Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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Remark and Example The pointwise limit of

continuous functions need not be continuous, as

can be shown by the following example:

(x) = x

, x 2 [0, 1]. We see that the limit function

f is not continuous:

f ðxÞ¼

0 x 6¼ 1

1 x ¼ 1

Definition 35 Let X be a metric space. A map

f : X !X is a contraction if there exists c < 1 such

that d(f (x), f (y)) cd(x, y) for all x, y 2 X.

Theorem 3 (Banach) If X is a complete metric

space and f is a contraction in X, then f has a unique

fixed point x 2 X, that is, f(x) = x.

Some Function and Operator Spaces

The spaces of functions and operators can be

equipped with different topologies, given by various

concepts of convergence and of norms (or sometimes

seminorms), very often with different such concepts

for the same space. As we saw earlier, a norm in a

vector space gives rise to a metric, and hence to a

topology. Similarly with the concept of convergence

for sequences of functions and operators, as one

then knows what the limit points, and hence closed

sets, are.

But before we do that, let us introduce, in a

slightly different context, a topology which is in

some sense the natural one for the space of

continuous maps from one space to another.

Definition 36 Consider a family F of maps from a

topological space X to a topological space Y, and

define W(K, U) = {f : f 2 F, f (K)  U}. Then the

family of all sets of the form W(K, U) with K

compact (in X) and U open (in Y) form a sub-basis

for the compact open topology for F.

Consider a topological space X and sequences of

functions (f

) on it. Let D  X. We can then define

pointwise convergence and uniform convergence

exactly as for functions on subsets of the real line.

Definition 37 Let X, D and (f

) as above.

(i) The functions f

converge pointwise on D to a

function f if the sequence of numbers

(x) !f (x), 8x 2 D.

(ii) The functions f

converge uniformly on D to a

function f if given >0, there exists N such that

for all n > N we have jf

(x)  f (x)j <, 8x 2 D.

Next we consider the Lebesgue spaces L

, that

is, functions f defined on subsets of R

, such

that jf (x)j

is Lebesgue integrable, for real

numbers p 1. To define these spaces, we tacitly

take equivalence classes of functions which are equal

almost everywhere (that is, up to a null set), but very

often we can take representatives of these classes

and just deal with genuine functions instead. Note

that of all L

, only L

is a Hilbert space.

Definition 38 In the space L

, we define its norm by

kf k¼

jf ðxÞj



1=p

Now we turn to general normed spaces, and

operators on them.

Definition 39 Convergence in the norm is also

called strong convergence. In other words, a

sequence (x

) in a normed space X is said to

converge strongly to x if

lim

n!1

 xk¼0

Definition 40 A sequence (x

) in a normed space X

is said to converge weakly to x if

lim

n!1

f ðx

Þ¼f ðxÞ

for all bounded linear functionals f.

Consider the space B(X, Y) of bounded linear

operators T from X to Y. We can make this into a

normed space by defining the following norm:

kTk¼ sup

x 2X; kxk¼1

kTxk

Then we can define three different concepts of

convergence on B(X, Y). There are in fact more in

current use in functional analysis.

Definition 41 Let X and Y be normed spaces and

let (T

) be a sequence of operators T

2 B(X, Y).

(i) (T

)isuniformly convergent if it converges in

the norm.

(ii) (T

)isstrongly convergent if (T

x) converges

strongly for every x 2 X.

(iii) (T

)isweakly convergent if (T

x) converges

weakly for every x 2 X.

Remark Clearly we have: uniform convergence ¼)

strong convergence ¼) weak convergence, and the

limits are the same in all three cases. However, the

converses are in general not true.

Homotopy Groups

The most elementary and obvious property of a

topological space X is the number of connected

components it has. The next such property, in a

certain sense, is the number of holes X has. There

Introductory Article: Topology 135

are higher analogues of these, called the homotopy

groups, which are topological invariants, that is,

they are invariant under homeomorphisms. They

play important roles in many topological considera-

tions in field theory and other topics of mathema-

tical physics. The articles Topological Defects

and Their Homotopy Classification and Electric-

Magnetic Duality contain some examples.

Definition 42 Given a topological space X, the

zeroth homotopy set, denoted 

(X), is the set of

connected components of X. One sometimes writes



(X) = 0ifX is connected.

