..., N) be as in 0.2.13. Then a = '2pla, and it follows that
la)=2 f
i J<p(
ff f
M j JM i \(U,) i J<p(U,)
Each of the integrals in this sum vanishes by the following argument.
Let fi be a compactly supported (n — l)-form onR", say fi = '2bJdxl

0 PRELIMINARIES 13
Л • • • AdxJ Л ■ • • Adx". Then
f
since b- vanishes outside of a compact subset of R". Now, set
<p7**p.a on (рЩ) and /3 = 0 on уЩУ. Then, 0 = /
0.2.15 Remark. Г/геге w a notion of an n-manifold M with an
(n — \)-manifold ЬМ for a boundary. The usual version of Stokes'
theorem that says jMda = JdMoc for compactly supported (и— \)-forms
a (and where dM has an orientation induced from that on M). We use
this result only once (in 10.4.13), and therefore we omit the proof (it can
be found in Spivak [1971], Sternberg [1963], etc.).
0.2.16 Theorem. If f: N->M is an orientation-preserving diffeo-
morphism of n-manifolds, then for any compactly supported aE.A"(M)
wehave j j
Proof. Let K= supp a, and let (p,: Ut,-> R" and p, E C°°(M) be as in
0.2.13. Then we have corresponding charts ^,-=<p,-°/: /1
and ai=plof^Cca(N), needed to define jNf*a. Then
2/
= 2/ «pr'IU0/0/-
= 2/ «P7U(P,«) = / a

14 GAUGE THEORY AND VARIATIONAL PRINCIPLES
0.2.17 Definition. A metric on M is a gE?T °i2(M) such that gx is a
metric on TXM. If M is oriented by с£Л"(М), then the volume element
of (M, g) is the juEA"(Af) such that jxx is a volume element of TxM
relative to gx and [xx is a positive multiple of vx. Let *x: Ak(TxM)-^
A"~k(TxM) be as in 0.1.6 or 0.1.8 relative to gx and цх. Define *:
Л*(М)-»Л"-*(М) by (*a)x = *xax. The relation aA*j8=g(a,j8)/i
(which follows from 0.1.6) implies that *f=fn and *(/м) = (— l)gf for
/EC°°(M) where (~l)g: M->{1,-1} is given by (- l)g(.x) = (- l)g*
(see 0.1.7). We define the codifferential 8: Ak(M)->Ak~\M) by
8a=-(-l)8(-l)"{k+v'>*d*a. ///EC°°(M) = A0(M), we set 5/=0.
Since **= ±1, we have 82 = ±*d2* = 0.
0.2.18 Theorem. Let M be an oriented n-manifold with metric g
and volume element ju. Let aEAk(M) and fi&Ak + ](M) such that
аЛ*/3 has compact support. Then we have
f g(a,
where g(a, 5/3) E C°°(M) is defined by g(a, 8j8)(jc)= gx(ax, 8/Sx).
Proof. Note that g(a, 8/S)tx = a A*8/3 = ~(-l)s(-l)n(k + 2)a
/\**d*fi= (using 0.1.9) -(-l)g(-iykaA(-l)g(-lY"-k)kd*IS =
-(-l)kaAd*IS. Thus, g(da,P)ix- g(a,8/3)ii = daA*p+(-l)kaA
d*[5 = d(aA*f5). Integrating this relation over M and applying
Stokes' theorem @.2.14), we obtain the result. ■
0.2.19 Theorem. Let M=Un with the usual coordinates x\...,xn
and volume element jj. = dxi A ■ ■ ■ Adx", with metric g such that
g,J = ±8IJ.Let
У R Л1' Л
where Bt , =^8C,,...,3, ). Then, we have
W=(-l)k+1jjlPil...,;,ldxl>A..-Adx^ A)
where we use the notation f t■ = ЗД/].

0 PRELIMINARIES 15
Proof. Let S;;;;;;* = (<&'■ Л • • • лЛс'*)(Э,.,..., dJk). For a =
,|...;4й?х'|Л ••• Лй?х''*ЕЛ*Aй"), we have
dald. ,...,3, ) = ут2а, ,, dxik+1 Adx'1 A - • ■
C,,...,3, )
V 7i' ' 7*+!-'
= —а ( — 11*8''•■■'■*+!
Thus, if a has compact support, we have
f g(a,Sp)n= f g(da,
2 «y,.. .л.л+, вл-.-.-^,^'1 ■ ■ • '*
k\ {k+\)\
where S/3 is the right-hand side of A). Thus, jR.g(a, 8^-Щц=0 for
all compactly supported аЕЛ*AЙ"). It follows that 8)8=8)8. ■
0.2.20 Definition. A form шбЛ'(М) is closed if Jw = 0, and is
exact if w=da for some aEAk~\M). Since d2 =0, every exact form
is closed. Let Zk(M) and Bk(M) be the vector spaces of closed and
exact k- forms, respectively. Then the quotient Hk(M) =
Zk(M)/Bk(M) is called the kth de Rham cohomology space of M. If
aEZk(M), then [a]EHk(M) denotes the class determined by a. If

16 GAUGE THEORY AND VARIATIONAL PRINCIPLES
[a]EH'(M) and [fi]EHJ(M), then we define a multiplication [a] A
[/?] E H'+ '(M) by [а]Л[/?] = [аЛ /3]. We must check that d(aAfi) =
0; and if [a\ = [a'] and [)8] = [j8'], we need [аЛ/?] = [а'Л/?'].
However, d(aA fi)=daA p+(-l)'aAdf}=0, and (for a'- a = da
and P'-/3 = db) we have a'A ft'-aA fi=(a+da)A(fi +db)~
aA/3 = daAl3 + aAdb + da A db = d(a A /8) + (- 1)'d(a A b) +
d(aAdb) = d(aAfi+(-\)'aAb + aAdb) whence [а'Л/?'] = [аЛ
P]. Note that [P]A[a] = (— l)tJ[a]A[fi]and the multiplication extends
(by linearity) to the entire space H*(M) — @k Hk(M), making H*(M)
an algebra.
0.2.21 Definition. A k-dimensional submanifold of an n-manifold M
is a k-manifold N that is a subset of M such that V is an open subset of
N iff V=Nr\V for some open subset V of M. Furthermore, it is
required that the inclusion i: N-+M is a map such that i^x: TXN'-»TXM
is one-to-one for all x EN. Thus, TXN can be identified with a subspace
of TXM.
0.2.22 Maxwell's Equations and Differential Forms. Let М-Шл
with coordinates (x°, x\ x2, x3) — (t, x, y, z) and metric g such that
g(d0,30)= 1, gC,., 3,.)= - 1 for i= 1,2,3, and g(8,, bj) = O for i*j (i.e.,
(M, g) isMinkowskispace). Consider the 2-form F—E]dxAdt+E2dy
Adt + E3dzAdt + Bx dyAdz + B2 dzAdx + B3 dx A dy. For dr =
(dx, dy, dz) and da — (dyAdz,dzAdx,dxAdy), we employ the
shorthandF=E'drAdt + B'da. By a simple computation, we obtain
VX£+y •daAdt+(V'B)dT
where dr=dxAdyAdz. Thus, dF=0 iff V XE + dB/dt = 0 and V 'B
= 0, which are two of Maxwell's four equations (where E is the electric
field and В is the magnetic field). Now, *F—E'da—B'drAdt, and so
d*F= (v -E) dr- Iv-B- -^
-duAdt.
Now 8=-(-l)g(-l)*ik+])*d* = *d* on A*(IR4). Thus, 8F=*d*F=
(VE)dt-(VXB-dE/dt)-dr. Let the maps p: R4-^IR andJ: R4->
R3 be the charge density and the current density, respectively. The
source 1-form jEA\R4) is defined by j= pdt—J'dr. Then 8F—j is

0 PRELIMINARIES 17
equivalent to the other two (inhomogeneous) Maxwell equations, V 'E
= p and \?XB — dE/dt=J. Thus, the four Maxwell equations are
summarized by dF—O and 8F=j. Applying 8 to 8F=j, we obtain
0 = 82F=8j=*d*j=*d(PdT-J-daAdt)
= * f -£-dtAdr- V 'JdrAdt) = - (^r + V •/) .
\ ot ■ I \ dt I
Thus, we obtain the so-called "continuity equation"
which says that charge is conserved.
0.2.23 Vector-Valued Forms. Let V be a vector space with a basis
u,,..., i5m. // a1,.. ., am E ЛА(М), then the function x н> a\vx
+ ■■■ +a™vm &Ak(TxM, V) is called a V-valued k-form, the space of
V-valued k-forms is denoted by Ak(M, V). For a = a1u1 + • • ■ +amvm
<EAk(M,V), we define da = {dax)vx + ■ ■ ■ +{dam)vm EAk(M, V),
and {in the case in which M is oriented and has a metric g) *a =
{*av)vx+ ■■■ +(*am)vm and 8 = (-l)g(-l)"(k+l)*d*. Let h be a
metric on V, and suppose that a, fiEAk(M,V). Then (gh)(a, /S)E
C°°(M) is defined by (gh)(a,P)(x) = (gxh)(ax,/3x) (see 0.1.5). //
a = 2aava and P = l£bvb and hah = h(va,vh), then (gh)(a,/3) =
1habg(a",fib). We have the following analogue of 0.2.18.
0.2.24 Theorem. Let M be as in 0.2.18, and let V be a vector
space with metric h. Suppose that aEAk(M, V) and @EAk + \M, V)
are such that suppBhaha" A*fib) is compact. Then, we have
f
Proof. For a = Iaava and P = lfibvb, we have fM(gh)(a,8f})ii
^hf8^2fHhfd^ Ш
0.2.25 Definition. Let M be an n-manifold and let N be an m-
manifold. We define the product manifold MXN of dimension m + n as
follows. Let {(p,: U,-+Rn} be an atlas for M and let {^: F; ->Rm} be
an atlas for N. Define <p,X^.: U,X V}->R"+m by (

18 GAUGE THEORY AND VARIATIONAL PRINCIPLES
^XlRm=IR"+m. Then {(р,Х-ф;: (^ X F;-^R"+m} will
be an atlas (for MXN), making MXN an (n + m)-manifold.
0.2.26 Definition. Let fECx(M) and suppose that df = 0 at some
xEM. then x is called a critical point of f. The Hessian of f
at such an x is a symmetric bilinear function (D2f)x: TXM XTXM~->
R defined by (D2f)x(Yx, Zx)=Yx[Z[f]\, where ZET(TM) is such
that ZX = ZX. Suppose YET(TM) with YX = YX. Then Zx[Y[f]] =
[Z,Y]x[f]+Yx[Z[f]] = Yx[Z[f]], since df = 0 at x. Thus, (D2f)x is
not only independent of the extension Z, but also symmetric.
0.3 LIE GROUPS AND LIE ALGEBRAS
0.3.1 Definition. Let G be an n-manifold and a group such that the
groups operation GXG-+G given by (g\, g2)^>gxg2 and the function
G-> G given by gh->g~] are (C°°) maps. Then G is called a Lie group.
0.3.2 Definition. Let Lg: G-^ G be defined by Lg(g')=gg'; Lg is a
diffeomorphism. Let e be the identity element of G, and let A ETeG.
Define A ET(TG) by Ag = Lgif(A)\ A is called the left-invariant vector
field determined by A.
0.3.3 Definition. Let §=TeG, and (for A,BE§) define [A,B]E§
by [A,B] = [A,B]e (see 0.2.4). Note that [A,B]=-[B,A] and
[A,[B,C]] + [C,[A,B]] + [B,[C,A]] = 0 (the Jacobi identity). Then g
(together with the bracket operation [ , ]) is called the Lie algebra
ofG.
0.3.4 Definition. For AE§, we can prove that A is a complete
vector field (see 0.2.7). Let {<pt} be the one-parameter group of diffeo-
morphisms generated by AE§. Let y: R->G be the curve through e
defined by y(t) — (pt(e). We prove that y(s+ t)=y(s)y(t) (group multi-
multiplication). Let sEU. be fixed and let yx(t) = y(s+ t), while y2(t) =
y(s)y(t). Then у1A) = у'^ + 1)=Ау{5 + 1) and y£t)= Ly{s)*(y'(t))
= Ly(s)*(AyU))=Ly(s)*(Ly(,)*A)=Ay(s)y(,v Th^s, Yi and y2 are in-
integral curves of the same vector field A, and (since y]@) = y(s) = y2@))
it follows thatyx(t)=y2(t) (i.e., y(s+ t) = y(s)y(t)). Thus, y: R-^G
is a homomorphism. Conversely, given a curve and homomorphism a:

0 PRELIMINARIES 19
U-^G, then \pt: G-^G (defined by \pt(g) — ga(t)) is a one-parameter
group of diffeomorphisms of G such that
defines the left-invariant vector field В determined by B=Be. Thus,
there is a one-to-one correspondence A<-*y. We define the exponential
map exp: §->G by exp(A) = y(l). Note that y(t) = exp(tA), and <pt(g)
0.3.5 Example. Let V be a vector space with dim V—m<oo, and let
GL(V) be the group of invertible linear functions F: V-*V. By
regarding GL(V) as a group of matrices, it is simple to see
that GL(V) (an open subset ofW) is a Lie group. Let IEGL(V) be
the identity, and denote T,(GL(V)) by §i(V). Note that §l(V) can be
identified with the vector space of all linear functions A: F-> V, the
correspondence being
l=0
ForAE§t(V), let
Exp(A)=I+A + ^A2 + ^A3 +
It is not hard to prove that the sum converges, and that Exp((t + s)A)
— ExptA Exp sA. Thus, Exp(A) Exp(—A)=I and so Exp(A)E
GL(V). Note that tn>Exp(tA) is a curve and a homomorphism with
ftExp(tA)
=A.
= 0
It follows from the discussion in 0.3.4 that Exp is the exponential map
for GL(V). In 0.3.10, we will prove that (for A,BE§t(V)) [A,B] =
AB-BA.
0.3.6 Definition. A Lie subgroup of a Lie group G is a submanifold
(of G) that is also a subgroup of G. A Lie subgroup H of G is itself a
Lie group. Since the homomorphisms y: f£ —*// are also homomor-

20 GAUGE THEORY AND VARIATIONAL PRINCIPLES
phisms into G, we have that exp: %-^H is just exp: §-> G restricted to
%. The next theorem implies that [ , ] on % is just [ , ] on § restricted to
0.3.7 Theorem. Let G and G' be Lie groups, and let F: G-> G' be a
C°° homomorphism. Then F^e: §->§' is a linear function such that
F^e([A, В]) = [Р^еА, F^eB] (i.e., Feif is a homomorphism of Lie alge-
algebras).
Proof. Note that F<> Lg(g') = F(gg') = Fjg)F(g') = (LF(s) о F)
(g'). Thus, F^Ag)=F^Lg^)=LF(g)t,AF^A) = (F^A)ng), and
so ^(Л) = (^еЛ). Using 0.2.5, we obtain F^([A,B]) = [
0.3.8 Definition. For gEG, let Adg: G->G be the C00 adjoint
isomorphism given by Adg(g')= gg'g~ '■ We let &bg: § -> % be the
g g
induced isomorphism of' § provided by 0.3.7 (i.e., $bg = &bgife). Let
gife)
&b: G-> GL(§) be the homomorphism gn> &bg. Then 0.3.7 gives us an
induced homomorphism ab: §-> §t(§) (i.e., ab = &b ).
0.3.9 Theorem. For A, В Е §, we have
Proof. Let {(pr} be the one-parameter group generated by A. By the
end of 0.3.4, wehave(p,(g) = gexpL4. Using LjB = [A, B] (see 0.2.7),
we have (at s = t — O)
di
——
= ~&b(cxp(
= ab(A)(B).

0 PRELIMINARIES 21
О.З.Ю Corollary. // G is any Lie subgroup of GL(V), then the
bracket operation on §C§i(V) is given by [A,B]=AB — BA.
Proof. By 0.3.6, it suffices to consider the case in which G = GL( V).
Using 0.3.9 with exp = Exp (see 0.3.5), we have
=AB-BA.
s,t = O
0.3.11 Definition. Let <?,,...,<?„ be a basis for the Lie algebra § of
G. The structure constants c,*- 6R are defined by [е„ eJ] — '2cfek. Note
that [ej,et]= — [ene}\ implies с^ = —с^. The Jacobi identity yields
2 l
p kTJ
>;mcZ)eh. Thus, 2mc*m<#
#
0.3.12 SU(n), the Special Unitary Group. The computation of the
Lie algebra of a Lie group of matrices is illustrated here for the group
SU(n), which is frequently used in elementary particle physics. Let
Ш(п,С) be the space of all nXn matrices with complex entries. For
A E§t(n,C), let A* denote the conjugate of the transpose of A. Recall
that the unitary group is U{n)-{A &§t(n,C)\AA*=I) and SU(n) =
{A E U(n)\detA-l}. If t^A(t) is a curve in U(n) with A@) = I, then
(at t = 0) we have
=A'@)A@)*+A@)A'@)*=A'@)+A'@)*.
Thus, for S= {Be§t(n,C)\B + B* = 0}, we have ^>D%(n) = the Lie
algebra of U(n). Conversely, if 5E§, then (ExpB)(ExpB)* =
(ExpB)(ExpB*) = Exp(B)Exp(~B)=I, and so ExpBEU(n). At
t = 0,
B= J-
whence %(«) = §. The Lie algebra S%(«) of SU(n) is the subalgebra
<?/%(«) consisting of matrices with trace 0 (i.e., §%(«)= {ВЕ%(п)\

22 GAUGE THEORY AND VARIATIONAL PRINCIPLES
tvB = O}). This follows immediately from the formula det(Exp B) = e'rb,
which is valid for any nXn matrix. We can prove this formula as
follows. Let f(t) = det(Exp tB). At h = 0, we have
= det(ExptB)-^det(l+hB)
= det(ExptB)tvB=(tvB)f(t).
Thus, f(t)=f(O)e{lrB)' = ei"'B)', and setting t- 1 yields the result.

CHAPTER
1
Principal Fiber Bundles and Connections
In the introductions to this and the following chapters, the topics
and results to be covered will be outlined, and some motivation will
be supplied to whet the grindstone, but no miracles are promised.
You may rest assured that you need not comprehend or agree with
the introductions in order to understand and accept the proper parts
of the chapters.
In this chapter, principal fiber bundles (PFBs) will be defined
and some nontrivial examples will be given (i.e., the double covering
of the circle and the frame bundle of a manifold). Three ways of
defining connections (ie., gauge potentials) will be proved to be
equivalent. The connection of a PFB with group U(l) over space-time
will be physically identified as the four-dimensional vector potential
of electromagnetism.
David D. Bleecker, Gauge Theory and Variational Principles ISBN 0-201-10096-7
Copyright © 1981 by Addison-Wesley Publishing Company, Inc., Advanced Book
Program/World Science Division. All rights reserved. No part of this publication
may be reproduced, stored in a retrieval system, or transmitted, in any form or by
any means, electronic, mechanical, photocopying, recording, or otherwise, without
the prior permission of the publisher.
23

24 GAUGE THEORY AND VARIATIONAL PRINCIPLES
In order to motivate the introduction of PFBs into physics, the
following considerations are offered in addition to the Prologue. In
the Prologue it was stated that a PFB could be regarded as a smooth
concatenation of reference frames. Since all measurements are made
relative to a choice of frame, and the measurement process can never
be completely divorced from the aspect of the universe being mea-
measured, we are led to the conclusion that the bundle of reference
frames should play a part in the very structure of the universe as we
perceive it.
In a different'vein, a PFB with a gauge potential provides a
natural way of geometrizing the forces of that potential. Indeed, we
can extend the following lesson from general relativity to show this.
Recall that two geodesies (straightest possible paths) in a possibly
curved space coincide if they are tangent at a point of contact. The
paths of massive objects in the three-dimensional space in which we
live can be tangent at a point without coinciding. In Figure 2, for
example, bullet 1 might escape the pull of the earth's gravity, while
bullet 2 (shot in the same direction, but with less speed) has fallen
back. We conclude that massive objects do not travel along geodesies
of some three-space geometry. In four-dimensional space-time, the
two bullets have paths that are not initially tangent, since the speeds
of the two bullets differ. Thus, the possibility that massive objects do
trace out geodesies in space-time (instead of in space alone) cannot
be ruled out by the example. Indeed, general relativity tells us that
the paths of objects that are subject to only the force of gravity are
geodesies relative to some geometry imposed on space-time. In other
words, adjoining the time dimension permits the geometrization of
the force of gravity.
Suppose that the earth carries a large positive electric charge,
bullet 1 has a negative charge, and bullet 2 is neutral. Even if the
paths of the bullets are initially tangent in space-time (i.e., shot in the
same direction with the same speed), the paths will not be the same.
Thus, the geometrization of electrical forces requires another dimen-
dimension, the "charge dimension." The resulting five-dimensional space
(the invention of Kaluza [1921] and Klein [1926]) is actually a PFB
with group [/A), the circle. A connection (or gauge potential) en-
endows the five-space with a geometry and allows us to speak of the
charge component of a geodesic in five-space. If the charge compo-
component of a certain geodesic is q, then the projection of this geodesic

1 PRINCIPAL FIBER BUNDLES AND CONNECTIONS 25
Figure 2. Here we have reduced space by one dimension. In deference to the
Flat Earth Society, the earth's surface at t = 0 is the x axis and the interior of the
earth traces out the half-space y<0. Curves 1 and 2 are the paths (or world
lines) of bullets 1 and 2 in space-time. Paths Г and 2' are the spatial projections
of paths 1 and 2 into the x-y plane. Paths Г and 2' cannot be geodesies relative
to some geometry imposed on the x-y plane, since they are tangent at @,0) and
yet are distinct. However, paths 1 and 2 are geodesies in a curved space-time.
They are still distinct, but they are not tangent at @,0,0) in space-time, since
bullet 2 was shot with less speed.
onto space-time is the nongeodesic path of an object of charge q
subject to the force of the gauge potential. The invariance of the
gauge potential under the U{\) action on the PFB implies the
conservation of charge. We will not verify these claims explicitly
until Section 10.1 (which can be read after studying Chapters 1 and 2
and Section 6.2). The purpose of bringing them up now is to
motivate the adjunction of extra dimensions to space-time (i.e., to

26 GAUGE THEORY AND VARIATIONAL PRINCIPLES
geometrize nongravitational forces via connections on PFBs in about
the same way that the adjunction of time to space permitted the
geometrization of the force of gravity). There are many other types of
charge that respond to various other kinds of forces (e.g., isospin;
hypercharge; red, blue, and green charge; weak charge). The associ-
associated geometrization of these forces requires PFBs with larger non-
Abelian groups, and so five dimensions are not nearly enough.
1.1 PRINCIPAL FIBER BUNDLES
1.1.1 Definition. A principal fiber bundle (PFB) consists of a mani-
manifold P {called the total space), a Lie group G, a base manifold M, and
a projection map it: P->M such that (A), (B), and (C) following hold.
(A) For each gEG there is a diffeomorphism Rg: P->P (we write
Rg(P)=Pg) s^ch that p(gxg2) = {pg\)g2 for al1 &\>&г eG and
pGP; and if eEG is the identity element, then pe=p for all
pEP. We require the function PXG->P given by (p, g)\-> pg to
be a map. We suppose that if pg—p for some p E.P and gEG,
then g — e. We summarize this paragraph by saying that G acts
freely (and differentiably) on P to the right.
(B) The map it: P->M is onto, andir^x(ir(p)) = {pg: gEG} (which
is, by definition, the orbit of G throughp). If x&M, then тт~\х)
is called the fiber above x. Note that for each p Елт~\х) there is
a map G-^ir^\x) given by gv+pg. This map is a diffeomor-
diffeomorphism by (A), but it depends on p. Thus, all the fibers it ~ \x) are
diffeomorphic to G, but there is no canonical identification of
тт~\х) with G, and hence no natural group structure on тт~\х).
(C) For each xEM there is an open set U with xEU and a
diffeomorphism Tu: тг "'([/)-> UXG of the form Tu(p) = (ir(p),
su(P)) where su- ir~\U)-^G has the property su(pg)=su(p)g
for all g£G,pEtt~~](U). The map Tu is called a local trivializa-
tion (LT), or (in physics language) a choice of gauge.
1.1.2 Remark. We abbreviate the foregoing by saying that it: P-^M
is a PFB with group G. If N is a manifold and G is a Lie group, we can
form a PFB it': NXG->N with group G by setting iT'(n,g) — n and
(n, g)g' = (n, gg'). This is called the product PFB of N with G. Note
that (C) states that for every xEM there is a neighborhood Uofx such

1 PRINCIPAL FIBER BUNDLES AND CONNECTIONS 27
that the restricted PFB it: tt~\U)-> U can be identified (via Tu) with
the product PFB of U with G. The requirement on su is necessary in
order that Tu respect the action of G (i.e., Tu(pg) = Tu(p)g).
1.1.3 Definition. Let Tu: tt~\U)->U XG and Tv: ir~\V)-+V XG
be two LTs of a PFB it: P -» M with group G. The transition function
from Tu to Tv is the map guv: UnV->G defined, for x = ir(p)EUr\V,
by guv(x) = su(P)sv(P)~]- Note that Suv(x) is independent of the
choice ofpEir-\x) because su(pg)sv(pg)~1 = su(p)g(sv(p)gyl =
su(P)gg ^v(p)~]=su(p)sv(p)'\ We have
@ g
(ii) g
(Ш) guv(y)gvw(y)gwu(y) = e № all y
The transition functions describe how the various products UX
G,VXG,... glue together to form the total space P. Indeed P may
be considered as the space obtained from the disjoint union (UXG)
U(FXG)U • • • by identifying the point (x, g)EUXG with (x, g')
EVXG if g—guv(x)g'. Because of (i), (ii), and (iii), this identifica-
identification is an equivalence relation. Thus, we see that a PFB can be
essentially recaptured by its transition functions.
1.1.4 Definition. We define a local section of a PFB it: P^M with
group G to be a map a: U^>P(UdM, Uopen) such that "n°o= \u =the
identity function on U(x\->x).
1.1.5 Theorem. There is a natural correspondence between local
sections and local trivializations.
Proof. If a: U^P is a local section, then define Tu: -u~\U)^UXG
by Tu(a(x)g) = (x,g). Conversely, given an LT Tu: it-\U)^UXG,
define a local section a: U-^P by o(x) = Tu~\x, e). ■
1.1.6 Remark. // Tu is a local trivialization with U—M (i.e., TM:
P-+MXG), then TM is called a global trivialization, and the PFB is
called trivial if such a TM exists. A local section a: U^P is called a
global section if U=M. In 1.1.5, global sections correspond to global
trivializations.

28 GAUGE THEORY AND VARIATIONAL PRINCIPLES
1.1.7 Example. Let 5'={zEC| |z| = l) be the unit circle in the
complex plane. Let it:S]^S] be given by тт(г)= z2; so it~\z2) =
{z,~ z). Let G = {e, g} = the two-element group, and let ze = z while
zg = - z for zES\ Set 17=5'-{1} and V=Sl-{~l), and let a:
U -» S1 be the local section defined by taking a( w) to be the square root
of w with Im(a(w))>0 while т. F-» S] is defined by taking t(w) to be
the square root of w with Re(T(w))>0. By 1.1.5, we have local
\ de-
detrivializations
termined by
\e
Thus, for w = z
т ■
if
if
2
ir-\U)
Im(
Im(
-*ux
>o,
<0;
G
y(z) = |*
77-1
if
if
Re(
Re(
F
XG
>o,
<0.
вт(»)=ФК(*) =[g if Im(w)<0.
Note that this PFB is not trivial, since the total space is S1 instead of
S1 X G, which is the disjoint union of two copies of Sx. Consequently,
there is no section defined on all of Sx; that is, there is no way to define
a continuous square root function on all of S1.
1.1.8 Example. Let M be an n-manifold. We will define a PFB it:
L(M)^>M with group GL(n,R), called the frame bundle of M. A
frame at xEM is a linear isomorphism u: U" -* TXM. Note that such a
frame determines a basis u{ex),..., u(en) of TXM where ex,...,en is the
usual basis ofR". Let L(M)X be the set of all frames at x, and set
L{M)= U L(M)x.
x<EM
For uEL(M)x,T7(u) = x. For AEGL(n,R), we define RA: L(M)-*
L(M) by RA(u) = u°A. This is a free, right action of GL(n,R) on
L(M), but we need to put a differentiable structure on L(M) before we
can speak of it: L{M)-*M or L(M)XGL(n,R)-*L(M) as being C°°.
Let W<ZM be a coordinate neighborhood with coordinates x\...,x"
and associated coordinate fields 3,,...,3n on W. Define a map a:
W-^L(M) by letting a(y): R"-*T M be the isomorphism such that

1 PRINCIPAL FIBER BUNDLES AND CONNECTIONS 29
a(y)(el) = (di)v. Let sw: ir"\W)^GL(n,R) be defined by sw(u) =
о(тт(и)) ]°u. Note that sw(u°A)=sw(u)° A. Define Tw: ir~\W)-^
WXGL(n,R) by Tw(u) = (tt(u),s^u)). Since WXGL(n,R) has a
differentiable structure, the maps Tw for various W<ZM define a
differentiable structure on L(M) provided that Tw,oT^]: (Wn W')X
GL(n,RH is C°° where W is another coordinate neighborhood {on
M) with coordinates x'\...,x'" and fields 3;,...,3^. From the equa-
equations Tw(u) = (tt(u), sw(u)) and Tw,(u) = (tt(u),sw,(u)), we obtain (for
y = iT(u)eWn W) Tw, о T~\y, A) = (y, sw,(u)sw(u)~ lA) =
(y,gW'W(y)A)- Hence, it suffices to prove gw,w: Wn W'-*GL(n,R) to
be C°°. For y = ir(u)^WDW', gw.w(y) = sw.(u)sw(u)~l =
(a'(y) ' о u)(o(y)~l о и) ' =о'(уухо(у), but this is just theJacobian
matrix with entires ЭД x"'] that are C°° on Wn W. The maps Tw, Tw,,...
are now diffeomorphisms, and hence are LTs. The maps it: L(M)-*M
andL(M)XGL(n,R)-*L(M) locally correspond (via the LTs) to the
C°° maps WXGL(n,R)-*W and (WXGL(n,R))XGL(n,R)-^WX
GL(n,R), and hence are C°°. By 1.1.6, the PFB it: L(M)-*M is
trivial iff there is a section M-*L(M), or in other words, a sequence of
vector fields Xx,...,Xn on M that are independent at each point. This
condition cannot be met on any compact surface, except for a torus; we
omit the proof (see Spivak [1971], Vol. 1).
1.2 CONNECTIONS
Connections can be defined in at least three ways. We will prove
that Definitions 1.2.1, 1.2.2, and 1.2.3 are equivalent. In the follow-
following we assume that it: P-*M is a PFB with group G; dim Л/=и.
1.2.1 Definition. A connection assigns to each pEP a subspace
HpCTpP such that for Ур={ХеТрР\тт^(Х)=О} we have TpP=Hp@
Vp. We require that Rgij!(Hp)—Hpg. Moreover, we assume that Hp
depends smoothly on p, in the sense that there are n vector fields
(defined on a neighborhood Uof p) that span Hq at each qEU. We call
Vp the vertical subspace of TpP, while Hp is the horizontal subspace
(see Figure 3).
1.2.2 Definition. Let § be the Lie algebra of G. A connection is a
§-valued \-form w defined on P such that the properties (a) and (b)
hold.

30 GAUGE THEORY AND VARIATIONAL PRINCIPLES
Vr
Figure 3
(a) Let AE§ and let A* be the vector field on P defined by
d
Then ьо(Л*)= A. A* is called a fundamental field.
(b) For gEG, let &bg: g-^g be defined as in 0.3.8. We require
"„*(**•*)= &bg-lUp(X) for all gEG, pEP, and XE TpP. In
other words, /?*w = &b -iw.
We call w a connection 1- form.
1.2.3 Definition. A connection assigns to each LT Tu: ir
X G {i.e., choice of gauge) a %-valued I-form wu on U. If Tv is another
LT and guv: U DV-^G is the transition function from Tu to Tv, then we
require iov(Yx)= Ll\x)Jf{guvJf{Yx)) + <$,Ъgm{x)-i{aJJx)) for all YXE
TXM and xEUГ) V. If G is a group of matrices, we can rewrite this
condition as follows. In matrix notation and with у a curve with

1 PRINCIPAL FIBER BUNDLES AND CONNECTIONS 31
= Yx, we have (at t =0)
— о (х) ' di (Y\
where dguv is d of the matrix-valued function guv (see 0.2.23, where V is
a vector space of matrices). For a group of matrices we have
d_
dt
dt
(see 0.3.8). Thus, 0bgMj(JC)-.o>u(Yx) = g;v\x)wu(Yx)guv(x). Conse-
quently, the transformation rule from ыи to u>v can be expressed as
1.2.4 Theorem. Definitions 1.2.1 and 1.2.2 are equivalent.
Proof. Suppose w is the connection 1-form of 1.2.2. Let Л/) =
TpP\icp(X) = 0}. We prove thatp^ Hp is a connection in the sense of
1.2.1. From 1.2.2(a) it follows that Hp®Vp = TpP. Also Rg^(Hp) =
Hpg, because (from (b) of 1.2.2) u(R X) = &bg-iu(X) = 0 for Xe
Hp. Conversely, suppose that p н> Нp is a connection in the sense of
1.2.1. For A* as in (a) of 1.2.2 and Xp EHp, define wp: TpP-^§ by
up(A* +Xp) = A. Condition (a) of 1.2.2 then holds. For (b) of 1.2.2,
we need to prove that upg(Rg^Y) = &bg-l(up(Y)) for all Г£Г/. If
YEHp, then Rgij!YeHpg, and so both sides vanish. If Y=A* for
some A Eg, then (at f = 0)

32 GAUGE THEORY AND VARIATIONAL PRINCIPLES
By linearity, we then have (b) of 1.2.2, and so w is a connection
1-form. ■
1.2.5 Theorem. Definitions 1.2.2 and 1.2.3 are equivalent.
Proof. Let w be a connection 1-form as in 1.2.2. If Tu: tt~\U)^>U
X G is an LT with associated local section au: U-^P given by 1.1.5,
then set wu=a*w. We prove that the assignment Tu^>u>u is then a
connection as in 1.2.3. Let Tv be another LT with local section av.
We need to check that the transformation equation of 1.2.3 holds
with uu = a*u and wv=a*w. Writing Tu(p) = (ir(p), su(p)), we
see that (for x = ir(p)eU) Tu(ou(x)su(p)) = (x,su(ou(x)su(p))) =
(x, su(ou(x))su(p)) = (x, esu(p)) = (x, su(p))=Tu(p). Thus, p =
ou(x)su(p), and similarly, p = ov(x)sv(p). Consequently, we have
av(x) = a«(x)su(p)sv(p)-l=ou(x)guv(x). Let YETXM, and suppose
that y: R -^M is a curve with y'@) = Y. Then (at f = 0)
dt °
— г -1
Now
by inserting the expression for ov^(Y) and applying (a) and (b) of
1.2.2. Thus, the assignment Ги^ьоц=а* is a connection as in 1.2.3.
Conversely, suppose that Tu\^>iou is a connection as in 1.2.3. Let
au: U^>P be the local section associated with Tu. Forp = a(x), xEU,
YETXM, and Л Eg, we define w": TpP^§ by tou{o^
p
+A. We extend w" to all of 77~\U) via the formula (for Xpg E TpgP)
tc"(Xpg) = &bg^u"(Rg-uXpg). It is left to the reader to verify that
w" is a connection 1-form on the restricted PFB 77: tt~\U)^>U. If Tv

1 PRINCIPAL FIBER BUNDLES AND CONNECTIONS 33
is another LT, then we can define w" on tt~\V) similarly. Once we
prove that u>" = wv on ir~\UnV), then the various w", w°,... piece
together to define a connection 1-form w as in 1.2.2. If w" and w"
agree on the set ov(UC\ V), then they must agree on all ofiT~l(Uf) V)
by 1.2.2(b). Now iou(A*)=A = icv(A*), and so we need only check
that o?(ovj) = tou(ovj) for YETxM,xEUnV. But wv(a^Y) =
uo(Y), while
which is woG), because of the transformation rule of 1.2.3. Thus
wu,wv,..- do piece together to define a connection w as in 1.2.2;
moreover, we see that icu = o*u>, tov = ao*w, etc. ■
1.2.6 Remark. Any PFB admits a connection. You can find a proof
of this fact in Kobayashi and Nomizu [1963].
1.2.7 Physical Interpretation. Physicists refer to the l-forms wu (in
1.2.3) as gauge potentials. In Chapter 2, we discuss (see 2.2.10) how the
field strength is computed from wu. For now, we consider the special
case of electromagnetism. Suppose that (M, g) is Minkowski space (as
in 0.2.22), and let it: P^M be a PFB with group U(l)= {eie\e<=R}
whose Lie algebra is %A) = {ia\ aEIR}. Suppose that w is a connection
1 -form on P, and let оu: U^> P be a local section. Thenwu=a*u>=—iAu,
where Au EA'((/,IR) is called the potential 1-form (or "vector poten-
potential"). The electromagnetic field strength relative to a: U-^P is then
Fu = —dAu EA2((/,IR). // av: V^P is another local section, then we
can prove that FV=FU. Indeed, according to 1.2.3, ">v=g~Jdguv +
8uvUuguv=guvi dguv + itu since U(\) is Abelian (i.e., commutative).
N™juvguv=l implies that d(g-v1)gUv+g^J dguv = 0 or d(g~v]) =
~SUJ dguvgj = ~guv2dguv. Thus, -d(gj dguv) = guv2 dguv/\dguv
= 0, and it follows that du>v =du>u or Fv =FU. Hence the Fu (for various
o: U->P) piece together to yield a well-defined FEA2(M,R), which is
interpreted as in 0.2.22. We will see that for nuclear forces, where the
group (typically SU(n),n>2) is non-Abelian, the field strength on
U<ZM is a nonlinear function of ыи, and it depends on the choice of
gauge ou: U-^P.

