Mukhi S., Mukunda N. Lectures on Advanced Mathematical Methods for Physicists

242

Chapter

16.

Representations

of

Compact

Simple Lie Algebras

there

is

no multiplicity problem - for instance

this

was

the

case in all

the

defining

UIR's.

For

Al

=

SU(l

+

1)

and

for

SO(n)

comprising

both

BI

=

SO(2l

+

1)

and

DI

=

SO(2l),

there

are

natural

or "canonical" ways of choosing these

~(r

-

3l)

additional diagonal operators:

they

are based

on

the

fact

that

if a

UIR

of

SU(l

+

1)

(respectively

SO(n))

is

reduced

with

respect

to

the

subgroup SU(l) x

U(l)

[respectively

SO(n

- 1)], each

UIR

of

the

latter

which occurs does so

just

once. So one

can

use

the

Casimir

operators

(i.e., basically

the

UIR

labels

{N})

of

the

chain of

sub

groups SU(l

+

1)

~

SU(l)

~

SU(l - 1)

...

~

SU(2) in

the

unitary

case,

and

of

SO(n)

~

SO(n

-

1)

~

SO(n

- 2)

...

~

SO(3) in

the

orthogonal case,

to

solve

the

multiplicity

and

state-labelling problem.

Apart

from this,

what

can

one say in general

terms

about

the

properties

of a weight

p,

E

W?

Clearly, by

the

application of

the

SU(2)(a)

and

SU(2)(a)

subalgebras,-

(16.69)

In

other

words, all

the

states

Ip,

...

)

for a given

p,

E

Ware

simultaneous

(a(ai)

-

eigenstates of

the

l

operators

J

3

.

Given

the

integers

{na},

l:!:.

is

uniquely

determined

just

as

.,1

was by

{N

a

}:

I

l:!:.'

ala)

=

na

{::}

l:!:.

=

L

na.,1(a)

(16.70)

a=l

But

for a given

UIR

{N

a

},

what

l:!:.

and

so

what

{n}

arise?

The

general answer

is

as follows. Every

p,

E

W is

obtained

from

.,1

by

subtracting

a unique non-negative linear

comb~ation

of

the

simple

roots

g(a)

with

integer coefficients:

.,1

=

highest weight,

l:!:.

E

W

'*

I

l:!:.

=

.,1

-

L

vag(a)

, Va

unique, integer,

~

O.

a=l

(16.71 )

The

uniqueness follows from

the

linear independence of

the

g(a).

These

Va

are related

to

the

na

(find

the

relation!). Now knowing

.,1,

how do we find

out

what

can

be

subtracted

to

get allowed weights present in

V?

There

is a

recursive process: we express W as

the

union of subsets,

W

=

W(O)

Uw(l)

UW(2)

U

...

(16.72)

16.7. Structure of States within a UIR

Here,

w(O)

=

L1

=

simple highest weight;

I

W(l)

=

{~E

WI

LVa

=

1};

a=l

I

W(2)

=

{~

E

WI

L

Va

=

2};

a=l

243

(16.73)

So

W(k)

is called

the

kth

layer

of

weights - every

f.L

E

W(k)

is k simple

roots

away from

L1!

-

How do we find

W(l)?

In

other

words: which

aJa)

can

be

subtracted

from

L1?

Evidently,

with

respect

to

each SU(2)(a)

we

s~

from Eq.(16.19)

that

L1

is

the

"maximum

m"

state:

L1

under

SU(2)(a)

:

m

=

ja

=

(1/2)N

a

.

(16.74)

Therefore

Na

?

1 is

the

necessary

and

sufficient

condition

for

L1-

Q(a)

to

be

present:

(16.75)

In

general, letf.L

E

W(k)

,

and

Q(a)

E

S.

The

question is: does

the

W(lc+l)

contain

~

Q(a)?

The

answer is given

by

an

application

of SU(2)(a)

to

~,

as

one would

of

course expect.

With

respect

to

SU(2)(a),

~

corresponds

to

a

magnetic

quantum

number,

We now see:

if

the

contents

of

W(O),

W(l),

...

,

W(k)

are

known; if

~

E

W(k),

~

+

Q(a)

E

W(k-l),

~

+

2Q(a)

E

W(k-2),

...