To define the fundamental group of X,or

(X),

we shall need the concept of closed loops, which we

shall find useful in other ways too. For simplicity,

we shall consider based loops (that is, loops passing

through a fixed point in X). It seems that in most

applications, these are the relevant ones. One could

consider loops of various smoothness (when X is a

manifold), but in view of applications to quantum

field theory, we shall consider continuous loops,

which are also the ones relevant for topology.

Definition 43 Given a topological space X and a

point x

2 X,a(closed)(based) loop is a continuous

function of the parametrized circle to X:

 : ½0; 2!X

satisfying (0) = (2) = x

Definition 44 Given a connected topological space

X and a point x

2 X, the space of all closed based

loops is called the (parametrized based) loop space

of X, denoted X.

Remarks

(i) The loop space X inherits the relative compact–

open topology from the space of continuous maps

from the closed interval [0, 2]toX. It also has a

natural base point: the constant function mapping

all of [0, 2]tox

. Hence it is easy to iterate the

construction and define 

X, k  1.

(ii) Here we have chosen to parametrize the circle

by [0, 2], as is more natural if we think in

terms of the phase angle. We could easily have

chosen the unit interval [0,1] instead. This

would perhaps harmonize better with our pre-

vious definition of paths and the definitions of

homotopies below.

Proposition 7 The fundamental group of a topo-

logical space X, denoted 

(X), consists of classes of

closed loops in X which cannot be continuously

deformed into one another while preserving the base

point.

Definition 45 A space X is called simply connected

if 

(X) is trivial.

To define the higher homotopy groups, let us go

into a little detail about homotopy.

Definition 46 Given two topological spaces X and

Y, and maps

p; q : X !Y

we say that h is a homotopy between the maps p, q if

h : X  I ! Y

is a continuous map such that h(x,0)= p(x),

h(x,1)= q(x), where I is the unit interval [0, 1]. In

this case, we write p ’ q.

Definition 47 A map f : X !Y is a homotopy

equivalence if there exists a map g : Y !X such

that g  f ’ id

and f  g ’ id

Remark This is an equivalence relation.

Definition 48 For a topological space X with base

point x

, we define 

(X), n  0 as the set of

homotopy equivalence classes of based maps from

the n-sphere S

to X.

Remark This coincides with the previous defini-

tions for 

and 

There is a very nice relation between homotopy

classes and loop spaces.

Proposition 8 

(X) = 

n1

(X) = = 

(

X).

Remarks

(i) When we consider the gauge group G in a Yang–

Mills theory, its fundamental group classifies the

monopoles that can occur in the theory.

(ii) For n  1, 

(X) is a group, the group action

coming from the joining of two loops together

to form a new loop. On the other hand, 

(X)

in general is not a group. However, when X is a

Lie group, then 

(X) inherits a group structure

from X, because it can be identified with the

quotient group of X by its identity-connected

component. For example, the two components

of O(3) can be identified with the two elements

of the group Z

, the component where the

determinant equals 1 corresponding to 0 in Z

and the component where the determinant

equals 1 corresponding to 1 in Z

(iii) For n  2, the group 

(X) is always abelian.

(iv) Examples of nonabelian 

are the fundamental

groups of some Riemann surfaces.

(v) Since 

is not necessarily abelian, much of the

direct-sum notation we use for the homotopy

136 Introductory Article: Topology

groups should more correctly be written multi-

plicatively. However, in most literature in

mathematical physics, the additive notation

seems to be preferred.

Examples

(i) 

(X  Y) = 

(X) þ 

(Y), n  1.

(ii) For the spheres, we have the following results:



ðS

Þ¼

0ifi > n

Z if i ¼ n





ðS

Þ¼0ifi > 1



nþ1

ðS

Þ¼Z

if n  3



nþ2

ðS

Þ¼Z

if n  2



ðS

Þ¼Z

(iii) From the theory of sphere bundles, we can

deduce:



ðS

Þ¼

i1

ðS

Þþ

ðS

Þ if i  2



ðS

Þ¼

i1

ðS

Þþ

ðS

Þ if i  2



ðS

Þ¼

i1

ðS

Þþ

ðS

Þ if i  2

and the first of these relations give the follow-

ing more succinct result:



ðS

Þ¼

ðS

Þ if i  3

(iv) A result of Serre says that all the homotopy

groups of spheres are in fact finite except 

)

and 

4n1

), n  1.