CHAPTER
Curvature and ^-Valued Differential Forms
The curvature of a connection corresponds to the physical notion
of the field strength of a gauge potential. While the local gauge
potentials transform in a fairly complicated way under a change of
gauge (see 1.2.3), the local field strengths obey a simpler transforma-
transformation rule (see 2.2.14). In the case of an Abelian group, the local field
strengths are invariant under a change of gauge. Thus, in the case of
electromagnetism, the local field strengths piece together to yield a
well-defined field strength on the base (see 1.2.7). For non-Abelian
groups the field strength is a well-defined ^-valued 2-form on the
total space P, but it does not give rise to a well-defined form on the
base. However, because of the simple transformation rule for local
field strengths, we can produce scalars from them that are gauge-
independent functions on the base. One such function is the self-
David D. Bleecker, Gauge Theory and Variational Principles ISBN 0-201-10096-7
Copyright © 1981 by Addison-Wesley Publishing Company, Inc., Advanced Book
Program/World Science Division. All rights reserved. No part of this publication
may be reproduced, stored in a retrieval system, or transmitted, in any form or by
any means, electronic, mechanical, photocopying, recording, or otherwise, without
the prior permission of the publisher.
34

2 CURVATURE AND ^-VALUED DIFFERENTIAL FORMS 35
Action, which is introduced in 5.2. For the electromagnetic field, the
KH2Hlll2
Both the gauge potential and the field strength are examples of
Lie algebra-valued forms. The algebra of all such forms is an
infinite-dimensional graded Lie algebra. Graded Lie algebras play an
essential role in physical theories of supersymmetry (i.e., Bose-Fermi
symmetry). Whether the graded Lie algebra of g-valued forms will
ever play such a role is uncertain. The reason for introducing this
algebra here is that calculations involving gauge potentials and field
strengths proceed with greater ease, and the use of structure con-
constants and indices can be avoided. The proof of the homogeneous
field equation (or Bianchi identity) is especially simplified (see 2.2.8).
The operation of exterior covariant differentiation of g-valued
differential forms on P, defined in 2.2.2, can easily be extended to
any vector-valued form on P. This is done in 3.1.3, but physicists
may not recognize this operation as that which replaces the ordinary
derivative via the principal of minimal coupling until we reach 3.1.5.
2.1 THE GRADED LIE ALGEBRA OF S-VALUED FORMS
Let TV be a manifold and § a Lie algebra. We denote the set of all
g-valued A:-forms on N by Ak(N, g).
2.1.1 Definition. Let (pEA'(N,§) and ^EA^iV, g). We define
where a ranges over the permutations of {1,2,..., i +j); (— l)a = ±\,
depending on whether a is even or odd; [, ] on the right-hand side is the
bracket of g; and Xx,..., Xi+J are arbitrary vector fields on N.
2.1.2 Component Formulation. Suppose that AE§ and <p is a
U-valued к-form on N; then we define <р®Л EAk(N, g) by (cp®
A)(Xl,...,Xk)=y(Xl,...,Xk)A for Xx,...,XkETyN. It is a simple
matter to check that {for another R- valued form \p on N and BE§)

36 GAUGE THEORY AND VARIATIONAL PRINCIPLES
,5]. Let Ex,...,Efbe a basis for g, with
structure constants c%p defined by [Ea, Ep] = '2cl^Ey. For q>E A'(N, g)
and \pE A'(N, g) there are unique R-valued forms y" and ^{a, /3 =
1,..., /) such that y = I(pa®Ea and ^ = 2^®£^. Then, we have
[]
а,/? ct.fi,у
2.1.3 Theorem. For (pEA!GV, g), ^eAJ(N, g)
we
(i) [^,«p]=-A)i7[«p,^];
B) (- iy*[[<p, ^], p] + (- l)^[[p, <p],^] + (- 1)>![[^ p], T] = 0.
In other words, the algebra of g-valued differential forms on TV is
a graded Lie algebra.
Proof. Relation A) follows (in the notation of 2.1.2) from у"
= (- l)'7^ Лф" and [Ea, Ep]= -[Ep, Ea]. For relation B), observe
that the left-hand side is
«,/S, у
which vanishes by the Jacobi identity for g. ■
2.1.4 Theorem. For <pEA'(N,§) and xpeAJ(N,§), we have
Proof. With the notation of 2.1.2, the result follows from d(ya Л

2 CURVATURE AND g-VALUED DIFFERENTIAL FORMS 37
2.2 CURVATURE
Given a connection 1-form w on a PFB тт: P^> М with group G,
we can write any XETpP as X=XV + XH where Xv is vertical (i.e.,
it (Xv) = 0) and XH is horizontal (i.e., w(XH) = 0).
2.2.1 Definition. If <pEAk(P,§), then we define q>H EAk(P, g) fey
2.2.2 Definition. The exterior covariant derivative of <p&Ak(P,§)
is Dw(p = (dup)H EAk+\P, g) where dip is the usual exterior derivative
of (p. Although the operator D" depends on w, it is customary to omit
the superscript w. Because we will consider functionals on the space of
connections and other situations where more than one connection is
involved, we will usually not observe this custom.
2.2.3 Definition. The curvature of the connection wEA'(P,g) is
Я" =D"wEA2(P, g). When u is regarded as a potential, Я" is called
the field strength of w.
The following structural equation will permit us to write an
expression for the field strength that looks more familiar to physi-
physicists.
2.2.4 Theorem (The Structural Equation). The curvature form is
given by fi"=dw + ^[w, w] (i.e., D"w=dw + ^[w, w]).
2.2.5 Lemma. Given a vector field X on M, there is a unique vector
field X on P such that u>(X) = 0 and чт^(Хр) = Х„1р) for all peP.
Necessarily, Rgij!X=Xfor all gEG. The field X is called the horizontal
lift of X.
Proof. The existence and uniqueness of X follow from the fact that
"■„,: H -*Tn, ^M is an isomorphism. We omit the proof that X is
smooth, but this is really clear from the smoothness of w. Observe
that TT*(RgJp) = (ir°Rg)il!(Xp) = iT^Xp) = X7T(p). Thus, RgJp =Xpg.

38 GAUGE THEORY AND VARIATIONAL PRINCIPLES
2.2.6 Lemma. IfA,BE§, then [A,B]* = [A*,B*] as vector fields
on P {see 1.2.2(a)).
Proof. Let (pt: P-*P be given by (p,(p)=pexp(tA). Then <pt is the
one-parameter group of diffeomorphisms generated by A*. Evaluat-
Evaluating all derivatives at zero, we have
dt ds
d d i л\ ( л\ ( л^~] d d
= — -rpexp{s&b tAB)
2.2.7 Lemma. If A Eg and X is a vector field on M, then [A*, X]-0
where X is the horizontal lift of X (see 2.2.5).
Proof. Defining yt as in the proof of 2.2.6, we have q>t~\X) = X by
2.2.5. Thus,
Proof of Theorem 2.2.4. Note that ^[w, w]G, Z) = \{[a{Y\ w(Z)]
-[w(Z), ш(У)]} = [ш(У), w(Z)]. Thus, we need to prove (for all
У, Z E Гр P) the equ ation
By linearity, we need only consider the following three cases.
Case 1 (У and Z horizontal). Equation (*) follows, since w(Y) =
u(Z) = 0, and УН = У and ZH=Z.
Case 2 (У and Z vertical). We may suppose that Y=A* and Z = B*
for Л,ЯЕ§. Then du(Y,Z)=A*[u(B*)]-B*[u(A*)]-
w([^*,5*]) = (since w(B*)=B = constant, etc.)= -w([A*,
В*\) = (Ъу2.2.6)-а([А,В]*)=-[А,В]=-[а(А*),а(В*)]
— — [u>(Y), w(Z)]. Hence, both sides of equation (*) vanish.

2 CURVATURE AND S-VALUED DIFFERENTIAL FORMS 39
Case 3 (Y vertical and Z horizontal). We may assume that Zp=X
where X is the horizontal lift of some vector field X on M,
and Y=A* for some A Eg. Now da{Y,Z)=A*[u(X)]-
Х[ы(А*)]-ы([А*,Х]) = 0, since ы(Х) = 0, и>(А*)=А =
constant, and 2.2.7 holds. Both sides of equation (*) then
vanish. ■
2.2.8 Theorem (Bianchi Identity or Homogeneous Field
Equation). If u> is a connection l-form on P with curvature Я", then
2ГЯ" =0. In fact, we have dQa=[Qa, w].
Proof. Observe that since w vanishes on horizontal vectors, D"u"
= 0 follows from dQa=[Qa,u]. Now dti" =d(du> + ^[w, «])
— d2w + \[du, w] — ^[w, dw] = (since u?2w = 0 and [w, с?ш]= — [die, w]
by 2.1.3 (l)) = [fi?u, u] = (since [[w, w], ш] = 0 by 2.1.3
2.2.9 Theorem. For allgEG, R*tt"=&bg-&u.
Proof. From the definition of [ , ] on ^-valued forms, it is evident
that [ , ] is preserved under pull-back (i.e., F*[y,\p] = [F*(p, F*\p]).
Thus, /?*fl"=/?*(Jw+i[w, u\) = d(R*gu+b[R*a, R*gu]) = d&bg-,tc
2.2.10 Local Expressions. Recall from the proof of 1.2.5 г/гаг ?/г<?
gauge potential icu of 1.2.3 и related to the connection l-form w o/1.2.2
fry ши =а*ш ЕЛ1 ([/,§). The field strength associated to wu w Яи =а*Я".
Of course, it is desirable to have a direct way of computing Яц in terms
2.2.11 Theorem. In terms of wu,Slu=dwu + ±[wu,wu].
Proof. We have Яи = а*(Я") = a*(du + i[«, ы]) = rf(o*«)
i
Since the groups used in physics are almost always groups of
matrices, it is convenient to have the following characterization
of [ , ].

40 GAUGE THEORY AND VARIATIONAL PRINCIPLES
2.2.12 Theorem. If N is a manifold and G is a matrix Lie group
(with matrix Lie algebra §), then for yEA'(N, <3) andipEhj(N, Q) we
have [<p, ф] = (рЛ-ф—(— 1)'Л//Л<р. Here <p and \p are regarded as
matrices of U-valued forms, and <рЛ^ is matrix multiplication where
the entries are multiplied via wedge.
Proof. Since [A,B]=AB-BA for A, В Eg, we have
1
X(p{xo0+V),...,xo0+J))
2.2.13 Corollary. // G is a matrix group, then
and пи = dcou + "„ Л сои.
Proof. Since coEA\P,§), we have JK(o] =
) Ao, etc. ■
The transformation rule for the local field strengths under a
change of gauge is relatively simple compared with the corresponding
rule (in 1.2.3) for the potentials.
2.2.14 Theorem. Let Tu and Tv be two LTs with transition function
guv: UHV^G. Then, on UnV,uv=&bg-v,Uu. In the case of a
matrix group, this becomes ttv =g~vl &uguv-
Proof. If ou and av are the local sections associated with Tu and Tv,
then (from the proof of 1.2.5) we have, for YETXM,

2 CURVATURE AND g-VALUED DIFFERENTIAL FORMS 41
y the definition of fi" in 2.2.3, we have that S2"(Z, Ж) = 0 if either
or И^ is vertical. Thus,
2.2.15 Theorem. Relative to an LT Tu: v~\U)^UXG, we have
the local form of the homogeneous field equation (or Bianchi identity)
</й„=[йц, wj. For a matrix group, we have duu = uu/\oou — o/u
Proof. This is evident from 2.2.8, because [ , ] is preserved under
pull-back. Also, [Йц,соц] = Й„ЛЫц-(-1J1ЫцЛЙц by 2.2.12. ■
2.2.16 Remark. At this point, one difference between Abelian and
non-Abelian gauge theory is manifest. For an Abelian group G, such as
U(\) in the theory of electromagnetism, we see that uv = gu~'^ugut) =fiM,
whence the local field strength is independent of the LT. Moreover,
since [fiM, wM] = 0, the homogeneous field equation is duu = 0 (or d2cou
= 0). However, for G non-Abelian, such as G = SUB) in classical
Yang-Mills theory, we have ^ =gj^'flugut) ^=ЯЦ in general. Instead,
we are moved to consider the field strength as being the well-defined
curvature form Й" that lives on P. Also, note that Й" =dco + j[co, со] is
no longer a linear function of со.
2.2.17 Remark. Let A be an arbitrary left-invariant vector field on a
Lie_group G (see 0.3.2). Define the %-valued one-form (p on G by
<p(A)=A =Ae ElTjG. Using the same computation as in Case 2 of the
proof of 2.2.4, we can prove that dy+ ^[у, <р] = 0. This is called the
Maurer-Cartan equation. The field strength fi" = dco+ j[co, со] mea-
measures the extent to which со fails to satisfy the Maurer-Cartan equation.
The assumption is sometimes made that a gauge potential be Maurer-
Cartan-like at infinity of U4. This just means that the field strength
tends to zero in some prescribed sense as we approach infinity.

CHAPTER
Particle Fields, Lagrangians,
and Gauge Invariance
A particle field can be regarded as a section of a vector bundle
associated to some PFB, or equivalently, as a vector-valued function
on P with certain transformation properties. Examples include the
Schrodinger wave function, the Klein-Gordon field, and the Dirac
electron field. Typically, a real-valued function (called the Action
density) on the base is assigned to each particle field according to
some fixed formula. The particle field obeys a differential equation
(Lagrange's equation) obtained by setting the first variation of the
integral of the Action density equal to zero. In other words, particle
fields obey the principle of least (or stationary) Action. Physicists use
the terms "Action density" and "Lagrangian" interchangeably for
the most part. However, we define (in 3.3.2) the Lagrangian as a
function (with certain invariance properties) on the finite-dimensional
David D. Bleecker, Gauge Theory and Variational Principles ISBN 0-201-10096-7
Copyright © 1981 by Addison-Wesley Publishing Company, Inc., Advanced Book
Program/World Science Division. All rights reserved. No part of this publication
may be reproduced, stored in a retrieval system, or transmitted, in any form or by
any means, electronic, mechanical, photocopying, recording, or otherwise, without
the prior permission of the publisher.
42

3 PARTICLE FIELDS, LAGRANGIANS, GAUGE INVARIANCE 43
space of 1-jets of vector-valued functions on P. The Action density
of a particle field is then defined by evaluating the Lagrangian on the
particle field and its differential (i.e., on its 1-jet).
Gauge transformations are defined in 3.2.1 as base-preserving
automorphisms of the PFB. Locally, they amount to a variable (i.e.,
space-time-dependent) change of "internal" reference frame or gauge.
There is little hope that the Action density is physically meaningful
unless it is invariant under gauge transformations of the particle
field. However, we will find (see 3.3.5) that this invariance is not
possible without introducing gauge potentials into the Action density
by replacing the ordinary differential of the particle field by its
covariant derivative. Physicists call this the principle of minimal
coupling (or minimal replacement). The gauge potentials (particu-
(particularly after quantization) can be interpreted as various forms of
radiation (photons, pions, gluons, intermediate vector bosons, etc.).
Thus, we can say that there can be no gauge-invariant Action
densities without "light."
3.1 PARTICLE FIELDS
Let it: P^M be a PFB with group G. Suppose that G acts on
some manifold F to the left. That is, for each gEG, there is a map
and the map GXF^F, given by (g,f)^ g-f, is C00. If F is a vector
space V and Lg: V^> V is linear, then the homomorphism G^GL( V)
given by g\-> Lg is called a representation of G. Two representations
G-*GL(V) and G^GL(V'), say g>->Lg and gh-> L'g, respectively, are
called equivalent if there is a linear isomorphism T: V-> V such that
L'=ToLoTl for all gin G.
5 5
3.1.1 Definition. // G acts on F as just stated, then we define
C(P, F) to be the space of all maps r. P^F such that T(pg)=g~l-
т(р). Readers who are familiar with associated fiber bundles will note
that C(P,F) is naturally isomorphic to the space of sections of the
associated bundle PXGF^>M with fiber F. Much of what follows can
be rephrased in terms of associated bundles, but we will not bother to do
so, since this point of view is foreign to most physicists and is also
notationally more difficult.
In the case where the action of G defines a representation
G^GL(V), the elements of C(P,V) are called particle fields.

44 GAUGE THEORY AND VARIATIONAL PRINCIPLES
3.1.2 Definition. Let Ak(P,V) be the space of V-valued differential
k-forms <p on P such that (for a given representation G-*GL(V))
(a) For Xx,...,XkETpP we have
<p(RgJfX],...,RgJfXk)=g-]-<p(Xu...,Xk)
(b) If one ofXl,...,Xk is vertical, then <p(Xv..., Xk) = 0.
Note that in the special case where V=§ and G-*GL(§) is the
adjoint representation gi-> &bg, we have (given a connection со on P)
Q" EA2(P,<3). Also observe that C(P,V) is the same as A°(P,V).
3.1.3 Definition. For a connection со on P we define Du: Ak(P,V)
-*~Ak + \P,V) by D"<p=(d<p)" (see 2.2.1). Note that R*Duq> =
R*g(dy)H = (R*gd<p)H_=(dR*<p)H = (dg~l-<p)H = g-l-(d<p)" = g~x-
Du(p, and so Du(pE Ak + l(P,V).
3.1.4 Definition. Let §-*§t(V) be the Lie algebra homomorphism
induced by the representation (/-> GL(V) (i.e., for AE% and vE V, let
A -v = — (exptA)-v
at
r=0
// <pEAk(P,V) and pEAJ~(P,<3), then we can define рЛсрЕ
AJ + k(P,V) by the formula
where a ranges over permutations of {1,..., j+k].
3.1.5 Theorem. For rEAk(P,V), we have D'V^
Proof. For vectors Xu..., Xk+, E TpP, we must verify that

PARTICLE FIELDS, LAGRANGIANS, GAUGE INVARIANCE 45
If X,,..., %k+i are aU horizontal, then X" = X,- and w(X,.) = 0, and so
both sides are the same. Suppose that two or more of X,,..., Xk+X
are vertical. Then, since т vanishes on vertical vectors and X,H =0 for
some i, equation (*) becomes 0 = u?t(X,,..., Хк+х). If we extend the
two or more vertical vectors to fundamental fields on P and extend
the other vectors arbitrarily, we obtain (see 0.2.10)
k+\
c/t(X],..., Xk+X) — 2j \~
+ У, (-\Y+J
The first sum vanishes, since at least one of Xx,..., Xl,...,Xk+x is
vertical, and (noting that if Xt and Xj are fundamental fields, then
[X,-, Xj] is also, by 2.2.6) the second sum vanishes for the same
reason. The remaining case is that for which all but one of Xx,..., Xk+x
are horizontal and the remaining vector, say Xx, is vertical. Extend Xx
to a fundamental field and X2,..., Xk+X to horizontal R^-invariant
fields by performing horizontal lifts on extensions of чт^Х2,...,
•п*Хк+х (see 2.2.5). From 2.2.7 we have 0 = [Xx, X2] = [XX, X3]
= ■ ■ ■ = [Xx, Xk+X]. It follows that dT(Xx,...,Xk+x) =
X\[t(X2,. .., Xk+X)].' Equation (*) therefore reduces to 0 =
Xx[T(X2,...,Xk+x)] + o,(Xx)-T(X2,...,Xk+x).LelXx=A*,AE§,Sind
write gt — exp tA. Then
ХХр[т(Х2,...,Хк + х)}
= -А-т{Х2,...,Хк + 1)
= -оз{Хх)-т{Х2,...,Хк +
Thus, equation (*) holds in this remaining case.
3.1.6 Corollary. Relative to the adjoint representation G -> GL(§)
j
given by g^&b wehave(forTEAk(P,<3)) D"t - di +[w, т].

46 GAUGE THEORY AND VARIATIONAL PRINCIPLES
Proof. Note that the Lie algebra homomorphism §-»§(!(§) induced
by the adjoint representation is given by A B = [A, B]. Thus, со/\т =
3.1.7 Remark. This corollary does not apply to со, since со & Л'( P, %).
It does apply to ЙШ6Л2(Р, §). Thus, D°1u01 = du01 + [со,п"], which
vanishes by 2.2.8.
3.2 GAUGE TRANSFORMATIONS
3.2.1 Definition. An automorphism of a PFB it: P^M is a diffeo-
morphism f: P^P such that f(pg)=f(p)g for all gEG, pEP. Note
that f induces a well-defined diffeomorphism /: M^>M given by
/G7-( p )) = 77-( /(/>))■ A SauSe transformation of a PFB is an automor-
automorphism f: P^P such that f=\M (i.e., тт(p) = тт(/(p))). We set GA(P)
= the group of gauge transformations.
3.2.2 Theorem. Let C(P,G) be the space defined in 3.1.1 where G
acts on itself via g-g' = gg'g~x {i-e., the adjoint action). There is a
natural (anti-) isomorphism C(P,G) = GA(P).
Proof. If tEC(P,G), then define /: P^P by f(p)=pT(p). Since
f(Pg)=PgT(Pg)=Pgg~lT(P)g=PT(p)g=f(p)g, it follows that/E
GA(P). Conversely, if fEGA(P), define т: P^G by the relation
f(P)=PT(p). Note that pgT(pg)=f(pg)=f(p)g=pT(p)g, whence
T(pg)=g~lT(p)g, and it follows that t6C(?,G). Finally, if/,/'E
GA(P) with f(p)=pT(p) and f'(p)=pr'(p), then (/»/')(/?) =
3.2.3 Caution, /и general, the maps Rg: P^P (p^pg) are not
gauge transformations because Rg(pg') = pg'g, while Rg(p)g'= pgg'.
Thus, unless G is Abelian (or g is in the center of G), we do not have
Rg(pg')=Rg(p)g'forallg'EG.
3.2.4 Misleading Misnomers. If -л: MXG^M is a product bun-
bundle, then the maps Lg. MXG^MXG given by Lg(x, g') = (x, gg') are
in GA(MX G). Physicists call such transformations global since g does
not depend on x. Local gauge transformations are those of the form

3 PARTICLE FIELDS, LAGRANGIANS, GAUGE INVARIANCE 47
(x, g')i-+(x' Hx)g') where h: M^G is not necessarily constant. We
emphasize that such gauge transformations make sense only for product
bundles (or for local trivializations), because in general there is no
natural way to define an action by G on P that commutes with the given
right action. Moreover, a gauge transformation that is global relative to
one LT may be local relative to another. We choose not to use these
terms, although they abound in the physics literature.
3.2.5 Theorem. If fEGA(P) and w is a connection \-form, then
f*co is a connection \-form.
Proof. Let A E§ and let A* be the corresponding fundamental field
on P. We have 1.2.2(a) for/*w, because
Since Rgof=foRg, we have Rgf*co = (foRg)*co = (Rgof)*oo =
f*R*a=f*&bg-ia = &bg-i f*a, as required by 1.2.2(b). ■
3.2.6 Theorem. Given a representation G-*GL(V) andfEGA(P),
the pull-back f* yields isomorphism /*: Ak(P,V)^ Ak(P, V), k =
0,1,2,....
Proof. For тЕАк(Р, V), we have Rgf*T=f*R*r=f*g~KT=g~l-
f*r. If A* is a fundamental field on P, then from the proof of 3.2.5
we have fjl*=A*. Thus-, (/*т)(Л*) = т(/„[4*) = т(Л*) = 0, and so
f*TEAk(P,V). Ш
3.2.7 The Space of Connections. _Let G be the space of all
connection \-forms on P. Note that G¥=A\P, §), but these spaces are
closely related.
3.2.8 Theorem. For a given coEG, the map Л'(Р, @)->в, given by
тн> t + 03, is one-to-one and onto.

48 GAUGE THEORY AND VARIATIONAL PRINCIPLES
Proof. li
while Rg(T + a) = R*gT + R*ga = &bg-iT + &bg-4a = &bg-i(T + a),
whence т + ибб. Conversely, ш'бб implies that w' — wEA\P,@).
The theorem then follows. ■
Note that for tEA\P, §) the curve y'- U^G, given_by y(t) =
It, through со has у'@) = т. In this way, we can regard Л'(Р, Q) as the
"tangent space" TjG to the "manifold" G at w.
3.2.9 Definition. The gauge algebra of a PFB with group G is the
space C(P,§), where the representation G^>GL(§) is the adjoint
representation g\-+&bg. The theorems to follow indicate the sense in
which C(P,§) can be considered to be the Lie algebra of C(P,G) =
GA(P).
3.2.10 Theorem. IfH,H'EC(P,§), then the map [H,H'\. P^G,
defined by [H,H'](p) = [H(p),H'(p)], is also in C(P,<3). Conse-
Consequently, C(P, §) has a Lie algebra structure.
Proof. Note that [H, H'](pg) = [H(pg), H'(pg)] = [&bg-,H(p),
&bg-< H'(p)] = &br<[H(p), H'(p)] = &bg-,[H, H'lp). Thus,
C(P,§) inherits a Lie algebra structure from %. ■
3.2.11 Theorem. There is a map Exp: C(P,§)^C(P,G) defined
by Exp(H)(p) = exp(H(p)) for HEC(P,§) such that t^Exp(tH) is
a one-parameter subgroup of C(P,G) with
dt
Moreover, if H, H' E C( P, §), then
s,t = L
Proof. Note that Exp(H)(pg) = exp(H(pg)) = exp(&bg-<H(p)) =
Ad g~>exp(H( p)) = Adg<Exp(H)( p). The other statements are also
easy to verify. ■

3 PARTICLE FIELDS, LAGRANGIANS, GAUGE INVARIANCE 49
3.2.12 Definition. We define exp: C(P, %)^GA(P) by exp(H)(p)
=:pexp(H(p)). Note that this is Exp of 3.2.11. followed by the
isomorphism of 3.2.2.
Theorems 3.2.5 and 3.2.6 tell us that GA(P) acts on the spaces &
and Ak(P,V), respectively. Now that we have a "Lie algebra"
C(P,§) for GA(P), we consider the effects on these spaces of
infinitesimal motions: For this, we first reformulate the action GA(P)
in terms of C(P,G).
3.2.13 Lemma. Let fEGA(P) and tEC(P,G) be related by f(p)
= pr(p). For XETpP, we have Д(X) = (^
Proof. Let у'- i^Pbea curve with y'@) = X. At t = 0,
3.2.14 Theorem. For wE6, feGA(P), and tEC(P,G) (with
f(p)=pT(p)),we have (^1
Proof. Simply apply w to each side of the equation of 3.2.13. ■
3.2.15 Theorem. For <pEAk(P,V),f(EGA(P), and tEC(P,G)
with Ар)=рт(р)), we have f*<p = T~l-<p.
Proof. Use the equation in 3.2.13, and recall that <p vanishes on
vertical vectors. ■
We are now ready for the infinitesimal versions of the actions in
3.2.5 and 3.2.6.

50 GAUGE THEORY AND VARIATIONAL PRINCIPLES
3.2.16 Theorem. Let соЕв andHEC(P, §). Then,
— (exp/tf)*w
r=o
= dH+[a,H]=DuHEAl(P,§)
dt
where exp tH EGA(P) was defined in 3.2.12.
Proof. Replacing/in 3.2.14 by exptH and applying
-0
dt{)
r=o
to the equation in 3.2.14, we get
3.2.17 Theorem. Let <pEAk(P,V) andHEC(P,§). Then,
— (exp?tf)*<p
r=o
Proof. Use 3.2.15 with /= exp tH and differentiate with respect to t.
3.3 LAGRANGIANS AND GAUGE INVARIANCE
Let 77-. P^M be a PFB with group G and let G^GL(V) be a
representation.
3.3.1 Definition. The space of 1-jets of maps from P to V is
J(P,V) = {(p,v,O)\pEP, vEV, and 0: TpP^V is linear). You can
verify that J(P,V) can be made into a manifold in a natural way.
3.3.2 Definition. A Lagrangian is a map L: J(P,V)^U such that
for all (p,v,0)EJ(P,V) and gEG, we have L(pg, g~Kv, g~x-
e°Rg-^) = L(p,v,6). As a result of this requirement, we have

3 PARTICLE FIELDS, LAGRANGIANS, GAUGE INVARIANCE 51
3.3.3 Theorem. Given a Lagrangian L: J(P,V)^>U, there is a
well-defined function £0: C(P,V)^>C°°(M) given (for xEM,->$/E
C(P,V), andpEP with чт(р) = х) by to(xp)(x) = L(p,^(p), d^p).
Proof. We must show that L(p,^(p),d^p) is independent of the
choice oipEir~l(x). Since ^ojR?=g~1-^5 we have d^pgoRgif=g^x
•d4>r, or d4/n=g~x-diioR, Thus,
3.3.4 Definition. A Lagrangian L:J(P,V)^>U is called G-invariant
if L(p, g-v, g-e) = L(p,v,e). Nearly all Lagrangians that arise in
practice have this kind of invariance.
Recall that the gauge group GA(P) acts on C(P,F) = A°(P,F)
via pull-back (i.e., (f*^)(P) = ^if(P)) for/E(J^(P),^EC(P,F),
pEP.)
3.3.5 Theorem. Let L: J(P,V)^R be a G-invariant Lagrangian.
Then it is not necessarily the case that £0(^) = £0(/~'*^) (i.e., £0 is
not necessarily gauge invariant).
Proof. Let т E C( P, G) be related to /E GA (P) by /(p) =p т( р). By
3.2.15, we havef*\p=T~l-\p. We compute d(T~l-\l>)p as follows: Let
XETpP and y. R^P be such that y'@) = X. Then (at / = 0),
Thus,

52 GAUGE THEORY AND VARIATIONAL PRINCIPLES
Now
Z0(f*+)(x)=L(p,{f*+){p),d(r+)p)
but because of the second term in the third slot, we cannot use the
(j-invariance of L to obtain L(p,\p(p), d\p ) or £0(^)(x). ■
In order to remedy the gauge noninvariance of £0, the physicists
sought an object that (when incorporated into £0 as a new variable)
produces a term (under a gauge transformation) that cancels with the
troublesome term /^(^„.(т)^-^/?)-^/?) in the proof of 3.3.5.
In this way, the concept of a connection was forced upon them. The
next theorem shows how nicely connections solve the gauge invari-
ance problem.
3.3.6 Theorem. Let L: J(P,V)^R be a G-invariant Lagrangian,
and let G be the space of connections on P. Define a function £:
С(Р,У)Хе^Сж(М) by e,(rf,,a)(x) = L(p,+(P),Duil>P) for xEM,
рЕтг~\х), \pEC(P,V), and шбб. Then t is not only well defined,
but also gauge invariant, in the sense that for fEGA(P), £(/*»/>, /*")
Proof. From 3.1.3, we know that Dw\pE A\P,V). Thus,
Then,

f 3 PARTICLE FIELDS, LAGRANGIANS, GAUGE INVARIANCE 53
and so £ is well defined. Now
3.3.7 Remark. By writing out d(f*\p)p+(f*<^)p-(f*^)(p) in terms
of т (using 3.2.14 and 3.2.15), you can directly see the cancellation of
the troublesome term in the proof of 3.3.5.
3.3.8 Definition. If L: J(P,V)^Uisa Lagrangian and t: C(P, V)
Xe^C°°(M) is defined as in 3.3.6, then £(^,co)EC°°(M) is called
the Action density of the pair (»//, со). The proof of 3.3.6 shows that L
need not be G-invariant in order that £(«/>, со) be well defined.

