Marathe K. Topics in Physical Mathematics

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Chapter 6

Theory of Fields, I: Classical

6.1 Introduction

In recent years gauge theories have emerged as primary tools for research

in elementary particle physics. Experimental as well as theoretical evidence

of their utility has grown tremendously in the last two decades. The isospin

gauge group SU(2) of Yang–Mills theory combined with the U(1) gauge group

of electromagnentic theory has lead to a uniﬁed theory of weak interactions

and electromagnetism. We give an account of this uniﬁed electroweak theory

in Chapter 8. In this chapter we give a mathematical formulation of several

important concepts and constructions used in classical ﬁeld theories. We be-

gin with a brief account of the physical background in Section 6.2. Gauge

potential and gauge ﬁeld on an arbitrary pseudo-Riemannian manifold are

deﬁned in Section 6.3. Three diﬀerent ways of deﬁning the group of gauge

transformations and their natural equivalence is also considered there. The

geometric structure of the space of gauge potentials is discussed in Section

6.4 and is then applied to the study of Gribov ambiguity in Section 6.5. A

geometric formulation of matter ﬁelds is given in Section 6.6. Gravitational

ﬁeld equations and their generalization is discussed in Section 6.7. Finally,

Section 6.8 gives a brief indication of Perelman’s work on the geometrization

conjecture and its relation to gravity.

The literature in the area covered in Chapter 6 is rather vast. We cite here

only a few references that may be consulted for additional information about

the material of this chapter and related aspects of gauge theories from the

physicist’s point of view. They are Cheng and Li [73], Faddeev and Slavnov

[120], and Quigg [322]. Classical ﬁeld theories need to be quantized before

they can be applied to study elementary particle physics. While there is

no mathematically satisfactory theory of quantization of ﬁelds, physicists

have developed several workable methods of ﬁeld quantization. The most

notable among these is Feynman’s method of path integration. For the path

integral method of quantization of ﬁeld theories and related topics, see, for

K. Marathe, Topics in Physical Mathematics, DOI 10.1007/978-1-84882-939-8 6, 169

 Springer-Verlag London Limited 2010

170 6 Theory of Fields, I: Classical

example, Feynman and Hibbs [125] and Schulman [340]. Standard references

for a general discussion of the mathematical aspects of gauge theories are

the books by Bleecker [43], Booss and Bleecker [47], Freed and Uhlenbeck

[135], and the papers [116, 273]. For discussion of modern geometry and its

relation to physics see, for example, the books by Frankel [132]andNovikov

and Taimanov [301].

6.2 Physical Background

In this section we give a brief account of some ideas and results from modern

physics which are important in elementary particle physics.Weusethe

physical terminology found in standard texts in modern physics. Elementary

particle physics is also referred to as high energy physics.Theterm“high

energy” refers to the kinetic energy of particles used in accelerators and

colliders to produce particle reactions. The principal experimental method

of producing elementary particles and studying their size and structure is

to accelerate them to high energies and shoot them against a given target,

which may be another accelerated beam. The distance and the time scale of

the resulting reaction is extremely small. The dispariy of the extremely small

scale (distances of less than 10

−13

cm and duration of the order of 10

−10

sec

or less) of elementary particles and the extremely high energies required to

produce and study them may be explained by two fundamental principles of

modern physics, now discussed.

First, Einstein’s well-known mass–energy relation

E = mc

where c is a universal constant (the velocity of light in a vacuum

∼

3 ·

cm/sec), allows us to calculate the energy E required for the creation

(or released with the annihilation) of mass m. It also allows us to express both

energy and mass in term of the same physical unit. In elementary particle

physics this unit is taken to be the electron-volt (eV), the kinetic energy

acquired by an electron in passing through an electric potential of 1 volt. In

terms of this unit the electron rest-mass is m

∼

0.5MeV(1MeV=10

eV).

The proton rest mass is m

∼

1GeV(1GeV=10

eV), while the mass

of the heaviest known elementary particle, the Z

, was ﬁrst conﬁrmed by

observations made in the early 1980s to be approximately 91 GeV.

Einstein is considered one of the greatest physicists of all time. His special

theory of relativity provides the foundation for both classical and quantum

ﬁeld theories. The general theory of relativity revealed the intimate connec-

tion between space-time geometry and gravity. Einstein received the 1921

Nobel Prize in physics for his many contributions to theoretical physics and

especially for his discovery of the law of the photoelectric eﬀect. The photo-

6.2 Physical Background 171

electric eﬀect provides conclusive evidence for the quantum nature of light.

Combined with the known wave properties of light, it gave an example of the

wave–particle duality.