Definition 49 Given a connected space X,amap

 : B !X is called a covering if (i) (B) = X, and (ii) for

each x 2 X, there exists an open connected neighbor-

hood V of x such that each component of 

1

(V)isopen

in B,and restricted to each component is a home-

omorphism. The space B is called a covering space.

Examples

(i) The real line R is a covering of the group U(1).

(ii) The group SU(2) is a double cover of the group

SO(3).

(iii) The group SL(2, C) is a double cover of the

Lorentz group SO(1, 3).

(iv) The group SU(2, 2) is a 4-fold cover of the

conformal group in four dimensions. This local

isomorphism is of great importance in twistor

theory.

Remarks

(i) By considering closed loops in X and their

coverings in B it is easily seen that the

fundamental group 

(X) acts on the coverings

of X. If we further assume that the action is

transitive, then we have the following nice

result: coverings of X are in 1–1 correspon-

dence with normal subgroups of 

(X).

(ii) Given a connected space X, there always exists a

unique connected simply connected covering space

X, called the universal covering space. Further-

more,

X covers all the other covering spaces of X.

For the higher homotopy groups, one has



ðXÞ¼

XÞ; n  2

One very important class of homotopy groups are

those of Lie groups. To simplify matters, we shall

consider only connected groups, that is, 

(G) = 0.

Also we shall deal mainly with the classical groups,

and in particular, the orthogonal and unitary groups.

Proposition 9 Suppose that G is a connected Lie

group.

(i) If G is compact and semi-simple, then 

(G) is

finite. This implies that

G is still compact.

(ii) 

(G) = 0.

(iii) For G compact, simple, and nonabelian,



(G) = Z.

(iv) For G compact, simply connected, and simple,



(G) = 0orZ

Examples

(i) 

(SU(n)) = 0.

(ii) 

(SO(n)) = Z

(iii) Since the unitary groups U(n) are topologically

the product of SU(n) with a circle S

, their

homotopy groups are easily computed using the

product formula. We remind ourselves that

U(1) is topologically a circle and SU(2) topolo-

gically S

(iv) For i  2, we have:



ðSOð3ÞÞ ¼ 

ðSUð2ÞÞ



ðSOð5ÞÞ ¼ 

ðSpð2ÞÞ



ðSOð6ÞÞ ¼ 

ðSUð4ÞÞ

Just for interest, and to show the richness of the

subject, some isomorphisms for homotopy groups

are shown in Table 1 and some homotopy groups

for low SU(n) and SO(n) are listed in Table 2.

Table 1 Some isomorphisms for homotopy groups

Isomorphism Range



(SO(n)) ﬃ 

(SO(m)) n, m  i þ 2



(SU(n)) ﬃ 

(SU(m)) n, m 

(i þ 1)



(Sp(n)) ﬃ 

(Sp(m)) n, m 

(i  1)



) ﬃ 

(SO(7)) 2  i  5



) ﬃ 

(SO(9)) 2  i  6



(SO(9)) ﬃ 

(SO(7)) i  13

Introductory Article: Topology 137

Appendix: A Mathematician’s

Basic Toolkit

The following is a drastically condensed list, most

of which is what a mathematics undergraduate

learns in the first few weeks. The rest is included

for easy reference. These notations and concepts

are used universally in mathematical writing. We

have not endeavored to arrange the material in a

logical order. Furthermore, given structures such as

sets, groups, etc., one can usually define ‘‘substruc-

tures’’ such as subsets, subgroups, etc., in a

straightforward manner. We shall therefore not

spell this out.

Sets

A [ B ¼fx: x 2 A or x 2 Bg union

A \ B ¼fx: x 2 A and x 2 Bg intersection

AnB ¼fx: x 2 A and x 62 Bg complement

A  B ¼fðx; yÞ : x 2 A; y 2 Bg Cartesian product

Maps

1. A map or mapping f : A !B is an assignment of

an element f (x)ofB for every x 2 A.

2. A map f : A !B is injective if f (x) = f (y)

¼) x = y. This is sometimes called a 1–1 map, a

term to be avoided.

3. A map f : A !B is surjective if for every y 2 B

there exists an x 2 A such that y = f (x). This is

sometimes called an ‘‘onto’’ map.

4. A map f : A !B is bijective if it is both surjective

and injective. This is also sometimes called a 1–1

map, a term to be equally avoided.