CHAPTER
Lagrange's Equation for Particle Fields
In this chapter, we formulate the principle of least (or stationary)
Action for particle fields under the influence of a gauge potential.
The particle field is then shown to obey this principle if and only if it
satisfies Lagrange's equation. Modulo some refinements in Chapter 7
for particles with spin, our approach is general enough to enable us
to obtain (as special cases of Lagrange's equation) the field equations
for quarks and their hadronic aggregates, as well as those for leptons.
The physicist may wonder why the derivation of Lagrange's
equation is more involved than usual. Often in physics books,
equations are established for free fields (without gauge potentials),
and then the ordinary derivatives are replaced by covariant deriva-
derivatives, according to the principle of minimal replacement, in order to
get the nonfree field equations. While this recipe always seems to
work, here we include the gauge potential throughout the derivation
David D. Bleecker, Gauge Theory and Variational Principles ISBN 0-201-10096-7
Copyright © 1981 by Addison-Wesley Publishing Company, Inc., Advanced Book
Program/World Science Division. All rights reserved. No part of this publication
may be reproduced, stored in a retrieval system, or transmitted, in any form or by
any means, electronic, mechanical, photocopying, recording, or otherwise, without
the prior permission of the publisher.
54

4 LAGRANGE'S EQUATION FOR PARTICLE FIELDS 55
of Lagrange's equation, just to make sure. Moreover, the base space
is not assumed to be flat Minkowski space. Thus, the integration-by-
parts step should be replaced by a coordinate-free version such as in
4.2.9. This requires some additional machinery. In spite of the fact
that gravity is so weak compared to other forces, its inclusion is
necessary in a complete theory, and it may play a crucial role in
some way even for small-scale phenomena. Another reason for the
length of the derivation is that our approach places Lagrange's
equation on the total space, instead of the base. In physics literature,
field equations live on space-time, and they explicitly involve local
gauge potentials. Thus, in general, the equations are not manifestly
gauge invariant, even though the equations are satisfied for all
choices of gauge, if they are satisfied for one choice. Here, we take
Lagrange's equation to be on the total space, where it is manifestly
gauge invariant and has a more natural appearance. Example 4.3.7
bears this out for charged scalar fields.
Finally, you should be aware that physicists sometimes add
non-gauge-invariant terms to the Action density. Since these terms
are always chosen to be divergences (i.e., codifferentials of 1-forms),
the new Action density still leads to the same field equation. In this
sense, the practice is harmless, and it is often convenient for the
purpose of applying canonical quantization in quantum field theory
and for handling certain boundary conditions in classical settings as
welL
4.1 THE PRINCIPLE OF LEAST ACTION
Let ir: P^M be a PFB with group G, and let G^GL(V) be a
representation. Let h be a metric tensor defined on M. For simplicity
we assume that M is oriented, so that there is a well-defined volume
form fi on M associated with h. Suppose that L: J(P,V)^R is a
Lagrangian, and w is a fixed connection on P. Recall that we defined
a function £: C{P,V)Xe^Cx(M) in 3.3.6. We write £"
C°°(M), where ta{^) = t^,a). Ideally, the Action of ^()
would be fM£"(ip)ii. However, there is no guarantee that this in-
integral exists, since M might be noncompact.
4.1.1 Definition. We use the notation UCCM to mean that U is
open with compact closure. For U(Z CM, we define the Action of
U to Ье£ /

56 GAUGE THEORY AND VARIATIONAL PRINCIPLES
4.1.2 Definition. For f£C(P,F), we define the projected support
of -^ to be the closure of the set {ir(
4.1.3 Definition. We say that feC(P,F) is stationary relative to
t01 if for all UCCM and also oEC(P,V) with projected support
contained in U, we have
Equivalently, we say that ty obeys the principle of least Action.
The remainder of this chapter is devoted to showing that
C(P,V) is stationary iff \p satisfies a certain differential equation
(Lagrange's equation). This amounts to mimicking the usual ap-
approach to calculus of variation problems, but the general setting here
forces some additional notions on us.
4.2 SOME MACHINERY
We continue to use the notation of Section 4.1. The metric hx on
TXM induces a metric h on the horizontal subspace H CTP
(pE-n~\x)) via the isomorphism m^. Hp^TxM (i.e., hp(X,Y) =
h^ir^X^^Y) for X, YETpP). In the same way a volume element p.
is induced on Hp from that on TXM, and thus we may define a star
operator *p: Ak(Hp)^A"~k(Hp) (« = dim M) such that *р(тт*т) =
■7т*(*хт) where it*: Ak(TxM)^Ak(Hp) is pull-back induced by тг,:
4.2.1 Definition. We define *: Hk(P,V)^A"~k(P,V) by setting
(for (pEAk(P,V)) (*(p)p equal to the unique extension of*p(<p\Hp) to a
V-valued (n~k)-form (on T P) vanishing on vertical vectors. In other
words, *<p is the unique form in A"~k(P,V) such that (*(p)\Hp=*p
4.2.2 Theorem. // aEAk(P,V) and a: U^P is a local section,
then o*(*a) — *o*(a).
Proof. If XEHp, then (о^Х)н =X since_тт„((о^*Х)н) =
^*(°*'n*x) = 'IT*(X). It follows that for any /3EA'(P,F), we have
(тт*а*Р)\Нр =Р\Нр. It suffices to prove that ir*(a*(*a)) agrees with

4 LAGRANGE'S EQUATION FOR PARTICLE FIELDS 57
(«)) on H Now ir*(a*(*a))\Hp =(*a)\Hp =%(a\Hp), while
({**\H) (\H) ■
4.2.3 Assumption. We will need to assume that the vector space V
has a metric h (not necessarily positive definite, but at least nondegen-
erate) such that the representation G -> GL(V) is orthogonal relative to
ft (i.e., h(g-v, g-w) = h(v,w) for all v, wEV). This assumption is
satisfied in all the physical examples we will consider. Moreover, h
always exists if G is compact, for we may set h(v,w)= jGh0(g~l-
v, g~]-w)dg, where h0 is an arbitrary positive definite metric on V and
dg is a volume element on G equal to the wedge of all components of the
Maurer-Cartan form B.2.17) for some basis of %.
4.2.4 Definition. Since we have a metric hp on Hp and a metric h on
V, we can define a metric (hph) on the space of V-valued k-forms
on Hp as in 0.1.5. For a, /3E~Ak(P,V), we define (hh)p(.ap,fSp) =
(hph)(ap\Hp,Cp\Hp).
4.2.5 Theorem. There is a well-defined function
(hh):Ak(P,V)XAk(P,V)^Cco(M)
given by(hh)(a,P) (x) = (hh)p(ap, flp) where тт(р) = хEM.
Proof. Note that apg=g~x-apoRg-ljf, and so on^Thus, (hh)pg(apg,
^) = (^Wg~4°Vv g-l-PpoRg~u) = (hh)p(ap,Pp), since
Rg^: Hpg^Hp is an isometry, and G^GL(V) is orthogonal rela-
relative to h. Ш
4.2.6 Note. Even more obviously, there is a function (hh):Ak(M, V)
k
4.2.7 Theorem. Let a, fi<E~Ak(P,V) and let a: U^P be a local
section. Then we have (hh)(a*a,a*P) = (hh)(a,/3).
Proof. Recall from the proof of 4.2.2 that {тг*о*а)\Нр =сс\Нр, and
also 77^: Hp -> TXM is an isometry relative to h and h. Thus, we have
(*^ h/**\H^

58 GAUGE THEORY AND VARIATIONAL PRINCIPLES
4.2.8 Definition. The covariant codifferential 5": Ak(P,V)-*
Ak~\P,V) is defined, for <p E Ak(P,V), by 8"(<p) =
-(- 1)A(- \yik+l)*D"(*<p) where (-1)A is the sign of the determi-
determinant of(h(dI,dj)), andn = dim M. Observe that when M is a space-time,
(—1)A = — 1 andn = 4; so then 8" = *DU*. In general, the factor ±1 is
necessary for the next theorem.
4.2.9 Theorem. Let UC CM {i.e., U is an open subset with com-
compact closure), and suppose that aEAk(P,V), while flEAk+\P,V).
Assume that the projected support of a is contained in U. Then
Jи Jи
Proof. We first assume that there is a local section a: U^P. Now
{hh){D"a,fi)n= [ {hh)(o*{Dua),o*P)n,
J и
by 4.2.7. From 3.1.5 and the preservation of d and Л under pull-back,
/ o*co)Ao*(a). Thus,
{hh){D»a,P)n= [
и J и
+ J (hh)(o*a>Ao*a,o*P)p.
By 0.2.24, the first term is
(hh)(o*a,8(o*C))ii, 8=-(-l)h{-\)k"*d*.
We need to do some algebraic work on the second term. Let
vx,...,vm be a basis for V and let eu...,efbe a basis for %. Using the
summation convention, we write a*co = co"elt, a*a = aava, o*/3 = /3hvh,
where со", аа, fih are real-valued forms on UCM. Then we have
(hh)(o*coAo*a, o*fi) = (hh)((uv f\aa)(ev-va), jihvh) = h{^ Л a",
jih)h{ev-va,vh). At /=0, we have
d

4 LAGRANGE'S EQUATION FOR PARTICLE FIELDS 59
Also, h(uv Aaa,y8fo)/x = (w" Лаа)Л*(Зь =(- 1) V Л (и" A*flh) =
(-_1)к(-1)л(-1)к<и-*>ао Л *(*(w" Л *flh)) = (-\)h(-\)"kh(aa,
*(w' Л *£V Then й( м' Л a", flh) = (-\)h(-\)nkh(aa, *( w" Л *£fo))
and h(ev-va,vh)=-h(va,ev-vh) together imply that
(hh)(o*co/\o*a,o*P)
= -{l)h(-l)nkh(a°,*(a>A*lil>))U{va,ev-vb)
= -(-\)h(-\yk(hh)(o*a,*{o*(a>)A*o*(C))).
We now have
X
f (
J и
= f {hh){o*a,o*{8"l3))ii= f' {hfi)(a,8uP)n.
J и J и
If there is no local section a: U^>P, then we may use the following
partition-of-unity argument (see Kobayaski and Nomizu [1963]). Let
К be the (compact) projected support of a. Let Ux,..., UN be an open
covering of К such that there are local sections a,: Ut -^P and Ц С U.
There are C00 functions/: M^R such that {xEM\ £(х)=^0}
has closure in U, and 2lNfl(x)=\ for xEK. Define ftaEAk(P,V),
where /,=чг*/„ by (fia)p=fi(p)ap, pEP. Since /a has projected
support in Ц-CU (and we have the local section а,: Ц^Р), our
previous result yields
f (hh){D»{f,a),C)n= f (h
Jи Jи
Noting that 2f /a = «, we then sum over i and obtain

60 GAUGE THEORY AND VARIATIONAL PRINCIPLES
4.3 LAGRANGE'S EQUATION
4.3.1 Definition. Let L: J(P,V)^R be a Lagrangian and let
A\P,V)p denote the spa
vertical vectors. For (i
A(P, V) by the equation
A\P,V) denote the space of linear maps T P->V that vanish on
vertical vectors. For (/>, t>,0) E/(P, F), define V 3Д/>, v, в) Е
For \f,EC(P,V), define a V-valued \-form dL/d(D>) on P by
4.3.2 Theorem. We have bL/b{D"^) ЕЛ'(Р, V).
Proof. Note that 9L/9(Da'^) vanishes on vertical vectors by defini-
definition^ We must show that R*dL/d(Duxp)=g~l-dL/d(DuxP). Let
PEA\P,V). Then
3L
R
3L
dt
\ >hp
I p )
Thus, g-dL/d(D"t)p oR =dL/d(D«t)p, etc

i 4 LAGRANGE'S EQUATION FOR PARTICLE FIELDS 61
4.3.3 Definition. For (p,v,0)EJ(P,V) we define v2L(P,v,0)E
У by the equation
d
T / Д \ \ j t
dt
If \pEC(P,V), we define a V-valued function дЬ/д\р on P by
4.3.4 Theorem. We have 3L/3^EC(P,F).
Proof. The idea here is the same as in the proof of Theorem 4.3.2,
and the execution (which is easier) is left to you. ■
4.3.5 Theorem. Suppose that Ud CM, and let tEC(P,V) have
projected support in U. Then at t = 0,
Proof. At t = 0,
£"(t+t)(())
Integrating both sides over U, applying 4.2.9, and noting that {hh) — h
on Л°(Р, V) = C(P, V), we obtain the result. ■
4.3.6 Theorem (Lagrange's Equation). The particle field ^e
C(P,V) is stationary (for a Lagrangian L: J{P,V)^>U and a fixed
connection со on P) iff the Lagrange equation holds: 5a>[3L/3(Da>^)}
+ 3L/3^O
Proof. Suppose that Lagrange's equation does not hold at some
pEP. Then, we can find a feC(P,V) such that /iEa'[3L/3(JDa'^)]
'"I, f)>0 at 7т(р). By continuity, this inequality persists in an

62 GAUGE THEORY AND VARIATIONAL PRINCIPLES
open set U (with compact closure) containing тт(р). Multiplying т by
a nonnegative C00 function/: P^U (constant on fibers) with pro-
projected support in U and with f(p)>0, we obtain r=ffEC(P,V).
Theorem 4.3.5 applies to т, and yet (by construction) the right-hand
side of the equation in 4.3.5 is nonzero. Thus,
and »// is not stationary. The converse is clear from 4.3.5. ■
4.3.7 Example (Spin-Zero Electrodynamics). Let тт: P-*M be a
PFB with group U(\) = {e'e\6EM} and fixed connection со: We sup-
suppose that M is a {general) space-time with Lorentz metric h. Let F=C
(regarded as a two-dimensional vector space over U), and suppose that
f/(l)-»GL(C) is the representation given by e'e-z = e'ez (complex
multiplication). The Lie algebra of U(l) is the space GlL(\) = {i6\6 EM}
of pure-imaginary numbers. Since
£«"•«1,-.=/*,
we see that %A) operates on С via complex multiplication. Let h be the
R-valued metric on С given by h(z, w) = \(zw + wz) (i.e., the ordinary
Euclidean inner product on R2sC). Recall D.2) that h induces
a metric hp on the horizontal subspaces Hp = {XETpP: cc(X) = 0}.
We_ define a Lagrangian L: J(P,V)^M by L(p,z,d) =
^(Щ^в^в")- \m2zz. //£■,,..., E4 is a_ basis of HP, в, = 0(E,),
and hij = hp(Et, Ej), then (hh)pFH, вн) = {h^Ofij + Ofr), which may
be more familiar to physicists. For any в' Е A\P, V) ,
whence V 3L(p, z, в) = вн. Similarly, V 2L(p, z,Q)— —m2z. Thus,
в в
{
and dL/d\p= —m2\p. Lagrange's equation is then 8"D"^ — m2^ = 0.
While this equation is perfectly elegant, physicists are used to thinking
of -^ as a function on M. This requires a local section a: U-+P (i.e., a
choice of gauge) to pull the equation back to UCM. We write \p' = \p°o:

4 LAGRANGE'S EQUATION FOR PARTICLE FIELDS 63
U-> P and — iA = а* со. Note that as со takes on pure - imaginary
values, A is a real-valued \-form on U. (A can be identified with the
- vector potential" up to a constant multiple depending on the charge.)
Using 4.2.2, we get
or
This is the equation for the "first-quantized " wave function of a spin-0
charged particle (e.g., а тт~ meson) of mass m under the influence of
an electromagnetic potential A (e.g., see Schiff [1968], p. 468, for the
special case where M is Minkowski space). Aside from the homoge-
homogeneous Maxwell equation d2A = 0, there is the inhqmogeneous Maxwell
equationS(-dA) =j' wherej' = h(d^', ixp')-A^'xP' = i(xP'd^'-dxp'xp')
—А-^/'-ф' is the current. This equation is derived from a principle of
least Action with respect to variations in со. Indeed, this is the subject of
the next chapter.

CHAPTER
The Inhomogeneous Field Equation
Given a particle field and a gauge potential to which it responds,
we will define a Lie algebra-valued 1-form on the total space. This
form is called the current. We give a number of equivalent char-
characterizations of the current. Conceptually, the best is 5.1.4, which
states that the current is the first variation of the Action with respect
to the gauge potential. The current can be pulled down to the base
by a choice of gauge, but the resulting local currents (like the local
field strengths) do not piece together to make a well-defined 1-form
on the base unless the group is Abelian. In the case of electromag-
netism, where we have the Abelian group U(l), the well-defined
pulled-down current is the usual 4-current occurring in Maxwell's
equations (see 0.2.22). Eventually we will consider the current of a
charged scalar field E.2.9), a charged Dirac electron field G.2), and a
David D. Bleecker, Gauge Theory and Variational Principles ISBN 0-201-10096-7
Copyright © 1981 by Addison-Wesley Publishing Company, Inc., Advanced Book
Program/World Science Division. All rights reserved. No part of this publication
may be reproduced, stored in a retrieval system, or transmitted, in any form or by
any means, electronic, mechanical, photocopying, recording, or otherwise, without
the prior permission of the publisher.
64

5 THE INHOMOGENEOUS FIELD EQUATION 65
oucleon field in the presence of a Yang-Mills potential G.3). Other
currents (e.g., various weak and hadronic currents) that are found in
the physics literature are also essentially special cases.
In the event that the Action density is gauge invariant, we prove
E.1.5) that the current is covariantly conserved whether or not the
inhomogeneous field equation (described later) holds. If the Action
density is not gauge invariant, we can use the inhomogeneous field
equation to obtain current conservation.
Gauge potentials are not particle fields, but in 5.2.1 we construct
from the gauge potential, a gauge-invariant function on the base.
This function is called the self-Action density of the gauge potential,
because it does not involve any particle fields. The total Action
density is the sum of the Action density and the self-Action density.
The particle field and gauge potential are stationary for the total
Action if and only if Lagrange's equation and the inhomogeneous
field equation hold (see 5.2.3). In the case of electromagnetism, the
inhomogeneous field equation is equivalent to the two Maxwell
equations that involve the charge density and current density (see
0.2.22 and 5.2.9). Returning to the general case, in the event that the
current is zero (e.g., consider a gauge potential free of particle fields),
then the inhomogeneous field equation is in fact homogeneous and is
then commonly called the Yang-Mills equation. If the group is
non-Abelian, the Yang-Mills equation is nonlinear in the gauge
potential, just like the homogeneous field equation (or Bianchi
identity of 2.2.8). However, unlike the Bianchi identity, which holds
for any gauge potential, the Yang-Mills equation holds only if the
gauge potential is stationary for the self-Action.
5.1 THE CURRENT
"We adopt_the notation used in Chapter 4." In order to define
the current/ЕЛ'(Р, g), we need a metric h on Vsuch that G-*GL(V)
is orthogonal, and we need a metric к on % such that &b: G-* GL(§)
is orthogonal. We then have a function
(hk):AJ'(P,$)XAJ(P,®)^C°°(M)
as a special case of 4.2.5.

66 GAUGE THEORY AND VARIATIONAL PRINCIPLES
5.1.1 Definition. For шЕв,^ EC(P,V), and pEP, the current Je
Л'(Р, §) is defined at p by the equation
= (hk)(jp,r),
required to hold for all тЕ_Л'(Р, §)p. Note that Jp exists and is unique
by the nondegeneracy of{hk)p. In 5.1.3, we show that JEA\P, §).
5.1.2 Theorem. Let eu...,ef be a basis for % and suppose that
(kafi) is the inverse of the matrix (kafj), kap=k(ea, ep). Then J(X)~
ка^ддХ
Proof. We calculate {hk)p{kalih\bL/b{Da^), ea -^)ep, туеу) =
kapkpyhp{h\bL/b{D^), еа'П Ti) = hp(h(dL/d(D»n ey ^), т*) =
(hh)p{dL/d(D"xP),T-xP) = (hk)p(Jp,Tyey), and the result follows. ■
5.1.3 Theorem. WehaveJEA\P,§).
Proof. For tEA\P,§), we have {hk)gg(&bg-,Jp°Rg-,.,Tpg) =
(hk)p(Jp, &bgWRg,) hkJ hhdd
p p_ p
Moreover, dL/d(Dy) E A\P, V) by 4.3.2. Thus, (hh)
d dd
ppp p pppg pgpgpg
and so &b-->JnoR ,. = J or R*J=&bp * J- Note that/ vanishes
on vertical vectors, by definition. ■
5.1.4 Theorem. Let £: C(P,V)X6-*CX(M) be as in 3.3.6. Let
Jw{4/) ЕЛ'(Р, §) be the current associated to the pair (\p, со). Then, for
all tETJ3=A\P, g), we have
Proof. For p E it ~ Xx), we have (at t=0)
) L{
= (hk)(J,r)(x).

I 5 THE INHOMOGENEOUS FIELD EQUATION 67
'k
5.1.5 Theorem (Conservation of Charge). Let L: J{P,V)-*R
be a G-invariant Lagrangian. For fixed со EG, suppose that \pEC(P, V)
is stationary relative to L in the sense of 4.1.3. Then the current JwD/)
obeys the "generalized continuity equation" 8"(J"(\p)) = O.
Proof. By 3.3.6, the G-invariance of L implies the gauge invariance
of t: С(Р,К)Хе^С°°(М). Thus, for any FEC(P,§) we have (in
the notation of 3.2.12) £((exp/F)*i//,(exp/F)*w) = £(i//, со) for all
t&U. Differentiating with respect to t and using 3.2.16 and 3.2.17,
we have (at / = 0)
(A) ftt^-tF^,a) + ft
Let UCM be an open set with compact closure, and assume that F
has projected support in U. Integrating (A) over U, we obtain (using
5.1.4)
(B) j-l°u^-
The first term vanishes, since ^ is assumed stationary. Applying 4.2.9
to the second term, we have
for all FEC(P, §) with projected support in U. It follows, as in the
proof of 4.3.6, that Sw(/w(i//)) = 0. ■
5.1.6 Theorem. // we do not assume that \p is stationary in 5.1.5,
then we still have kafih{8w[dL/d(Dw\l>)] + dL/d4>, ea -\p)ep =5W(^)
where ex,...,efis a basis for % and (kaP) is the inverse of (kap), ka/3 =
Proof. If \p is not necessarily stationary, then the first term of
equation (B) in the proof of 5.1.5 is

68 GAUGE THEORY AND VARIATIONAL PRINCIPLES
by 4.3.5. The integrand can be expressed in the form k(kaPh(Sw[dL/
d(Da\p)] + dL/d\p, ea-ip)efi, F), and the result follows by combining
the two transformed integrals in (B). ■
5.2 DERIVATION OF THE INHOMOGENEOUS FIELD EQUATION
In order to deduce the inhomogeneous Maxwell equation(s) from
an Action principle, it is necessary to add (to the Action density
£•(»//, w) of 3.3.8) the Action of the electromagnetic field itself.
Classically, this new term is i(\\E\\2-\\B||2) or - %g(F, F) where F
is the 2-form of 0.2.22 and g is the metric on 2-forms (see 0.1.4) on
Minkowski space. The next definition is the natural generalization.
5.2.1 Definition. Suppose that there is a metric к on % such that &b:
G->GL(g) is orthogonal. Define §: 6-* CX{M) by S(w) =
- i(A/c)(fiw, fiw). §>(w) is called the self-Action density of со. Then we
have (£+§>): C{P,V)Xe^C'x{M) where (£+§>)(,//, <o) = £(»//, w) +
§( w) is the combined Action density of the pair (\p, to).
5.2.2 Definition. We say that the pair (-ty, to) is stationary relative to
£+e> if, for_all open sets UCM with compact closure and crE
C(P,V), тЕЛ'(Р, §) with projected supports in U, we have
at t=0.
5.2.3 Theorem. The pair (\l>,co) is stationary relative to (£+§) iff
conditions (A) and (B) hold:
(B) 8"fl"=/"(^) (inhomogeneous field equation).
Proof. Let oEC(P,V) and тЕЛ'(Р, g) each have projected sup-
support in the open subset UCM. For flr = ua+tT = d(co + 1т) + ^[
tT, co +tr] we have

5 THE INHOMOGENEOUS FIELD EQUATION 69
at Ja
f
J и
Since a and т can be chosen independently, (A) and (B) follow if
(»/>, со) is stationary relative to (£•+§>). The converse is evident. ■
5.2.4 Remark. In 5.2.3 we do not assume that the Lagrangian L:
J{P,V)^R is G-invariant or that t: C{P,V)XQ^CX{M) is gauge
invariant. This assumption was crucial in 5.1.5. We devote the re-
remainder of this chapter to proving that 8"(/"(^)) = 0 is a consequence
of the field equation 8aua=Ja(\p). However, this does not give us an
alternate proof of 5.1.5, in which no assumption on со was made (in
contrast to 5.2.3).
5.2.5 Theorem. Let it: P-+M be a PFB with group G, and suppose
that G^GL(W) is a representation. IfTEA\P,W) and со Ев, then
D
Proof. We have £ы(_£ыт)
— соЛй?т + соЛй?т+соЛ(соЛт) = й?соЛт+соЛ(соЛт). By a simple
computation (using Definitions 2.1.1 and 3.1.4), we have со Л (со Лт) =
2 [со, со] Л т. The result then follows. ■

70 GAUGE THEORY AND VARIATIONAL PRINCIPLES
5.2.6 Theorem. For <p,p6AJ(P, §), we have [p,*<p] = — [<p,*p]. In
particular, [<p, *<p] = 0.
Proof. Write <p = (paea and p = ppep where ex,...,efis a basis for g.
Then [p, *<p] = p« Л *</К, ^] = /Г(ря, <р")/1[ев, ер] =
5.2.7 Theorem. We have 5"Eий") = 0.
Proo/. Note that
S"(S"ft") = ±*D"**D"Sft" = ±*D"(D"Sfi") = (by 5.2.5)
snw]) = 0 6y 5.2.6. ■
5.2.8 Corollary. //8ЫПЫ =/"(>//),/Леи 8ЫGЫ(>//)) = 0.
5.2.9 The Spin-Zero Case. Же can now justify the inhomogeneous
Maxwell equation stated in 4.3.7. In the notation of 4.3.7, we take
i = / — 1 /o fee a fray's vector of %A) аий? we choose the metric к on
%A) 50 that k(i,i) = \. According to 5.1.2, the current is J=
h{dL/d{Duip),i\f/)i. We found that дЬ/д(Па-ф) = Оа-ф in 4.3.7, and
so J = h(D"\p, i\p)i=h(d\p + com\p, i^)i. For a local section a, we have
o*J = h(d\l>'-iA\l>',i\l>')i = h\d4>',i\l>')i-Axl>'^'i. Applying_o* to the
equation 8"fl"=/, we obtain 8(-dA) = h\dip', м//)-Л»//»//=/ after
dividing by i.

CHAPTER
Free Dirac Electron Fields
The main goal of this chapter is to define properly the notion of a
free Dirac electron field over a curved space-time, and to exhibit
Dirac's equation for this particle field as a special case of Lagrange's
equation of Chapter 3. To do this in complete generality, starting
with the basics, requires the introduction of several concepts that are
of independent interest. These include the Lorentz group (and its
universal cover SXB,C)), the Levi-Cevita connection, spin struc-
structures, and the algebra of Dirac (or gamma) matrices (i.e., Clifford
algebra).
Physicists will probably be quite familiar with the results of
Section 6.1. However, they may find the bundle-theoretic determina-
determination of the topology of the Lorentz group and the related group
5LB,C) refreshing compared to the awkward discussions (if any)
David D. Bleecker, Gauge Theory and Variational Principles ISBN 0-201-10096-7
Copyright © 1981 by Addison-Wesley Publishing Company, Inc., Advanced Book
Program/World Science Division. All rights reserved. No part of this publication
may be reproduced, stored in a retrieval system, or transmitted, in any form or by
any means, electronic, mechanical, photocopying, recording, or otherwise, without
the prior permission of the publisher.
71

72 GAUGE THEORY AND VARIATIONAL PRINCIPLES
found in many physics books. Some facts about the irreducible
spinor representations of SLB,C) are provided (without proof) in
6.1.11 for your information. Also, we give an explicit explanation
why spinors with noninteger spin (such as the electron bispinor) are
changed to their negatives under a spatial rotation of 360°.
The Levi-Cevita connection (defined in Section 6.2) is a standard
connection on the PFB of orthonormal frames over a (pseudo)-
Riemannian manifold (e.g., curved space-time). This connection is
not usually introduced in physics books in conjunction with Dirac's
equation, since that equation is typically considered only on flat
Minkowski space. On a general space-time, the Levi-Cevita connec-
connection is used to replace coordinate differentiation by covariant differ-
differentiation, which has meaning independent of coordinates. The Levi-
Cevita connection also plays a crucial role in Chapters 8 and 9 (and
Section 10.1), which involve general relativity and its extensions.
A spin structure is essentially a PFB (with group SLB, C)) that is
a twofold cover of the oriented o. n. frame bundle, whose group is
the proper Lorentz group, over an oriented space-time. A spin
structure is needed because electron fields and other noninteger
spinor fields are particle fields associated to representations of
SXB,C) instead of the proper Lorentz group.
Physics students must eventually master the algebra of Dirac (or
gamma) matrices. Only the bare essentials are introduced in Section
6.3, since our primary use is only in the construction of the Lagrangian
for electron fields and in the expression of Dirac's equation. They
will also appear in the nucleon field constructions of Chapter 7.
Mathematicians may be surprised by the widespread use of Clifford
algebra in physics.
In Section 6.4 Lagrange's equation for electron fields is revealed
to be Dirac's equation in disguise. The unmasking requires some
work when carried out over a curved space-time. Essential use of
properties of the Levi-Cevita connection is made in the reduction.
Dirac's equation also appears more natural on the spin bundle than
on space-time itself because Christoffel symbols (or local connection
forms of the Levi-Cevita connection) arise when the equation is
pulled down to space-time by a choice of gauge (i.e., a choice of local
spinor frame field).
Finally, a brief account of the evolution of Dirac's equation, in its
simplest form, is given in 6.4.10. You are urged to consider this
account before immersing yourself in the details of this chapter.

6 FREE DIRAC ELECTRON FIELDS 73
6.1 COVERING THE LORENTZ GROUP
6.1.1 Definition. For x=(xo,...,x3) and y = (yo,...,y3) in R4, we
define (x, y) = xQyQ-ххУх-х-2у2-хъуу
The Lorentz group L is the group of all linear transformations B:
r4^R4 such that (Bx, By) = (x, y). Let tj be the 4X4 diagonal
matrix with entries 1, — 1, — 1, — 1 on the diagonal. Regarding x, yE
R4 as column matrices, we have хт<цу— (х, у); while taking BEL to
be a 4X4 matrix, we have (Вх)тт)Ву = хтт)у or хтВтт)Ву = хтт\у for
4. Thus, 5ELiff
6.1.2 Theorem. The Lorentz group L has four connected compo-
components. Writing В EL as (Я^.),0< i, j<3, they are
Ll = {BEL\detB=-l,B00>\),
Li+ = {BEL\detB=\,B00<-l),
Ll_ = {BEL\detB=-l,B00<-l}.
Proof. Let BEL. Then - 1 = det tj = det(BTr)B) = det(BT)
det(T))detE) = -(det BJ. Thus, det5 = ±l. Also (for e0 -
(l,O,O,O)),l = (eo,eo) = (Beo,Beo) = B^-B2o-B2o-Blo. Thus, B&
>\ (i.e., Доо>1 ог5ш<-1). It follows that L=L\uV_ULl+ULi_
(a disjoint union of open subsets). Moreover, /=diag( 1,1,1,1)EL'+,
Is = t,= diag(l, - 1, -1, - 1) ELI, I, = diag(- 1,1,1,1) ELi, Iu =
ISI = -IEL1+, and L\ =IsV+,Ll =ItV+,Ll+=IstL]+. Thus, it
suffices to show that L\ is connected. Let
Certainly (хо,...,х3)^(х1,х2,х3) defines a diffeomorphism of H
with R3. If v°EH, then (t>°,t>°)= 1, and we can complete v° to an
o.n. basis of R4, say t>°,...,t>3 with the matrix of column vectors
[t)°,...,D3]6Lf+ (note that v§>\, since v°EH; and we can always
change u3 to -t>3 to get det[t>0,..., t>3]= 1). If eo=(l,O,O,O)E#, then
[vo,...,v3]eo=v°. Thus, the map -л: L\^>H given_by_77E)=Be0 is
onto. We see that -7T~\eo) = {BEL\\Be0=e0} = SOC), and also
■!T~\vo) = [v0,..., v3]SOC). Indeed, 5OC) acts on L\ to the right in

74 GAUGE THEORY AND VARIATIONAL PRINCIPLES
such a way that -n: L\^>H is a PFB over H=U3 with group
549CJ= S0C). Any bundle over U" is trivial (see Steenrod [1951], p.
53), and so L\ is topologically U3XSOC), which is connected. ■
6.1.3 Notation. Let H{2,C) be the space of 2X2 Hermitian matrices
A(AT=A). A basis for HB,C) is given by
o = [l 0] ,_[0 Г
7 [о ij' T~[i oj'
2_[0 ~i] з Г 1 0
T "I,- oj' T-[o -i
There is an isomorphism U4 -» HB,C) given by x\^x = хот° + х,т' +
х2т2 + х3т3. Later we will have need for another isomorphism R4->
7/B,C), namely, хн>x = хот° — х,т' — х2т2 — х3т3.
6.1.4 Theorem. The following hold for x6R4:
(A) detx = det x = (x,x>, (B) xx = xx = (x, x)I.
Proof. Compute with
co + x3 xx ix2
x =
X j i IX 2 -Л-q .X^
^Q "^3 "^ 1 2
X1 IX 2 Xq 1 X^
Recall that SXB,C) is the group of 2X2 complex matrices A
with det^ = l.
6.1.5 Theorem. There is a homomorphism A: SLB,C)-*L^+ given
by (A(A)(x))^=AxA* where A*= AT, A E5LB,C), and xER4.
Moreover, A is onto with A~\l)—±I.
Proof. Certainly AxA* is linear in x, and so A(A) is linear. By
6.1.4(A) we have (A(A)x,A(A)x) = det[{A(A)(x))^] = det(AxA*) =
(detA)(detx)(detA*) = detx = (x,x). Thus, A(A)EL. We can prove
that SXB,C) is connected.'lndeed, let -n: SXB,C)^C2-{@,0)} be
defined by

6 FREE DIRAC ELECTRON FIELDS 75
that is,
Note that
•
-
([*]) =
1 b
0 1
tec HCs
We see that чг. SXB,C)^C2 - {@,0)} is a PFB with group C. There
is a (global) section a: C2 - {@,0)} ^5LB,C) defined by
a ~
J
b a/y
where y = aa+bb. By 1.1.6 we have SLB,C) diffeomorphic to
(C2 - {@,0)} XC or (since C2-{@,0)} = R4-{0} = 53XIRM'3XIR3,
which certainly connected. Thus, A(SLB,C))CL]+. Suppose that
A{A)= I. Then x = AxA* for allxER4. Writing
and setting х = т° =/, we obtain ad+bb= 1,cc + dd= 1. Then setting
х=т3, we get aa — bb=\,cc — dd= — \. Hence, we must have b = 0
and c = 0, whence \ = detA=ad and aa= 1. Thus й?=а, and so
Settingx = T' yields a2= 1, whence a2-ad= 1, and so Л~'(/) = {±/}.
Note that Л(Л5) = Л(Л)ЛE), because (Л(Л5)(х))^=Л5х(Л5)*
=ЛEх5*)Л*=/4(ЛE)(х))^* = (Л(Л)(ЛE)(х)))^. Thus" Л is a
homomorphism, and in particular, Л sends the curves /н>ехр(/Л) of
5LB,C) to the curves /^ехр^Л^ДЛ)) (without degeneracy, since
Л~'(/)={±/}). Hence, Л#/: SfB,C)^£ is one-to-one. Moreover,
Л , is onto, since dim(SeB,C)) = dim(S'3X|R3) = 6 = dim(S'0C)X
R ) = dim(Lt+) = dim£. Since AHc-4=LA(-4))|coA)|c/oL-4-i)|c, it follows
that Л is a local diffeomorphism onto an open subgroup Lo of V+.
Now V+ is the disjoint union of open cosets of Lo in V+, whence the
connectedness of V+ implies that V+ =L0=AELB,C)), and so Л is
onto. ■

76 GAUGE THEORY AND VARIATIONAL PRINCIPLES
6.1.6 Remark. In the course of the preceding proof we showed that
SLB,C) is diffeomorphic to the simply connected space S3X|R3. [t
follows that A: SLB,C)-*V+ is the so-called universal covering homo-
morphism.
6.1.7 Definition. We define a representation p: SZB,C)-> GLD,C)
by
a*-
Clearly p is the direct sum of two irreducible representations, commonly
denoted by Z^1/2-0»: SLB,C)->GLB,C) and D^/2): SLB,C)->
GLB,C), given by D°/2'°\A)=A and D^x/2\A)=A*-\ In Section
6.2 we will introduce a PFB P^M with group SLB,C). The Dirac
electron fields will be the particle fields in C(P,C4) = {\p: P->C
^А А'}
4|
6.1.8 Theorem. The representations £)A/20) and D@A/2) are not
equivalent.
Proof. We must show that there is no 5EGLB,C) such that
BAB'' =A*~X for all Л ESXB,C). Taking the trace of both sides, we
have tv A=tv A*~\ but this does not hold for
-2i 0
0 //
6.1.9 Theorem. There is no representation p': L^GLD,C) such
that p'(A(A)) = p{A),A ESXB,C).
Proof. If p' exists, then we have p'(/)=/, and yet p\I) —
р'(Л(-/)) = р(-/)= -/, since Л(-/)=/. ■
6.1.10 Remark. Here we examine the physicist's somewhat para-
paradoxical statement that the Dirac electron field is transformed into its
negative when space undergoes one complete rotation. Let

6 FREE DIRAC ELECTRON FIELDS 77
show that A(A): R4^IR4 is a rotation about the x3 axis by 26
oing x0 fixed). Indeed,
Thus, we see that A(Ae) leaves x0 and x3 fixed, while xx and x2
change according to x, + bc2i-» е2в'(хх + ix2) (a rotation by 26 in the
x,-jc2 plane' or equivalently, about the x3 axis). If 6 — m, then Л(А„)
= Iisa rotation by 2тт, butp(A7r) = — I.Accordingly, for фЕ C(P, С4)
asm6.1.1,4>{pA,)= *
6.1.11 Remark. The irreducible representations of SLB,C) (with
complex representation space) form a doubly indexed family D(*'v)
where /x and v run independently over the set @,\, \,\,2, ... }. We
have already defined £>(I/2-0) and D{0A/2). The representation D^'p) is
just the subrepresentation of
£H/2'0)B) .2/\ (g)/)(l/2,0)^)£)@,l/2)^) ,2.". g,2)@,1/2)
with representation space being the space S(^'p) of tensors (referred to
as spinors) symmetric in the first 2/x slots and in the last 2v slots. The
dimension of this representation space is B/x+ 1)B^+ 1). When Z?(|i•"'
/5 restricted to SUB)CSLB,C), it decomposes into a direct sum of
irreducible representations of SUB), the largest of which has dimension
2(v + ijl)+1. Consequently, the number v + ц is called the spin of the
representation D(>l-V). The representation p=D°'/2'0)®D@'l/2) in 6.1.7
« the direct sum of spin-^ representations, and so the electron is said to
have spin ^. Finally, note that the number of slots of a spinor in S^1"'
is 2y.+2v, and so D^-'X-1) = (- lJ^+l/)I, which is I if the spin ц + v
is an integer and —I if \i+v is half-integral. Thus, the property
explored in 6.1.10 is characteristic of particles with half-integral spin.
6.2 THE LEVI-CEVITA CONNECTION
Let M be an и-manifold with metric h of signature (r, s). For the
most part, we will be concerned with the case in which r— 1,5 = 3,
but the general case is no more difficult and will be of use to us later.
Let ( , ) denote the standard scalar product on R" with signature

78 GAUGE THEORY AND VARIATIONAL PRINCIPLES
(r,s)(i.e.,(v,w)=v[wl + ■■■ +vrwr-vr+lwr+] vr+swr+s). Let
O(r, s) denote the subgroup of GL(n,R) preserving ( , ),
in the sense that A EO(r, s) iff (Av, Aw) = (v, w) for all v,wEU".
If т) is the nX n diagonal matrix with diagonal entries
A, ■ ■ ■, 1, — 1, •••,— !), then for v,wER" regarded as column vec-
vectors, we have (v, w) ~vTi]w and A EO(r, s) iff ATy\A —т\. If t\-> A(t)
is a curve in O(r, s) with A@) = I, then (A'@)v, w) + (v, A'@)w)=0
follows from differentiating (A(t)v, A(t)w) =(v,w). Thus, the Lie
algebra of O(r,s) is 6(r, s) = {BE§f(n,R)\(Bv, w) + (t>, Bw) =0}
r
6.2.1 The Orthonormal Frame Bundle. An o.n. frame at xEM is
a frame uEL(M)x,u: W->TXM, such that h(u(v),u(w)) = (v,w).
The set of all o.n. frames at x is denoted by F(M)X. We set F(M) =
UxeMF(M)x and define -n: F(M)->M by tt(u) = x ifuEF(M)x. Note
that ifuEF(M)x and AEO(r,s), then uA=u°A EF(M)X. Indeed,
using the same ideas as in 1.1.8, we can prove that it: F(M)->M is a
PFB with group O(r,s); the essential difference is that o.n. vector
fields must be used in place of coordinate fields in proving local
triviality.
6.2.2 Definition. The canonical 1-form on F(M) is the U"-valued
form yEk\F(M),Un) defined {for XUETUF(M)) by <p(Xu) =
u~\tt,(Xu)). Note that this l-form is the restriction of a form фЕ
A\L(M),R") defined by the same equation (for XUETUL(M)).
6.2.3 Theorem. Relative to the usual representations O(r,s)^>
GL(R") (resp. GL(n,R)^GL(R")), we have (in the notation of 3.1.2)
<pEA\F(M),R") (resp. <pEA\L(M),R"))-
Proof. For XUE TUF(M) (resp. XUE TUL(M)) and A E O(r, s) (resp.
AEGL(n,R)), we have y(RAif Xu) = (uA)-\iTJf(RAjfXll))
^ p. ф
tnitXu =0, both <p and ф vanish on Xu.
(resp. <p(RAjfXu) = А~1-ф(Хи)). When
6.2.4 Definition. // w /5 a connection on F(M), then the torsion
form of со is the 2-form 0" =D"<peA2(F(M),R"). Analogously, we
may define the torsion form of a connection on L(M).