,

~

+

PQ(a)

E

W(k-

p

),

~

+

(p

+

1)Q(a)

tJ.

W,

and

if

m>

-(m+p)

or

m>

-p/2,

(16.76)

i.e.: m is

not

yet

the

least possible value

of

the

magnetic

quantum

number,

then

~

_

Q(a)

E

W(k+l)

More formally one

can

express

the

situation

thus:

~EW(k),

~+PQ(a)EW

for

p=1,2,

...

,p,

~

+

(p

+

1)Q(a)

tJ.

W,

~.

(i(a)

+

p

>

0

:::}

~

_

Q(a)

E

W(k+l)

(16.77)

244

Chapter

16.

Representations

of

Compact

Simple Lie Algebras

In

principle,

then,

once

A

is chosen,

and

knowing S, all

the

weights

present

in

V

are

known.

The

level

of

a weight

J1

is

the

same

as

the

"layer

number"

k

above,

namely

the

"number

of

simple roots"

to

be

subtracted

from

A

to

get

to

it.

This

is

of

course

not

the

same

as

the

multiplicity

of

J1,

though

all

the

weights

of

a given

layer

or

at

a given level do have

the

same

multiplicity.

This

multiplicity

is given

by

a

formula

due

to

Freudenthal

(quoted

for

instance

in

the

book

by

Wybourne).

The

"highest

layer" is

the

one

with

maximum

k,

and

this

maximum

value is

denoted

by

T(.,1).

It

is also called

the

"height"

of

the

VIR.

The

behaviour

of

multiplicities shows a

pyramidal

or,

better,

"spindle-shaped"

structure.

Multi-

plicity

at

level

k

is

the

same

as

at

level

T(.,1)

-

k.

The

highest

layer

WeD)

has

a

simple weight

A,

so

multiplicity

one;

then

the

multiplicity

keeps

increasing

for

a while as we come

down

to

lower layers,

reaching

a

maximum

at

k

=

~T(.,1).

After

that

it

keeps

decreasing

again,

until

for

the

lowest weight

!!.

it

becomes

unity

again

.. Exercise:

Construct

the

simple

roots

S

and

the

weight

and

multi-

plicity

diagrams

for

the

UIR's

(10),

(0, 1), (1, 1),

(2,0),

(0,2)

of

SU(3).

In

the

process

many

of

the

general

results

described

above

become

clear.

Exercises

for

Chapter

16

1.

For

the

orthogonal

groups

VI

=

SO(2l),

Bl

=

SO(2l

+

1), prove

that

the

generators

belong

to

the

second

rank

antisymmetric

tensor

representations

V(2),

and

that

this

is

the

adjoint

representation.

2.

Work

out,

in

the

SU(3) case,

the

simple

roots

S,

and

the

weight

and

multiplicity

diagrams

for

the

VIR's

(1, 0), (0,1), (1, 1), (2, 0)

and

(0, 2).

3.

Is

the

adjoint

representation

(1, 1)

of

SU(3) a faithful

representation?

If

not,

why

not?

Chapter

17

Spinor

Representations

for

Real

Orthogonal

Groups

We have seen in

the

that

among

the

fundamental

VIR's

for

the

groups

Dl

=

SO(2l)

and

Bl

=

SO(2l

+

1)

there

are some "unusual" rep-

resentations which

cannot

be

obtained

by

any

finite algebraic means from

the

familiar defining vector representations of these groups.

These

are

the

spinor

representations. For

Dl

we saw

that

there

are two inequivalent spinor VIR's,

which we denoted by

..1(1)

and

..1(2);

their

descriptions as fundamental

VIR's

,

and

their

highest weights, were found

to

be

(see Eqs.(16.43),(16.45)):

'T"\(l-1)

- A(1) -

{O

0 1

O}

.

A(l-l)

_

1 ( )

v

=

Ll

= , ... , " . _ -

"2

~1

+

~2

+

...

+

~1-1

-

~l

=

(1/2,1/2

,

...

,1/2,

-1/2);

'T"\(I)-A(2)_{O

01}

.A(

I)_1(

)

v

=

Ll

= ,

...

" . _

-"2

~1

+

~2

+

...

+

~1-1

+

~l

=

(1/2,1/2,

...