Second, Heisenberg’s uncertainty principle states that in a simulta-

neous measurement of two conjugate observables A, B of a quantum system

in a given state ψ, one has

AΔ

B ≥ /2,

where Δ

C denotes the standard deviation in the measurement of the ob-

servable C in the state ψ and 

∼

−27

erg sec is a universal constant related

to the Planck constant h by the relation  = h/2π.

Werner Heisenberg

(1901–1976) was one of the greatest physicists of the

twentieth century. He is considered one of the founders of quantum mechan-

ics. Heisenberg received the Nobel Prize in physics in 1932. The uncertainty

principle is a cornerstone of quantum physics. In particular, applying the un-

certainty relation to the conjugate observables position Q and momentum P

yields that quantum states in which the precision of the measurement of the

position is high are also states with large uncertainty in the measurement of

momentum. Thus, the more precisely the position of a particle is determined,

the less precisely its momentum is known. A similar situation exists with re-

spect to the simultaneous measurement of the conjugate observables time

and energy. This explains in part the requirements as well as the limitations

of high energy experiments. Indeed, the energies required for the production

of some conjectured elementary particles such as the Higgs boson are so high

that they do not seem to be reachable in the near future. In the remaining

part of this section we touch upon some basic aspects of the development of

elementary particle physics.

Elementary particle physics is the science that studies the ultimate con-

stituents of matter, the interactions among them, and the fundamental forces

acting on them. In this broad sense elementary particle physics deals with

questions that have been asked since ancient times. For example, it is well

known that the classical Greek philosophers discussed such questions. Indeed

the concept of atom as an indivisible fundamental constituent of matter can

be traced to early Greek writings. Of the four fundamental forces known to-

day, the force of gravity and some of its properties have been recongnized

since ancient times. However, the beginning of a theory of the gravitational

ﬁeld was made only in the sixteenth century by Galileo, and the ﬁrst com-

plete theory of gravity was worked out by Newton in the following century.

Newton’s main motivation was to obtain a mathematical explanation of Ke-

pler’s laws of planetary motion, one of the most important instances in which

experimental physics required and led to the development of a new area of

I had the opportunity to visit Prof. Heisenberg at the MPI in Munich in 1970 and cherish

the autographed copy of his autobiography Der Teil und das Ganze, which he gave me at

that time.

172 6 Theory of Fields, I: Classical

mathematics. Newton developed the calculus to provide an appropriate tool

for studying Kepler’s laws. The second law is a conversion law, which provides

a ﬁrst integral of motion. The equation of an ellipse in polar coordinates with

origin at one of the focal points was already well-known. Using this and the

second law Newton could easily calculate the radial accelaration of the planet.

This calculation led him to his law of gravitation. If one starts with this law

then one sees that orbits depend on initial conditions and are, in general,

conic sections. The elliptic orbits arise from special initial conditions.

Some properties of electric and magnetic forces were also known for a long

time, but a systematic experimental and theoretical study of these forces

started only in the eighteenth century, culminating in the uniﬁed theory of

the electromagnetic ﬁeld developed by Maxwell. The discovery of the periodic

table and radioactivity led scientists to suspect that in spite of their name,

atoms may not be the indivisible, ultimate constituents of matter. This sus-

picion was conﬁrmed with the discovery of the electron and the nucleons

(proton and neutron) as subatomic particles.

The foundations of quantum theory were established during the ﬁrst quar-

ter of the twentieth century, The theory immediately met with great success

in explaining a wide range of phenomena in molecular, atomic, and nuclear

physics. This, coupled with the discovery of subatomic particles, created

a new and rapidly growing branch of physics, namely, elementary particle

physics. The principal theoretical tools for studying elementary particles and

their interactions are provided by (quantum) gauge ﬁeld theories. We now

discuss two well-known examples of classical gauge theories. The problem of

their quantization is discussed from a physical perspective in Chapter 7.

Maxwell’s electromagnetic theory provides the simplest example of a gauge

theory, with the ﬁeld equations being given by Maxwell’s equations.We

therefore begin with a brief review of Maxwell’s equations, which in their

classical form are given by:

div B =0

curl E = −∂B/∂t

div E = ρ

curl B = J + ∂E/∂t,

where the electric ﬁeld E and the magnetic ﬁeld B are time-dependent vector

ﬁelds on some subset of R

and ρ and J are the charge and current densi-

ties, respectively. These equations uniﬁed the separate theories of electricity

and magnetism and paved the way for important advances in both experi-

mental and theoretical physics. With the introduction of the 4-dimensional

Minkowski space-time M

it became possible to describe both the electric

and magnetic ﬁelds as parts of a skew-symmetric tensor ﬁeld, or a diﬀeren-

tial 2-form F ,onM

as follows. Using the standard chart on M

and the

induced bases of the tensor spaces, the tensor F has components given by:

6.2 Physical Background 173

= E

, 1 ≤ k ≤ 3 F

= B

Maxwell’s equations written in terms of F (regarded as a 2-form) are

dF =0,δF= j, (6.1)

where j =(J, ρ) is the current density 1-form and δ = ∗d∗ is the codiﬀerential

operator on 2-forms of M

. The source-free ﬁeld equations are obtained by

setting j = 0 and can be expressed as

dF =0,d∗F =0. (6.2)

This is the modern form of Maxwell’s source free ﬁeld equations. It is well-

known that Maxwell’s equations are globally invariant under the conformal

group and in particular, under the Lorentz group. Their Lorentz invariance

is the starting point of Einstein’s theory of special relativity. They are also

invariant under a local group of transformations known as gauge trans-

formations. This invariance arises as follows. On M

the equation dF =0

implies that the ﬁeld F is derivable from a 4-potential or 1-form A, i.e.,

F = dA. However, the potential A is not uniquely determined. If B is an-

other potential such that F = dB,thend(B − A) = 0. Every closed form on

is exact and this implies that B − A = dψ, ψ ∈F(M

). Thus, we may

think of the potential B as obtained by a gauge transformation of A by ψ.

We can write this transformation as

A → A

:= A + dψ, ψ ∈F(M

). (6.3)

Since ψ is real-valued Weyl considered ψ(x) to be the choice of the scale

at x. The inﬁnitesimal change dψ = ψ(x + dx) − ψ(x) corresponds up to

ﬁrst order to a change of local scale. This is not related to the conformal

invariance of the Maxwell equations. In Einstein’s theory of gravitation the

base manifold M is a 4-dimensional Lorentz manifold, usually referred to as a

space-time manifold, and its pseudo-metric or the corresponding Levi-Civita

connection plays the role of “gravitational potential.” Now Maxwell’s equa-

tions written by using F admit immediate generalization to the case of the

Minkowski space replaced by an arbitrary 4-dimensional Lorentz manifold.

However, in contrast to the geometric description of a gravitational ﬁeld on

M, the electromagnetic ﬁeld is put in “by hand” as a 2-form on M satisfying

the generalized Maxwell’s equations. Thus, it was natural to look for a uniﬁed

geometric theory of gravity and electromagnetism. Weyl sought to incorpo-

rate the electromagnetic ﬁeld into the geometric structures associated to the

space-time manifold as arising from local scale invariance. He referred to

this scale invariance as “eich-invarianz” and this is the origin of the mod-

ern term gauge invariance. In fact, with slight modiﬁcation, replacing local

scale by local phase taking values in the unitary group U (1), one obtains a

174 6 Theory of Fields, I: Classical

formulation of Maxwell’s equations as gauge ﬁeld equations

with the gauge

group U(1). The gauge transformation with the real-valued function ψ of

equation (6.3) replaced by the gauge transformation g := e

iψ

∈ U(1) allows

us to rewrite equation (6.3)as

iA → iA

:= g

−1

(iA)g + g

−1

dg, g = e

iψ

∈F(M

,U(1)). (6.4)

Note that the gauge potential iA takes values in the Lie algebra u(1) = iR

of the gauge group U (1). We study gauge transformations with arbitrary

gauge group G later in this chapter. Equation (6.4) is a special case of the

equation (6.18) of a general gauge transformation. We discuss this formulation

of Maxwell’s equations and the interpretation of the form F as a U(1) gauge

ﬁeld on M

in Chapter 8.

Another approach to a uniﬁed treatment of electromagnetic and gravita-

tional ﬁelds leads to the Kaluza–Klein theory, which uses a 5-dimensional

pseudo-Riemannian manifold and a suitable (4 + 1)-dimensional decompo-

sition to obtain Einstein’s and Maxwell’s equations from the 5-dimensional

metric. The ideas of Kaluza–Klein theory have been applied to higher di-

mensional manifolds for studying coupled ﬁeld equations and the problem of

dimensional reduction. For a modern treatment of Kaluza–Klein theories, see

Hermann [190] and Coquereaux, Jadczyk [85].