6 FREE DIRAC ELECTRON FIELDS 79
The following is known as the "fundamental theorem of pseudo-
Riemannian geometry."
6.2.5 Theorem. There is a unique connection в {called the Levi-
Cevita connection) on F(M) with vanishing torsion form (i.e., Dey = 0).
Proof. Let со be any connection on F(M) (see 1.2.6). By 3.2.8, we
need to prove that there is a unique oEA\F(M), 0(r, s)) such that
D"~°<p=0. By 3.1.5, Dw~~a(p = d(p+(oj-o)/\(p = @w -o/\<p. Thus, it
suffices to prove that there is a unique oEA\F(M), 0(r, s)) such
that @ы = оЛ<р. Note that (o/\<p)(X, Y) = o(X)<p(Y)-o(Y)<p{X),
according to 3.1.4. Let us assume that there is a a such that
0ы=аЛф. We have
(A)
for all X,Y,Z<ETUF(M) and all u<EF(M). Let 2(*, У, Z) =
. By (A),
(<p(Y),o(Z)<p(X))-(<p(Y),o(X)<p(Z))
Since o(X)EG(r,s), we have <«p(Z),a(^)«p(Y)) = -(
<P(Y)), and so on. The resulting cancellations then yield
(B) 2(*,Y,Z) = 2(<p(Z),a(J0«p(Y)>.
Since <p(Z) and a( Y) range independently over U" as Y and Z range
over TUF(M), we see that a(X) is uniquely determined by (B). Thus,
assuming a exists, we have that a is unique. Now suppose we define
о by (B). Then a(X) is linear in X, since 2(X Y, Z) is linear in
X For ЛЕО(г, 5) we have ( q>(RA#Z), @(RA*X, RA^ Y)> =
(y /4~'0(Х, Y)> = (<p(Z), @(X, Y)). Thus, SCi?^^,

80 GAUGE THEORY AND VARIATIONAL PRINCIPLES
RAtY,RAtZ)=~2(X,Y,Z), and so
Thus, we have a(RA,X)=A'la(X)A = &bA-,a(X). Since 2(*, Y, Z)
vanishes for X, Y, or Z vertical, we see that a(X) = 0 for X vertical
(and a is well defined). Thus, oEA\F(M), 6(r,s)). Finally, from (B)
we have (<p(Z), o(X)<p(Y))-(<p(Z),a(Y)<p(X)) = $B(X, Y, Z)-
2(Y, X, Z)) = («p(Z),0"(X Y)>, and so 0м = аЛ<р. Thus, the re-
required a exists. ■
There is a "local" version of 6.2.5, which will be of use to us later.
Let £,,..., En be o.n. vector fields defined on some open UCM, and
let <p',...,<p" be the 1-forms dual to Ex,...,En (i.e., <p'(Ej) = 8j).
6.2.6 Theorem. There is a unique matrix в —{в') of real-valued
l-forms 6' EA'(f/,IR) such that
(A) втг1 + т]в = 0, Tj = diag(l, • • • , 1, — 1, •••, — !);
(B) dip' = ~2-6^ Лф7 (or ш matrix form dy= —^Лф).
Proof. Let e,,..., en be the standard basis of W. We can define a
local section T: U-*F(M) by letting т(х): IR"^7XM be given
by j(x)(ej) = Eix for г = 1,...,и. We prove т*(ф) = ф where ф is
the canonical 1-form of 6.2.2. Indeed, T*(y>)(Ei) = ц>(т*Е() =
T(x)-\ir.a,Ei) = T(xr\Ei) = e,=(^(Ei),...,^(El)) = ^(El) for i
= 1,...,и. Let в be the Levi-Cevita connection on F(M). Then
d(p= —в/\(р, and applying т*, we have й?ф~= — т*@)Л<р. Thus, set-
setting в = т*(в), we have (B); and since в is ©(/•,s) valued, we have
(A). Let в' satisfy (A) and (B). Then 6' induces aconnection в' on
ir-\U) (as in the proof of 1.2.5) such that т*в' = в'. Now T*@e') =
т*(</ф + в'Лф) = </ф + в'Лф = О. So в6*' ЕЛ2(т7~\U), 0(r, 5))
vanishes on т(Г/), and hence throughout чт~\U\ Thus_6' must be
the Levi-Cevita connection, and в' = т*в' = т*в = в (i.e., в is unique).

6 FREE DIRAC ELECTRON FIELDS 81
6.3 SPIN STRUCTURES AND THE LAGRANGIAN
We specialize to the case where M is a 4-manifold with metric of
lature A,3) (i.e., a space-time).
!$.3-1 Orientability. Let it: F(M)^>M be the o.n. frame bundle.
)f]ote that tt~\x) = F(M)x has four components, corresponding (in an
iinspecified way) to the four components of L=O(\,3). It can happen
that F(M) has fewer than four components, because it is conceivable
that a point in one component of F(M)X could be joined to another
component by a curve in F(M) that leaves F(M)X and then returns.
Upon return, an observer who had traveled along the projection of such
a curve might find that the world is a mirror image of what it was, or
that time is running backward, or bothl You may contemplate these
possibilities, but for the sake of simplicity, we will assume that F(M)
has four components. In this case M is called space and time orientable.
A choice of one component of F(M) is called a space and time
orientation. Let F0(M) be such a choice (i.e., F0(M) is some compo-
component of F(M)). Note that тт: F0(M)-*M is a PFB with group Lr+,
instead of L = 0A,3).
6.3.2 Definition. A spin structure for F0(M) consists of PFB tts:
S(M)-*M with group SLB,C) and a map X: S(M)-*F0(M) such that
ir(\(p)) = TTs(p) for all p^S(M), and \(pA) = \(p)A(A) for all
pES(M),A<ESLB,C), where A: SZB,C)-*LT+ is the homomor-
phism 0/6.1.5. If tt: Fo(M)-*M is trivial, a spin structure for F0(M)
clearly exists. In the case where M is noncompact, Geroch [1968]
proved the converse. For compact M, the converse may not hold (see
Geroch [1968]). We assume throughout the rest of this chapter that there
is a spin structure X: S(M)-*F0(M).
6.3.3 Dirac matrices. In the notation 0/6Л.З, we define a linear
map y: U4 -*§£D,C) by
о
The matrices Y/ —y(et), i = 0,..., 3, are referred to as Dirac matrices,
although sometimes {у/} is replaced by Ey,5"'} for some matrix B,
depending on the book used.

82 GAUGE THEORY AND VARIATIONAL PRINCIPLES
6.3.4 Theorem. For all x, y£R4 we have y(x)y(y) + y(y)y(x) =
2(x,y)I.
Proof. Since both sides are symmetric and linear in x and y, it
suffices to verify the result in the case in which у—х, but this case is
clear from 6.1.4(B). ■
6.3.5 Remark. The Clifford algebra C(R4)®C, relative to < , ), is
the 16-dimensional algebra over С generated by the unit vectors
eo,ei,e2,ei with relations e^j+e]el■=2{ei, Sj)\, where 1 is the unit.
Note that у extends to a representation of C(R4)®C. Up to equiva-
equivalence, this is the only irreducible representation of C(R4)®C.
6.3.6 Theorem. Let X: S(M)-*F0(M) be a spin structure; let Л,:
S£B,C)-> £ be the isomorphism of Lie algebras induced by A: SLB,C)
-*L\; and suppose that в is the Levi-Civita connection on FQ(M).
Then §=Avl °\*в is a connection on S(M).
Proof. For p<ES(M) and А&ЫB,С), we have X(pexptA)-
X(p)A(exp tA) = \(p)exp(tA*(A)). It follows that X*(A*) -
(A*(A))t(p), and so_ в(А*) = А-14в(Х^(А*р))] = A-^(A^(A))= A.
For g£ SLB,C), R*e = 6oRgjf = A~ lX*6oRgjf = Л^ ' ° во Л* ° r^ =
(since ЛоЛ, = ЛЛ(г?)°А) = Л,1о0оЛл(^оЛ,= Л,1овЬЛ(;?) ->0<>\,
6.3.7 Theorem. Let ф — Х*(р where <p is the canonical 1- form on
F0(M) (see 6.2.2). Forg£SLB,C) we haveR*<p = A(g)~'-f, and Ф
vanishes on vertical vectors. Equivalently, ф£Е A\S(M),R4) relative
to the representation SXB,C)-* GL(R4) given by g-v — A(g)(v).
Proof. Since <p vanishes on vertical vectors and Л^ restricts to an
isomorphism of vertical subspaces, it follows that ф — q>° X^ vanishes
on vertical vectors. Moreover, R *ф = ф ° R я# = ф ° Л^ ° R gif =
6.3.8 Definition. Let H: C4XC4-*C be the "twisted" Hermitian
form given by H((zl,...,z4),(wl,...,w4))=z{w3+z2w4+z3wl+z4w2.
Regarding z,w£=C4 as column matrices, we have H(z,w) = zTyow,
where

6 FREE DIRAC ELECTRON FIELDS 83
las in 6.3.3. The reason for the "twist" is that otherwise the next
theorem is false.
■ 6.3-9 Theorem. For A eGLB,C) (in particular, for A G5LB,C))
and
\A 0
.0 A*'1.
we have H(p(A)z, p(A)w) = H(z,w) for all z,w (EC4.
Proof. Since Я(р(Л)г,р(Л)уу) = ггр(Л)гуор(Л)уу, it suffices to
check that p(A)Tyop(A)=yo; this is a simple computation. ■
6.3.10 Definition. We define a real-valued metric h on C4 by
h(z,w) = j(H(z,w) + H(w,z)). From 6.3.9, it is evident that p:
SLB,C)-*GL(C4) is orthogonal relative to h (i.e., h(p(A)z, p(A)w)
= h(z,w)).
6.3.11 The Lagrangian for Free Electron Fields. A free electron
field is a particle field ^eC(S(M),C4) where the representation is p:
SLB,C)-*GL(C4) 0/6.1.7. We define a Lagrangian L: J(S(M),C4)
-*R as follows. Let p<ES(M), and suppose that a^A\S(M),C4)p-
Let Eo, ...,E3(=T S(M) be horizontal vectors (relative to the connec-
connection в of 6.3.6) such that ф(Е1)=е1 =ith standard basis vector of R4,
where <jp = \*(p(see 6.3.7) or (equivalently)
TpS(M). Define yXoeC4 by
YXa= 2 1чу(е,)
where T]-diag(\, -\, - 1, - 1). Then, for deC4, we let L(p,v,a) =
h(i(yXo),v)-mh(v,v). In order to prove that L: J(S(M),C4)-*R
is a Lagrangian, we need some lemmas.
6.3.12 Lemma. For all A&SLB,C) and x&R4, we have
(A) = p(A)y(x)p(A)-1.
Proof. By continuity, we need only prove this for (x,x)jk0.
Using 6.1.4(B), we have xx =(x, x)I, or (assuming that (x,x)
¥=0)x = (x, x)x~l. Since < A(A)x, A(A)x ) = < x, x ) ^0, we
have (A(A)x)~ ~= (x, x)(A(A)x)Zl = (x, x)(AxA*)~l = A* ~ (x, x)

84
GAUGE THEORY AND VARIATIONAL PRINCIPLES
, (A(A)x)~ =
and so
p(A)y(x)p(A) ' =
О
А*
0
(A(A)x)
0 х
х 0
АхА*
0
(А(А
0
А '
0
0
A*
(A
6.3.13 Lemma. Let Eq,..., E^TpS(M) be horizontal vectors
such that <р(Ец),...,ф(Еъ) are o.n. relative to ( , ). For all crG
A\S(M),C%, we then have yX а =1т]чу{ф(Е'1))а(Е'1).
Proof. There is an A&L = O(\,3) such that ei = '2.Akifp{E'k), and
hence E=^AmjE'm. Then
6.3.14 Theorem. 77ге junction L: /E(M),C4)
L(/>, и, а) = /г(гухог, и) —т/г(и, и), is a Lagrangian.
g/uen
Proof. We must prove L(pg, g~lv, g~l ■ o° Rg-^) = L(p, v, a).
By 6.3.9 and 6.3.10, it suffices to prove that yX.(g~KooRg-li>t) =
g
g~K(yXo). Let E0,...,E3(ETpS(M) be as in 6.3.11. According to
6.3.7, we have <p(/?^) = A(g)-1-<p(£,) = A(g)-1e,- Thus R^E,
may serve as £/6E TpgS(M) as in Lemma 6.3.13, and it follows that
l[°(Rg-'*(Rg*EJ))] =
a) = g-l-(yXa), as
J
required.
6.4 DIRAC'S EQUATION
Here we find Lagrange's equation for the Lagrangian L of 6.3.11
and prove that it reduces to Dirac's equation for a free electron field.

6 FREE DIRAC ELECTRON FIELDS 85
0,4.1 Lemma. For all xeR4, y(x) is self-adjoint relative to h (i.e.,
h{y(x)v,w)=h(v,y(x)w)).
Proof. Note that h(y(x)v,w) = vTy(x)Tyow, while h(v,y(x)w) =
vTyoy(x)w. Thus, we must prove Y(*)rYo =YoY(*)- Since y(x) is
linear in x, we need only check this for x = eo,el,e2,ei. For x =
еоМео) = Уо> and so Y(eo)rYo=YoYo=YoY(eo)- For x = et,i¥=0, we
havey(e,)Yo+YoY(e,)=O by 6.3.4. Thus, Y(^)rYo+YoY(^)r = O, and
it suffices to check that y(ei)T— —y(ei) for i^O. For z — zxex +z2e2
+ z3e3, note that z= —z, and
f
while
l
Since z is Hermitian, zT~z=— z and zT — z=— z, we see that
y(z)r= —y(z), as needed. ■
6.4.2 Theorem. For L as in 6.3.11, we have that (in the notation of
4.3.1) V3L(/>,t>,a): TpS(M)-*C4 is given by V 3L(p, v, a)(X) =
Proof. For /? G Л'( S(M))p we have
(hh){p,v3L(p,v,o))
etc.
6.4.3. Theorem. For L^as in 6.3.14, Lagrange's equation is
-i8e[(yoy)(^)\ + i(y<<De^)-2m^=Q where ^<EC(S(M),C4) and
в is as in 6.3.6.

86 GAUGE THEORY AND VARIATIONAL PRINCIPLES
Proof. From 6.4.2, it follows that 9L/9(#fy) = - i(y°q>)(\py
From the definition of L in 6.3.11, we see that 9L/9^ = /(yX £>fy)
— 2m\l>. Noting that 8e commutes with any fixed linear operator on
C4 (e.g., multiplication by /), we find that the desired result follows
from 4.3.6. ■
The reduction of the equation in 6.4.3 to Dirac's equation F.4.8)
is done with some lemmas.
6.4.4 Lemma. For В<еЫB,С), x<EU4, and pEC4, we have
Proof. Set A =exp(tB) in the equation of 6.3.12. The result follows
from applying both sides of that equation to v, and then differentiat-
differentiating with respect to t at / = 0. ■
6.4.5 Lemma. Let ^<EC(S(M),C4) and recall that X: S(M)->
F0(M) is the spin structure of 6.3.2. We have
where
Proof According to 3.1.5 and 4.2.8, we have 8в[(уоф)(ф)] =
*{d[(yo *<р)(\р)] + вЛ[(уо *ф) (»//)]}. Using 6.4.4, we can rewrite the
second term:
Also, d[(y о *ф)(ф)] = [y ° (d*<p)](\p) — (у ° *ф) Л di>. Combining

6 FREE DIRAC ELECTRON FIELDS 87
uese results yields 8в[(уоф)(ф)] = *{[уоX*(d*<p + 0Л
6.4.6 Lemma. For the Levi-Cevita connection в and canonical form
Pwo/. Since 5*<p= ±*.D9*<p, it suffices to prove that De*<p = 0. Now
*<р = (£<р',..., *<p") where *«р' = (-1)'"'т),|(р'Л ••• Лф'Л •••Л<р".
Applying й? to *<p (componentwise) and using d(pJ — —ieJkA(pk, we
see that d(*(p) vanishes on horizontal subspaces, because of the
factors 0Jk. Thus, De*<p=(d*(p)H = 0. Ш
6.4.7 Lemma. For ^GCE(M),C4), we have *[(y°*q>)/\(De\p)]
Dep
Proof. Let ф — (ф°,..., ф3) = 2 ф'е,. Suppose that Eo,..., E3
TpS(M) are as in 6.3.11. Then
6.4.8 Theorem. The Lagrange equation of 6.4.3 reduces to the
Dirac equation yXDe\p+im\p = 0.
Proof We need to prove that_86'[(Y°<p)(^)] = -yXDe\p. Now
8<>[(УОФ)D>)]= -*[(Y°(*9))A(Z>fy)]> by 6.4.5 and 6.4.6. The result
follows from 6.4.7. ■
6.4.9 Remark. Let М—Ш4 be Minkowski space with the usual
coordinates (x0, x,, x2, x3) and coordinate fields 30,3,, 32,33. We may
then take S(M) to be the product MXSLB,C). The horizontal sub-
space at (x, A) (relative to the connection в) is just the tangent space of

88 GAUGE THEORY AND VARIATIONAL PRINCIPLES
the submanifoldMX{A} at (x, A). If a: M-*MXSLB,C) is defined
by a(x) = (x,I), then аД=£, (defined in 6.3.11). Let \p' = o*4> for
^eC(S(M),C4). Then
3>', and so yXDV 7^ yy
2t)j■■y(ei)d-\p'. Writing y' =1,r]jky(ek), we //zen see //га/ ?/ге Z);>ac
equation on MXSLB,C) pulls down (via a) to the equation 2y/3/^'
+ im\p' = O found in physics books.
6.4.10 Historical Note. The relativistically invariant Klein-Gordon
equation 2tj jkd<dk\p+m2\p=0 is unlike Schrodinger's equation because
of the second time derivative. In order to remedy this (and other related
problems) Dirac sought a first-order equation of the form 2a*3^=
-/mf which upon iteration yields the Klein-Gordon equation:
— im^) = { — im) \p= —m2\p
or
This is the Klein-Gordon if j(aJak + akaJ) — f],kI. The smallest (nec-
(necessarily square) matrices ak(k = 0,1,2,3) satisfying these relations are
4X4 (e.g., the Dirac matrices yk). Thus, Dirac found his equation, and
deduced that \p must have at least four components. If m =0 (unlike the
electron), then a simpler possibility exists. Multiplying 2 a* 3^=0 by
(a0), we obtain Э0^ = 2-/?7Э7^ where /?7=(a°)~ la'(j = 1,2,3).
Thus, 3O2 4> = 2 {(Pkfij + PJ(lk)dk д/4>, which is the Klein-Gordon equa-
equation (for m=0) provided that {(fikfi> + (lJPk) = SkjI(\^ j, к <3).
The 2X2 matrices т1, т2, т3 of 6.1.3 (/.e., the Pauli matrices) will meet
this condition (use 6.1.4E)). The corresponding two-component equa-
equation 30^ + 2t'3,^=0 was noticed by Weyl (see Bjorken and Drell
[1964]), but was dismissed because of the lack of a so-called parity
symmetry. The Weyl equation was vindicated in 1956 with the discovery
of parity violation by neutrinos in experiments proposed by T. D. Lee
and C. N. Yang et al. Indeed, the Weyl equation is satisfied by neutrino
fields in C(S(M),C2) where the representation is Z)A/20) or D@A/2),
depending on the "handedness." Naturally, the ambidextrous electron
arises from Z)('/2-°>©Z)(°.'/2).

CHAPTER
Interactions
In Chapter 6 we developed the basic theory of the free electron
field. In reality, however, the electron field is always attended by the
electromagnetic gauge potential (i.e., four-dimensional vector poten-
potential of 1.2.7) with which it interacts. This gauge potential is defined
on the total space of a PFB with group U(\), while the free electron
field is defined on a different PFB, namely, the bundle of spinor
frames, with group SXB,C). However, in Chapters 3-5 the particle
field and the gauge potential to which it responds were always
defined on the total space of the same PFB. Hence, the results of
these chapters do not apply directly to the case of electron fields and
electromagnetic gauge potentials. To remedy the problem, in Section
7.1 we describe a general, straightforward procedure for splicing two
PFBs together to form a new PFB whose group is the product of the
groups of the original PFBs (see the diagram in 7.1.1). Then a free
David D. Bleecker, Gauge Theory and Variational Principles ISBN 0-201-10096-7
Copyright © 1981 by Addison-Wesley Publishing Company, Inc., Advanced Book
Program/World Science Division. All rights reserved. No part of this publication
may be reproduced, stored in a retrieval system, or transmitted, in any form or by
any means, electronic, mechanical, photocopying, recording, or otherwise, without
the prior permission of the publisher.
89

90 GAUGE THEORY AND VARIATIONAL PRINCIPLES
particle field that lives on P, and a gauge potential that lives on p
can both be lifted to the total space P, о Р2 of the spliced bundle,
where they may interact essentially as in Chapters 3-5, as shown in
7.1.15.
In Section 7.2, the general procedure of 7.1 is applied to the case
of the interaction of an electron field with an electromagnetic gauge
potential. In Section 7.3 we apply it to the case of a nucleon field
interacting with a classical Yang-Mills gauge potential (i.e., a con-
connection on a PFB with non-Abelian group SUB)). The procedure of
7.1 generalizes to the case where more than two PFBs are spliced, as
when a particle field responds to several different gauge potentials
(e.g., electro-weak interactions). It is noteworthy that while Lagrange's
equation for nonfree particle fields resides on the total space of the
spliced bundle, it is shown in 7.1.15 that the inhomogeneous field
equation and the current can be pushed down to the total space of
the PFB on which the gauge potential was originally defined. Finally,
in 7.3.13-7.3.22 we make a detailed effort to show that our formula-
formulation of the field equations for nucleon fields and Yang-Mills poten-
potentials is equivalent to that in the original Yang-Mills paper [1954].
7.1 BUNDLE SPLICING
The constructions of this section will be used in both 7.2 and 7.3.
7.1.1 Definition. Let -nf. P^M be a PFB with group G,, / = 1,2.
We define P, о P2 ={(;,„ Pl) eP, XP2 h(/>,)=*2(/>2)}. Let ira:
P,oP2-^M be given by тт12(Р[, />2) = *,(/> ,)=w2(/>2). For (g,,g2)E
G^XGj and (/>,,/>2)EP,oP2, define (p,,p2)(g1,g2) = (p1g1,/?2g2).
You can verify that тт[2: Р, oP2 -*M is a PFB with group G\~XG2. We
say that the bundles irf. Pt-*M have been spliced. There are also
projections it': P\°P2-*Pl given by ir'ip^ Р2)=Р>, / = 1,2. Certainly
' 2
77': P, о p2 -*p, is a PFB with group {1}XG2 =G2, and it2: P, о Р2 -*P
is a PFB with group Gx X{1} =GV We have the diagram
2

7 INTERACTIONS 91
JM.2 Theorem, //w, is a connection for <n{. Pt->M (/=1,2), then
_i*tf| is a connection for к2: P, »Р2 -*P2,77-2*w2 /s a connection for тт1:
p op2 -*Pi, and tt1*co[®tt2*co2 is a connection for чт12: P, о p2 ->M.
proof. This is routine. Note that we have identified g, with g, X {0}
§ ^ SO ОП- ^
7.1.3 Theorem. Let pf. G{-*GL{V) (/=1,2) fee representations
such that Pl(gl)°p2(g2) = P2(g2HPi(gi) M all (gl,g2)BGlXG2.
Then (p^Xpj)- GXXG2-*GL(V), given by (g,, g2)^ P,(g,) °P2(g2),
/s a representation, and so C(PX° P2,V) makes sense.
Proof. We have
= (by assumption) =P,(g,) °P2(g2) °p,(gO °p2(g2)
= (p,xp2)(g1,g2)o(p,xp2)(g;,g2). ■
7_.1.4 Definition. Let Ak(Pl о Р2, g.) fee //ге subspace of
Л/с(Р| оР2, §,Ф§2) consisting of those forms with values in g. С§,Ф§2,
/=1,2. 5/nce g, anc/ §2 are invariant subspaces_of the representation
(% G, X (J2 -> GL(g,©g2), ?/геге are projections §,: Л*(Р, ° P2, §,©§2)
-* Л/с(Р, о p2, g.) induced by the projections §,©g2 -> g,.
7.1_.5 Theorem. For anj^ aGA*(P,oP2, g,®g2)_we /гаие a = g,(a)
+ S2(a)._ Moreover, there is a unique form a. GA*(P,-, g,-) smc/z
-Proo/. The first statement is obvious, by definition. For
T(P),P2)Pi°P2, define aXp^X) = (Qxa){X). We check that alpi is
well defined (i.e., independent of the choice of X and p2). If X'&
T(P> p,)P^Pi with <*=*;*', then wl2l»( A--A") = w,X(^-^") = 0.
whence X-X' is vertical for mx2\ P,oP2-^M. Hence a(X-X')=0,
and §,(«)(X) = §,(«)( X'). For any g2G{l}XG2, wehave77-^(«g2!(cX)
= <(X), and (S,a)(/?ft#A') = abg2-(9la)(X) = (9la)(^)' since
6Eb -i acts trivially on §,. Thus, a, is well defined. If ii\X is vertical

92 GAUGE THEORY AND VARIATIONAL PRINCIPLES
for 77,: P,^M, then 0 = 77|!(с77^=7г|2!(сХ, and so al;,iG7^) = (§l
= 0. Finally, for g, EG,, we have a,(/?giX^) = «i(<^gl*^) =
(S1a)(/?Jfi^) = abJfi-.(S1a)(^)=SbJfi-.a1(^). The proof is the
same for г = 2. ■
7.1.6 Theorem. Let u, fee a connection for <n{. Pf^>M, and let
ш, =77-'*w, vw//z w,+w2 the connection for_тт[2: PX°P2-+ М provided by
7.1.2. 77геп £)"| +  sen* /orm5 m Л*(Р, о P2, §.) /0 /orms in
Л* + 1(Р, oP2, §.), f =1,2. Moreover, in the notation of 7.1.5,
(Z> *■ + *>( a)),=D "■(«,).
Лчю/. For aE Л*(Р, о Р2, §,), we have £>"' + a = rfa + (w, + ш2)Л
a=c/a + w,Aa, since the Lie algebra action of 0®§2 on §,©0 is
trivial. Both da and ш,Аа have values in §,®0, and so £>"l + aG
Л* + 1(Р, °P2' ^i)' and so on- To prove the second statement, we need
771*(Z)a1))=§1(£)"l + U2(a)), but TTl*(Da'(al))=TLl*(da[+ool/\al)
+ iT^co^AiT1*^ = d(§la)+ w, A(S,a) = §,( rfa + ( w, + a2)
7.1.7 Theorem. Assume that M is an oriented Riemannian n-
manifold in order that star operators * , * , and * can fee defined on
the spaces Л(Р„§,), Л(Р2,§2), апс? Л(Р,о.Р2Д©§2). Г/геп *
respects the decomposition Ak(Pl ° P2, §, © §2) = ^(P, ° P2,
ФЛ*(Р, о Р2, §2) (f.e., *l2(§,-a) = §,-(*l2a)). Moreover, for aE
Ak(P[oP2, §,©§2), we /гаие (*|2а), =*.(а,), ш the notation of 7.1.5.
Proof. By definition of * in 4.2.1, it is evident that §, commutes
with * . The last statement follows from the fact that 77^: H{p^Pi) -»
/f is an isometry of horizontal subspaces where the metrics on these
subspaces are induced by that on T (Pi)M via ■77-12He and чты. ■
|2
7.1.8 Theorem. In the notation ofl.X.6 and 1Л.1, we have (for
aEAk(P]oP2^§j)) that 6i|+b2(«NAbl(?,oP2,§.). Moreover, in the
notation 0/7.1.5, (S"' + (a)). =8ы'(а,.).
Proof. This is clear from 7.1.6 and 7.1.7. ■
7.1.9 Note, /n order /0 motivate the rest of Section 7.1, we mention
that in Sections 7.2 and 7.3 тт,: P, ->M wi7/ fee the PFB S(M)^M with
connection в of 6.3.6. The bundle tt2\ P2^>M will carry the potential

I 7 INTERACTIONS 93
i{i.e., connection) that interacts with the free Dirac-type fields on P,
when everything is lifted to the arena P, о P2. The remaining considera-
considerations ofl.\ also apply to other situations of physical interest.
7.1.10 Definition. Referring to 3.3.1, we define J(P, V) = {(p, v, a)
&J(P, V)\oE:K\P,V)p}. In other words, we require a to vanish on
vertical vectors. We could have required this in 3.3.1, but then the
expression L(p,^(p),d^p) in 3.3.3 would not be defined. Note that
L(p,\p(p)^ D"\pp) does make sense, even if L is only defined on
J(P, V). Indeed, with the exception of 3.3.3 and 3.3.5, all that we have
done so far makes sense for Lagrangians defined only on J(P, V).
7.1.11 Definition. Suppose that p^. Gt^>GL(V) are commuting rep-
representations as in 7.1.3, and let L: J(Pl,V)^>U be a Lagrangian (see
7Л.10). For aeA'(P,°P2,F)(;,|i;>2), define ^o ЕЛ'(Р„ V)Pi by
(tt1o)('7t\.X)=o(X) (well defined, since a vanishes on vertical vectors).
Define L: 7(P, oP2, F)->R by L((px, p2),v,o)=L(pvv,Tflo).
7.1.12 Theorem. The function L: 7(P, о Р2, F)->R is a Lagrangian,
provided L is G2-invariant in the sense that L(p{, g2-v, g2-a) =
L(px, v,a) for all(px, v, a)E/(P,, F) and g2 EG2.
Proof. First note that for аЕЛ'(Р, °P2,V)(pt pi) and (g,,g2)EG,
XG2, we have ?(a» R^^ ) = (^a) 0 R , since wl((p
:p,g, =77-'(Yp,, P,))g,. Then we have
= L((pl,p2),v,a). Ш
7.1.13 Notation. Suppose that §, has a metric /c, such that &b:
Gi^>GL(§i) is orthogonal (г = 1,2). From /c, and k2 we can form a
metric kl2 on §]®§2 (such that §^±§2) in the obvious way. Assuming
that L o/7.1.11 /5 Gj-invariant as in 7.1.12, we have a Lagrangian L:
J(P{ oP25J/)->R. Let ш,+ш2 be cis in 7.1.6, and suppose that \[>E
C(P,oP2,V). Let/=/"|+(^/)ЕЛ1(Р|оP2,§,©§2) be the current, as

94 GAUGE THEORY AND VARIATIONAL PRINCIPLES
defined in 5.1.1 or 5.1.2. Accordingjo 7.1.5, J=QXJ+\J. In applica-
applications we will not have much use for §, /, but (in the notation of 7.1.5) we
7.1.14 More Notation. Let в(Р2) be the space of connections for
тт2: P2^M, and let §2: e(P2)^Cx(M) be the self-Action {i.e.,
§2(Ы2)= ~ 2(h2k2)(®Wl,QWl)). For a fixed connection w, on P,, we
define {t+%2): C{P,o P2,V)X6(P2)^ CX(M) by (£+S2)(,//, ^)(x)
^^ %2{^2){x) where x =
7.1.15 Theorem. Let /(^) be as in 7.1.13, and (£+§2) as in
7.1.14. The pair (»//, w2)E C(P, »P2, F) X6(P2) /5 stationary relative
to (£+§2) (z'n a sense similar to 5.2.2) iff conditions (A) and (B) hold:
(A)
(В)
Proof. The Lagrange equation (A) holds if \p is stationary for £. (or
£+S2) when w2 is fixed (i.e., w, +co2 is fixed). Thus, (A) is necessary.
To prove the necessity of (B), we follow the proof of 5.2.3. Let §|2:
6(P, oP2) ^ C°°(M) be the self-Action (i.e., §|2(ш,+ш2) =
~ 2(hnk\2)(®"' + ,®"'+))- Note that every connection onP,oP2
is of the form ш, +ш2. The proof of 5.2.3 almost applies to £+§|2.
Since we are keeping w, fixed, in the proof we must assume that
тЕЛ'(Р, op2, g2). We are then led to
0= /
Since т has values in §2, we can replace the forms in the inte-
integrands by their %2 projections. Using 7.1.8 and the identity
(hakl2)(<32a,§2(l) = (h2k2)(a2,(}2), we then have
0= f
Ju
It follows that (B) is necessary. The sufficiency of (A) with (B) is
clear, as in the proof of 5.2.3. ■
7.1.16 Remark. As a consequence of 8fl =J(\p), we have
8J(\p) = 0 (see 5.2.8). However, this continuity equation can be