,

1/2,1/2)

(17.1 )

For

Bl

,

there

is only one spinor VIR, which

we

wrote

as..1 (see Eqs.(16.50,16

.5

2)):

1

V(l)

==

..1

==

{O,

...

,

0,

I} :

J(l)

=

"2(~1

+

~2

+

...

+

~l)

=

(1/2

,

1/2,

...

,

1/2)

(17.2)

As

part

of

the

general

Cartan

theory

of

VIR's

and

fundamental

VIR's

of

compact

simple Lie groups

and

algebras, these spinor representations more or

less fall

out

automatically, once

the

root

systems

and

simple

roots

for

Dl

and

Bl

are

understood.

At

the

same

level one also

understands

and

accepts

that

for

the

unitary

and

the

symplectic groups

there

is

nothing

analogous

to

spinors.

But

from

the

point

of view of physical applications, spinors

are

unusually interesting

and

useful quantities,

and

it

is therefore

appropriate

that

we

understand

them

in some detail from a somewhat different

starting

point.

In

doing so, of course,

we

can

be guided by

the

general

theory

outlined in

the

246

Chapter

17.

Spinor

Representations

for

Real

Orthogonal

Groups

In

the

succeeding sections, we shall deal

at

first

with

the

spinor

VIR's

for

DI

=

SO(2l),

and

later

for

BI

=

SO(21

+

1).

In

each case we also clarify

some

of

the

relationships between spinors

and

antisymmetric

tensors which were

briefly alluded

to

in

the

The

entire

treatment

is

based

on the

representations

of

the

Dirac algebra,

many

aspects

of

which were first worked

out

by

Brauer

and

Weyl.

It

is possible,

and

probably

most

elegant,

to

base

the

entire

analysis

on

an

unspecified

representation

of

the

Dirac algebra, i.e.,to develop

everything in a

representation

independent

way. However,

taking

inspiration

from

Feynman,

we

shall do all

our

calculations in a specific representation,

and

believe

there

is no sense

of

defeat

in

doing

so!

The

SO(2l)

and

SO(2l

+

1)

treatments

in this

Chapter

will form a basis for

studying

spinors for

the

pseudo-orthogonal groups SO(p,

q)

in

the

17.1

The

Dirac

Algebra

in

Even

Dimensions

The

introduction

of

spinors

in

the

context

of

the

Lorentz group,

based

on the

Dirac

anticommutation

relations which arise

in

the

relativistic electron

equation

is

quite

well-known. We

start

from a similar

algebra

for arriving

at

spinors

with

respect

to

SO(21).

Consider

the

problem

of

finding

hermitian

matrices

I

A

of

suitable

dimension

obeying

the

algebraic relations

bA,IB}

==

IAIB

+

IBIA

=

20AB,

A,B=1,2,

...

,21

(17.3)

The

range

of

indices is

appropriate

to

SO(2l),

and

as

in

the

previous dis-

cussions

of

this

group, we will use split index

notation

when

necessary.

In

fact,

in

such

notation,

Eq.(17.3)

appears

as

{lar,lbs}

=

2o

ab

o

rs

;a,b

=

1,2,

...

,l;r,s=

1,2.

(17.4)

It

is

an

important

fact

that,

upto

unitary

equivalence,

there

is only one

irreducible

hermitian

set

I

A

obeying

these

relations,

and

the

dimension is

21.

The

proof

is given, for instance,

in

Brauer

and

Weyl. We will give

an

explicit

construction,

but

it

is

instructive

to

recognise immediately

the

"source"

of

the

spinor

VIR's

of

SO(21)

-

it

lies in

the

essential uniqueness of

the

solution

to

(17.3). For, if

S

is

any

matrix

in

the

defining

representation

D

of

SO(2l),

i.e.,

a real

orthogonal

unimodular

21-dimensional

matrix,

then

(17.5)

is also a solution

to

Eq.(17.3), if

IA

is.

By

the

uniqueness

statement,

there

exists

then

a

unitary

matrix

U(S) -

determined

up

to

a

phase

factor - which connects

IA

to

lA,

and

as is easily seen, which again

up

to

a

phase

gives a

representation

of

SO(2l):

S

E

SO(2l):

U(ShAU(S)-l

=

SBAIE

(a)

U(S')U(S)

=

(phase)U(S'S)

(b) (17.6)

17.1. The Dirac Algebra in Even Dimensions

247

The

U(S)

contain

the

spinorial

VIR's

of

SO(2l),

and

this

is

their

origin.