5. For any map f : A !B and any subset C  B, the

inverse image f

1

defined, although, of course, it can be empty. On

the other hand, the map f

1

is defined if and only

if f is bijective.

6. A map from a set to either the real or complex

numbers is usually called a function.

7. A map between vector spaces, and more particu-

larly normed spaces (including Hilbert spaces), is

called an operator. Most often, one considers

linear operators.

8. An operator from a vector space to its field of

scalars is called a functional. Again, one con-

siders almost exclusively linear functionals.

Relations

1. A relation  on a set A is a subset R  A  A.

We say that x  y if (x, y) 2 R.

2. We shall only be interested in equivalence relations.

An equivalence relation  is one satisfying, for all

x, y, z 2 A:

(a) x  x (‘‘reflexive’’),

(b) x  y ¼) y  x (‘‘symmetric’’),

3. If  is an equivalence relation in A, then for each

x 2 A, we can define its equivalence class:

½x¼fy 2 A : y  xg

It can be shown that equivalence classes are

nonempty, any two equivalence classes are either

equal or disjoint, and they together partition the set

A. Subgroup equivalence classes are called cosets.

4. An element of an equivalence class is called a

representative.

Groups

A group is a set G with a map, called multiplication

or group law

G  G !G

ðx; yÞ7!xy

satisfying

Table 2 Some homotopy groups for low SU(n) and SO(n)



SU(2) Z

SU(3) 0 ZZ

0 Z

SU(4) 0 Z 0 ZZ

120

þ Z

SU(5) 0 Z 0 Z 0 ZZ

120

SU(6) 0 Z 0 Z 0 ZZ

SO(5) Z

0 Z 00Z

120

SO(6) 0 Z 0 ZZ

120

þ Z

SO(7) 0 0 0 ZZ

þ Z

SO(8) 0 0 0 Z þ ZZ

þ Z

SO(9) 0 0 0 ZZ

þ Z

SO(10) 0 0 0 ZZ

Z þ Z

138 Introductory Article: Topology

1. (xy)z = x(yz), 8x, y, z 2 G (‘‘associative’’);

2. there exists a neutral element (or identity) 1 such

that 1x = x1 = x, 8x 2 G; and

3. every element x 2 G has an inverse x

1

, that is,

1

= x

1

x = 1.

A map such as the multiplication in the definition

is an example of a binary operation. Note that we

have denoted the group law as multiplication here.

It is usual to denote it additively if the group is

abelian, that is, if xy = yx, 8x, y 2 G. In this case, we

may write the condition as x þ y = y þ x, and call

the identity element 0.

Rings

A ring is a set R equipped with two binary

operations, x þ y called addition, and xy called

multiplication, such that

1. R is an abelian group under addition;

2. the multiplication is associative; and

3. (x þ y)z = xz þ yz, x(y þ z) = xy þ xz, 8x, y, z 2 R

(‘‘distributive’’).

If the multiplication is commutative (xy = yx) then

the ring is said to be commutative. A ring may

contain a multiplicative identity, in which case it is

called a ring with unit element.

An ideal I of R is a subring of R, satisfying in

addition

r 2 R; a 2 I ¼) ra 2 I; ar 2 I

One can define in an obvious fashion a left-ideal and

a right- ideal. The above definition will then be for a

two-sided ideal.

Modules

Given a ring R,anR-module is an abelian group M,

together with an operation, M  R !M, denoted

multiplicatively, satisfying, for x, y 2 M, r, s 2 R,

1. (x þ y)r = xr þ yr,

2. x(r þ s) = xr þ xs,

3. x(rs) = (xr)s, and

4. x1 = x

The term right R-module is sometimes used, to

distinguish it from obviously defined left R-modules.

Fields

A field F is a commutative ring in which every

nonzero element is invertible.

The additive identity 0 is never invertible, unless

0 = 1, so it is usual to assume that a field has at least

two elements, 0 and 1.

The most common fields we come across are, of

course, the number fields: the rationals, the reals,

and the complex numbers.

Vector Spaces

A vector space, or sometimes linear space, V, over a

field F, is an abelian group, written additively, with

a map F  V !V such that, for x, y 2 V, ,  2 F,

1. (x þ y) = x þ y (‘‘linearity’’),

2. ( þ )x = x þ x,

3. ()x = (x), and

4. 1x = x.