7 INTERACTIONS 95
Meduced if we assume only that \p is stationary for £ relative to a fixed
^connection cc2. We prove this next, by a slight modification of the proof
О/5.1.5.
7.1.17 Theorem, //^e C(P[ о P2, V) is stationary relative to tfor a
fixed connection w,+w2 (see 4.1.3), then 8(/^) = 0.
Proof. Implicitly we assume that L is G2-invariant so that L is a
Lagrangian (see 7.1.12). H fEGA(Pl °P2) is of the form f(p,, p2) =
(pvf(p2)), then (from the proof of 3.3.6) fc(f*4>, /*w) = £(»//, со). In
the notation of 3.2.12, exp(F) is such a gauge transformation iff
FEC(Pl°P2,Q2). Proceeding as in the proof of 5.1.5 with FE
Р2Д)> we find
'и
Thus, §2E"'+Ч/)=0; and so (using 7.1.5 and 7.1.8) 8(/2) = 0, but
/(^)=/2 (See 7.1.13). ■
7.1.18 Remark. 77ze results of this section generalize to the situation
where more than two PFBs are spliced. This is needed when a particle
field interacts with several potentials.
7.2 THE (NONFREE) DIRAC ELECTRON FIELD
We derive Lagrange's (or Dirac's) equation and the inhomoge-
neous Maxwell equation, which govern the interaction of the Dirac
electron field with an electromagnetic potential. Much of this amounts
to specializing the results of Section 7.1.
7.2.1 The Preliminary Setup. Let M be a space- and time-oriented
Lorentz manifold with oriented frame bundle F0(M)^>M and spin
structure X: S(M)^>F0(M). Let в be the Levi-Civita connection on
F0(M), and suppose that в is the related connection on S(M) defined in
6.3.6. The PFB S(M)^M with group SXB,C) and connection в will
serve as the PFB чт^.Р^М with group Gx and connection со, of
Section 7.1. We suppose that it2: P2^>M is a PFB with group U(\) =
{e'e\6ER} and co2 is a variable connection (i.e., electromagnetic
potential) on P2. With F=C4, we define the representation p,: Gx ->

96 GAUGE THEORY AND VARIATIONAL PRINCIPLES
GL(V) to be p: SLB,C)^GL(C4) of 6.1.7 and we suppose that p2-
G2^GL(V) is p2: U(\)^GL(C4), given by p2(ei$)(v) = eiev. Note
that for AESLB,C), we have pl(A)(p2(eie)(v)) = pl(A)(ewV) =
ewpl(A)(v) = p2(e'e)(pl(A)(v)), as required in 7.1.3.
7.2.2 Definition. A Dime electron field (nonfree) is a particle field
in C(P\ °P2, V) (see 7.1.3) where Pv P2, p,, andp2 are given in 7.2.1.
7.2.3 Theorem. Let L: J(S(M),C4)->U be the Lagrangian of
6.3.11. Then L is G2-invariant in the sense o/7.1.12. Moreover, the
Lagrangian L: J(S(M)<>P2,C4)->R (supplied by 7.1.12) is explicitly
given by
L((pl,p2),v,o) = ti(iyX(;iflo),v)-mh(v,v)
where -n^a is defined in 7.1.11.
Proof. We have L( p{, e'ev, ei6a) = h(i(y X e'ea), e'ev)-
mh(e'%,e'ev). From the definition of yXa in 6.3.11, we see that
уХешд=ешуХд. Thus, it remains to prove that h(eww,ewv) =
h(w, v), but this follows from the definition of H in 6.3.8 and h in
6.3.10. The rest is clear. ■
7.2.4 Definition. For аЕЛ'(Р, о P2,C4)(Pi PiV we define yXa
= 2 Чуу(е,)(а(£у)) G C4 where Ej E T^pi)P, о Р2 satisfies
MPi) X'n\2*Ej) = ej=thejth standard basis vector in R4, X: S(M)^>
F0(M) being the spin structure. Note that E- is not unique, but if a
connection w2 on P2 is given, then E- is unique if (в + <Ь2)(Е1)=0 is
imposed. Also, if E-(ET S(M) is defined as in 6.3.11 and 6(E ) — 0 is
imposed, then it^E ) = E . Then we have у X W]a =
2vijY(ei)[(vlo)(Ej)] = 2r,ijy(ei)(o(Ej))=yXo. The use of yXa
rather than yXir]a is a matter of aesthetics.
7.2.5 Theorem. For ^ECE(M)oP2,C4), Lagrange's equation
relative to the Lagrangian L of1.2.3 and connection в + &2 is (for ф as
in 6.3.7) -i
Proof. Noting that чг1*ф(Е-) = e- is an equivalent characterization
for Ej as in 7.2.4, we see that V3L((pvp2)v,a)= — /(уо^'*)()

7 INTERACTIONS 97
a computation as in the proof of 6.4.2. Thus, we have
/d(De+4')= -/(y°77-'*9)(^). From the equation for L in 7.2.3,
see directly that dL/d4>=iyX(De+D>))-2m4>. The result fol-
follows. ■
7.2.6 Theorem. The Lagrange equation of 125 reduces to the
Dirac equation for an electron field \p influenced by the electromagnetic
potential oo2, namely, уХ(й?»^+@+<Ь2)-^) + гт^ = О.
Proof. We must prove that 8в + [(уотт1*<р)D/)] = -уХ De + (\p).
This is done by using 6.4.6 and analogues of 6.4.5 and 6.4.7. The
analogue of 6.4.5 is S +W2[(y ° тт1*ф)(\р)] = [у ° тт1*Х*(8 <р)](»^)
— * [(y°(* ■7Tl*<p))A(De + W2\p)]. The proof of this equation is simi-
similar; note that co2 A[(y отт1*ф)(\р)]= —(уотт]*ф)Аоо2-\р since the rep-
resentation p2: [/A)^ (jL(C4) commutes with the gamma matrices
(i.e., y(x)(e'ev) = e'ey(x)(v)). The analogue of 6.4.7 is
*
12
in the notation of 7.2.4. Again the proof is similar. The result then
follows, since 8e<p = 0 by 6.4.6. ■
♦
7.2.7 Theorem. Let §2 =%A) = {/a|aElRJ have the metric k2 with
k2(i,i) = \. Then the current component %2J {defined in 7.1.13),
relative to the Lagrangian L of 7.2.3, of the pair (\p, со) in
CE(M)oP2,C4)Xe(P2) is given by §2J- -й((уочг1*ф)(\р), \p)i.
Proof. In 5.1.2, let ex — i be the o.n. basis for §2, and note that §2J
is independent of the remaining o.n. basis vectors of §,Ф§2 that^span
S, (relative^ to some metric on §,). Then we see that %2J—
h(dL/d{De+"i\l>), i\p)i, which (from the proof of 7.2.5) is
h{ — г(у°77-'*ф)(^), i\p)i, and so on. ■
7.2.8 Remark. In 7.1.13 we found that_ there is a unique form
/^)EA'(P2,g2) such that tt2*JW2{^) = §2J. In the case at hand,
%2J {and hence /(^)) is independent of cc2 by 7.2.7. Let a: U^P2 be
a local section. Since U{\) is Abelian, the next theorem shows that

98 GAUGE THEORY AND VARIATIONAL PRINCIPLES
a*Jai(^)EA\U,6H(\)) yields a well-defined form (on M) that is
independent of the choice of local section a. We denote this form bv
7(^еЛ'(М%A)) (iej(t)i\U=o*J"^)
7.2.9 Theorem. Let it: P-^M be a PFB with group G and let
Gh>GL(V) be a representation. Suppose that au: U^P and av: V~>p
are local sections related by av(x) = au(x)guv(x) for some guv: UC\ V~>
G. Then for any aE~Ak(P,V) we have (ov*a)x=guv(x)-l-(ou*a)x. ln
particular, if V-% and G -> GL(§) is ®b, then (ov*a) =
Proof. The proof is a clear generalization of the proof of 2.2.14. ■
7.2.10 Theorem. Let a: U-^S(M)oP2 be a local section of the
bundle щ2: S(M)oP2^M. For ^ECE(M)oP2, C4), let ^' = a*^
(i.e., \p'(x) = \p(o(x))). Note that X ° ттх ° a /5 an oriented frame field on
U, say XQ,..., Хъ. Let ф=(ф0,..., ф3) be the coframe (i.e., <p'(Xj) =
S/). Then the form j(\l>)eAl(M,U), introduced in 7.2.8, is given by
Proof. Note that 77^00: U-*P2 is a local section. Thus jD>)i\U=
G72oCT)*/(^) = a*(g2/) = -/г([уо(а*о77-1*ф)](^/), ^')i = -h(y°
[(Лотт-1 оа)*<р]D,'),-фу = (Ъу the proof of 6.2.6)= -h(y[v]{\f/f),ty.
7.2.11 Theorem. Let a: U^> S(M)° P2be a local section, and write
co2 = -iA for co2 Ев(Р2). Define the "vectorpotential" A'EA\U,R)
byA' = (iT2oa)*A. Then the pair (ф, co2) in C(S(M)o Р2,С4)Хв(Р2)
is stationary for £+§2 (see 7.1.15, where L is given in 7.2.3) iff
(a) yX(d4> + F-iA)-4>) + imxp = O (Dirac equation);
(b) 8( — dA') —j(\p) (inhomogeneous Maxwell equation).
Proof. We show that this theorem is a special case of 7.1.15. By
virtue of 7.2.6, we have that (a) is equivalent to (A) of 7.1.15 in the
case at hand. Applying G72oa)* to both sides of (B) in 7.1.15, we
obtain (b). ■

7 INTERACTIONS 99
I7.3 THE NUCLEON IN A YANG-MILLS POTENTIAL
fflere we find Lagrange's equation and the inhomogeneous field
ation for a nucleon (proton-neutron doublet) subject to a Yang-
Is potential (i.e., connection on a PFB with group SUB)). The
d physicist will be quick to point out that this section is only
first step toward a physically realistic model (see 10.3).
^7.3.1 The Setup. We proceed as in 7.2.1. Now, however, тт2: Р2^>М
is a PFB with group G2=SUB). We set F=C4®C4 and define p,:
SLB,C)^GL(V) by Pi(A)(vl®v2) = p(A)(v1)®p(A)(v2) where p is
as defined in 6.1.7. The representation p2. SUB)^>GL(V) is defined,
for
by
Writing
we can express this definition in terms of matrix multiplication. The
linearity of p(A) gives us P\(A) op2(B)= p2(B) °P\{A) {for AE
SLB,C), BESUB)), as required in 7.1.3.
7.3.2 Definition. A nucleon field is a particle field in
C(S(M)oP2,C4®C4) (see 7.1.3 where p, and p2 are defined in 7.3.1).
We may write a nucleon field \p in terms of its C4 components
D' = -ф1®4/2) with \pl and \p2 being the proton and neutron components,
respectively. Note that SLB,C) respects these components, but SUB)
scrambles them.
7.3.3 Definition. Let h be the metric on C4®C4 given by h{V\®
v2,wx®w2) = h~(v[,wl) + h~(v2,w2) where h was defined in 6.3.10. We

100 GAUGE THEORY AND VARIATIONAL PRINCIPLES
defin_ef-U4 ^§i(C4®C4) byy(x)(v]®v2) = y(x)(v,)®y(x)(v2). For
44
p we set у Xa = (yXa,)©(y Xa2)
where a = ox®a2 and yXa, is defined in 6.3.11. Finally, we define a
function
)^R byL(p,v,o)=h(iyXo,v)-mh\v,v).
With the proof of 6.3.14, it is clear that L is a Lagrangian.
7.3.4 Theorem. The Lagrangian L is fi 2-invariant (G2 = SUB)) in
the sense of 7.1.12. The Lagrangian L: J(S(M)o P2,C4®C4)^R
(supplied by 7.1.12) is explicitly given by
Proof. The G2 -invariance of L follows from the next two lemmas,
and the rest is immediate from the definitions. ■
7.3.5 Lemma. For x£R4, AeSUB), u£C4®C4, and a£
A\S(M), C4©C4)p we have y(x)(A-v)=A-(y(x)(v)), and conse-
consequently yX(A-o)=A-(yXa), where we have used our customary
shorthand A -v = p2(A)(v) and A -o = p2(A) «a.
Proof. Let
'a b
с d
and write v=vl®v2. Then y(x)(A ■v) = y(x)(av] +bv2)®y(x)(cv] +
dv2)=A -(y(x)(v)) by the linearity of y(x). ■
7.3.6 Lemma. For A<=SUB) and u,w£C4©C4, we have h(A-
v, A -w) = h(v,w).
Proof. From the symmetry of h, it is enough to check the case in
which v = w. Let

7 INTERACTIONS
write v=vx®v2. Then
Г h(A-v, A-v)=h(avx +bv2, avx +bv2) +h(cvx +dv2,
=H(avx +bv2, avx +bv2)+H(cvx +dv2,cvx +dv2)
in the notation of 6.3.8 and 6.3.10. This equals (aa + cc)H(vx,vx) +
(ab+cd)H(v \,v2) + (ba+dc)H(v2,v,) + (bb + dd)H(v2, v2) — H(vx,
vx)+H(v2,v2) = h(v,v). ■
7.3.7 Definition. For теЛ1E(М)оР2,С4ФС4)(рь(,2) feryXrGC4
ФС4 be defined in the same manner as у Ха was defined in 7.2.4 (i.e.,
replace у by у).
7.3.8 Theorem. For ф£СE(М)оР2,С4©С4) and в the connec-
connection on S(M) induced by the Levi~Civita connection on F0(M) (see
6.3.6) we have that Lagrange's equation (relative to the Lagrangian L
of 13 A and the connection в + ы2оп S(M) ° P2) reduces to у X(d\j/ +
Proof. The proof is just a matter of checking that the proofs of
7.2.5 and 7.2.6 carry over to the case at hand. Note that one step in
7.2.6 requires that p2 commute withy (i.e., y(x)оp2(A) = p2(A)°y(x)
for xeR4, A £S£/B)), but we have this from 7.3.5. ■
7.3.9 Definition. We define the metric k2 on S2=S%B) to be that
positive definite metric such that a1 = — \irx, a2= —jir2, and
аъ= — \1тъ form an o.n. basis (see 6.1.3). Note that [a',aJ] = 1,ejjkak.
We can check that k2 is then invariant under &b: S£/B)->GL(S%B)).
In fact, k2 is determined up to a constant multiple by this invariance
property.
7.3.10 Theorem. The current component S2/G A](S(M) о Р2,
S%B)) (defined generally in 1.1.13) for the pair (\p,w2), relative to the
Lagrangian L of 7.3.4 and metric k2 of 7.3.9, is given by
k=\
which is independent of co2.

102 GAUGE THEORY AND VARIATIONAL PRINCIPLES
Proof. We obtain
as in the proof of 7.2.5. The rest follows from 5.1.2 and the definition
of §2J. ■
7.3.11 Remark. 5у_7Л.13, there is a unique J(xP) eA'(P2, g,)
such that tt2*JuD>) = §2J. In the case at hand, J(xp) is independent
of co2. // o2: U2 ->P, ° P2 is alocal section of тт2: P, °P2 ->P2, then
JU2D>)\U2=o2*tt2*JU2D>) = o2*§2J, which is independent of a* since
J(xp) is independent. However, because SUB) is non-Abelian, the
form JU2D>) does not pull back (via a local section of 7т2: P2^>M) to a
form independent of that gauge. The next theorem clarifies the situa-
situation .
7.3.12 Theorem. Let au: U^S(M)°P2 be a local section. For
^GC(S(M)oP2,C4©C4), let ^и=о*ф (i.e., ^H(x) = ^(aH(x))). Note
that Ao7r'oau: U^>F0(M) is a frame field on U. Let qpu be the
corresponding dual coframe field on U. Note that ■7T2°ou: U^>P2 is a
local section. We then have (for
-! 2 л(у(
If av: V-*S(M)°P2 is another local section with GT2°ai;)(x)
(it2 °ou)(x)guv(x) for some guv: UHV^ G2=SUB), then
Proof. The expression for the pulled-back current follows from the
proof of 7.2.10. The transformation equation is a special case of
7.2.9. ■
7.3.13 Remark. Physicists sometimes call (it2 ° au)*/(v//) e
A'(M, S%B)) (after dividing by i~^J— 1 ) the isospin 4-current. It
depends on the choice of gauge au, but tranforms nicely under a change

7 INTERACTIONS
103
g (as in 7.3.12). Perhaps it is preferable to think of the isospin
rent as being the well-defined (gauge-independent) form J(xp) on
$ At this point, we could merely state that the Lagrange equation and
homogeneous field equation for the nucleon in a Yang-Mills potential
given (as a special case of 7.1.15) by yX(dxp+(e+co2)-xp) + imxp
JteO and5fl =J(\p). In the case where M is Minkowski space, we
twill proceed to check that these equations are equivalent to the corre-
Isponding equations in the Yang-Mills paper [1954].
7.3.14 Notation. For w2 G6(P2), we write
where У &A\P2,U) (or b<=A\P2,U3)). We define a form
A2(P2,№y) by the equation (bXb)(X,Y) = b(X)Xb(Y) where the
right-hand side is the cross product of vectors in R3.
7.3.15 Theorem. In the notation of 7.3.14, we have
Proof. By definition,
т, —ib-т]. Now
jk
Я = do>2 + \[u2, co2] = -/ db • т+ {[-ib •
jkm
= -'■2
jk
= -iBbXb)-r.
7.3.16 Notation. The equations in the Yang-Mills paper are defined
on Minkowski space M=R4. Since any PFB over U4 is trivial, we can
pull our equations down to M by means of a (global) section a:
, M-*S(M)°P2. Moreover, we can assume that Х°тт^°а is the usual
t coordinate vector frame field on U4. Using 6.2.6, we find that 0 = в —

104 GAUGE THEORY AND VARIATIONAL PRINCIPLES
^). We write o*(b) = (тт2о o)*(b) = b', and
7.3.17 Theorem. In the notation above, Lagrange's equation (when
pulled down to M via a) is
where ^'^=Ц'/Ъх^Ь'^Ь'^), and у^ = 2т)^у(е,).
Proof. Pull the equation of 7.3.8 down to M via a, noting that
a*@*) = O and £
7.3.18 Notation. We define the form f £ Л2( P2, U3) by fi = if • т.
By 7.3.15, f=-(db+2bXb). We write /' = (тг2°а)*( /), /;„ =
7.3.19 Theorem. /« the notation above, the inhomogeneous Yang-
Mills equation 8fl =/(ф) (when pulled down to M via о°тт2)
becomes
a,13
where
Proof. By 4.2.8 and 3.1.5, we have 8fi = *2(d*pf- r) + [-ib-
T,* (if'7)])- The result follows from the next two lemmas along with
7.3212. ■
7.3.20 Lemma. We have (*
Proof. See 0.2.19. ■
7.3.21 Lemma. We have *[*'*т,*/'-т](Эм) = 2/2а/3 r^(b'a X/^)'
Proof. Note that

7 INTERACTIONS 105
HIS,
*[А'-т,*/'-т]=2*(б'УА(*/'*))[т/,т*]
Jk
= 2i2^m*F'M(*/'*))T-,
and it remains to prove that *{b'JЛ ^)(Эм) = 2аДт)а^/;*. Now
This is the p,p2p3-component of the star of the 1-form with ju-
component ff^b'Jf^. Since **= 1 on 1-forms, the result follows. ■
7.3.22 Remark. The equations in 13Л1 and 7.3.19 differ slightly
from those in the Yang-Mills paper. One difference can be attributed to
a different choice of k2 on §2 =S%B). In the Yang-Mills paper, т1,
t2, t3 are orthonormal. Moreover, we have absorbed the constant e into
b. Finally, the gammajnatrices in the Yang-Mills paper are (~i) times
ours, and the metric h is expressed differently.

CHAPTER
8
Calculus on the Frame Bundle
In this chapter, the essentials of tensor analysis on a (pseudo-)
Riemannian manifold are developed. These tools will enable us to
formulate a principle of stationary Action that leads to the Einstein
field equation for a gravitational field under the influence of the field
strength of a gauge potential. The Action density involves a certain
tensor called the Riemann-Christoffel curvature tensor. In Section
8.2 we show how this tensor is essentially the curvature of the
Levi-Civita connection of some given metric. In Section 8.3 we find
formulas describing how the Riemann-Christoffel curvature tensor
changes under an infinitesimal change of metric. These formulas are
used in Chapter 9 to compute first variations of functionals defined
on the space of metrics on a given manifold (e.g., to derive the
Einstein field equation from an Action principle).
David D. Bleecker, Gauge Theory and Variational Principles ISBN 0-201-10096-7
Copyright © 1981 by Addison-Wesley Publishing Company, Inc., Advanced Book
Program/World Science Division. All rights reserved. No part of this publication
may be reproduced, stored in a retrieval system, or transmitted, in any form or by
any means, electronic, mechanical, photocopying, recording, or otherwise, without
the prior permission of the publisher.
106

? 8 CALCULUS ON THE FRAME BUNDLE 107
In Section 8.1 we show that tensor fields on a manifold can be
regarded as certain vector-valued functions on the frame bundle of
the manifold. One purpose for doing this is that physicists are used
to working with components of tensors, whereas "modern" differen-
differential geometers prefer to work with invariant notation (e.g., indepen-
independent of coordinate systems). On the frame bundle, components of
tensors have invariant meaning; hence there is some possibility of
pleasing everyone equally. Also, complicated computations involving
many contractions, index raisings, covariant differentiations, and so
on are much easier to carry out by using components of tensors. We
can take advantage of this greater facility, without sacrificing invari-
ance, by working with components on the frame bundle.
8.1 TENSOR FIELDS ON ЦМ)
Let M be an «-manifold with m: L(M)^> M the frame bundle of
1.1.8 with group GL(nM).
8.1.1 Definition. Let K" be the dual space of R", and for integers
p,q&*0, let Tp-q be the vector space of all multilinear functions f:
R"X -P■ XR"XR"X •?• XR"-*R. Define a representation
GL(n,U)^GL(Tp-q) (for A<=GL(n,U), f^Tp-q, e,,..., t^ER",
f, w?eR") by (A-f)(vb...,vp, wl,...,wq)=f(v]oA,...,vpoA,
A~]wl,...,A~]wq). Note that A (B ■ f) — (AB) ■ ffollows because of the
use of A~xWj instead of Awr The space of L-tensors of type (p, q) is
C(L(M),Tp-q) = {f: L(M)^Tp-q\f(uA)=A~x -f(u) for all
L(M) andA<=GL(n,[
8.1.2 Theorem. There is a natural one-to-one correspondence be-
between the space of L-tensors of type (p, q) and the space of tensor fields
oftype(p,q) on M (i.e., C(L(M), Tp-q)s$p-q(M)).
Proof. For f<=C(L(M),Tp-q) we define f<=6Sp'q(M) by
l(ll)(Xi,...,Xp,Y,,...,Yq)=f(u)(XioU,.._.,u-\Yq)) where X, E
Т„(и)М, y.Grw(u)M, and u^L(M). Now/is well defined, since

108 GAUGE THEORY AND VARIATIONAL PRINCIPLES
Conversely, given f<=$p-q(M), we define f<=C(L(M),Tp-q) by
f(u)(vi,...,wq)=f7l(u)(v]°u-\...,u(wq)); note that it follows that
f{uA)—A~x-f(u). These correspondences are inverses. ■
8.1.3 Definition. Let еи...,е„ be the standard basis for U" with
dual basis ё\...,ё". For fE:Tp-q we define the components of f by
8.1.4 Theorem. Forf<=Tp-qandA<=GL(n,U),wehave
(A-f))'"i!=Ail\---A's(A-iy}--- (Л~хУ'Н\У"
^ J ' J\ Jq >\ lpX ' J\ V ' JqJ 7l ' ' 'J
where A'j=e'{Aej) are the matrix entries of A.
Proof. We have
8.1.5 Theorem. Let Ql(n,U)^Ql(Tp'q) be the homomorphism of
Lie algebras derived from GL(n,U)^GL(Tp-q). For В<=Ш(п,П) and
-q, the derived representation is given by
- BJ[ f'x '"'p.— ■ ■ ■
Proof. In 8.1.4, set A=e'B, and apply d/dt\r=0. ■
8.1.6 Definition. The components of f<=C(L(M),Tp'q) are the
functions fjx\\':Ip: L(M)^U given by fj[l.'.'.'jp(u)=f(u)j\'-'.'-'jpll- We write
f=(fj'r.'.'j'p). rfote that the components of j are defined without refer-
reference to any coordinate system. Observe that f/r.'.'j'p(u) are the compo-

I 8 CALCULUS ON THE FRAME BUNDLE 109
nents offv(u) (in the proofof"8.1.2) relative to the basis w(e,),..., u(en)
8.1.7 Theorem. Let f/'.'.'.'Jp- L(M)-*U be a collection of functions
!</,,..., jq<n. There is an L-tensor f^C(L(M),Tp-q) such that
f^(fh;;;jp) if and only if we have (for all и ^L(M) and A
the rule
Proof. For f<=C(L(M), Tp-q) we have fj?.'::}"(,uA) =/(мЛ)''' ;;>=
{A~^-f{u))l}\"'p, and then the rule follows from 8.1.4. The converse
is now clear. ■
8.1.8 Theorem. Let /=(/V;.^)eC(L(M),P'«). // we set an
upper index of fj'^.'.'J" equal to some lower index and sum over that
index, we obtain the components of a new L-tensor of type (p—l, q— 1),
called a contraction of f; there are pq different ways of contracting.
Proof. In the transformation rule of 8.1.7, note that if we contract
with respect to, say, z, andy',, then we hae (A~l)^AJJ = 8^ as a factor
on the right-hand side. Hence we obtain the required rule for the
collection of contracted components. ■
8.1.9 Theorem. Iff<=C(L(M),TP-i)andh<EC(L(M),Tr-s),then
there is an L-tensor fh £C(L(M),Tp + r-q + s) with components
fi''.'.'.''ph'p + i' V + '. The new L-tensor fh is called the tensor product off
and h.
Proof. Upon multiplying the transformation rules for / and h, we
obtain that for fh. ■
8.2 PSEUDO-RIEMANNIAN GEOMETRY
Here we describe the covariant differentiation process and
Riemann-Christoffel curvature tensor of pseudo-Riemannian geom-
geometry in terms of the frame bundle formalism of Section 8.1. We work
on L(M) instead of F(M) because (in the next section) we consider

110 GAUGE THEORY AND VARIATIONAL PRINCIPLES
what happens to the curvature tensor under a variation of the metric.
By using L(M), we need not cope with the problem of F(M)
changing with the metric.
8.2.1 Theorem. Let g be a metric on M and let g(EC(L(M), T0-2)
be the corresponding L-tensor given by 8.1.2. Let <p£A'(L(M),[R") be
the canonical \-form of 6.2.2. Then there is a unique connection §
on L{M) such that Deg=0 and £>"ф=0.
Proof. Let в be the Levi-Civita connection on F(M) (see 6.2.5). We
define в at points uGF(M)CL(M) by ви(А*)=А for A<=GL(n,U)
and ви(Х) = Ои(Х) for X<=TUF(M). This uniquely determines §u on
TUL(M), for uGF(M). AnypGL(M) is of the lormp = uA for some
u<=F(M) and A<=GL(n,U) (not unique). We define 0p on TpL(M),
for X<=TUL(M), by 9p(RAitLX) = &bA-^u(X). A simple argument
shows that §p is independent of the choice of и and A. Observe that
the horizontal subspaces of в at points u^F(M) are tangent to
F{M). Also, the components gtJ of g^C(L(M), Г02) are constant
on F(M) (i.e., gij = r\ij °n F(M)). Thus, Deg=dg"-0 on F(M).
Since z/g£A'(L(M), Г0-2), it follows thati/g = 0 on all of L{M).
Also note that D*qp = 0 implies that De<p = 0 on F(M), and hence
8.2.2 Remark. 77z/s proof clearly generalizes to give the fact that
any connection on a "sub-PFB" extends uniquely to a connection on
the ambient PFB. Also, there is no harm in calling в (as well as в) the
Levi-Civita connection associated to g.
8.2.3 Definition. For v GU" and u£L(M),the standard horizontal
vector in TUL(M) associated to v is the unique horizontal vector
vu &TUL(M) (relative to 9) such that <p(uu) = u (i.e., u~\'iTijivu) = v).
The assignment ut-*vu defines the standard horizontal vector field on
L(M) associated to v. This definition makes sense relative to any
connection on L(M), but we do not need this generality.
8.2.4 Theorem. For v^W andA£GL(n,W), we have RA^(v) =

8 CALCULUS ON THE FRAME BUNDLE 111
Proof. Note that RAJ[vu) is a horizontal vector in TuAL(M), and
8.2.5 Theorem. There is a natural isomorphism A'(L(M), Tp-q)ss
C{L{M),Tpq+x) (depending on в) which we denote by /н> / where
f{u){vx,...,vp,wx,...,wq+x) = fu{wq+x){vx,...,vp,wx,...,wq) ("w? + 1E
TUL(M), as in 8.2.3).
Proof- We check that f(uA) = A'l-f(u). Using 8.2.4, we have
f(uA){vx,...,wq + x) = fuA(wq + x)(vx,...,wq)
Since / vanishes on vertical vectors, the map /W / is one-to-one, and
this calculation reveals surjectivity. ■
8.2.6 Notation. Given f <= C{L{M), Tp'q), the form Def^
~ti{L(M),Tp<q) gives rise {via 8.2.5) to an L-tensor Def<=
C(L(M), Tp-q+{). The components of this new tensor are denoted by
Jj\
:L(M)
For those who know of covariant differentiation of tensor fields on M,
we mention that the tensor field on M corresponding to Def is the
covariant derivative of that corresponding to f. Since everyone has a
different definition of covariant differentiation, we leave it to you to
show that this is correct for your definition. Finally, if the operation
ft-*Def is repeated к times, we write
Jj\-Jq |У„+1 •••/, + !
for the components of the resulting L-tensor.

112 GAUGE THEORY AND VARIATIONAL PRINCIPLES
8.2.7 Theorem. For f<=C(L(M),Tp-q) and ek the standard hori-
horizontal vector field on L(M) corresponding to the vector ekE:R", we
have
which is the directional derivative of fj ['.'.'jp: L(M)->R along the
field ek.
Proof. We have
8.2.8 Raising and Lowering Indices. The matrix of components
gtj of the metric tensor gE:C(L(M),T0-2) can be inverted to give a
matrix of functions gIJ. Using 8.1.7, we can easily check that the g'J are
components of some tensor, say g~'GC(L(M), Г2'0). For /G
C(L(M),Tp'q), we can form the tensor product gf (or g~xf) as in
8.1.9, and then we can contract an index of g (or g~x) with an upper
(or lower) index of f as in 8.1.8. This process changes f to an L-tensor
of type (p — \,q+ 1) (or (p+ \,q— 1)). When performing such opera-
operations, it is a good idea to write the components of f in the form
f'x 'Oi •••>' t>ecause we WM use the following shorthand (where the
symbol ik means that ik is to be erased, etc.)
and
f'\---ip " =s»Jkfh-'p
8.2.9 Theorem. For L-tensors f and h we have De (fh) = De (f)h
+ fDe (h). In terms of components,

8 CALCULUS ON THE FRAME BUNDLE 113
Proof. This follows from 8.1.9 and 8.2.7, using the product rule for
differentiation. ■
8.2.10 Note. The symbols fr-i...im...j4{ii and р-^.^...^
seem ambiguous because it is impossible to tell whether the raising or
lowering of indices occurred before or after the covariant differentiation.
Fortunately, it does not matter, since glJ\ll = 0 and g'J^ = 0, as a
consequence of Deg=Q {see 8.2.1); that is, the gtJ and g'j behave like
constants under covariant differentiation.
8.2.11A Theorem. Let /н> jf be the isomorphism of 8.2.5, and suppose
that Se-. A\L(M),Tp'q)^C(L(M),Tp'q) is the covariant codifferen-
fia/o/4.2.8. Then for /G A\L(M),Tp'q) we have
(«•>)r-i=-/''-"v ■■>>
Proof. Let ф=(ф',...,фл)еЛ1(ЦМ),[йл) be the canonical 1-form.
For the standard horizontal fields e,,..., e~n we have ф'(ё7) = Ц- Thus,
the iormf<=A\L(M), Tp-q) can be written as/ = 2/(e,)<p,. Recall
that g is the metric on M, and note that (for u^L(M))
(^Ш^,, ej) = g(nji, v^j)=g(u(et), u{ej))=g(u)(el, ey)= gu(u).
Thus, we have */ = 2g'V(e,)|g|1/2ei,-i...1.n_^'>® • • • ®ф''»->. Note
that Z)V = 0- Also< since ёт[8*А= gij\m'=°"™e have em[g;7t] = 0 and
] = o. It follows that
We see that r>'(*/) = *Bg'*ej/(e,-)])- Thus, 8e/=-(-l)**Z)V=
-Bg'^J/(e,)]) EC(L(M), Г'-»), since *2 = (-\y on 0-forms. The
result now follows. ■
8.2.12 Corollary. For s£C(L(M),T0]) with compact projected
support, we have
>
m l
Proof. Since roo=IR with GL(/!,R)-> GL(T°'°) being trivial (i.e.,
ix = i VxGR), we have я1',, GC(L(M), Гао) is constant on the

114 GAUGE THEORY AND VARIATIONAL PRINCIPLES
fibers and hence JMs',,/л makes sense. The obvious metric on Г°-° is
g(x^y) = xy. Applying 8.2.11 and 4.2.9 and writing s=f for some
/GA'(L(M), Г00), we obtain
м
8.2.13 Definition. Let fl = De§£ A2(L(M),GL(n,U) be the curva-
curvature of в, and let e,,..., en be the usual basis ofU" with el,...,en being
standard horizontal vector fields as in 8.2.3. Note that we have a
function Щёрек): L(M)-+ §Цп,П)- Define functions Rhljk: L(M)^U
by the relation u(ej,ek){ei) — ldhRhiJkeh or equivalently RhiJk —
eh(u(eJ,ek)(ej)) where ёх,...,ёп is dual to ex,...,en. The functions
Rhijk are the components of an L-tensor (see 8.2.14) called the
Riemann-Christoffel curvature tensor.
8.2.14 Theorem. The Rhijk are components of an L-tensor in
C(L(M),TU3).
Proof. We use 8.1.7. For u<=L(M) and A eGL(n,R), Rhljk(uA) =
h
eh(Q(uA)(en ek)(el))=eh{(A-] AJj AUh
Now Aei = Ai'iei,,ehoA-x=(A-xfh.eh', and (by 8.2.4) RA-i
AJ'jer. Thus, we see that Rhljk(uA) = (A-])hh,Aii'Aj'AkkRh'I,J,k,(u).
8.2.15 Theorem. For the standard horizontal vector fields e,,..., en
(relative to the Levi-Civita connection в), the Lie brackets [ej,ek]
satisfy -eh(9([ejrek])(eI)) = Rhljk and [e,,ej"=0.
Proof. We have Щёр ёк) = йЩ, ёк) = ё^в(ёк)} - ёк[Щ)} -
в([еу, ек])= — e([ej,ek]). The first equation then follows from 8.2.13.
Now 0=&в(ёрёк)=О°ф(ё^ёк) = ё;.[ф(ёк)]-ёк[ф(е^]-ф([ё;;ёк])
= — ф[(еу, ek}) since ф(еу) and ф(ек) are constant. Thus, [ву,ек]н = 0.
8-2.16 Note. Although we will have no use for connections со (on
L(M)) other than Levi-Civita connections, Definition 8.2.13 still
makes sense for со and 8.2.14 is still true. From the proof of 8.2.15 we

8 CALCULUS ON THE FRAME BUNDLE 115
see that in general в"(еу, ёк)~ — ф([еу, ёк}). So the torsion form
measures [ё^,ёк]н, while the curvature form measures [ej,ek]v.
8.2.17 Theorem (First Bianchi Identity). For an arbitrary con-
connection со on L(M) with torsion form ®u=Dw<p, we have DU@U-2W
Лф. In the case in which в" = 0 (e.g., co = #), this identity is equivalent
Proof. Since Du@u=DuDu<p, the equation Ои&и=пи/\ф is a spe-
special case of 5.2.5. If в"=0, then й"Лф=0 implies that 0 = (fi"A
ргк ^ 1^}
Now ф(е,) = е,. So applying eh to both sides and using Rhijk=
eh(ttu(ej, ek)(et)) gives the result. ■
8.2.18 Theorem (Second Bianchi Identity). In the case where
в" = 0 (e.g., co=0), the identity DT = 0 is equivalent to Rhljkl,n +
h h0
Proof. Note that Dwuu = 0 as a special case of the general Bianchi
identity of 2.2.8. By the invariant definition of exterior derivative and
the fact [ё,.,ё/.]// = 0 of 8.2.15, we have (DaQa)(em,ej,ek) =
em[Qa(ej,ek)] + ej[Qa(ek,em)] + ek[Qa(em,ej)]: which at each point
u^L(M) is an element of GL(n,U). Letting it act on en and letting
the result be acted upon by eh, we obtain the result. ■
8.2.19 Theorem. Let Rhijk be the components of the curvature
tensor associated to the Levi-Civita connection of some Riemannian
metric g. Recall that Rhijk —ghmRmijk are the components of an
L-tensor in C(L(M), T0A). We have the identities:
A) RhiJk = -Rhik/, B) Rhljk+Rhkll+Rhjki = 0;
C) Rhuk = -RlhJk; _ D) Rhljk=RJkhl\
E) Rhijk\m+ И-ИШАк^ Rh,km\j -Q-
Proof. Identity A) follows from П(ёу, ёк)= ~Щёк, ёу), while B)
and E) follow from^.lAl and 8.2.18 (see also 8.2.10 for E)). For
C), note that 0= De"(D°g)= U-g by 5.2.5. Recalling (in 8.1.5) the
way Ш(п,М) acts on Г0-2, we then have 0 = (Щё ,ёк)-g)hi =

116 GAUGE THEORY AND VARIATIONAL PRINCIPLES
, ek)h'gh'i - fi(^' ek)khv = -Rihjk -Rhijk> whence we have C).
id h ri
For D),
consider the matrix
Rhtjk
Rkh,j
Rjkhi
Rijkh
R
R
R
R
hkij
kjhi
jikh
ihjk
R
R
R
R
hjki
kijh
jhik
ikhj
Note that the first column is obtained by cyclically permuting the
four indices, while the rows are obtained by cyclically permuting the
last three indices of the initial entries. Let p, be the sum of the entries
in the z'th row. Since p,=0 by B), we have 0 = p, +p2 — p3 — p4 =
2Rhijk -2Rjkhi, using only A) and C). Thus, D) holds. ■
8.2.20 Definition. The Ricci tensor is that tensor in C(L(M), Г0'2)
with components Rjk=Rjijk. The scalar curvature is the tensor in
C(L(M), T°'°) with component R=Rkk =gikRik. Note that Rik=Rki,
since (using 8.2.19 D)) we have Rik=RJljk=ghjRhijk=ghJRjkhi =
RJ'kjj—Rki- Using the identities of 8.2.19, we can verify that the Ricci
tensor and scalar curvature are the only tensors (up to sign) that can be
obtained by contracting the "full" curvature tensor (Rhijk) after possi-
possible index raising or lowering.
8.3 METRIC VARIATIONS
For the next chapter, we derive formulas in this section that give
the change of the Riemann-Christoffel curvature tensor under an
infinitesimal change of metric.
8.3.1 Notation. Let g,<=C(L(M),T0-2) be a (smooth) family of
metric tensors. Let 9t be the Levi-Civita connection (for gr) on L(M).
We set
= 0
a £, d
and 9—-J-
dt
= 0
8.3.2 Theorem. We have в' E A'(L(M), Ql(n, U)) a A'(L(M),
Ги). Hence 9' determines (via 8.2.5) a tensor в'EC(L(M), Г1'2),
which we denote by 9' for the sake of simplicity.

