数学方法论物理学家高级讲座：拓扑、李群与表示

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"Lectures on Advanced Mathematical Methods for Physicists" 是一本专为理论物理学家和数学家设计的书籍，旨在介绍拓扑学、微分几何、李群与李代数及其表示等高级数学方法。这本书由Sunil Mukhi和NMukunda合著，由HINDUSTAN U LQJU BOOKAGENCY和World Scientific Publishing出版。本书分为两个部分。第一部分（Part I）主要涉及基础拓扑学和同伦论，然后深入到微分流形的微积分，包括引入黎曼度量、连接和曲率等关键概念。最后，书中介绍了单纯同调和德·拉姆共形，以及纤维丛，特别是切丛和余切丛，这些都是研究引力和弦理论等领域不可或缺的知识。第二部分（Part II）首先回顾了群论，接着深入到表示论的基础。详细探讨了李群和李代数的结构常数和线性表示，重点是根系统及其分类，以及简单李代数的表示，特别是正交和伪正交群的旋子表示。这部分内容在规范理论、粒子物理、可积系统和核物理等领域有广泛应用。本书的写作风格简洁精确，不包括对物理学生来说非核心的复杂数学证明，旨在让读者能够接触到当前文献中的多种资源，同时作为物理学家学习高级数学方法的教科书。书中的内容深度适中，既适合自学也适合课堂教学。通过阅读这本书，读者可以系统地掌握数学工具，为理解和应用现代物理学，如引力理论、弦理论、规范理论和粒子物理等领域的前沿问题打下坚实的基础。此外，对于想要拓宽数学视野和理解物理之间紧密联系的数学家，这本书同样具有很高的价值。

1.2.

Topological

Spaces

words, a

subset

will

called

open

every

point

inside

can

enclosed

open

interval

(a,

lying

entirely

inside it.

Examples:

(i)

Every

open

interval

(a,

open

set, since

any

point

inside

such

interval

can

enclosed

a smaller

open

interval

contained

within

(a,

b).

(ii) X

= {

0 } is

open

set.

This

not,

however,

open

interval.

(iii)

itself is

open

set.

(iv)

The

empty

set

open

set.

This

may

seem

tricky

first,

but

one

merely

has

check

the

definition, which goes "every

point

inside

can

...

Such

sentence

is always

true

there

are

points

the

set!

(v) A closed

interval

not

open

set.

Take

[a,

bj.

Then

and

cannot

enclosed

any

open

interval

lying

entirely

[a,

bj.

All

the

remaining

points

the

closed

interval

can

actually

enclosed in

open

interval

inside

[a,

bj,

but

the

desired

property

does

not

hold

for

all

points,

and

therefore

the

set

fails

open.

(vi) A single

point

not

open

set. To see

this,

check

the

definition.

Next,

we define a closed

set

IR.

Definition:

closed set

the

complement

any

open

set.

Examples:

(i) A closed

interval

[a,

is a closed set.

the

complement

the

open

set

= {

b }

U {

a }.

(ii) A single

point

{a}

is a closed set.

the

complement

the

open

set

= {

a }

U {

a }.

(iii)

The

full

set

and

the

empty

set

are

both

closed sets, since

they

are

both

open

sets

and

are

the

complements

each

other.

We see

that

set

can

open,

closed,

neither,

both.

For

example,

[a,

neither

closed

open

(this

the

set

that

contains

the

end-point

but

not

the

end-point

b).

Since

cannot

enclosed

open

set,

[a,

not

open.

its

complement,

cannot

enclosed

any

open

set. So

[a,

not

closed

either.

IR,

one

can

check

that

the

only

sets

which

are

both

open

and

closed

according

our

definition

are

and

cp.

important

emphasize

that

so far, we

are

only

talking

about

open

and

closed

sets

in the real line

IR.

The

idea

extract

some key

properties

these

sets

and

use

those

define

open

sets

and

closed

sets

abstract

setting.

Continuing

this

direction, we

note

some

interesting

properties

open

and

closed sets in

IR:

(a)

The

union

any number

open

sets

open.

This

follows from

the

definition,

and

should

checked carefully.

(b)

The

intersection

finite number

open

sets in

is open.

Why

did

we have

specify a finite

number?

The

answer is

that

taking

the

intersection

infinite

number

open

sets

IR,

can

actually

manu-

Chapter

1. Topology

facture a set which is

not

open. As

example, consider

the

following infinite

collection of

open

intervals:

{

(-~,~)

=1,2,3,

...

}

(1.2)

As n becomes very large,

the

sets keep becoming smaller.

But

the

point

contained in each set, whatever

the

value of n. Hence it

also contained in

their

intersection. However any

other

chosen point

will lie outside

the

set

once n becomes larger

than

I!I'

Thus

the

infinite intersection over

all

n contains only

the

single

point

0, which as we have seen

not

open

set.

This

shows

that

only finite intersections of

open

sets are

guaranteed

be open.

Having discussed

open

and

closed sets on

JR,

are

now in a position

extend

this

a more

abstract

setting, which

that

of a topological space.

This

will allow us

formulate

the

concepts of continuity, limit points, compactness,

connectedness etc. in a

context

far more general

than

real analysis.

Definition:

topological space

set

together

with

a collection

of subsets

called

open sets,

satisfying

the

following conditions:

(i)

(ii)

The

union of

arbitrarily

many

is again in

(thus

the

union of

any

number

open

sets is open).

(iii)

The

intersection of finitely

many

subsets

again in

(thus

the

intersection of finitely

many

open

sets is open).