In 1954 Yang and Mills [413,412] obtained the following now well-known

non-Abelian gauge ﬁeld equations for the vector potential b

of isotopic spin

in interaction with a ﬁeld ψ of isotopic spin 1/2.ThesearetheoriginalYang–

Mills equations for the SU(2) gauge group

∂f

μν

/∂x

+2(b

× f

μν

)+J

=0, (6.5)

where the quantities

μν

= ∂b

/∂x

− ∂b

/∂x

− 2b

× b

, (6.6)

are the components of an SU(2)-gauge ﬁeld and J

is the current density of

the source ﬁeld ψ. There was no immediate physical application of these equa-

tions since they seemed to predict massless gauge particles as in Maxwell’s

theory. In Maxwell’s theory the predicted massless particle is identiﬁed as

photon, the massless carrier of electromagnetic ﬁeld. No such identiﬁcation

could be made for the massless particles predicted by pure (i.e., source-free)

Yang–Mills equations. This diﬃculty is overcome by the introduction of the

Higgs mechanism [192], which shows how spontaneous symmetry-breaking

can give rise to massive gauge vector bosons by a gauge transformation to a

I discussed this in my talk at the Geometry and Physics Workshop organized by Prof.

Raoul Bott at MSRI, Berkeley in 1994. After my talk Bott remarked: “We teach Harvard

students to think of functions as Lie algebra valued 0-forms so that they know the distinc-

tion between scale and phase.” When one of his students said he never learned this in his

courses, Bott gave a heary laugh.

6.2 Physical Background 175

particular local gauge. This paved the way for a gauge theoretic formulation

of the standard model of electroweak theory and subsequent development of

the general framework for a uniﬁed treatment of strong, weak and electro-

magnetic interactions.

The Yang–Mills equations may be thought of as a matrix-valued general-

ization of the equations for the classical vector potential of Maxwell’s theory.

The gauge ﬁeld that they obtained turns out to be the curvature of a connec-

tion in a principal ﬁber bundle with gauge group SU(2). The general theory

of such connections was developed in 1950s by Ehresman [117]. However,

physicists continued to use the classical theory of connections and curva-

ture that was the cornerstone of Einstein’s general relativity theory. In fact,

using this classical theory, Ikeda and Miyachi

[199, 200] essentially linked

the Yang–Mills theory with the theory of connections. However, this work

does not seem to be well-known in the mathematical physics community,

and it was not until the early 1970s that the identiﬁcation between curva-

ture of a connection in a principal bundle and the gauge ﬁeld was made.

This identiﬁcation unleashed a ﬂurry of activity among both physicists and

mathematicians and has already had great successes, some of which were

indicated in the preface. Since the physical and mathematical theories have

developed independently each has its well-established terminology. We have

used the notation that is primarily used in the mathematical literature but

we have also taken into account the terminology that is most frequently used

in physics. To help the reader we have given Appendix A, a table indicating

the correspondence between the terminologies of physics and mathematics,

prepared along the lines of Trautman [378]. The physical literature on gauge

theory is vast and is, in general, aimed at applications to elementary particle

physics and quantum ﬁeld theories. The most successful quantum ﬁeld theory

is quantum electrodynamics (QED), which deals with quantization of elec-

tromagnetic ﬁelds. Its predictions have been veriﬁed to a very high degree of

accuracy. However, there is as yet no generally accepted mathematical theory

of quantization of gauge ﬁelds.

After this brief look at two classical gauge theories, we now turn to a

discussion of some major developments in elementary particle physics in the

last 30 years. As we observed earlier, the aim of elementary particle physics

is to study the structure of matter in terms of some fundamental system of

constituents. The last three decades have seen a dramatic increase in our

knowledge of both the theoretical and the experimental aspects of elemen-

tary particle physics. A host of new experimental results have come out es-

pecially from a new generation of particle accelerators. We have identiﬁed

two types of elementary particles, namely the leptons and the hadrons.

In addition to these there are the carriers of the forces of interaction. Each

particle also has a corresponding anti-particle with the same rest mass and

spin, but with opposite charge. All particles are subject to two kinds of statis-

We would like to thank Prof. Akira Asada of Shinshu University for introducing us to

Prof. Miyachi and his work.

176 6 Theory of Fields, I: Classical

tics. Those subject to Fermi–Dirac statistics have half-integer spin and are

called fermions, while those subject to Bose–Einstein statistics are called

bosons. Leptons come in three pairs: the electron,themuon,andthetau

particle, each with its corresponding massless neutrino. The hadrons are

divided into two groups called baryons (with half-integer spin) and mesons

(with integer spin). The well-known nuclear particles proton and neutron are

baryons, while pion and kaon are examples of mesons. A large number of

other hadrons have been discovered. We now have strong evidence that the

properties of hadrons can be explained by considering them as composed of

quarks u (up), d (down), c (charm), s (strange), t (top or truth), and b

(bottom or beauty). Each quark also carries a color index corresponding

to the color gauge group SU

(3). Each quark has its anti-quark part-

ner. The hadrons are made up of combinations of quarks and anti-quarks.