8 CALCULUS ON THE FRAME BUNDLE 117
Proof. We omit the full proof that 0' exists. Let it suffice to say
that 0, is uniquely determined by the equations dф + 0rAф = O and
dg, + $ ^g, = 0; thus, since g, depends smoothly on ?, it is reasonable
to expect that 0, does, too. The isomorphism in 8.3.2 follows, once we
prove that the representations
@,b:GL(n,R)-+GL(§l(n,RJ) and GL(n,U)^GL(TXA)
are equivalent. Define an isomorphism $£(h,IR)->7'u by B^>B
where B(v,w)~v(B(w)). For A£GL(n,U), we have Г ~
which proves that B->B is an equivalence of representations. The
rest is clear. ■
8.3.3 Theorem. // 0 is the Levi-Civita connection of g=g0 (i.e.,
? = 0), then we have (in the notation of 8.3.1) Deg'=~0'Ag and
0'Лф О
Proof. Since 0, is Levi-Civita for gn we have Q = dgl+0,/\gl. Dif-
Differentiating, at t = 0, we obtain O = dg' + 0'Ag+0Ag' or Deg'=~0'
Ag. Similarly, the equation dy + 0t Лф = 0 yields 0'Аф = О. ■
8.3.4 Theorem. Let 0'kl} be the components of 0'EC(L(M), 7"-2)
fl^ ш 8.3.2; /e? g=g0; fl«cf /e? g' be as in 8.3.1. 77ге« we /zaue
/. We write the equation Deg'=—0'Ag of 8.3.3 in terms of
components by using the formula for the action of Ql(nM) on Тол.
The result is equation A); B) and C) are obtained by cyclic permuta-
permutation of m, i,j.
8>ns, 0)
(з)

118 GAUGE THEORY AND VARIATIONAL PRINCIPLES
The equation 9'Лф=0 of 8.3.3 yields
or 9"mk = 9"km. Thus, subtracting C) from the sum of A) and B)
yields
g' -f«' —а' = 79'S V
&mi\j &jm\i &ij\m ij&ms'
Multiplying by gmk and summing over m yields the result. ■
8.3.5 Caution. The symbols 0'*. are not Christoffel symbols. How-
However, they can be related {via a local section of L(M) induced by a
local coordinate system on M) to the t-derivative of the Christoffel
symbols.
8.3.6 Theorem. Let Rh,jk(t) be the components of the curvature
tensor ofgt, and set Rhjjk' = Rhijk'@). We have
Proof. Let ej(t) be the standard horizontal vector field relative
to в,. Then Rhijk(t) = eh(tt(t)(eJ(t),ek(t))(ei)), where Й(Г)= fif. For
ej = ej@), we have q>(ej(t) — eJ-) = ej—ej=Q (i.e., eJ(t) — eJ is vertical).
Since fi(f) vanishes on vertical vectors, RhiJ.k(t) = eh(Q(t)(eJ,ek)(ei)),
and so RhIjk' = ё"(й'@)(ё^ё-к)(е()). Now Q(t) = d6t+\[§tJt], whence
fi'(O) = dO' + [9,9'] = D59'. Thus, ti'(O)(ej,ek) = (d6')(ej,ek)
= ё\9\ёк)]-ёк[9'(ё^], since [ёу, eJH =0 by 8.2.15. From 8.2.7 and
8.3.2, it then follows that R^'^ ёи(Щ0)(ё^ёк)(е,))=9^ки-9%lk.
By 8.3.4, this last quantity is
28 r\8'rk\ijJt~g',r\kj~8'ik\rj~8'rj\ik~8',r\jkJr8ij\rk) •
By the next theorem, g'ir]kj-8',r\jk= ~RS,jk8'Sr-Kjk8',s, and the
result follows. ■

8 CALCULUS ON THE FRAME BUNDLE 119
8.3.7 Theorem. Using the notation of 8.2.6, we have (for /G
C{L{M),TP-q)) the "Ricci identity"
if'p —fif-'p —f'\--'pj?s + . . . +/■'! ■■ ■ <p Ds
J\--Jq\km Jj\---Jq\mk Jsjf-jq )\km Jj\--Jq-\S J
km
Jjr-Jq skm Jjr-Jq ^ skm-
Proofs By 5.2.5, we^have D\Def) = пвЛ/. Now De\D'f )(ёк,ёт) =
ё^вУ(ё„)]-ёт[^вУ(ё^] = ёк[ёт[П]-ёт[ё^Г]]. Thus, the com-
components of De(Def)(EA2(L(M),TP-ci), regarded as an element of
C(L(M), Tp-«+1) (iterate 8.2.5), are /^//^-/Д.:^. Using 8.1.5
with В = пв(ёк,ёт) and Bhi=Qe{ek,em)hi = Rhikm, we see that the
components of tie A/(regarded as in C(L{M),TP-q+2)) are given by
the right-hand side of the desired equation. ■
8.3.8 Theorem. Let Rhijk(t)= g'm(t)Rhmjk(t), and set Rh'jk' =
Rhijk@). Then we have
where the indices on the right-hand side have been raised using the
initial metric g = g0.
Proof. Note that gij{t)gJS{t)= 8/, and so gIJ'@)gJS@) + g'7@)gy/@)
= 0, whence g""'@)= - g%sgsm. Hence, Rh'jk'= g'm'@)Rhmjk +
g"nRh,nJkW=-g%sgsmRhmjk+g"nRhmjk'- The first term is
— RhsJkg"s, while the second is computed by using 8.3.6. Note that
there is a term within the second term that would cancel with the
first term were it not for the factor of j in 8.3.6. ■

CHAPTER
Unification of Gauge Fields and Gravitation
The Einstein field equation for empty space-time (Л, — jRgt ■• = 0)
arises from setting the first variation (with respect to the space-time
metric) of the integral of the scalar curvature equal to zero. In
Section 9.1, a coordinate-free method is given for computing first
variations of integrals of scalars formed from the curvature tensor by
means of tensor product, index raising or lowering, and contraction.
This method not only yields the Einstein field equation from a
variational principle, but also gives a nice proof of the Gauss-Bonnet
theorem (see 9.1.10).
In Section 9.2 we investigate a consequence of the fact that
integrals of scalars, as above, are invariant under a change in metric
via pull-back (see 9.2.1) by a diffeomorphism. We already found in
Chapter 5 that the current of a particle field is covariantly conserved
David D. Bleecker, Gauge Theory and Variational Principles ISBN 0-201-10096-7
Copyright © 1981 by Addison-Wesley Publishing Company, Inc., Advanced Book
Program/World Science Division. All rights reserved. No part of this publication
may be reproduced, stored in a retrieval system, or transmitted, in any form or by
any means, electronic, mechanical, photocopying, recording, or otherwise, without
the prior permission of the publisher.
120

9 UNIFICATION OF GAUGE FIELDS AND GRAVITATION 121
because of the gauge invariance (or internal symmetry) of the Action
density. In the case of invariance of metric-dependent integrals under
pull-back by diffeomorphism (i.e., external symmetry), we arrive at
another conservation law. A special case of such a law is the
conservation of the energy-momentum tensor in general relativity. A
more explicit description of the basic idea is given in 9.2.1.
The main goal of this chapter is the natural unification of gauge
fields and gravitation. The first model of this type goes back to the
five-dimensional model of Kaluza [1921] and Klein [1926] for the
unification of gravity and electromagnetism. Their model extends in
a reasonably straightforward way to the case of gauge potentials on
PFBs with arbitrary (e.g., possibly non Abelian) groups. This exten-
extension is carried out in great detail in Section 9.3, but the basic idea is
outlined in the next paragraph.
Let t: P->M be a PFB with group G over a space-time M with
metric g. If со is a connection 1-form (i.e., gauge potential) on P and
к is some fixed (Jb-invariant metric on §, then we can construct a
metric h (depending on g, k, and со) on P. It happens that for all A in
G, R/. P->P is an isometry of (P, h). As a consequence, the scalar
curvature R: P->R of h is constant on the fibers, and hence yields a
function on the base M. Since this function depends on both g and
со, we denote it by R(g, со): M->U. If U is an open subset of M with
compact closure, let Iv(g, со) denote the integral of R(g, со) over U.
We will prove that if (for all U), Iyig, со) is stationary for variations
(with support in U) of g, then the Einstein field equation holds for g,
where the energy-momentum tensor (i.e., source of the gravitional
field) depends on the field strength fi". Moreover, /^(g, со) is sta-
stationary (for all U) for variations of со (with projected support in U)
if со satisfies the Yang-Mills equation. Thus, the Einstein field
equation and the Yang-Mills equation arise simultaneously from a
single variational principle that derives from the scalar curvature of
the metric h on P. In Section 10.1 it is shown that the geodesies of
(P,h) project down to generally nongeodesic paths of charged
particles on M, where the charge is essentially the vertical component
of a tangent vector of the geodesic in P. In view of this, we can
hardly deny the physical significance of the geometry of (P, h).
In spite of the foregoing, these Kaluza-Klein type models are not
regarded as solutions to the unified field theory problem. One reason
is that these models do not predict the values of certain universal

122 GAUGE THEORY AND VARIATIONAL PRINCIPLES
constants, such as the ratio of the strength of the gravitational to the
electrical force between the proton and electron. This ratio is around
jq-40 -р^е pOSSibility that such a small constant can ever be pre-
predicted on purely theoretical grounds seems remote. It is perhaps
more likely that such constants are just arbitrary parameters that
were frozen by some spontaneous symmetry-breaking mechanism at
some initial stage of the big bang. Since the relative abundances of
atomic elements and their chemical properties are sensitive to the
values of these constants, it is conceivable that their values might be
characterized as those at which life intelligent or curious enough to
contemplate them could evolve. Thus, we might conjecture that a
unified field theory of reasonable proportions does not exist.
9.1 GRADIENTS OF METRIC-DEPENDENT FUNCTIONALS
Here we formulate a calculus to compute gradients of functionals
defined on the space of all metrics on a manifold. The results we
obtain are of independent interest, and they exceed that which is
necessary to obtain the Einstein field equations.
9.1.1 Notation. We let M be an oriented n-manifold, and let Ш
denote the space of all (nondegenerate) metrics on M. The notation
Ud CM means that U is an open subset of M with compact closure.
For UC CM, we denote the space of all restrictions to U of metrics in
9li by 'Dli17. The space of all symmetric tensors on M is denoted by
S2(M), while the space of restrictions to U of such tensors is denoted by
S2(U).
9.1.2 Heuristic Motivation. For t/c CM, ge<9lu, ands<ES2(U)
we have g + ts€zc$\Lu for sufficiently small t. Thus, it is convenient to
think of S2( U) as the "tangent space of the manifold <Жи at g," and we
sometimes write S2(U) as T3\lu. Note that for M noncompact with
gtE^and s<ES2(M), it is possible that g + ?s£9H for all t¥=0. This is
one reason for introducing the spaces ty\lu.
9.1.3 Definition. For r, s G S 2( M) and gG9lL we let r-LJ, sip and g,7
be the components of r, s, and g considered as L-tensors in
C(L(M),T0-2) (see 8.1.2). The scalar gh'gJkrhJsik&C(L(M),T0-0)
induces a function on M that we denote by (r, s) : M->R. Let ц be the

9 UNIFICATION OF GAUGE FIELDS AND GRAVITATION 123
volume element on M determined by g and the orientation of M. For
f/CCM, we write {r, s)^= jи(г, s)gn . In the notation of 9.1.2, we
can regard ( , )L% as a "metric" on Tp\LL', and hence the assignment
gt-*( , )и„ defines a "pseudo-Riemannian metric on вЖи."
9.1.4 Definition. Let t: 9И^С°°(М) assign to each gG9H a scalar
formed from g and its curvature tensor by means of tensor product,
index raising or lowering, contraction, and/or covariant differentiation.
For example, £(g) might be the scalar curvature of g, or perhaps
something more exotic (e.g., Rh'jkRjk,in\hpR'"n)- For such £ we define
Lu: GJliu^U by Lu(g) = ]u£(g)txg. The gradient of Lu at gG9H is
that VgLu &Т^и = 82(и) such that for all s<ES2(U) with compact
support in U, we have (at ?=0)
We say that gG9H is stationary for L if VgLu = 0 for all L/C CM.
From the examples we will consider, it will be clear that in general
VgLu always exists and is unique. Moreover, the V%LVpiece together
to form a well-defined tensor vgLGS2(M).
9.1.5 Theorem. //ju.g is the volume element on M determined by g,
then for any s&S2(M) we have (on any f/C CM)
Proof. The volume element jug corresponds to an L-tensor
in C(L(M),T°-n) with components e,v..Jg|l/2 where \g\ is
the determinant of the matrix g,. of components of g^C(L(M),
T0-2) (i.e., Ы = 2(/^1у1--^я,Л1...л). Now ixg + ts has components
and (at ? = 0)

124 GAUGE THEORY AND VARIATIONAL PRINCIPLES
We have
Jt\g+tS\= 2 &\n'"Skh-"&nJ*h-J.
= 2
,• Е-
Thus, we have
as desired. ■
9.1.6 Definition. Let t: 9IL-^ C^M) be as in 9.1.4.
s&S2(M) with compact support, we define £4(j)eC°°(M)
For f/CCM, the partial gradient of Lu (see 9.1.4) a? ge^ll/7 м
(unique) dLu <ES2(U) such that
\U
/or all s£S2(M) with compact support in U. In our examples, we will
see that dgLu exists and that there is a dgLES2(M) such that
(dgL)\U=dgLu for all f/C CM. The next theorem reduces the compu-
computation of gradients to that of partial gradients.
9.1.7 Theorem. In the notation c/9.1.4 and 9.1.6, we have VgLu =
dgLu+\t(g)g.

9 UNIFICATION OF GAUGE FIELDS AND GRAVITATION 125
of. For s£.S2(U) with compact support in C/C CM we have (at
4=0)
g + 's
Since s is arbitrary and ( , ) is nondegenerate, we have v Lu =
3 Lu+^t(g)g by a standard argument (as in the proof of 4.3.6.).
■
9.1.8 Remark. Ift(g) is a scalar (as in 9.1.4) that does not involve
the covariant derivatives of the curvature tensor, then the formula of
8.3.8 reduces the computation of dgL to a mechanical procedure, which
we illustrate in Examples 9.1.10 and 9.1.11. // £(g) involves covariant
derivatives of the curvature tensor, then you can find formulas for
Rh'ijk\m', and so on (as in 8.3.6) in order to compute 3^L.
9.1.9 Notation. For /<=C(L(M), Г'-0 or T0J), the scalar /',, (com-
(computed relative to the Levi-Civita connection of some gG9H) is called
the divergence of f. Given UC CM and scalars a,fi£C(L(M), T°-°)
= CX(M), we write a— /? if a~p—f\, for some f with support in U.
Note that a —ft implies that /(yajug=/(y/8jug by 8.2.12. As an example,
note that for s&C(L(M),T0-1) and t&C(L(M),T20) (with s or t
having support in U), we have f4s,-, — —t'JySj, since (t''s:)y = t'JySi +
t'J'si{j by 8.2.9.
9.1.10 Example. Let £(g) be the scalar curvature o/ge9H (i.e.,
£(g)=Rhih,(g) = R(g))- We derive (VgLH- -Rlf(g)+ ^R(g)gij,
or equivalently, C L), = —Л, (g) = components of the Ricci tensor of
g. For U(Z CM andf£S2(M) with support in U, we have (using 8.3.8

126 GAUGE THEORY AND VARIATIONAL PRINCIPLES
with g'=f)g {lh
f'hlh-f\',)-l(-Rh\,fs-Rslh,fhs), since /% = (/*,.,% and so on
are divergences. Using 8.2.20 and 8.2.19 A), C), we obtain
Д,7/^ = (-(Д,7),Д, whence
s, 3gL(y= -(Д,7) or{\LL=-R,j. Note that g<E9H/s stationary
for t(see 9.1.4) *// «,/g)- i«(g)g,7 = 0. W^e« M w аи "empty"
space-time, this is the Einstein field equation. Also, it is worth mention-
mentioning that when M is a 2-manifold, this equation is automatically
satisfied. Indeed, R,j = RhihJ=RhkhJgik, and (for dim M = 2) Rhkh; = 0
unless k=j¥=h. In this case, Rhjhj = Rl2l2 = R2\x = h{Rn\i +
R2l2\)=2R- Thus> R,j=2Rg,j holds for any ge9H. It follows that if
M is a compact Riemannian 2-manifold, then JMRfig is independent of
the metric g. This is a qualitative statement of the Gauss-Bonnet
theorem.
9.1.11 Example. Here we take t(g) = RhiJkRJ'khl(g), and derive
(VgL)ah=-2RhhjkR^ha + 4R'J{ji + ^(g)gah. The computation is
more representative of the general case than that of 9.1.10. Let fE.
S2(M) have support in UC CM. We have
The covariant derivatives on f may be "transferred" to Rjkhl modulo
divergence {e.g., Г\^'\^- V\V
, we obtain ^
id j /_ d/ h i n к h _ n к i\— fab( — I R*l -, RJ,k +
bui\j bha\j b hci\k b ш\к ' J V ^7" "^
using 8.2.19 ш the final step. As in the previous example,
;sion within the parentheses is dgL, and the result follows.

9 UNIFICATION OF GAUGE FIELDS AND GRAVITATION 127
9.2 CONSERVATION LAWS FROM EXTERNAL SYMMETRY
9.2.1 Notation and Motivation. We denote the group of all diffeo-
morphisms of M by ty(M). If f^p-q(M) and X&%M), then we
define the pull-back of f by X, X*f^p-q(M), by (X*f)x(Xl,...,
Xp,Yl,...,Yq)=fMx)(Xlo\-\...,\^Yq) for Xx,...,Xp^fxM and
Yx,...,Yq<ETxM. Observe that X*: $°-2(M)^$°'2(M) leaves the set
911 of metrics on M invariant (i.e., X*: 911^911). In this way, ty(M)
acts on 9H. // gG^lt and X*g=g, then X is called an isometry of
(M,g). In this section, we will prove that for £: 91L-» C°°( M) any
scalar as in 9.1.4, we have
f
J
MU)
where AG ^(M) and U С С М. Hence we may say that the variational
problems of 9.1 possess the "external" symmetry group <Ф(М). In
contrast, we considered variational problems in earlier chapters that
possess an "internal'"' symmetry group (i.e., the group of gauge trans-
transformations). There we found that gauge invariance led to conservation
of charge. Here we will prove that external invariance implies the
conservation law (vgL)kJy = 0 where vg^£ S2(M) was defined in
9.1.4, the index was raised using g, and the covariant derivative is that
of the Levi-Civita connection of g. In general, for /GS2(M), the
I- form with components fL, is called the divergence of f relative to the
metric g.
In the special case where £(g) is the scalar curvature of g, we
obtain the well-known identity (RkJ — 2^Sk) \j ~0 (Ле Einstein tensor
Gjj=Rij — ^Rgij has divergence zero). Actually, GkJ{j=0 can be
derived directly from 8.2.19, but it is more satisfying to obtain it
from external symmetry. The Einstein field equation is G,7 = Erj
where (£,-.-) is the so-called energy-momentum tensor due to non-
gravitational sources. Consequently, EkJy-Gjy = 0, meaning that
energy-momentum is conserved.
9.2.2 Definition. For any AG6D(M) there is an induced diffeomor-
phism X: L(M)^L(M) defined by X(u) = X^(u)o u: R"->7\()r(H))M.

128 GAUGE THEORY AND VARIATIONAL PRINCIPLES
Observe that for u<EL(M) and A £GL(R") we have X(uA) = X(u)A
and тт(Х(и)) = Х('гт(и)).
9.2.3 Theorem. Let f<E<5p-q(M) correspond to /(EC(L(M), Tp-«)
as in 8.1.2. Then X*fcorresponds to /° Л (i.e., /°Л = Л*/).
Proof. For «^....UpGR" and w w?eR", we have
(foX)(u)(vu...,Wq)=f(X^U)(vl,...,Wq)
as required (see proof of 8.1.2). ■
9.2.4 Theorem. Let <peA'(L(M),R") fee ?/ге canonical \-form
(ф(Хи) = и-\^(Хи))). For Ле^(М) a«rf Л: L(M)^L(M) as in
9.2.2, we have A*qp = qp.
Proo/. For le TUP we have
9.2.5 Theorem. // 9(g)<EA\L(M), §t(n,U)) is the Levi-Civita
connection for ge9It, then \*F(g)) is that for X*g (i.e., 6(X*g) =
Proof. For BGgP(«,R) and B* the fundamental vertical field on
L(M), we have (at t=0)
Thus, X*(e(g))(B*) = e(g)(B*) = B. Moreover, for A&GL(n,U),
ЩХ*9(8)) = Х*(К*9(8)) = Х*(&ЪА->в(8)) = &ЪА-1(Х*в(8)). Thus,
X*6(g) is a connection on L(M). Regarding gG9H as being in
C(L(M), T0-2), we have dg+0(g)Ag = O. Applying Л* to this equa-

9 UNIFICATION OF GAUGE FIELDS AND GRAVITATION 129
tion yields d(g°\)+\*e(g)/\(go\) = 0. Applying A* to the equa-
equation d<f> + ()(g)/\<f> = 0 and using 9.2.4 gives d<p+ \*(e(g))A<p=0.
In view of 9.2.3, it follows that A*@(g)) is the Levi-Cevita connec-
connection of A*gG 9lt. ■
9.2.6 Theorem. // fi(g)(EA2(L(M), Ш(п,Щ is the curvature of
9(g), then A*(fi(g)) is that of 9(\*g) (i.e., fi(A*g) = A*fi(g)).
Proof. We have Sl(g) = d$(g) + ±[$(g)J(g)\. Applying A* to this
equation (and noting that A* commutes with d and [ , ]), we obtain
A*fi(g) = fi(A*g) by 9.2.5. ■
9.2.7 Theorem. Let el(g),...,en(g) denote the standard horizontal
fields (relative to 0(g)) associated to the usual basis e,,..., en of R"
(see 8.2.3). Then, ^\(
Proof. Certainly A~' =(Л"'). Thus, by 9.2.4, we have
= <J>(e,(g))=e,.
Moreover,
and the result follows. ■
9.2.8 Theorem. Let Rhi]k(g)( u) be the components of the curvature
tensor of^<=9H at u<EL(M). For\<E%M) and\: L(M)^L(M) as
in 9.2.2, we have Rh,jk(X*g)(u) = Rhljk(g)(\(u)).
Proof. Using 9.2.6 and 9.2.7, we have fi(A*g)u(e/A*g),^(A*g)) =
Letting both sides act on et (and then be acted upon by eh), we find
that the result follows from 8.2.13. ■
9.2.9 Theorem. For f<EC(L(M),Tpq), let fj[\'.':jqp\k(g) denote the
components of the covariant derivative of f with respect to 6(g). For

130 GAUGE THEORY AND VARIATIONAL PRINCIPLES
AG^D(M), we havefl\:;;r[k(g)(Mu)) = (fo\yr/:'flk(\*g)(u) for all
u<EL(M). Note that fo\<=C(L(M),TP'i) by 9.2.3,
Proof. By 8.2.7, we have
9.2.10 Theorem. With notation as in 9.2.9, we have
Proof. This result follows from 9.2.8 and repeated application of
9.2.9. ■
9.2.11 Theorem. If £(g)<ECco(M) = C(L(M),T0-0) is a scalar as
in 9.1.4, then (for \ВЩМ)) £(g)(A(u)) = e(A*g)(u) for all u<E
L(M).
Proof. Recall that £(g) is a contraction of a product of components
of the form g,7, g4 and Rhljk^... m{g) (possibly s = 0). By 9.2.3 and
9.2.10, these components satisfy g,7(A(u)) = (A*g),7(u), g'^(A(u)) =
(A*g№), and Rhi]k^...m{g){ku)) = Rhl]klmx...m{\*g){u). Thus,
9.2.12 Theorem. //AG6D(M) is orientation preserving and jugG
Л"(М) is the volume element of'gG 9H, then fxx*g = A*ju?.
Proof. Let X\,..., Xn€=TxM be an oriented o.n. basis relative to g.
Since ('K*g)(^\Xi,X-\XJ)= g(X,, Xj), we know that \~\XX,...,
\~\Xn is an oriented o.n. basis of Tx-\(x)M relative to A*g.
Thus, \ = Н^{\-\ХХ,...,\-\ХП)=\-'*{11Х*Я){ХХ,...,ХП). Thus,

9 UNIFICATION OF GAUGE FIELDS AND GRAVITATION 131
9.2.13 Theorem. 7/AG6D(M) and £(g) is a scalar as in 9.1.4, then
JMU) Ju g
for any U(Z CM.
Proof. By 0.2.16, 9.2.11, and 9.2.12, we have
\(U)
9.2.14 Definition. Let X be a compactly supported vector field on
M, and let Xt €Е<ф(М) be the one-parameter group of diffeomorphisms
generated by X. The Lie derivative of f<E<T\ p'q{M) is L'^
defined by
9.2.15 Theorem. Supposef^p'q{M) corresponds to f<EC(L(M),
Tp'q). For X and A, as in 9.2.14, we let X be the vector field on L(M)
generating the group ArG6D(L(M)) induced by \t as in 9.2.2. Then,
Lxf<E$p-q(M) corresponds to X[f](EC(L(M),Tp'q).
Proof. According to the proof of 8.1.2, we must check that
Vu---,»>q) = {Lxf)ir(u){vxou~\...,u{wq)). AW = 0, we have
X[f\u{v{,...,wq)=j-tf(\t(u)){vx,...,wq)

132 GAUGE THEORY AND VARIATIONAL PRINCIPLES
9.2.16 Theorem. Let g£9H and let X be a vector field on M.
Regardingg andLxg as in C(L(M), Тол) andXas in C(L(M), 7м-0),
we have (Lxg)i]=Xk^gkj + Xkyg:k where the covariant derivatives
come from the Levi-Civita connection of g.
Proof. From 9.2.15 we have (Lxg),7=/[g,7] = afg,7_(X)- К 9 is the
Levi-Civita connection of g, then dg(X) = — 9(X)Ag. Using 8.1.5,
we have {-9{X)f\g)ij = 9{X)klgkj + 9{X)kjglk. Thus, it remains to
prove that 9(X)k=Xklr Let X be the horizontal vector field on
L(M) such that тт^(Х)—Х, от equivalently X=XH. Suppose that
?н> ut is the integral curve of X through m=m0GL(M). Define
At£GL(n,U) by Xt(u) = ^t*°u=utoAf Then at r~° we have
=9u{X+A>@)*)=A>@).
Note that A= u~' ° ЛГИс ° и. Thus, for e
9u(X)(e,)=A'@)(e,)=jtu;<{\Ju(e,)))
at t=0. Let У be a vector field on M that agrees with the field
ЛГИс(м(е,)) at XtGr(u)) for all t in some small neighborhood of OK
At x = 7t(u) and ?=0,
Let У be the horizontal field on L(M) such that n^(Y)= Y, and note
that u~\\ti):(u(ei))) = <f>u(Yu). Thus, from above, we have
at r = 0. Now Xu[<p(Y)] = (dy)u(X,Y)+Yu[<p(X)]+<pu([X,Y]): but
the first term vanishes since De<p = 0, and фи([Х, Y])= u~\irJ_X, Y])
= u-\[X, Y]x) = 0. Thus, 9u(X)k, = Хы[фк(¥)]=¥и[фк(Х)] =
9.2.17 Theorem (Conservation Law). Let £(g) be a scalar with
"gradient" v»LeS2(W) as in 9.1.4. Considering vxL as in

9 UNIFICATION OF GAUGE FIELDS AND GRAVITATION 133
C(L(M),T0-2), we have (vgL)^ = O where the index was raised
using g, and the covariant derivative is that of the Levi-Civita
connection of g.
Proof. Let U С С M and let X be a vector field on M supported in
U. Let ArG6D(M) be the one-parameter group generated by X. To
within first order in t we have X*g — g + tLxg. Since X has support
in U, we have Xt(U) = U; and so by 9.2.13 we have
Lu(g)=f £(g)ixg = f £(
Ju Ju
Hence at t=0,
but the integrand is (vgL)'\Xkligkj +Xkljglk) = 2(vgLyJXkug,k
since vgL is symmetric. Moreover, (s7gL)'JXk,jgjk ——\s7gL)kJ^Xk
Thus
and it follows that on U, (vgL)kJy = 0, since X is sufficiently arbi-
arbitrary. Since UC CM is arbitrary, (vgL)^|y = 0 on M. ■
9.3 THE EINSTEIN-YANG-MILLS ACTION PRINCIPLE
Here we pursue the program outlined in the introduction to this
chapter. We easily concoct an Action density that leads to both the
Einstein equation and the Yang-Mills equation. With some persever-
perseverance, we then show that this density is the scalar curvature of a
certain metric on P, as in the introduction.
9.3.1 Definition. Let к be an B b-invariant metric on §, and let Qw
be the curvature of some connection со on P, as in the introduction to
this chapter. There is a tensor field on M of type @,4) denoted by

134 GAUGE THEORY AND VARIATIONAL PRINCIPLES
) and defined as follows. Let x€LM, and suppose that чт(р)~
x,peP. Let X,Y,Z,WeTxM and let X,Y,Z,W be the horizontal
vectors in TpP projecting to X, Y, Z, W. We set к(п,п) (X, Y, Z, W) ~
k{uw(X,Y),uw(Z,W)). Observe that к(п,п) is well defined (inde-
(independent of the choice of p), since (for A&G) k(9,"(RAij!X,
RA^Y),Q"{RA^Z,RJmW)) = k(&bA-lQa(X,Y),&bA-lQa(Z,W)) =
k(u"(X,Y),u"(Z,W)). We write the components of the associated
element of C(L(M), T0-4) as k(uhj,uJm), but note that uhj makes no
sense by itself. Finally, note that the self-Action of со
(see 5.2.1) is then given by - ighJg'mk(Uhi,UJm) or -
9.3.2 Definition. Let G(P) denote the space of connections on P,
and 91L the space of metrics on M. Suppose that R(g) is the scalar
curvature function for (M,g), and let c£R. We define a function
Л + S + c: G3\LX6(P)^CX(M) by (R + S + c) (g,u)=R(g) + §(g,u)
+ c. We say that the pair (g, со) is stationary relative to /?+§> +с if(at
t=0) we have
jf f R(g+
for all UC CM, alls<ES2(M) with support in U, and all т<=Л'(Р, S)
with projected support in U.
9.3.3 Theorem. The pair (g, со)<е91х6(Р) is stationary for Л + S
+ ciff
(A) RtJ - ±RgiJ - ±g.j = $ghmk{Qhi, QmJ) + ^(g, a)giJ (Einstein
field equation);
(B) 8"fi" = 0 (Yang-Mills equation).
Proof. By 9.1.10, we have (at t=0)

9 UNIFICATION OF GAUGE FIELDS AND GRAVITATION 135
Also,
A
Now
A
dt
from which it follows that
Piecing these results together, we see that (A) holds iff g is stationary
relative to /? + §> +с for a fixed со. Since /{,§>( w, g)ju is the only part
of the integral J(y^(g) + S(co, g) + cju depending on со, we see that
(B) holds iff со is stationary for the self-Action (see the proof of
5.2.3). ■
9.3.4 Definition. With notation as in the introduction to this chapter,
we define a metric on P (depending on g, со, and к) as follows. There is
a symmetric tensor field к со of type @,2) on P defined by (ko>)( X,Y) =
k(u(X),u(Y)) for X, Y(ETpP. Another such tensor field is ir*g {i.e.,
Ti*g(X,Y) = g('!Til.X,tn^Y)). Adding these yields a (nondegenerate)
metric h = n*g+kcc called the bundle metric associated to g, со, and k.
9.3.5 Theorem. The maps RA: P^P(A^G) are isometries of P
with the bundle metric h = TT*g+kco.
Proof. For X, Y(ETpP, we have
= h(X,Y). ■
9.3.6 Notation and Remarks. As a consequence of 9.2.11 and
9.3.5, the scalar curvature R of (P, h) is constant on the fibers of P.