A vector space is then a right (or left) F-module.

The elements of V are called vectors, and those of F

scalars.

Algebras

An algebra A over a field F is a ring which is a

vector space over F, such that

ðabÞ¼ðaÞb ¼ aðbÞ;2 F; a; b 2 A

Note that in some older literature, particularly the

Russian school, an algebra of operators is called a

ring of operators.

Further Reading

Borel A (1955) Topology of Lie groups and characteristic classes.

Bulletin American Mathematical Society 61: 397–432.

Kelly JL (1955) General Topology. New York: Van Nostrand

Reinhold.

Kreyszig E (1978) Introductory Functional Analysis with Applica-

tions. New York: Wiley.

Mc Carty G (1967) Topology: An Introduction with Application

to Topological Groups. New York: McGraw-Hill.

Simmons GF (1963) Introduction to Topology and Modern

Analysis. New York: McGraw-Hill.

Introductory Article: Topology 139

Abelian and Nonabelian Gauge Theories Using Differential Forms

A C Hirshfeld, Universita

t Dortmund, Dortmund,

Germany

Introduction

Quantum electrodynamics is the theory of the

electromagnetic interactions of photons and elec-

trons. When attempting to generalize this theory to

other interactions it turns out to be necessary to

identify its essential components. The essential

properties of electrodynamics are contained in its

formulation as an ‘‘abelian gauge theory.’’ The

generalization to include other interactions is then

reduced to incorporating the structure of nonabelian

groups. This becomes particularly clear when we

formulate the theory in the language of differential

forms.

Here we first present the formulation of electro-

dynamics using differential forms. The electromag-

netic fields are introduced via the Lorentz force

equation. The y are recognized as the components of

a differentia l 2-form. This form fulfills two differ-

ential conditions, which are equivalent to Maxwell’s

equations. These are expressed with the help of a

differential operator and its Hermitian conjugate,

the codifferential operator. We consider the effects

of charge conservation and introduce electromag-

netic potentials, which are defined up to gauge

transformations. We finally consider Weyl’s argu-

ment for the existence of the electromagnetic

interaction as a consequence of the local phase

invariance of the electron wave function.

We then go on to present the nona belian general-

ization. The gauge bosons appear in a theory with

fermions by requiring invariance of the theory with

respect to local gauge transformations. When the

fermions group into symmetry multiplets this gives

rise to a gauge group SU(N) involving N

1 gauge

bosons mediating the interaction, where N is the

dimension of the Lie algebra. The interaction arises

through the necessity of replacing the usual deriva-

tives by covariant derivatives, which transform in a

natural way in order to preserve the gauge

invariance. The covariant derivatives involve the

gauge potentials, whose transformation properties

are dictated by those of the covariant derivative.

Whereas for an abelian gauge theory such as

electromagnetism scalar-valued p-forms are suffi-

cient (actually only p = 1, 2), a nonabelian gauge

theory involves the use of Lie-algebra-valued

p-forms. These are introduced and used to construct

the Yang–Mills action, which involves the field

strength tensor which is deter mined from the gauge

potentials. This action leads to the Yang–Mills

equations for the gauge potentials, which are the

nonabelian generalizations of the Maxwell equations.

Relativistic Kinematics

The trajectory of a mass point is described as x



(),

where  is the invariant proper time interval:

d

¼ dt

 dx dx ¼ dt

ð1  v

Þ½1

with v = dx=dt. With the abbreviation  = (1  v

)

1=2

this yields d = (1=)dt.

The 4-velocity of a point is defined as u



=d = ( dx



=dt). The quantity

¼ g









d

¼ 1 ½2

is a relativistic invariant. Here



1000

0 100

0010

0001

½3

is the metric of Minkowski space.

The 4-momentum of a particle is p



= m



, m

v), and p



= m

. The 4-force is



d

¼ 



¼ 

; f



½4

with the 3-force

f ¼

dðm

vÞ

½5

Differentiate p

= m

with respect to , this yields



¼ 2m



 f v



¼0 ½6

¼ f v ¼ f 

½7

This says that

¼ f dx ¼dW ½8

where W is the work done and p

is the energy.