We will

construct

their

infinitesimal

generators

in

the

sequel.

To

construct

a

solution

to

(17.3)

in

a convenient form, we

take

l

independent

(i.e.

mutually

commuting)

sets

of

Pauli

matrices

ud

a

),

j

=

1,2,3,

a

=

1,2,

...

, l.

Since

each

Pauli

triplet

needs a

two

dimensional

vector

space

to

act

upon,

the

entire

set

is defined

on

a

vector

space

V

of

dimension

21.

The

construction

of

1'A

is

best

expressed

in

split

index

notation

as

= _

(a)

(a+1) (a+2)

(I)

1'A

-1'ar

-

U

r

U

3

U

3

...

U

3

,

T

=

1,2,a

=

1,2,

...

,l

(17.7)

Notice

that

1'ar

contains

no

contributions

(factors) from

~(1),

~(2),

...

,

~(a-1)

(However

the

2 x 2

unit

matrices

in

those

slots

are

not

indicated

but

left im-

plicit) .

The

hermiticity

of

1'A

is obvious from

that

of

the

u's.

The

irreducibility

is

also

quite

easy

to

prove: we will

be

showing

in

a

later

section

that

each

u~a)

can

be

expressed

as

a

product

of

two

of

the

1"s (in fact

this

is

already

evident

from

Eq.(17.7));

from

this

fact we

then

see

immediately

that

ui

a

)

and

u~a)

can

also

be

expressed

as

a

product

of

1"s. Since

this

is so,

and

since

the

complete

set

of

all

l

Pauli

triplets

is

certainly

irreducible,

the

irreducibility

of

the

1'A

follows.

Checking

that

the

Dirac

algebra

is satisfied is easy;

it

is

best

to

look

at

a

<

b

and

a

=

b

separately:

b

(a)

(a+1)

(b-1)

(b)

a

< :

1'ar1'bs

=

U

r

U

3

...

U

3

U

3

Us

,

'Y 'Y

=

~(a)

~(a+1)

~(b-1)

~(b)

-"

,bs,ar

u

r

u3

",u3

Us

u3

......,-

1'ar1'bs

=

-1'bs1'ar;

a

=

b:

'Y

=

uta)

=}

Jar

,as

r s

(17.8)

'For uniqueness

up

to

unitary

equivalence,

the

reader

may

refer

to

Brauer

and

Weyl.

All

our

further

calculations

will

be

based

on

the

above

specific

21

dimen-

sional

solution

for

1'A

acting

on

V

of

dimension

21.

In

some

contexts

it

is

necessary

to

pass

from

the

Dirac

algebra

for

one

value

of

l

to

that

for

the

l

+

1.

For

this

purpose

we

may

note

that

(1+1) _ (I)

(l+1)

_ .

1'A

-1'A

uA

,

A-1,2,

...

,2l,

(l+1)

_ (1+1)

1'21+1

or

21+2 -

u

1

or

2

(17.9)

Here

for

clarity

we

have

indicated

as

a

superscript

on

l'

the

relevant "l-value".

248

17.2

Chapter

17.

Spinor Representations

for

Real Orthogonal Groups

Generators,

Weights

and

Reducibility

of

U(5) -

the

spinor

VIR's

of

Dl

The unitary

operators

U(S)

acting

on

V

are

not

completely defined by Eq.(17.6).

To

the

same

extent,

the

corresponding

generators

Jt1AB

are

also known only

upto

additive

constants.

It

turns

out

that

both

U(S)

and

MAB

can

be

so chosen as

to

achieve

at

most

a sign ambiguity in

the

representation

condition in Eq.(17.6)

- so we have here a

"two-valued"

representation

of

80(2l).

The

generators

of

U(S)

are

(17.10)

These

are

hermitian,

reducible (as we shall soon see),

and

obey

the

commutation

relations (14.6) as a consequence

ofthe

Dirac

anticommutation

relations (17.3).

In

checking this, one finds

that

the

numerical factor

±

is essential.