136 GAUGE THEORY AND VARIATIONAL PRINCIPLES
Thus R determines a well-defined function R(g,u) on M. The next
theorem states that R(g, u) — R(g) + S(g, io) + RG where RG is the
scalar curvature of the fibers of P with the metric induced by h. Note
that all of the fibers _of P are isometric to the group G with the
(bi-invariant) metric к defined as follows. For A&G and X, FG TaG,
we set_k(X,Y) = k(LA^X, LA^Y). Certainly LA is an isometry
of(G,k). Since RAl X = &bA(LAl X), we have k(X,Y) =
k{R~lX, RAl Y), using the &b-invariance of k. Thus, RA: G-> G is
an isometry of (G, k). For any p €E P, the map gn> pg is an isometry of
(G,k) onto the fiber тт~ \тт(р)). It is remarkable that (as a conse-
consequence of 9.3.7 and 9.3.3) both the Einstein equation and the Yang-
Mills equation arise from a single Action principle based on the
geometry of the bundle metric. The significance of the bundle metric is
enhanced in the next chapter, where we show that the geodesies of the
bundle metric project to paths of "charged test particles" on M.
9.3.7 Theorem. The scalar curvature of P with the bundle metric
h = "ir*g + кы projects to the function R(g) + S(g, w) + RG on M,
where RG is the (constant) scalar curvature of (G, k) described in 9.3.6.
Proof. The calculation of the curvature tensor (at some fixed p&P)
is simplified if we use 6.2.6 with a judicious choice of o.n. vector
fields in a neighborhood of p. We suppose that Ex,...,En are o.n.
vector fields on (M, g) defined on a neighborhood U of x = it(p) in
such a way that the local section a: U->F(M) determined by
E{,..., En is tangent to the horizontal subspace of Ta(x)F(M) relative
to the Levi-Civita connection 9(g). This way, we have e(g) = a*6(g)
= 0 at x. Let £,,..., En (defined on w~\U)CP) be the horizontal
lifts of Ev...,En relative to the connection to (i.e., to(£,) = (), ^^(E,)
= Ei). Let e,,...,ey be an o.n. basis of § relative to k. Set En+a=e*
for a=l,...,/; these are fundamental vertical fields on P. Then
Ev...,En, £„+,,..., En+f are o.n. vector fields (defined on ir~\U))
relative to the metric h. In the expressions to follow, we use the
Einstein summation convention. Moreover, 1=5/, j,k,... *Sn, and
l<a, ft,y,... *£/, and Ijia, b,c,... =£и+/аге the ranges on indices.
We write gIJ=g(El,EJ) = h(El,EJ) = ±8ij and кар=к(еа,ер) =
h(En+a,E,l+fi)=±8afi. Let <p\..., <ря+/ be the 1-forms dual_ to
Eu...,En+f. If fl is the curvature of со, we can write со = 5Йа,7(<р'Л

9 UNIFICATION OF GAUGE FIELDS AND GRAVITATION 137
qiJ)ea for some functions паи on w~ l(U). The indices in fi",- can be
raised or lowered by using gjJ or kafj (e.g., Й„'у =я""^аДЙдту).
Finally, we define structure constants саду by [e^, ey] = capyea. In the
following lemmas, we compute the components of the curvature
tensor of (P, h).
9.3.8 Lemma. The matrix 9{h) = {e(h)uh)\^a,b^n+f provided
by 6.2.6 relative to the above choice of o.n. fields £,,..., En+f (and
dual forms ф„...,Фя+/_) is given by 0(A)^*@(g)I,)r£fie'.^'I+<\
ri^M^-ia^', and e(h)n+all+y
Proof. Let ф^,...,ф^ be the 1-forms on M dual to £,,...,£„.
According to 6.2.6, we have dy1 M = — 0( g)'■ Л <pJM. Since ir*4>M' = ф',
we obtain dip' = — ir*(j(g)'j ЛфЛ Writing со = соаеа, we see that
-я + а = ы«_ ThuSj ^« + «ea = ufw = -^[w,w]+fi = (-^ca/Syw/JAw1'
+ fia)ea. Hence, </фя + в = -^с%уфя + " Лфя + т + ^Пв17ф''ЛфЛ For
■qah = h{_Ea,Eh) \^a,b^n+_f, recall that the matrix e(h)\_must
satisfy 0(/г)гт) + т,0(/О = О or ^(/i)"ar,,c+ -qj(h)hr = 9{h)ca +
= 0. Note that
at r = 0. Thus, it is natural to try 9(h)n+an+y = icafiyyn+fi. The other
formulas are then forced upon us by the requirement 0(h)ac =
anc^ 1Ье expressions above for dip' and dip"+a. ■
9.3.9 Lemma. 77ze components (relative to Eu...,En+f) of the
curvature tensor forthe metric h on P are given by 2^"л^ф'' Aipd =
e
Proof. Let а: тт~ \U)->L(P) be the local section determined by the
o.n. fields £,,..., En+f. Note that R"bcdat q^ir' \U) is пв(к)\{ес, ed)
evaluated at 5(q) (see 8.2.13). Since ЙЙ(Л) vanishes on vertical vec-
e ^ e

138 GAUGE THEORY AND VARIATIONAL PRINCIPLES
9.3.10 Lemma. We have the formulas
nn + a — L~a ~Y
^ n+p n + S n+v 4C y/3c Si/)
n+a — ± a qy _ I / Qa Q A: _ Пи
P—2c « i«« W
and
Л « + /3я+у; Л
+p. Now (yn+/iyfiy
<p, and в(А)я+вя+уАв(_А)я+1'„+/,= 1сввус^фвЛф',апAв(А)я+вА Л
0{h)kn+p= — iuaklupkj(p'A(pJ. Evaluating these forms on (En+S,
En+V) yields
— _ X(ra „y
ljra ry _ ra у \ — _\ а у
Л\с SyC vp C vyC Sp)~ 4C ypc 8
_ ra у \
since cySv = — cyvS and the second parenthetical expression is cay/icySl/
by the Jacobi identity 0.3.11. Evaluation on (Et,E) yields the
expression for Rn + an + PlJ, and clearly Rn + an + p,n + y =0. ■
9.3.11 Lemma. For Я™,_а = £а[П™..], we have
pn + a —I/O" О" ^
and
and
n + a — A

9 UNIFICATION OF GAUGE FIELDS AND GRAVITATION 13
Proof. We have d9{h)n+aj = ^ji,u^u /\^', and 9{h)n + an + y /
Л<р"+Р. Using 9.3.9 and evaluating on appropriate pairs (E ,
we obtain the desired result. ■
9.3.12 Lemma. Let R'jkm be the components of the curvature tensi
of(M, g) relative to the o.n. fields El,...,En, and let flay, a = Ea[uaf
At p&P, R'jkm —R'Jkm +{tiyjtiymk -д(ЙДЯат — йа'тпа к), аг
Proof. Here we use the fact that Ex,...,En were chosen s
that 0(g)'j = O at x = ir(p); and so в{к)'"= **0(gYj =0 at p. W
calculate at p, to obtain rf0(A)'y=7r*(rf0(g)';)-^na'y аф"Лф'
t Л фт -
'"'. Also alp
and 9(h)'n + a A9~(h)" + a/ =-^а'^ау„,фАЛф"'. The 'result follow
from 9.3.9. ■
9.3.13 Lemma. The Ricci tensor of (P, h) at p, relative to £,,..
En+f, is given by RJm=R]m- ^„^V,, R,,,+y = - 2^y'y,„ at
Proof. From 9.3.12, we obtain R'j:m = R]m + lQy'jWmi, since Ua',
g'Juaij =0 because пш] = - uaJ1. From 9.3.11, we have R" + ajn + a
= 2®am п + а + А^ау®7m + ffikm^J'j- Observe that the first tv
terms are antisymmetric inj and m while R"+aj n+a m = Rjm — R'j,m
symmetric in j and m (see 8.2.20), whence the first two terms mu
sum to zero. Thus, Л"+" „+а „, = \пактпакг and we haveRJm = R'jim
R"+aj n+am=R.,m~ 2^akj^akm- From 9.3.11, we have Л"+ау„+а „+y=
while 9.3.12 gives R) , n+y= ~^у),г Thus, Rj „+y= 4 %.,■ No
R"+an+y n+a „+, = -\caPycpay by 9.3.10. Also R'n+y t „+v
e' >R
n+v
jn+ у i n + v й -■>■ n + у j n+ v i ft -yo"
^ n+у n+a n+v' К n + y i n+v 4C jiyC av ' 4*"yA " J'
Now we complete the proof of 9.3.7. The scalar curvature <
(P,h) is (using 9.3.13) R"a =RJ]-+Rn+\+y =R-^akj'Qak

140 GAUGE THEORY AND VARIATIONAL PRINCIPLES
~\ку"саРус^ау. We know that S(g, a) = -\uakjuakj. It remains to
prove that Rc = -\ку"саРусрау. Note that the restrictions of
<ря+',...,<ря+/ to a fiber 7г~'Gг(/?)) are dual to the o.n. fields
En+ u..., En + ,on tt~Xit(p)). Thus, the computation of the curvature
tensor RGapyS of the fiber is much like that of 9.3.10, noting also that
the connection forms on the fiber are the restrictions of 0(h)n+an+p
to the fiber (see the proof of 9.3.8). Indeed, we have RGapy8 =
follows that Rc = k^R"+\+y n+
by 9.3.10.
Rn+an+p n+y n+s, and it follows that Rc = k^R"\+y n+a n+v
As a by-product, we have a geometric interpretation of the
Yang-Mills equation fiwnw=0.
9.3.14 Theorem. The equation 8МЙМ =0 holds iff the vertical and
horizontal subspaces at each qE:P are orthogonal relative to the Ricci
tensor of the bundle metric h = ir*g+ kcc.
Proof. The horizontal and vertical subspaces at the point p&P are
orthogonal relative to the Ricci tensor for h iff 0 = Rjn+y= -jfiy'. ,=
$Qyjj. We have П<|>=Па1-|1-2Ф1'|®Ф1'2ев. Recall that the fields^,,...,' En
were chosen so that Oig)^ =0 at n (p). Thus, d' = тг*(- %)^)Л^
= 0 at p. By a computation similar to that in 8.2.11, we then have
-Гйи = й";;,ф^а, which vanishes iff ^y/, vanishes for all у and j.

CHAPTER
10
Additional Topics
10.1 GEODESICS AND FORCES ON CLASSICAL PARTICLES
Here we define the acceleration of a curve in a manifold with a
metric; geodesies are then curves of zero acceleration. Let n: P^>M
be a PFB with group G. For a metric g on M, a connection со on P,
and an (Jb-invariant metric к on §, we can form the bundle metric
h = 7r*g+ku on P (see 9.3.4). We will prove that the geodesies on
(P, h) project to curves on M that can be nicely interpreted as the
paths of "charged" particles that are accelerated by the field strength
аи.
10.1.1 Definition. Let тт: P^>M be a PFB with connection со.
Suppose that y: [a,b]^>M is a curve, and let рЕ:тт ~'(y(a)). The
horizontal lift of у through p is that curve u: [a, b]^P such that
David D. Bleecker, Gauge Theory and Variational Principles ISBN 0-201-10096-7
Copyright © 1981 by Addison-Wesley Publishing Company, Inc., Advanced Book
Program/World Science Division. All rights reserved. No part of this publication
may be reproduced, stored in a retrieval system, or transmitted, in any form or by
any means, electronic, mechanical, photocopying, recording, or otherwise, without
the prior permission of the publisher.
141

142 GAUGE THEORY AND VARIATIONAL PRINCIPLES
7тои=у, u(a)=p, andu'(s) is horizontal [u(u'(s)) = O] for alls£[a, b].
The existence and uniqueness of horizontal lifts is fairly clear; a proof
may be found in Kobayashi and Nomizu [1963]. In the special case
where M has a metric g and P = F(M) with the Levi-Civita connection
of g, the horizontal lift и of у determines a set of o.n. vectors
u(s)(e,),..., u(s)(en)€ETy{s)M for each sG[a, b]. This basis is said to
be the parallel translate of w( a)(e,),..., u( a)(en) along у from у (a) to
y(s). In Euclidean space, this is just ordinary translation, but in
general, parallel translation is path dependent.
10.1.2 Definition. Let M have a metric g, and let u: [a, b]^
be the horizontal lift of y: [a, b]->M, as in 10.1.1 Suppose that
$и X(s)€= Ty{s)M is a (smooth) vector field along y. Let f: [a, ft]->[R"
be given by f(s) = u(sy\X(s)) (i.e., X(s) = 2f>(s)u(s)(ej). We de-
define a new vector field DX/ds along у by DX/ds=u(s)(f'(s)); DX/ds
is called the covariant derivative of X along y. Note that DX/ds is
independent of the horizontal lift u; if A &g = O(r, s), then RA°u is the
horizontal lift through u(a)A and
as needed. If X(s) = y'(s), then Dy'/ds is called the acceleration of y,
and у is a geodesic if Dy'/ds^O.
10.1.3 Theorem. Let Ev...,En be o.n. vector fields defined on
some open UdM. Suppose that (#'.) is the pull-back of the Levi-Civita
connection 9 on F(M) via the section a'. U->F(M) determined by
Ev...,En (i.e., a(x)(e,) = Ej(x)). Let y: [a, b]->UCM be a curve,
and define functions y": [a, b]^>R by y'(s) = '2y"(s)Ej. For a vector
field $и X(s)€E Ty(s)M along y, we have (where terms are evaluated at
s or y(s))
where X(s) = 2X'(s)E, defines X': [a, b}^ R.

10 ADDITIONAL TOPICS 143
Proof. From 10.1.2, it is clear that
ds 2 ds
Thus, it suffices to prove that
у Q E yE
k.J
Select an arbitrary point s0E:[a, b], and let u: [a, b]->F(M) be the
horizontal lift of у such that M(so) = a(y(so)). Then, at s=s0,
Thus, we have
at s0. Letting both sides act upon ^ER", we obtain
at s = s0. Applying u(s0) to both sides yields
DF
Since both sides are now independent of u, we have

144 GAUGE THEORY AND VARIATIONAL PRINCIPLES
10.1.4 Corollary. With the notation of 10.1.3, we have
ds \
Proof. Set X=y' in 10.1.3.
10.1.5 Theorem. With notation as in the opening paragraph of
Section 10.1, let y: [a, b]->P be a geodesic relative to the bundle metric
h = 7T*g+kco. Then co(y'(t))&§ is independent of t.
Proof. We take £,,...,£„, En+V..., En+f to be the o.n. vector fields
on tt~\U) CP that were introduced in the proof of 9.3.7. We use the
conventions and other notations in that proof. As a consequence of
10.1.4, we have (with the summation convention):
'а)' + в (h)"b(Ec)y'by'c]Ea=0.
Applying со to both sides, we obtain (see 9.3.8)
Since cayfi and uaJt are antisymmetric in the lower indices, we obtain
(y'«+«)' = 0. Thus, W(y')' = W(Y'"+a£a)'=(Y'"+aR = 0. ■
10.1.6 Theorem. Let у be a geodesic in (P,h) and let Q = co(у') G§
be the (constant) element provided by 10.1.5. Continuing to use the
notation in the proof of 9.3.7 and setting у=7гоу, we have (relative to
the metric g on M)
where Q= Qaea and П" = Пв1.,ф/

10 ADDITIONAL TOPICS
145
Proof. Taking 7^ of equation (*), using 9.3.8, and noting that
y" = y", we have
10.1.7 A Physical Interpretation. Let M be a {possibly curved)
space-time. The curve у=7то у of 10.1.6 is the path of a particle (of rest
mass m0 and "charge" q=Q/mo&§) under the influence of the
velocity-dependent four-force with components qaua'Jy'J,0*^1*^3. (i.e.,
mo(Dy'/ds) = qaua'Jy'JEj). Note that the charge q depends on the
choice of the initial point p = y(a). Indeed, for A €E G, RA о у is another
geodesic such that 7roRAoy = y1 and yet a>((RAoy)') = (t}(RA^y') =
&ЬА-чс(у'). Thus for G a non-Abelian group, charge is a gauge-
dependent concept, just as current was found to be, when pulled down to
M (e.g., see 7.3.12).
10.1.8 Electrically charged particles. As a simple check on the
physical interpretation of 10.1.7, we consider the case of a charged
particle in an electromagnetic field in Minkowski space M = R4 with
the metric g = dt2 — dx2 — dy2 — dz2. Here m: P -> M is a PFB with
group U(\) = {e'e\0SU} (and Lie algebra %A) = {i0|0GR}) and
connection u> = — iA. Then the field F= — dA = d(— /co)= — ifl", when
pulled down to M via a section a: M -> P, becomes the (a-independent)
form E{ dxAdt + E2 dyAdt + E3 dzAdt + fi, dyAdz + B2 dzAdx +
B3 dx A dy where E and В are the electric and magnetic fields, respec-
respectively. With the convention (x0, x,, x2, x3) = (t, x, y, z), we use the
o.n. coordinate fields 30, 9,, 32, 33 (and their horizontal lifts to P) when
referring to components. The single o.n. basis vector for %A) will be
ex = i = /zzT. Thus, we have FJk = F(dj, dk)=- tt\k. In matrix nota-
notation, we have
о
-Е, -
0
в3
в.
-в
0
в
B2
-в,

146 GAUGE THEORY AND VARIATIONAL PRINCIPLES
whence
О Ех
Е[ О
Е2 -Въ
Е, В,
В3
О
-B2
0
Write y(s) = (t(s), x(s), y(s), z(s)), and suppose that g(y'(s), y'(s))-
1 E0 that s is proper time). Set
15 =
1
t'(s)
and note that t'(s) = (\ — \v|2) 1/2 = ^8,
a column matrix. Multiplying the matrices (й,"'л)
J E-v 1
[E+vXB\
Note that for r=(x, y, z), we have
and r"(
t"(s)
r"(s)
while
Thus, the equation

10 ADDITIONAL TOPICS 147
is equivalent to the pair
and ~
The first equation states that the rate of the particle's energy increase is
equal to the rate at which the field does work on the particle. The
second equation is the relativistic analogue of Newton's equation, where
the force is the so-called Lorentz force.
10.1.9 Remark. Given a geodesic y: [ — a, a]->P, consider the geo-
geodesic y:[ — a, a]->P defined by y(s) = y(~s). Note that y'(s) =
— y'( — s). If у is the trajectory in P of some classical particle x, then у
is the trajectory of a particle x that, relative to x, is "traveling" in the
opposite spatial direction, backward in time, and has opposite charge
(i.e., x is the antiparticle of x). Apparently, the correspondence y<->y is
the PTC (parity, time, charge) symmetry. The PTC principal in
physics says that if all the particles in a physically possible interaction
undergo the PTC symmetry, then the result is also a physically possible
interaction. That space, time, and charge should be related by such a
principle seems natural from the bundle viewpoint; note that PTC
leaves trajectories (in the total space P) setwise fixed, whereas P, T,
and С individually do not. Finally, note that conservation of energy-
momentum and charge in particle interactions can be considered as a
single conservation of'"momentum" in the total space.
10.2 UTIYAMA'S THEOREM
Not long after the appearance of the Yang-Mills paper [1954], R.
Utiyama [1956] generalized their work and provided a plausible
proof of an important theorem that now goes by his name. Essen-
Essentially Utiyama's theorem states that any gauge-invariant Action
density defined on the space of connections on a PFB (and con-
constructed from the -jet") must be an (Jb-invariant function of the
curvature of the connection (e.g., the self-Action w-> — \(gk)(uw, fi")
is such an Action density). In what follows, we develop the machin-
machinery needed to provide a precise statement and proof of Utiyama's
theorem.

148 GAUGE THEORY AND VARIATIONAL PRINCIPLES
10.2.1 Definition. Let it: P^M be a PFB with group G, and let в
denote the space of connection \-forms on P. For p E.P, let J°(G)p =
[up: TpP^§\u(Ee} = the set of all linear maps TpP^§ such that
A*» A for A eg. Set J°(G) = UpBPJ°F)p; /°(в) is called the space
of 0-jets of connections. It is a simple matter to show that J°(в) can be
made into a manifold in such a way that тт°: J°(&)^P (taking J°(Q)p
to p) is C°°. For each gGG, there is a map Rg: 0
defined by
Thus, G acts on J°(Q), and we write ccp-g = Rg(up). A connection can
then be regarded as a map со: Р-> J°(P) such that тт°оы = \р and
up-g = upgfor allp^P (i.e., R°gou = uoRg).
10.2.2 Definition. The differential of a connection со: P^J°(G) at
p£P is the linear map ump: TpP^Taj\e). Then Р(е)р = {шшр\ыЕ:
в}, and J\G)= UpePJl(G)p is called the space of 1-jets of connec-
connections. Again, /'(в) can be made into a manifold such that the
projection 7Г1: J\G)^P is C°°. Let R°gm: TwJ°(e)^TWp.gj\e) be the
differential of Rg at up^J°(Q). We define an action of Q on /'(в) by
K("*p)=K*o"*PoRg-'*&Jl(e)Pg> we write R\^*P)="*p-8- v
then u>v-g=u>vr
10.2.3 Definition. A (first-order) Lagrangian for connections is a
C°° function S: J\G)^U such that 5(co^-g) = S(co^) for all co^ <E
/'(в). Associated to S, there is a function §: &->Cco(M) given by
Note that § is well defined, since S(co^g) =
10.2.4 Definition. For k = 0,l,2,---, we define J°(Ak(P,§)) =
l-)pePAk(P,§)p where Ak(P,§)p is the space of all k-linear antisym-
antisymmetric functions тр: TpPX ■ ■ ■ XTpP->§ that vanish whenever one
argument is vertical. There is an action of G on J°(Ak(P, §)) given by
&b<R)£Ak(P§)
'PS'
10.2.5 Theorem. There is a well-defined map U: У
J°(A2(P,§}) given by^ U(u^p) = (du}p + !,[up,up} = U"p for
Moreover, й(сс^р-g) = U(coifp)-g, and U is onto.

10 ADDITIONAL TOPICS 149
Proof. Since fi"p depends only on u>p and u>^p, we have that U
is well defined. For to ев, we have &(.u>^p-g) = U(u>^ps) = uwpg =
&bg-iR*-&ap) = Qap-g = U(amp)-g. To prove that U is onto,
we need to show that for each тр еЛ2(Р, §)р we can find со ев such
that (du>)p + [u>p, и>р]=тр. Let UCM be a coordinate neighborhood
about q=7r(p) with coordinates x\...,x" vanishing at q. We can
choose Usuch that there is a local section a: U-*P with a(q)=p. Let
т =о*тр, and suppose that we can find a>uE.Al(U, §) such that
Then there is a unique connection to on тт '(£/) with a*to = cou, and
applying m* to equation (*) yields (du)p + j[up, ыр]= тт*о*тр = тр,
since тр vanishes on vertical vectors. Thus, it suffices to find an
appropriate u>u. Let t?C,, Зу) = 2ата;7еа where e{,...,efis a basis for
§ and 3,,...,3n are the coordinate vector fields of x\...,x". Let
0еЛ'(£/, §) with 0а1 = ва(д/) and 0а, у = ЭД0",]. For [еа,ед] =
2 cya/iey, the equation (*) then becomes (at g) Oa.J—eajj
+ 2с<хв (Q^fi^j ~ ^^j^yi)= T<Xij- In terms of coordinates, set
ва,(х1,..., xn)= -\Taikxk. Then 9a, =0 and 0a,,y= -^та,.у at 4, and
equation (*) is solved. ■
10.2.6 Definition. Let f: M^N be a map between manifolds. Then
f is called a submersion iff^x- TxM^Tf{x)N is onto for each хёМ. It
is a consequence of the implicit function theorem that for each у е
f(M), f~\y) is a submanifold of M of codimension equal to dim N
(i.e., dim f ~\y) = dim M~dimN).
10.2.7 Theorem. For each p <=P, the map &p: J\e)p^A2(P, §)p
is a submersion.
Proof. Let /'(Л'(Р, §))р be the vector space of all differentials (at
p) of maps о: Р^/°(Л'(Р, §)) such that o(p'g) = a(p')-g for all
p'GP; of course, such a a may be regarded as an element of
A'(P,g). Note that J\7L\P,§))p can be regarded as the tangent
space of J\Q-)p at any (owe/'F)f via the isomorphism
— (

150 GAUGE THEORY AND VARIATIONAL PRINCIPLES
Under this isomorphism (/'(Л'(Р, §))p^T^ /'F)), the differential
of Up at «„ is a linear map J\A\P, §))p^ Щ(^ }Л2(Р, §)pSs
A2(P,§)P given by
■*,'
The equation (do + [u, a])p = 0 imposes 2n(n~ 1)/ (n = dimM, f=
dimG) independent linear conditions on a . Thus, the kernel of the
differential of Up at wv has codimension 2>i(n — l)f, which is the
same as dim(A2(P, §)p). Thus, this differential is onto. ■
10.2.8 Theorem. For each тр еЛ2(P, §)p we have that Up '(Tp) is a
connected submanifold of J\Q)p of codimension ^n(n— 1)/. Moreover,
the tangent space ofUp\rp){at w^ Е.й~\тр)), regarded as a subspace
ofJ\K\P, §))p, consists of all a^p such that (D"a)p-0.
Proof. All assertions, except for the connectedness of й~\тр),
follow from 10.2.7 and its proof. Recall the following equation (*) in
the proof of 10.2.5: ваА1-ваи + ^саРу(вр1в^-вр/ву1)=та1Г We
must show that for given components т",., the set of solutions for
9ai j,9yj is a connected subset of U{ + n)f. There is the one special
solution 0au = - ^та,,, ва,• = 0. A solution @вл/; ва,) such that в",=0
is called normal. Any normal solution (в", ■; 0",) can be deformed
into the special solution: set 9attJ(t) = (\ -[)ва1 j-h^ij and ^",@
= 0 for 0^ t ^ 1. Suppose now that (9a, /, 0™.) is any solution of (*).
By defining TatJ = TatJ - ^сару(в^9^ - вр/ву1), observe that (ва,-j =
9al: y, 0™,. =0) is a normal solution of (*) when та( is replaced by та,у.
(Call this new equation (*).) We can deform this normal solution of
(*) into the special solution of (*) (keeping в",=0). Having deformed
the first component of @а, у, в"^ in this way, we perform a second
deformation by setting 0a,(?) = (l — ?Ha, and letting 0a,/O be the
special solution of (*)(t), which is equation (*) when та,7 is replaced
^« 0
7 y^yy^
the solution obtained after the first deformation, while at t = 1 we
have the special solution of (*). Thus, we have exhibited a path of

10 ADDITIONAL TOPICS 151
solutions of (*) linking the arbitrary solution @я, /,ва.) to the
special solution (i.e., й~\тр) is connected.) ■
10.2.9 Definition. Recall that the gauge group GA(P) acts on в via
pull-back (see 3.2.5). This induces an action of GA(P) on /'(в),
defined (for f^GA(P)) by (umpyf=(f*^r'(PY For ^GP' let
GA(P)p be the subgroup of all f(EGA(P) such that f(p)=p. Then
GA(P)p acts on J\S)p, and (for /еСА(Р)р)ш^-/=(/*а) only
depends on the first and second differentials of f at p. By identifying /,
and f2E:GA(P) if the first and second derivatives of'/, and /2 are equal
at p,we arrive at a finite-dimensional Lie group denoted byJ2( GA(P) ).
By 3.2.9, the Lie algebra ofJ2(GA(P)p)is then seen to beJ2(C(P, §)p),
consisting of equivalence classes of elements A £C(?, §) with A(p) = 0
where A is equivalent to A' if their first and second derivatives agree
at p. The group J2(GA(P)p) acts on /'(в), and the set О(со^р) =
{<% -f\f^j2(GA(P)p)} will be called the gauge orbit through to As
a consequence of the smoothness of the action ofJ2(GA(P)p) on У (в),
we have that O(co^) is a submanifold of'/'(в).
10.2.10 Theorem. The tangent space of the gauge orbit O(w )
consists of all а^е/'(Л'(Р, §))р s^ /'F) such that а<=Л'(Р, S)
is of the form a = dA+[u,A]= DWA for A(EC(P,§) with A(p) = 0.
Moreover, dimO(coij!p) = Bn(n + 1) + n)f, where f= dimG and n —
dim M.
Proof. According to 3.2.12, for A (EC(P, §) with A(p) = 0, we have
exp(tA)£GA(P)p. Tangent vectors to O(co^) are of the form (at
r=0)
by 3.2.16. Let xx,..., xn be coordinates on some U С М such that
there is a local section a: U-> M with o(q)= p. For Au = a*A, the
components of a*DwA relative to the coordinate fields 3,,...,3n and
basis e,,..., ef of § are Aua,■, + c^ccf^A J. Thus, (DaA)mp&
JX(AX(P, §))p is determined by the parameters Aua (.. and Aau t at q.
Since А"и = Aua Jt, there are (^n(n + 1) + n)f independent parame-
parameters, and this number is consequently dimO(co^). ■

152 GAUGE THEORY AND VARIATIONAL PRINCIPLES
The crux of Utiyama's theorem is
10.2.11 Theorem. The gauge orbit O(u^p) is precisely п~ \®шр).
Proof. Uf(EGA(P)p, then f(p')=p'T(p') for some t<EC(P,G) with
T(P) = e=identity element of G. Ву^З.2.15, we then have f*uup =
&bT{p)-;U"p = U"p.^We also have Qf*a = df*a+$[f*u>, /*ы]=/*Яи.
Thus, il(co -f) = il((f*o}) ) = &' u' =/*йк' — йш =£1(сб ). Hence,
we have O(co )См!~'(йи' ). Moreover, dimO(co ) = B«(« + l) +
")/= (n2 + n)f— \n(n — \)f= dim Jl(G)p — codim Й^ '(й^) =
ШтЙ^^П^) (see 10.2.8). Thus O(co^) is an open subset of
U~\u"p). Suppose that п^рЕ:йр1(п"р) — О(ы^р). Then О(п^р) is an
open subset of й~1(пар) such that О(п^р)ПО(и^р)= 0, since orbits
coincide or are disjoint. Thus, O(co^) and йр\пир) — О(ш#р) are
open. The connectedness of Й1"'(йы ) (see 10.2.8) then implies that
10.2.12 Theorem. Ler S: e^C°°(M) fee associated to the
Lagrangian S as in 10.2.3. Then § /5 gauge invariant (i.e., S(/*co) =
S(co)/or allu(Eeandf(EGA(P)) iff S is invariant under GA(P) (i.e.,
Proof. According to 10.2.3,
Hu)(
and
Thus, the left-hand sides of these equations are equal iff the right-
hand sides are equal (note T(f~l(p)) = 7r(p)). Ш
10.2.13 Definition. A curvature Lagrangian is a function K:
J°(A2(P,§))^№ such that K(rp-g) = K(Tp) for all g<EG (see 10.2.4).
We say that К is &b-invariant if K(&bgTp) = K(rp) for all g(EG and
\2§
10.2.14 Theorem. //К is a curvature Lagrangian and S: j\6)-^U
is the composition Ко fl, then S is a Lagrangian as in 10.2.3. Moreover,
§>: в->С°°(М) is gauge invariant iff К is &b-invariant.