For a charged particle, the Lorentz force is

f ¼qðE þ v BÞ½9

where q is the charge of the pa rticle, E is the electric,

and B the magnetic field strength. Since f v = qE v ,

we have the four-dimensional form of the Lorentz

force:



¼ qðE v; E þ v BÞ½10

The Lorentz Force Equation with

Differential Forms

We write the Lorentz force equation as an equation

for a differential form f = f



, with f



= g





. The

velocity-dependent Lorentz force is

f ¼qi

F ½11

with

u ¼ 

þ v



½12

the 4-velocity and F the electromagnetic field

strength:

F ¼E^dt þB ½13

where E is a 1-form in three dimensions,

E¼E

dx þ E

dy þ E

dz ½14

and B is a 2-for m in three dimensions,

B¼B

dy ^dz þ B

dz ^dx þ B

dx ^dy ½15

The symbol i

indicates a contraction of a 2-form

with a vector, which is defined as

FðvÞ¼Fðu; vÞ½16

for an arbitrary vector v. The contraction of a

2-form with a vector yields a 1-form.

It is easily seen that a 2-form can be expressed in

terms of a polar vector and an axial vector: if it is to

be invariant with respe ct to parity transformations

with

t !t; x !x; y !y; z !z ½17

the fields in eqn [13] must transform as

E !E; B !B ½18

Now we check the validity of eqn [11]. We have

f ¼qi

¼ qðv EÞdt  q½ðE

þðv BÞ

Þdx

þðE

þðv BÞ

Þdy þðE

þðv BÞ

Þdz½19

in agreement with eqn [10] . We remember to change

the signs in E

= E

, B

= B

, etc.

The Codifferential Operator

The space of p-forms on an n-dimensional manifold

is an



n  p



ðn  pÞ!p!

½20

dimensional vector space. The space of p-forms is

thus isomorphic to the space of (n  p)-forms. The

Hodge dual operator maps the p-forms into the

(n  p)-forms, and is defined by

 ^ ¼h;  idx

^^dx

½21

Here h,  i is the scalar prod uct of two p-forms:

h;  i¼

i



si

½22

where 

si

are the coefficients of the form ,

 ¼ 

i

^^dx

½23



sj

are the coefficients of the form ,

 ¼ 

j

^^dx

½24

and



i

¼ g

g



j

½25

The indices satisfy i

<  < i

and j

<< j

The basis elements are orthogonal with respect to

this scalar product, and

hdx

^^dx

; dx

^^dx

¼ g

g

½26

142 Abelian and Nonabelian Gauge Theories Using Differential Forms

The Hodge dual has the property that

 dx

ð1Þ

^^dx

ðpÞ



¼ g

ð1Þð1Þ

g

ðpÞðpÞ

ðsign Þ

 dx

ðpþ1Þ

^^dx

ðnÞ



½27

where  is a permutation of the indices (1, ..., n),

(1) < <(p), and (p þ 1) < <(n). We also

have

 dx

ðpþ1Þ

^^dx

ðnÞ



¼ g

ðpþ1Þðpþ1Þ

g

ðnÞðnÞ

ð1Þ

pðnpÞ

ðsign Þ

 dx

ð1Þ

^^dx

ðpÞ



½28

We therefore find that the application of the

Hodge dual to a p-form twice yields

 dx

ð1Þ

^^dx

ðpÞ



¼g

ð1Þð1Þ

g

ðpÞðpÞ

ðsignÞ dx

ðpþ1Þ

^^dx

ðnÞ



¼g

ð1Þð1Þ

g

ðnÞðnÞ

ð1Þ

pðnpÞ

ð1Þ

^^dx

ðpÞ

½29

 ¼ ð1Þ

pðnpÞ

ð1Þ

Ind g

Id ½30

where Ind g is the number of times (1) occurs along

the diagonal of g.

Now let  be a (p  1)-form, and  a p-form.

Then d   is an (n  p þ 1)-form, and

dð^Þ¼d  ^ þð1Þ

p1

^d 

¼d^ þð1Þ

ðp1Þ

ð1Þ

ðnpþ1Þðp1Þ

ð1Þ

Indg

^ðÞd 

¼d^ þð1Þ

nðp1Þ

ð1Þ

Indg

^ðd Þ½31

We then have

ðd; Þð; d



Þ¼

dð ^Þ½32

with



¼ð1Þ

nðp1Þ

ð1Þ

Ind g

 d ½33

We are here using the scalar product of two p-forms

ð; Þ:¼

ð ^Þ½34

With the help of Stokes’ theorem the last integral in

eqn [32] may be turned into a surface term at

infinity, which vanishes for  and  with compact

support. d



is the adjoint operator to d with respect

to the scalar product ( , ). Whereas the differential

operator d maps p-forms into (p þ 1)-forms, the

codifferential operator d



maps p-forms into (p  1)-

forms.