In

addition

to

the

Dl

commutation

relations, between

MAB

and

,e

one

has

the

useful

relation

(17.11)

This

is

in

fact

the

infinitesimal form

of

Eq.(17.6a).

For

later

purposes

we will need

the

analogue

to

Eq.(17.9), a relation between

the

MAB's

generating

80(2l),

and

the

set

of

MAB's

generating

80(2(l

+

1».

If

we once again indicate

the

l-value as a

superscript,

we have,

using Eq.(17.9)

and

leaving implicit

the

unit

matrices

in relevant subspaces:

A, B

=

1,

2,

...

,

2l

M(Hl)

_

M(l)

AB

-

AB

M(I+I)

_

1

(I)

(HI).

A,2Hl

-

-2"'A

(72

,

M(Hl)

_

+

1

(I)

(HI)

A,21+2 -

2"'A

(71

M(I+l)

_

1

(1+1)

21+1,21+2

-

-2"(73

(17.12)

Going

back

to

the

MAB

for a given fixed

l,

we

can

identify

the

Cartan

sub-

algebra

generators

Ha:

H

- M -

i"V

"V

- 1

".(a)

a-I

2

1

a -

al,a2

-

2"

,al

,a2

-

_2"V3

, - , ,

...

,

(17.13)

Therefore, in

the

usual

representation

of

Pauli

matrices,

the

Ha

are

already

simultaneously diagonal,

and

we immediately have

an

orthonormal

basis for

V

given by vectors

with

definite

nondegenerate

weights. We use

Ca

for

the

two

possible eigenvalues

of

each

H

a

,

so we describe

this

basis for

V

in

this

way:

I{c})

=

l{cl,c2,

...

,cl}),ca

=

±1;

1

Hal{c})

=

2"cal{c})

(7~a)l{c})

=

-cal{c})

(17.14)

17.2. Generators, Weights

and

Reducibility of

U(S) -

the

spinor UIR's of

Dl

249

The

weight vector!!:.

associated

with

I {

f})

is

evidently

(17.15)

So

the

complete

set

of

weights

occurring

in

V

is

and

each

being simple,

the

total

number

21

is precisely

the

dimension of

V.

Now we come

to

the

question

of

the

reducibility of

the

unitary

representa-

tion

U(S)

of

D

I

.

Symbolically

the

relation

between

these

finite

unitary

operators

and

the

generators

is

U(S)

rvexp

(-~WABMAB)

WAB

= -

WBA

=

real

parameters

. (17.16)

The

operators

U(S)

happen

to

be

reducible because

there

exists a

nontrivial

matrix

IF

which,

by

virtue

of

anticommuting

with

each

lA,

commutes

with

the

generators

M

AB

.

All

this

happens

because we

are

dealing

with

an

even

number

of

dimensions,

2l.

(In

analogy

with

the

properties

of

the

matrix

15

in

the

Dirac

electron

theory

,

we

write

F

as

a

subscript

on

IF

,

to

remind

us

of

FIVE).

We

define

·1

IF

=

~

11/2

...

121

_ (_1)1

(1) (2) (I)

-

0'3 0'3

..

·0'3

,

(17.17)

and

immediately

verify all

the

following basic properties:

t

T

*

-1

IF

=

IF

=

IF

=

IF

=

IF

(a)

bA"F}

=0,

[MAB"F]

=

0,

U(ShF

=

IFU(S);

(b)

IFI{f})

=

(11

fa)

I{c})

(c)

(17.18)

Since

IF

is

already

diagonal, we call

the

construction

(17.7) a Weyl repre-

sentation

of

the

I's.

We

can

split

V

into

the

direct

sum

of

two

2

1

-

1

dimensional

subspaces

VI,

V

2

corresponding

to

IF

=

±

1:

V

=

V

1

EEl

V2,

IF

=

-1

on

V

1

,+1

on

V

2

(17.19)

250

Chapter

17.

Spinor

Representations

for

Real

Orthogonal

Groups

Clearly, a basis for

VI

consists of all

I {

E})

with

an

odd

number

of

E'S

taking

the

value

-1

, while for

V

2

we

use

I{E})

with

an

even

number

of

E'S

being

-1.