The

pair

(S, U)

called a

topological space.

The

collection

of subsets of

called a

topology

The

reader will notice

that

the

set

happens

coincide

with

the

real

line

JR,

then

the

familiar collection of

open

sets on

satisfies

the

axioms above,

and

together

with

its

usual collection of

open

sets provides

example of

a topological space.

But

as we will soon see, we

can

put

different topologies

the

same

set

points, by choosing a different collection of subsets as

open

sets.

This

will lead

more general

(and

strange!) examples of topological spaces.

Having defined

open

sets, it

natural

define closed sets as follows:

Definition:

a given topological space, a

closed

set

a subset of

which is

the

complement of

open

set.

Thus,

then

{

xES

rf:.U

}

closed.

Examples:

(i)

Let

any set whatsoever. Choose

the

collection of two sets

{ ¢,

S}.

This

the

smallest collection we

can

take

consistent

with

the

ax-

ioms!

Clearly all

the

axioms are trivially satisfied.

This

is called

the

trivial

sometimes

indiscrete

topology on

(ii)

Let

again

any

set. Choose

the

collection of

all

subsets of

This

is clearly

the

largest collection we

can

possibly take.

this

topology, all

subsets

are

open.

But

they

are

all closed as well (check this!).

This

another

1.3.

Metric

spaces

trivial example of a topology, called

the

discrete

topology, on

We see, from

this

example

and

the

one above,

that

is possible

define more

than

one

topology on

the

same

set. We

can

also guess

that

the

topology has

too

few

too

many

open

sets, it

liable

trivial

and

quite

uninteresting.

(iii)] Let

the

real line

JR.

the

collection of all subsets

such

that

EX=}

(a,

eX.

(1.3)

This

our

old definition of "open set in

JR".

We realise now

that

was

not

the

unique choice of topology,

but

it was certainly

the

most

natural

and

familiar.

Accordingly,

this

topology is called

the

usual topology

JR.

(iv)

a finite set, consisting of, say, six elements. We write

{a,

(1.4)

Choose

{¢,S,{a,b},{b},{b,c},{a,b,c}}

(1.5)

This

defines a topology. Some of

the

closed sets (besides

and

are

{d,

and

{a,

fl.

This

example shows

that

can

very well define a topology

a finite set.

Exercise:

we leave

out

{a,

still get a topology?

What

if we

leave

out

{b}?

What

if we

add

{d}?

1.3

Metric

spaces

A topological space carries no intrinsic notion of metric,

distance between

points. We

may

choose

define

this

notion on a given topological space if

we like. We will find

that

among

other

things, introducing a metric helps

generate

many

more examples of topological spaces.

Definition:

metric

space

is a set

along

with

map

which assigns a real

number::::

each

pair

of points in

satisfying

the

conditions below.

xES,

yES

then

d(x, y)

should be

thought

of as

the

distance between

and

The

conditions on

this

map

are:

(i)

d(x, y)

and

only if

(ii)

d(x, y)

d(y, x)

(iii)

d(x,

:::;

d(x, y)

d(y, z)

(triangle inequality).

The

map

-7

+ is called a

metric

S .

We see from

the

list of axioms above,

that

are

generalising

abstract

topological spaces

the

familiar notion of distance

Euclidean space.

Later

on we will see

that

Euclidean space

really a "differentiable manifold

with

metric", which involves a lot more

structure

than

we have encountered so far.

useful

keep in

mind

that

a metric

can

make sense

without

any

that

other

structure

- all

need

a set, not even a topological space.

fact we

Chapter

Topology

will see

moment

that

defining a

metric

set

actually

helps us define a

topological space.

Examples:

may

choose

the

usual

metric

d(x,

yl.

can

similarly choose

d(x,

111-

The

reader

should check

that

the

axioms

are

satisfied.

Exercise:

Does

d(x,

(x -

define a

metric

lR?

Given

any

metric

set

can

define a

particular

topology

called

the

metric topology.

Definition:

With

respect

a given

metric

set

the

open disc

the

point

xES

with

radius a

0 is

e set

Dx(a)

= {

yES

d(x,

a }

The

open

disc is

just

the

set

points

strictly

within

a distance

the

chosen point. However,

must

remember

that

this distance is defined

with

respect

abstract

tric

that

may

not

necessarily

the

one familiar

us.

Having defined

open

discs via

the

metric, we

can

now define a topology

(called

the

metric topology)

as follows.

subset

XeS

will

called

open

set

if every

can

enclosed by some

open

disc

contain

ed entirely

open

EX=}

Dx(a)

(1.6)

for some

This

gives us

our

collection

open

sets

and

together

with

the

set

we claim

this

defines a topological space.

Exercise:

Check

that

this

satisfies

the

axioms for a topological space.

We realise

that

the

usual topology

just

the

metric

topology

with

d(x,

The

metric

topology also gives

e usual topology

dimensional Euclidean space

IRn.

The

metric

there

is:

d(x\

yi)

2)xi

yi)2

(1.7)

i=l

where

Xi,

yi,

1,2,

...

are

the

familiar

coordinat

points

lRn.

The

open

discs

are

familiar too. For

they

are

the

familiar discs

the

plane (not including

their

boundary),

while for

they

are

the

interior

solid ball

about

the

point.

higher dimensions

they

are

generalisations

this,

so we will use

the

term

"open ball"

describe

open

discs in

any

lRn.

Note

that

for a given

set

points

can

define

many

inequivalent

met-

rics.

lR,

for example, we could use

the

rather

unorthodox

(for physicists)

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