However, they are supposed to remain conﬁned inside hadrons and hence

are unobservable. This phenomenon is called quark conﬁnement.Thefact

that all searches for free quarks since 1977 have had negative results strongly

supports the hypothesis of quark conﬁnement. In view of this we will not call

quarks fundamental particles. In view of the indirect yet rather strong ex-

perimental evidence now available we can consider quarks as virtual particles

that form the fundamental constituents of hadrons.

Table 6.1 displays what we consider to be the fundamental constituents

of matter at this time. Each constituent is identiﬁed by its name, symbol,

charge (in units of the proton charge), and mass or range of mass (in units

of GeV), in that order. The information in the table is a summary of our

knowledge of the fundamental constituents of matter at this time. These

particles are subject to the various fundamental forces, which act via their

carrier particles. Three of them have interpretation as gauge ﬁelds. They are

combined to obtain the standard model of fundamental particles and forces

described later. The data in Table 6.1 and Table 6.2 are taken from [11], with

2009 web updates by the Particle Data Group. They are not needed for the

mathematical formulation discussed in detail in Chapter 8.

All matter is subjected to one or more of the four fundamental forces.

They are the well known classical, long range forces of electromagnetic

and gravitational ﬁelds and the more recently discovered short range forces

of weak and strong interactions. Leptons are subject to all but the strong

interaction while hadrons participate in all of them. In classical ﬁeld theory

it was assumed that each particle generates a set of ﬁelds that extend over

entire space. A neutral, massive particle generates gravitational ﬁeld while a

charged particle also generates electromagnetic ﬁeld. Thus, for example, two

particles with charge of the same sign exert a repulsive force on each other

as a result of the interaction of their ﬁelds. In quantum ﬁeld theory it is pos-

tulated that all forces act by exchange of carrier particles or quanta.For

the gravitational ﬁeld this carrier particle is called the graviton.Ithasspin

2 related to the representation of the Lorentz group on symmetric tensors

of order 2. This is consistent with the observation that gravitation is always

6.2 Physical Background 177

Tab le 6. 1 Fundamental constituents of matter

Leptons Quarks

electron electron neutrino up down

e ν

u d

−1 0 2/3 −1/3

0.0005 0(< 10

−17

) 0.0015−0.0033 0.0035−0.006

muon muon neutrino charmed strange

μ ν

c s

−1 0 2/3 −1/3

0.105 0(< 10

−3

) 1.2−1.3 0.6−1.3

tau tau neutrino top (or truth) bottom (or beauty)

τ ν

t b

−1 0 2/3 −1/3

1.78 0(< 0.25) ? 171 4.1−4.4

an “attractive force.” There is no experimental or theoretical evidence for

the graviton. For the electromagnetic ﬁeld, the prototype of Abelian gauge

ﬁelds, this particle has been identiﬁed as the photon. The beta decay of

unstable, radioactive nuclei provided an early example of weak interaction.

Its analysis led Pauli and Fermi to conjecture the existence of the neutrino

(Italian for “tiny neutron”). The prediction and detection of the carrier par-

ticles W

−

of the weak force is one of the triumphs of the electroweak

theory (a U (1) ×SU(2)-gauge theory), which provides a uniﬁed treatment of

electromagnetic and weak interactions. The particles W

−

are called

weak intermediate vector bosons.TheU(1) factor of the electroweak

gauge group is called the weak hypercharge gauge group and is de-

noted by U

(1). The SU(2) factor of the electroweak gauge group is called

the weak isospin gauge group and is denoted by SU

(2). Hence, the elec-

troweak gauge group is often denoted by U

(1) ×SU

(2). The subscript

L in the term SU

(2) denotes the action on left-handed fermions. In fact, the

electroweak theory is left-right asymmetric. For further details see [11].

The energy of the carrier particles and the range of corresponding inter-

action are related by Heisenberg’s uncertainty principle. The inﬁnite

range of electromagnetic interaction is consistent with the zero rest mass of

its carrier particle, the photon. The observed short range of the weak inter-

action requires that its carrier particles be massive. A very readable account

of the problems and triumphs of electroweak theory is given in [330]. The

strong force corresponds to the color gauge group SU

(3) and has eight ex-

change particles, called gluons. The corresponding quantum ﬁeld theory is

called quantum chromodynamics or QCD for short. The success of the

electroweak theory has led scientists to a uniﬁed theory of strong, weak, and