10 ADDITIONAL TOPICS 153
Proof. Using 10.2.5, we have S(«w-g) = K(u(u^,-g))= K(Ci(u )
• g)= K(U(o>^p))= S(o}^p), and so S is a Lagrangian. Suppose К is
&b-invanant.Let f&GA(P)withf(p)=p ■h(p).Then,S((f*cc)i):f-l{p))
= K(U((f*a).f-,{p))) = K((f*Qa)f-,(p)) = (see 3.2.15 and note that
p (p)
= h~\p))= K(&bh-4p)Q»p)= K(Q"p)=
4p)p p mp)
whence § is gauge invariant by 10.2.12. Conversely, if § is gauge
invariant, then from the foregoing we conclude that K(&bh(pr-iu" )
= K(Qap) for every w;<= 6 and/G G4(P). Since every т?еЛ2( Р, § ^
is of the formfi'Jp =Й(ш^) for some со G 6by 10.2.5 (and since /z(/?)
ranges over all of G as / ranges over GA(P)), we have that К is
B b -invariant. ■
10.2.15 Theorem (Utiyama). Let S: J\G)^U be a Lagrangian,
and let §: 6-> CX(M) be associated to S as in 10.2.3. Then § is gauge
invariant (i.e.,<i>(f*u) = §((o) for all f£:GA(P) and @^6) iff S = Kou
for some &b-invariant curvature Lagrangian K.
Proof. The "if part is 10.2.14. Conversely, if § is gauge invariant,
then S: Jl(G)^U must be invariant under the action of GA(P) on
/'F) by 10.2.12. Since S is then constant on the gauge orbits
O(co^) of 10.2.9, we deduce from 10.2.11 and 10.2.5 that S induces
(via й) a curvature Lagrangian К: /°(Л2(Р, $))->R such that
S = Ko&. Since o> is gauge invariant, we have that К is (Jb-invariant
by 10.2.14. ■
10.2.16 Construction of Curvature Lagrangians. We examine a
method for producing some (lib-invariant curvature Lagrangians. Let
F: §X - ■ XS^R be a multilinear function that is &b-invariant (i.e.,
F(&bgAl,...,&bgAn) = F(Al,...,Am)forallgSGandAl,...,Alf[E:
§). If F is symmetric, then F is called a (polarized) Weil polynomial.
An excellent treatment of the determination of the Weil polynomials
(for G=U(n), O(n), SO(n), and Sp(n)) is found in Section 2,
Chapter 12, of Kobayashi and Nomizu [1969]. In any case, given a
connection со on a PFB тт: Р^>М with group G, we can define a tensor
field f(fi)G?r@-2m)(P) (for Q = fl"w) by F(U)(X,,..., X2m) =
F(U(XvX2),...,U(X2m^l,X2ni)). Just as in 9.3.1, we can prove that
there is a unique tensor field F(u)^{0-2m)(M) such that n*F(u) =
). Using a fixed metric on M, we can raise m indices of F(U) and

154 GAUGE THEORY AND VARIATIONAL PRINCIPLES
contract those with the remaining indices to form a scalar, say CF(Q,).
The function J°(A2(P,§))^U given by up>-*CF(u)w(p) is then a
curvature Lagrangian. The self-Action is derived from a curvature
Lagrangian of this type where F: §X§->[R is an &b-invariant metric.
10.3 SPONTANEOUS SYMMETRY BREAKING
Here we study the so-called .Higgs mechanism, which has been
applied extensively by physicists in building more realistic models for
the behavior of elementary particles. For example, one problem with
the model of Section 7.3 is that the mass (a term we define later in
this section) of the gauge fields b\b2,b3 (where co= — ib-т, as in
7.3.14) is zero. Apparently it is fine if the gauge field of electromag-
netism has zero mass because there the force is mediated by photons,
which are massless. However, Yang-Mills type forces must arise
from the exchange of massive particles because of the observed short
range of these forces. The Higgs mechanism helps in two ways. First,
gauge fields can acquire mass by the symmetry breaking. Second, the
undesirable Goldstone bosons (which arise in the symmetry-breaking
process) can usually be gauged away. We will not apply the Higgs
mechanism in a detailed fashion to the model in Section 7.3; rather,
we describe the mechanism in general terms, and refer you to the
physics literature for the many specific applications.
10-3.1 Notation. Let M be an n-manifold with metric h. Suppose
that 7Г. P -» M is a PFB with compact group G. Let V be a vector space
with positive definite metric h, and let G-> O(V) be some orthogonal
representation. Assume that L: J(P,V)^>R is a Lagrangian of the
form L(p,v,9)={(hh)(9,9)-F(v) where F:V^U is (necessarily) a
G-invariant function (i.e., F(g- v) = F(v) for all gE:G,vE:V). We
think of F(v) as being the potential energy of v.
10.3.2 Definition. // uoe V is a local minimum for F, then we call
v0 a vacuum, and the set Gv0 = [gvo\ gfE G) is called a vacuum orbit.
The unbroken subgroup of G relative to v0 is Go = {g €E G \ g ■ v0 = v0}.
Note that Gv0 is a submanifold of the sphere of radius h(v0, uo)l/2 '" ^
and Gv0 can be identified with the quotient space G/Go. In^ the physics
papers, F is often given by — F(v) = ^m2h(v, v) — ^X2h(v, vJ. At

10 ADDITIONAL TOPICS 155
t=0,
—r-F(v + tw) = m2h(w, v)~ X2h(v,v)h(w, v)
= h(w,(m2 — X2h(v,v))v),
which is zero for all w when h(v, v)l/2 = m/X. In this case, the set of
vacuum points is the sphere of radius m/X, but the vacuum orbits may
be smaller, depending on the representation G->O(V). Often, the
unbroken subgroup is just some U(\)CG corresponding to electric
charge.
10.3.3 Theorem. Referring to 0.2.26, we let D2F: TVVXTVV^№
be the Hessian of F at the vacuum v0. For d=dim(Gv0), there is an
o.n. basis u{,...,ud,ud+v...,um of TVV {with metric induced by h)
such that uv...,ud spans the subspace Tv(Gv0) СTv V, and the mXm
matrix with entries Mah—D2F{ua,uh) is diagonal with Mu= ■ ■ ■ =
Proof. Let TVo(Gvo) be the subspace of TVV orthogonal to TVo(Gvo).
First, we prove that D2F(wl,w2) = 0 if w, &TVo(Gvo) and w2°(ETvV.
Select A eg such that
— (expL4)-«0=w,
at t = 0, and let y: [— \,\]->V he a curve with y'@) = w2. Define Я:
[ —1,1]X[ —l,l]^Fby ЯE, t) = (exptA)-y(s). Then at 5=r=0,
while
Hence,
since F is G-invariant. Let ux,...,ud be an arbitrary o.n. basis of
Tv£Gv0), and let ud + l,..., um be an o.n. basis of Tv^Gv0)± (provided
by a well-known result) such that the matrix D2F(ua, ub) d + \^a,b
«£ m is diagonal. Then, и,,..., um is the desired basis. ■

156 GAUGE THEORY AND VARIATIONAL PRINCIPLES
10.3.4 Definition. If\pE:C(P,V) is a particle field, then xP' = xP~v0
is called the shifted field of \p (relative to v0). Note that \p'&C(P,V),
since xP'(pg)=g-lxP(p)-v0^g~ixpr(p) ifg&G0. We think of V as
the deviation of \p from the chosen vacuum v0. Moreover, it is conve-
convenient to think of \p' as having values in TVV under the identification
F= TvY Siven ЪУ
d i
There are real-valued functions £,,..., £rf and t]d+ v...,T\mon P defined
byV(p) = UP)«i+ ■■■+Vm(p)um- We call iv...Лd the Goldstone
bosons of \p (or \p') and rjd+,,..., i\m the scalar mesons. Collectively,
£,,..., rjm are known as Higgs fields. Let ma^0 be defined by m2u~
D 2F( ua,uh), a=\,...,m. Note that to within second order in the Higgs
fields, we have F<>xp=Fo(v0 +\p') = F(vo)+ ^D2F(^', xP')+ ■ ■ ■ =
F( uo) + 2f гт1И +2^+1 Ki)!+ ' ' ' > and (in analogy with the
Lagrangian of 4.3.7) we call ma the mass of the Higgs field associated
to ua. By 10.3.3 the Goldstone bosons have mass 0, while the scalar
mesons may have positive mass.
10.3.5 Definition. Let H: P^M. be a C°° function. We define the
fiber derivative of H atp&P to be the linear functional (dGH)p: S->R
given by
(dcH)p(A)=~H(pcxptA)
at t=0. The next result is an observation of Weinberg [1973].
10.3.6 Theorem. Ler u0 be a vacuum and let \p£C(P,V). Define
H: P^U by H(p) = h(xP(p),v0). Then the Goldstone bosons of xp
vanish at p iff (dGH)p =0.
Proof. For A €E §, we have
—h

10 ADDITIONAL TOPICS 157
at t = 0. Since h{vo,A-vo) = 0 for all Л eg, we have {dGH)p{A) =
h{\p'{p),A-v0), and the result follows, because Tv{Gv0) is identified
with {A-vo\A<E§}. Ш
10.3.7 Definition. For uoeF and \P<EC{P,V) we call UD*,vo) =
[p €=P\{dcH)p = 0} the unitary set of \p relative to vQ. Since the fiber
tt~i{x) is compact, we know that Н\чт~х{х) has a critical point, whence
U{\p, ьо)Г\тг~*(х)ф-0. Goldstone bosons are considered to be spurious
{or unphysical) states. Thus, it is important to know when they can be
"gauged away," in the sense that there is a local gauge a: W^>P such
that the Goldstone bosons vanish on a{W) {or equivalently a{W)C
U{\p,v0) by 10.3.6). Such a a is called a unitary gauge. Some may
think that a {global) unitary gauge can be defined by taking o{x) to be
"the" point where Н\чт~х{х) achieves its maximum. This procedure is
doomed to fail somewhere if тт. Р->М is nontrivial, since there can
never be a global gauge {unitary or not) for a nontrivial bundle.
Indeed, it is not even certain that local unitary gauges exist about every
x E.M, although a sufficient condition will be given in 10.3.10.
10.3.8 Definition. Since G is compact, we can introduce a positive
definite &b-invariant metric к on g. Let §^ be the subspace of g
orthogonal to §0. We define another metric к on §^ by k{A, B) = h{A-
v0, B-v0). Note that к is positive definite, and we can find an o.n. basis
e{,...,ed of Q^ {relative to k) such that k\ea,ep) = 0 for афE, while
k{ea,ea) = M^>0, l«£a, /?=£<! The vectors ev...,ed are referred to as
broken generators.
10.3.9 Definition. For p&U{\p,v0), we define the broken Hessian of
H at p to be the bilinear form {D^H)p: §^ X g^ ->R given by
{D^H){AB)
at s = t = 0. A short computation yields {Dj;H)p{A, B)= h{\p{p),A-В
■ v0), which is symmetric in A andB, since h{\p{p),[A, B]-vo) = 0.
10.3.10 Theorem. For p&U{\p, v0), we have a local unitary gauge
a: W ^ U{\p, vo)CP {for some neighborhood W of х = тт{р)) such
that a{x) = p, provided that {D^H)p is nondegenerate.

158 GAUGE THEORY AND VARIATIONAL PRINCIPLES
Proof. Let §^ be the dual space of g^. Define /: P-^ by
f(p)={bcH)p\^. Note that LW,«o)=/~'(O). Thus, £/(*,«<>) will
be a submanifold (of codimension d) in some neighborhood of p if
(see 10.2.6) Д: TpP^T0(§^)^§^ is surjective. Moreover, кегД is
the tangent space TpU(\p,v0) in this case. For A, B&§^, we have
{bH){B)
at s=t=O. Thus, the surjectivity of/^ is equivalent to the nondegen-
eracy of {DlH)p. Hence, we see that not only is U(\p,v0) a submani-
submanifold in a neighborhood oip, but also TpU(\p,v0) has a d-dimensional
complement in the vertical space of P alp. Thus, ir^(TU(\p,vo)) =
TXM, and this is enough to give us the unitary gauge a. ■
10.3.11 Remark. Observe that U(\p,v0) is invariant (as a set)
under the action of Go on P, as a consequence of the G0-invariance of
H. Thus, U(\p,vo)C\4r~\x) is typically a manifold of dimension
dim(G0). Aside from this, there is a compelling analogy between the
situation described in 10.3.10 and catastrophe theory. The points x^lM,
such that (D^H)p is singular for some p Е:тт~1(х), are catastrophe
points. About such points, the existence of a unitary gauge is not
ensured, even if a different vacuum in Gv0 is chosen. The locus of
catastrophe points in M should be of some physical interest. Finally,
differential geometers should note that for pE:U(\p,vo),(DJ;H) is
essentially the second fundamental form of Gv0 at v0 in the normal
direction \p( p).
10.3.12 Massive Vector Bosons. The physicists claim that those
gauge potentials corresponding to the broken generators acquire mass,
thereby deserving to be called "massive vector bosons." Here we make
some sense out of this claim. Let к be the (lib-invariant metric on § as
in 10.3.8, and suppose that (\p, со) GC(P, V)XG. Consider the Action
density (£+§)(xp, со) <EC°°(M) given by

10 ADDITIONAL TOPICS 159
Let \p' = \p — v0 be the shifted field of \p relative to the vacuum v0, and
suppose that a: W->P is an arbitrary local section. We introduce the
notation \|/ =a*\p' = \p'oo and ww = a*w. Let ev...,ed be the broken
generators of 10.3.8, and extend these to an o.n. basis ex,...,ef of §;
necessarily, ed+v...,ef is an o.n. basis of §0. We write tow=2w"ea,
where the to" are real-valued \-forms {or vector potentials) on WCM.
Take ux,..., um G TVV as in 10.3.3 and consider i|/ as having values in
T F= V. We write °
v0
b=\ c=d+\
Now,
where we have retained only those terms that are less than third order in
the "variables" i|/, d^'w, cow, and du>K. Note that none of the retained
terms, except the cross term {hh){d\p^, uw-v0), involve both ccw and i|/,.

160 GAUGE THEORY AND VARIATIONAL PRINCIPLES
If a: W^P is unitary, then this cross term vanishes. Indeed,
Id m f \
(hh)(d^,cow-vQ) = (hh) 2#X+ 2 difwuc, 2 <ea-v0\
\b=\ c=d+\ a=\ I
m d
= 2 2h(d^,uaJh(uc,Maua) = 0,
c=d+\ a=\
where we have used the facts £*=0, ea-v0=Maua (see 10.3.8), and
h(uc, ua) = 0 for c>d and a<d. Then, assuming that a: W^>P is a
unitary gauge, the expression for (£+§)(»//, со) (up to second order in
r\cw, drfw, u>l, du>l) becomes
c = d+\
d
- 2
a=\
2
This is essentially the Klein-Gordon type Lagrangian density for
m — d scalar mesons ifw of mass mc, together with d massive vector
bosons and f—d massless vector bosons. Moreover, without the
higher-order terms, we see that none of these fields is coupled to any
other in the unitary gauge. It is hoped that you have gained not only
an understanding of how gauge potentials acquire mass, but also a
feeling for the significance of the unitary gauge.
10.3.13 Remark. You may wonder how the foregoing applies to the
model in Section 7.3, because there the Lagrangian was of Dirac type
instead of Klein-Gordon type. The answer is that in addition to the
spin-\ nucleon field, we must introduce the scalar Higgs fields
(transforming according to some representation of SUB)). Then we add
the Dirac type Lagrangian involving the nucleon field to the Klein-
Gordon type Lagrangian for the Higgs fields. The rest of the Lagrangian
consists of the self-Action of the connection along with terms coupling
the nucleon field to the Higgs fields. For further details on such models
(e.g., the Weinberg-Salam model), we refer you to Abers and Lee
[1973], Salam [1968], and Weinberg [1973].

10 ADDITIONAL TOPICS 161
10.4 CHARACTERISTIC CLASSES, MONOPOLES,
AND INSTANTONS
Here we define certain real-valued forms /(fi") on M that are
constructed from the curvature form fi" of a connection со for a PFB
•n: P ^> M with group G. The forms /(fi") are closed and depend on
со. However, /(Я") —/(fi"') is exact. We say that two forms are in
the same class if their difference is exact. Thus, the form /(fi")
determines a class [/(fi")] that is independent of со and is called a
characteristic class of it: P -» M. If the PFB is nontrivial, then there
might be nontrivial characteristic classes, in which case Йш Ф0 for all
со. Thus, we see that a nonzero field strength can be forced upon us
by the topology of the PFB (i.e., solitons arise). As a simple example,
we consider magnetic monopoles. Finally, we make some elementary
comments about instantons.
10.4.1 Definition. Recall A0.2.16) that a Weil polynomial of degree
m is a symmetric multilinear map f: § X • • • X § -»R (or C) such that
f(&bgAl,...,&bgAJ = /(/!„..., AJ for all gGG and A,G§. IJ'we
set g = exp tB (BE§), then applying (d/dt)\l = Q to this equation yields
We denote the space of such Weil polynomials by lm(G). IffEI'(G)
andf'EIJ(G), thenff'Er+J(G) is defined by
where a ranges over all permutations of {1,...,/+_/}. In this way,
I(G)=®klk{G) is a graded algebra, the Weil algebra.
10.4.2 Definition. Let со be a connection for the PFB it: P^M with
group G. For fElm(G) and fl" = dcc+{[ic, со] we define f{U")E
A2m(P,IR) by

162 GAUGE THEORY AND VARIATIONAL PRINCIPLES
Note that г/ Й" = 2 tt"ea (ex,...,efa basis for §), then
10.4.3 Theorem. For each /£/m(G) there is a unique closed 2m-
form f(Q") on M such that ir*f(Qu)= f(Qu). Moreover, we have
> /Я /Я), for fe'f(G), /' e P(G).
Proof. Since со is fixed, we write Я" = Я. If ЛГ,,..., X2k £TpP
are horizontal, then we set /(Щ„(Р)(^^Хи . . . , "п^Х2т) =
f(u)p(Xu...,X2m). Since R*u = &b'g-,u and / is £b-invariant, we
see that /(Я)х is independent of pE7T~\x). Since п vanishes on
vertical vectors, we have/(fi) = 7r*/(fi). In the notation of 10.4.2, we
have
Thus,
and so df(u)H =0, since JflH =0 by 2.2.8. Now •n*df{u)= dir*f(u)
= df(Q). Thus, df(U)=ir*df(tt) = (TT*df(U))H = df(U)H=0, "and
<//(Я) = 0 follows. For fel'(G) and feiJ(G), we have
Note that fi Л • • • ЛЙ"'+; is symmetric in a,,..., aj+J, since Я"* is а
2-form. Thus,
(i.e., the symmetrization of/®/' is effected by Я Л • • • Afi"-^). The
right-hand side is Шеп/(Я)Л/'(Я). ■
10.4.4 Definition. Леса// the definition of Hk{M) in 0.2.20. For a
fixed connection со on P we define a homomorphism of algebras W(u>):
®mIm{G)^@kHk{M) by W(u)(f) = [f(Qa)]. Note that W(u) dou-
doubles the degree, and preserves the multiplicative structure by 10.4.3.
Our immediate goal is to prove that W(oi) (the Chern-Weil homomor-
homomorphism) is actually independent of the choice of со.

10 ADDITIONAL TOPICS 163
10.4.5 Lemma. Let ф0 and ф, be any %-valued 1-forms on P (or any
manifold), and let fEl"'(G). Set а=ф,— <р0, ф,=фо + ?а, а«^/Ф,=^ф(
+ НфеФг]- ^е? /(ф() *е ?^e 2m-form defined as in 10.4.2, ал</ let
f(а, Ф,,..., Ф,) йе the analogously defined Bm— \)-form. Then we have
Proof. Note that
dt
Hence
-r<b=da+[<pt,a].
m dt
It suffices to prove that the right-hand side is <//(а,Ф,,..., Ф,). Now
d<bt—— [ф,,Фг] (see the proof of 2.2.8), we have df(a, Ф,,...,Ф,) =
Ф,,...,Ф,)}. The expression within the braces is (in the notation of
10.4.2)
by the в Ь-invariance of /(see 10.4.1). ■
10.4.6 Theorem. For any two connections со and со' о« Р we have
W(w)=W(u'). That is, [/(Йш)] = [/(Йш')]еЯ2т(Л/) for all /G
Proof. We apply 10.4.5 with Ф0 = со and ф,=со'. Then /(Йы)
) = mdp, where

164 GAUGE THEORY AND VARIATIONAL PRINCIPLES
Since со-со'£Л'(Р, §), Qu- EA2(P,§), and / is (£b-invariant, we
have /?*/? = /? and fi vanishes on vertical vectors. It follows that there
is a unique form /?'£Л2т~'(М) such that tt */?' = /? (as in the proof
of 10.4.3), an
10.4.7 Definition. The various classes [/(Йш)]еЯ2т(М) for fE
I"'(G) are called characteristic classes. Since they are independent of
со, they depend only on the PFB it: P^>M. If it': P'^>M is another
PFB with group G, then it': P'^M is equivalent to it: P^M if there is
a map Л: P^>P' such that \(pg)=\(p)g (for allpEP andgEG) and
10.4.8 Theorem. The Chem-Weil homomorphisms ®mI
©kHk(M) for equivalent bundles are the same.
Proof. For it: P^M and it': P'^M as in 10.4.7, let со' be a
connection on P'. Note that со = Л*со' is a connection on P, and
ЯШ = Л*ЯШ'. Thus (for fElm{G)) /(ПШ)=Л*/(ЙШ), and we have
), since Л induces \M: M^M. ■
10.4.9 Corollary. The characteristic classes for a trivial PFB all
vanish.
Proof. By 10.4.8, it suffices to show that the characteristic classes
for a product bundle {P — MXG) vanish. On MXG there is the
trivial connection со with horizontal subspaces tangent to the slices
MX{g). Since co|MX{g} vanishes, so does d<x>\MX{g), whence
) = 0 for
10.4.10 Remark. // G is a compact semisimple Lie group and r is
the dimension of a maximal Abelian subgroup (i.e., torus) of G, then it
is known (see Chern [1972]) that the ring ®mIm(G) is generated by r
elements, say/,,..., fr. The Chern-Weil homomorphism for a PFB with
group G is then determined by [/-(й")], i=\,...,r. When G~U(n),
then r — n, and for a certain choice of generators f EI'(U(n)), we
obtain the Chern classes [/.(Ои)]е#2'(М), i=\,...,/?. If G = SOBn
+ 1), then we obtain the Pontryagin classes, and if G = SOBn), then we
obtain not only the Pontryagin classes, but also the Euler class in

10 ADDITIONAL TOPICS 165
H2"(M). For further details, see the excellent treatment in Kobayashi
andNomizu [1969].
10.4.11 Theorem. Let [/(Йш)]еЯ2т(М) be a characteristic class
of it: P^> M. Suppose that N is any orientable compact submanifold (of
M) of dimension 1m. Then ^/(fi") is independent of со.
Proof. Note that jNf{u") is an abuse of notation for jNi*f(u")
where i: N^M is inclusion. Now, i*f(Qu)-i*f(Qu') = i*~dP' =
d(i*/3'), in the notation of the proof of 10.4.6. Integrating this
relation over N, and applying Stokes' theorem, we are done. ■
10.4.12 Definition. Let N and N' be oriented submanifolds of M.
We say that N can be deformed into N' if there is a map H:
NX@,3)^M such that (Я|ЛГХ{1}): NX{\}^N and (tf|JVX{2}):
NX{2}^Af' are orientation-preserving diffeomorphisms. We write N
<-+N' if N can be deformed into N'.
10.4.13 Theorem. // Л^ and N' are Im-dimensional orientable sub-
manifolds ofM and N<-*N', then (for any /G/m(G) and connections со
and со' on P) we have
/7(яи)=/дяи')-
JN~ JN'-
Proof. Since
/ /
JN'~ JN'~
by 10.4.11, we need only consider the case where co = co'. Let H:
@,3)XJV^M be as in J0.4.12. Then tf*(/(fi")) is a closed C°°
2m-form on NX[1,2] = N, and so
0= [_dH*(f(№))
JN
= Г_Я*(/(ПИ))= f Я*(/(ПИ))- f Я
JdN ' JNX{\} ' JNX{2)
= (№")-[ /(яи),
•'Л'" JN"
using the version of Stokes' theorem in 0.2.15 and 0.2.16.

166 GAUGE THEORY AND VARIATIONAL PRINCIPLES
10.4.14 Remark. Typically, the generators /,,..., fr of 10.4.10 are
normalized so that the various integrals JNfj(&"), for compact N with
dim N=deg ft(u"), are in tegers.
10.4.15 Magnetic Monopoles. Here we apply the foregoing to
show how magnetic monopoles arise from nontrivial PFBs with group
U(\). Let mF: F(R3)^IR3 be the bundle of oriented o.n. frames of R3
with the Euclidean metric and usual orientation. For any xER3, there
is an identification TXU3 = U3, and so we can write F(U3) = U3 X SOC).
Let PdF(U3) be the submanifold consisting of all (x, A)EU3 XSOC)
such that x=£0 and A(e3) = x/\\x\\. We can identify SOB) with
{BESOC)\B(e3) = e3}. If (x, A)EP and ВESOB), then(x,A)-B =
(x,AB)EP, and so we see that it: P^R3 = R3 -{0} (ir = iTF\P) is a
PFB with group SOB). For r>0 and Sr = {xEU3\\\x\\ = /■}, note that
it: it ~\Sr)->Sr can be identified with the oriented o.n. frame bundle of
Sr. Let e = (e'j) be the Levi-Civita connection on F(U3). Let
0 0\\P
[в\\Р 0
an §><QB)-valued \-form on P. It is not hard to see that со is a
connection for it: P^IR3. Let
'LI s])=-
and note that fEl\SO{2)). We have f(u")=f(du+ ^[u, u])=/(rfu)
= ^F>'2|Р)=-F'1з|Р)ЛF'32|Р), since de=-B/\B+Ue and fl"=0
for Euclidean IR3. Let <p = (<p\ ф2, ф3) be the canonical \-form
on F(U3). We prove that (в\\Р)= \\x\\ "'( ф1 \Р) at (x,A)EP. In-
Indeed, let y(t) = (x(t), A(t))EP be a curve in P with y(Q) = (x,A).
Then 0'3(Y'(O)) = ( ex, в(у'@))(е3)) = {ех,А~ U'@)(e3)> =
( ex, A -\\\x\\ -'jc)'(O)> = IIjcH -' < e,, A ~lx'@)) + (
е],А-\\\х\\~и@)х)) = \\х\\ "'(e,, Ф(у'@))> =ф'(у'@)), where we
have usedA(ei) = x/\\x\\. Similarly, в\\Р= \\x\\ ~ У \Р. Thus,f{U")
- \\xII ^2(ф'|^)Л(ф2\P) at (x, A)EP. Let ju. be the 2-form on R3 such
that n\Sr is the area element of Sr and ix(X, Y) = 0 if X is radial.
At (x,A)EP, we prove that ttf*ix = ф1 Л ф2. Indeed, for ex and e2
standard horizontal fields in T(x A)F(U3), we have (ттр1л)(ёх,ё2) =

10 ADDITIONAL TOPICS 167
lx(iTFJi,7TFJ2) = ix(A(el),A(e2))=\, since А(е^ = х/\\х\\ implies
A{e^),A(e2) is an oriented o.n. frame of TxSr. It is clear that
and we then have f{U")= ||дс1ГУ Note that
We consider the physical interpretation. Note that there is an
isomorphism U( 1) -> SOB),
,„ . cos a sin a 1
sin a cosaj
with induced Lie algebra isomorphism
0 fi
-fi or
Thus we see that it: P^>R3 can be regarded as a PFB with group
U(\), and /(fi") is the (time-independent) electromagnetic field
strength of the %(l)-valued connection corresponding to со. Also,
/Y О ш\ — II у (I ^2 — || v||~3/vl Jv2 д jv3 _j_ v2 Jv3 д Jv 1 _j_ v3 Jv! д
у V *& ^ v — |[a|[ jx — пли \Х их i \ их т x их i \ их i л йл / \
dx2) is (according to 0.2.22) a "purely magnetic" field due (presuma-
(presumably) to a magnetic monopole at 0. Actually, [(l/27r)/(£2t0)] is the
Chern class of it: P^IR3 (with group U(\)) and the number
f 1 _
/s, 27r"
is the (magnetic) charge of the monopole. By 10.4.13, we cannot
change the magnetic charge by deforming Sr or by changing the
potential со. The monopole is a manifestation of the topology of тт:
P^> M and not the choice of со.
In this example, the charge was 2. In order to change the charge
to 1, we must change the PFB. Let S(IR3) = IR3 XSUB), and let Л:
S(IR3)^F(IR3) = IR3X5'<9C) be given by X(x,A) = (x,A(A)) where
Л: SUB)^SOC) is the universal covering homomorphism of 6.1.5
restricted to SUB)CSLB,C). Let S(P) = \~\P), and note that tts:
5(P)^R3 is a PFB with group A~\SOB))=U(l). Since A~'(S0B))
covers SOB) twice, we have that со' = ^Л*(со) is a connection on
S{P). Also, f(u"') = \f{uu), and so tts: S(P)^IR3 gives us a mono-

168 GAUGE THEORY AND VARIATIONAL PRINCIPLES
pole of charge 1. Note that the cyclic group ZpС U(\) of order n acts
on S(P), and gives rise to a PFB S(P)/Zn->R3 with group U(\)/Zn
= GA). Since the fibers of S(P) cover those of S(P)/1n n times, the
charge of the monopole given by S(P)/Zn^> IR3 is n. By reversing the
direction of the U(\) action on S(P)/Zn, we can achieve charge —n.
10.4.16 Instantons. Let к be an &b-invariant metric on § (such
always exists if G is compact). Observe that kEl2(G). For a PFB
it:P^M with group G and connection со, we have k(u")EA4(P,R) as
in 10.4.2. Writing п" = ^паеа for a basis <?,,..., ey of §, we have
к(п") = 2карпаЛ пр, where ка/} = к(еа,ер). If a: U^P is a local
section, then k(U")\U=lka/iUauA Upu where Uau = o*Sla. Suppose
that M is an oriented Riemannian 4-manifold with (positive definite)
metric g. Then we have **—\ on 2-forms. Using the notation Яои =
2кар$1аи, we have
(see 0.2.17).
Since **= 1 on 2-forms and *: A2(M)x -*A2(M)x is self-adjoint in
the sense that gx(*a,/3) = gx(a, */3), it follows that A2(M)X decomposes
into orthogonal + 1 and — 1 eigenspaces of *, say A2(M)X = Л2(М)ХЬ
ФЛ2(M)x . For ctEA2(M) we then have a decomposition a~a + +a~,
and a is called self-dual if ct~ =0 and anti-self-dual if a+ =0. Simi-
Similarly, we have a decomposition A2(P, §) = Л2(Р, §) + ®A2(P, §)~, and
we can write ПШ=ПШ++ПШ'. Now, g(Upu, *Upu)n = g(U+pu +
Q-pu,Qp+u-Qp-u)^=g(Q+Pu,Qp+u)^-g(Q-pu,Qp-u)li. Summing
over p, we see that к(пш) = к(п"+)~к(п" ), where the right-hand
side can be defined independently of a, using the fact that the decom-
decomposition of A2(P, §) is preserved under R* (A EG), and so on. Define
the Action density of со to be &(o>)ii=(gk)(tt",tt")n=(k(tt"+) +
к(п"~)). If M is compact, then 10.4.13 says that jMk(u") is a
constant, independent of со. Calling this constant c, and defining
a = jMk(U"+) and b=jMk(U"~), we have a~b=c and a + b=
jM&(ic)ix or fM&(o>)n=2a-c=2b+c. Thus, /w(£(u)ju. will be mini-
minimized for those со such that fi" is self-dual (i.e., b = 0) or fi" is
anti-self-dual (i.e., a = 0). Such connections are called instantons or

10 ADDITIONAL TOPICS 169
anti-instantons, respectively. For arbitrary со, we have
where с depends only on the PFB.
Unlike 10.4.15, we will not consider the physical significance of
instantons, since this is currently beyond the author's scope. Let it
suffice to say that the case of most interest is that in which M is
the ordinary sphere in R5 and G = SU(n). There are a number of
nice results in this area (see Atiyah, Hitchin, and Singer [1978];
Bouguignon, Lawsoh, and Simons [1979]; Schwarz [1979]; etc.).

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't Hooft, G. 1976. Symmetry-breaking through Bell-Jackiw anomalies, Phys. Rev.
Lett. 37, 8-11.
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Plenum Press, New York, 287-307.
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Physics. University Press of Hawaii, Honolulu, Hawaii, 489-561.

Index of Notation
In the following, each notation is accompanied by the section(s)
in which it was introduced. The order of the index is by section
number.
aA/3
g
*
\g\
(-\у
тхм
Yx[f]
[Y,Z]
/..*
9,
Ak(M)
die
/*co
5a
Ak{M,V)
Ad_
0.1.2
0.1.4
0.1.5
0.1.6,0.2.17
0.1.7
0.1.7
0.2.3
0.2.3
0.2.4
0.2.5
0.2.6
0.2.8
0.2.9
0.2.10
0.2.11
0.2.17
0.2.23
0.3.8
ab
PFB
LT
§
A*
D"
п"
C(P,V)
Ak(P,V)
рЛф
GA(P)
J{P,V)
UCCM
hp
*
P
*
(hh)
0.3.8
0.3.8
1.1.1
1.1.1
1.2.2
1.2.2
2.2.2,3.1.3
2.2.3
3.1.1
3.1.2
3.1.4
3.2.1
3.3.1
4.1.1
4.2.0
4.2.0
4.2.1
4.2.4
175

176 Index of Notation
(hh)
8"
V гЬ
dL/d(D"t)
V 2L
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X, X
weA](F(M) U")
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9"
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FQ(M)
<ns: S(M)^M
A: S(M)^ F0(M)
y: R4^§CD,C)
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4.2.6
4.2.8
4.3.1
4.3.1
4.3.3
4.3.4
5.1.4
6.1.3
6.2.2
6.2.2
6.2.4
6.2.5
6.3.1
6.3.2
6.3.2
6.3.3
6.3.6
Ф
yXa
Р\°Рг
J(P,V)
•jfxa
yXo
fP.q
_
/
в
f
~nh
Kijk
Rik,R
a = /3
X*f
Л: L(M)^L(
6.3.7
6.3.11
7.1.1
7.1.6
7.1.10
7.1.11
7.2.4
8.1.1
8.1.2
8.2.1
8.2.5
8.2.13
8.2.20
9.1.9
9.2.1
M) 9.2.2

Index
Acceleration, 142
Action, 55
density, 53
self-, 68
Adjoint actions, 20
Automorphisms of PFBs, 46
Base manifold, 26
Bianchi identity, 39
first, 115
second, 115
Broken Hessian, 157
Canonical 1-form, 78
Characteristic classes, 164
Charge density, 16
Chart, 7
Chern-Weil homomorphism, 162
Choice of gauge, 26
Clifford algebra, 82
Closed form, 15
Codifferential, 14
Compact subset, 11
Connection, 29-31
Conservation law
charge, 17, 67
energy-momentum, 127
external, 132
Continuity equation, 17, 67
Contraction of tensors, 109
Coordinate system, 7
Coordinate vector fields, 8
Covariant codifferential, 58
Covariant derivative, 37, 111, 142
Critical point, 18
Current, 65-66
Current for E-M, 16
Curvature
of connection, 37
as field strength, 37, 39
see also Riemann-Christoffel
curvature tensor
Curve, 8
de Rham cohomology space, 15
Diffeomorphism, 7
Differential of map, 8
Dirac equation
free, 87
nonfree, 98
Dirac matrices, 81
Divergence, 125, 127
Duality for 2-forms, 168
Einstein field equation, 127, 134
Einstein tensor, 127
Electron field, 83, 96
Electric field, 16, 145
Electromagnetic field, 33, 145
Exact form, 15
177

178
Index
Exponential map, 19, 48
Exterior derivative, 11
covariant, 37, 44
Fiber, 26
Fiber derivative, 156
Field equation
Einstein, 127, 134
homogeneous, 39
inhomogeneous, 68, 94, 98, 103-104
Field strength, 37, 39
Forms
equivariant, 44
real-valued, 10
vector-valued, 17
Frame bundle, 28, 78
Fundamental field, 30
Gauge
algebra, 48
choice of, 26
orbit, 151
potentials, 33
transformations, 46
unitary, 157
Gauss-Bonnet theorem, 126
Geodesic, 142
Global
section, 27
trivialization, 27
Goldstone bosons, 156
Graded Lie algebra, 36
Gradient of metric functional, 123-124
Hessian, 18
broken, 157
Higgs fields, 156
Horizontal lift
of curve, 141
of vector field, 37
Instantons, 168-169
Integration of forms, 11-12
Isospin current, 101-102
Jacobi identity, 8
Jets, 50, 93, 148
L-tensors, 107
components, 108-109
contraction, 109
covariant differentiation of, 111
raising and lowering indices, 112
tensor product, 109
Lagrange's equation, 61, 94
Lagrangian,
connection, 148
curvature, 152
electron, 83, 96
G-invariant, 51
nucleon, 99-100
particle field, 50
spin-zero, 62
Left-invariant vector field, 18
Levi-Civita connection, 77, 110
Lie algebra, 18
Lie derivative, 9, 131
Lie group, 18
Lie subgroup, 19
Local section, 27
Local trivialization, 26
Lorentz group, 73
Magnetic monopoles, 166-168
Magnetic field, 16, 145
Manifold, 7
Map, 7
Mass of Higgs field, 156
Massive vector bosons, 158-160
Maxwell's equations, 16-17, 63
Metric
on manifold, 14
on vector space, 3
Minkowski space, 16
Nucleon field, 99
Nucleon field equation, 103-105
One-parameter group, 9
Open subset, 6
Open covering, 11
Orientation, 3, 11, 81
Orthonormal frame bundle, 78
Parallel translation, 142
Particle fields, 43
Principal fiber bundles, 26

Index
179
Principle of least action for
metrics, 123
with connections, 134
particle fields, 56
with connections, 68
Product manifold, 17
Product bundle, 26-27
Projected support, 56
Pull-back, 11, 127
Representation, 43
Ricci identity, 119
Ricci tensor, 116
Riemann-Christoffel curvature tensor,
114
identities, 115
infinitesimal changes, 118-119
Scalar curvature, 116
Scalar mesons, 156
Self-Action, 68
Shifted field, 156
Source 1-form, 16
see also Current
Special unitary group, 21-22
Spin structure, 81
Spin-zero electrodynamics, 62, 70
Spinors, 76-77
Spliced bundles, 90
Spontaneous symmetry breaking, 154-
160
Standard horizontal fields, 110
Star operators, 4, 14, 56
Stationary
metric, 123
with connection, 134
particle fields, 56
with connection, 68
Stokes' theorem, 12
Structure constants, 21
Structural equations, 37, 80
Submanifold, 16
Submersion, 149
Support, 11
Tangent vector, 8
Tensor field, 10
Tensor product, 109
Torsion form, 78
Total space, 26
Transition function, 27
Trivial bundle, 27
Twisted metric, 82
Unbroken subgroup, 154
Unitary gauge, 157
Unitary set, 157
Utiyama's theorem, 153
Vacuum, 154
Vector field, 8
Vector-valued forms, 17
Vertical subspace, 29
Volume element, 3, 14
Wedge product, 3
Weil algebra, 161
Weil polynomial, 153, 161
Yang-Mills equation, 65, 134