The relation d

= 0 leads to

ðd



/ðdÞðdÞ / d

¼0 ½35

This fact plays an essential role in connection with

the conservation laws.

Finally, we want to obtain a coordinate expres-

sion for d



. Indeed d



 = Div  for

ðDivÞ

@

½36

where K is the multi-index of the coeffecients in

 = 

, and K indicates that K = (k

, ..., k

)isin

the order k

< < k

. We will show that

(, d



) = (, Div) for an arbitrary (p  1)-form

. It is a fact that

ð; d



Þ¼ðd; Þ¼

ðdÞ



 1 ½37

Now we have the coordinate expres sions

d ¼ðd

Þ^dx

½38

and (dx

)

= 

. It follows that

ðdÞ

¼ðd

^dx

¼ 

@



½39

ðdÞ

¼ 

@

½40

Here we use

ð ^Þ

¼ 





½41

where



1 if (KL) is an even

permutation of I

1 if (KL) is an odd

permutation of I

0 otherwise

½42

Use of the Leibnitz rule yields

ðdÞ



 1 ¼



@



 1

@ð





 1







@

 1 ½43

Abelian and Nonabelian Gauge Theories Using Differential Forms 143

The first term corresponds to a surface integration

and we can neglect it. We then have 



= 

from

the antisymmetry of , so that

ð; d



Þ¼



@

 1 ¼ð; DivÞ½44

The Maxwell Equations

The Maxwell equations become remarkably concise

when expressed in terms of differential forms, namely

dF ¼ 0; d



F ¼j ½45

where F is the field strength and j is the current

density. We wish to demonstrate this. We use a

(3 þ 1)-separation of the exterior derivative into a

timelike and a spacelike part:

d ¼ d þ dt ^

½46

We then get

dF ¼ dEþ



^dt þ dB¼0 ½47

By comparing coefficients, we arrive at

dE¼

; dB¼0 ½48

In vector notation

curl E ¼

; div B ¼ 0 ½49

the usual form of the homogeneous Maxwell

equations.

By direct application of the formula [27],onefinds

F ¼?B^dt þ ?E½50

where ? means the Hodge dual in three space

dimensions. One finds

d  F ¼ d ? E d ? B

@?E



^dt ½51

Therefore,

d  F ¼ðdiv EÞdx ^dy ^dz

þðcurl BÞ





dy ^dz ^dt

þðcurl BÞ





dz ^dx ^dt

þðcurl BÞ





dx ^dy ^dt ½52

We apply again the Hodge dual:

d  F ¼ðdiv EÞdt þðcurl BÞ





þðcurl BÞ





þðcurl BÞ





dz ½53

In Minkowski space the expression  d equals the

codifferential. Therefore, the equation d



F = d 

F =  j holds, with j given by j



= (, J), which is

equivalent to

div E ¼ ; curl B 

¼ J ½54

the inhomogeneous Maxwell equations.

Current Conservation

The electromagnetic 4-current is



¼ 



¼ð

; 

vÞ¼ð; JÞ½55

where  is the charge density and J the current

density. This corresponds to a 1-form

j ¼ dt  J

dx  J

dy  J

dz ½56

The Hodge dual is j = 

 j

^dt, with the 3-form



= dx ^dy ^dz, and the 2-form

¼J

dy ^dz  J

dz ^dx  J

dx ^dy ½57

From the Maxwell equation d



F = j , it follows

that

ðd



F ¼d



j ¼ 0 ½58

that is

dðjÞ¼dð

 j

^dtÞ¼ðd

 dj

^dtÞ

¼

@

þ div J



dt ^dx ^dy ^dz

@

þ div J ¼ 0 ½59

This is the ‘‘continuity equation.’’

The total charge inside a volume V is Q =

dV,

therefore



¼

 dV ¼

J n dS ½60

where @V is the surface which encloses the

volume V, dS is the surface element, and n is the normal

vector to this surface. This is current conservation.

144 Abelian and Nonabelian Gauge Theories Using Differential Forms