The

unitary

representation

U(S)

of

Dl

restricted

to

VI

then

gives us

the

UIR

..1(1)

with

the

highest weight being

~y-l)

=

~(~1

+

~2

+ ... +

~1-1

-

~l)

(17.20)

We see here

the

effect of having

to

include

at

least one minus sign

among

the

E'

S.

A spinor (an element of

Vd

transforming according

to

this spinor

UIR

of

SO(21)

is

called a

left handed

spinor.

On

the

other

hand,

on

V

2

the

U(S)

furnish

us with

the

other

spinor

UIR

..1

(2)

characterised by

the

highest weight

(17.21)

Elements of

V

2

are

then

called

right handed

spinors. We

can

express

this

reduc-

tion

of

the

representation

U(S)

of

Dl

symbolically as:

U(-)

=

reducible spinor representation of

Dl

on

V

generated

by

MAE

=

left

handed

spinor

UIR

L1(1)on

VI

EEl

righthanded

spinor

UIR

..1

(2)

on

V

2

(17.22)

While

the

MAE

and

U(S)

are

thus

reducible

the

IA

are of course irreducible

on

V:

in fact since

they

each

anticommute

with

IF,

the

only non-zero

matrix

elements of

I

A

are those connecting

VI

to

V

2

and

vice versa!

17.3

Conjugation

Properties

of

Spinor

VIR's

of

Dl

Each

of

the

representations

U(-),

..1(1)

and

..1(2)

is unitary, i.e., self adjoint. We

would like

to

know how

they

each behave

under

complex conjugation. Here

the

pattern

keeps changing systematically from one I-value

to

the

next, which

is

why

we

assembled

the

Eqs.(17.9),(17.12).

We ask if

the

reducible representation

U (.)

is equivalent

to

its complex

conjugate, i.e., whether

there

is a

matrix

C

acting

on

V

such

that

U(S)*

=

CU(S)C-

1

,

i.e.,

(U(Sf)-1

=

CU(S)C-

1

,

i.e.,

(17.23)

It

turns

out

that

such a

matrix

C

does exist,

and

it

can

be

constructed

recursively from one value of

I

to

the

next.

Of

course since

U(·)

is

reducible

C

will

not

be unique. We will

construct

one possible

C

matrix

in a convenient

form, see how its properties change

with

I,

and

also examine

its

properties

with

17.3. Conjugation Properties of Spinor UIR's of

Dl

251

respect

to

IF.

This

last

will

then

disclose

the

conjugation

behaviours

of

the

VIR's

..1(1)

and

..1(2).

Since

the

building

up

of

e

varies from

l

to

l+

1,

momentarily

let

us

reinstate

the

l-value

as

a

superscript

on

all relevant

matrices

and

factors.

The

desired

behaviour

of

M~1

under

conjugation

by

e(l)

suggests

that

we

ask

for

the

I~)

to

transform

as

follows:

(17.24)

For

either

value

of

101,

we do

get

M~1

transforming

correctly. Suppose

e(l)

has

been

found; let us see if

e(lH)

can

be

taken

as

e(l)

times

a

suitable

2

x

2

matrix

in

the

space

of

the

additioiIal

Pauli

triplet

Q:(l+1):

e(l+l)

=

e(l)

A (1+1),

A

(l+1)

=

function of

Q:(IH)

The

requirement

given Eqs.(17.9),(17.12),(17.24),(17.25), leads

to

conditions

on

A(l+l)

as

(a)

A,

B

::;

2l:

no

conditions;

(b)

A ::;

2l,

B

=

2l

+

1:

A (1+1)

(7~1+1)

=

EW~I+l)

A

(l+1);

(c)

A::;

2l,

B

=

2l

+

2:

A(lH)(7i

lH

)

=

-El(7i

l

+

1

)

A(l+I);

(d)

A

=

2l

+

1,

B

=

2l

+

2:

A

(

lH)(7~l+I)

=

_(7

~

1+1)

A(l+I)

;

These

conditions

fix

A(l+I)

depending

on

El:

El

=

+

1:

A

(l+I)

=

i(7~l+I

)

El

=

-1:

A

(1+1)

=

(7il+

1

)

(we have

opted

to

make

A(l+I)

real).

But

then

what

about

E1H

in

Now we discover:

(a)

A::;

2l:

101+1

=

-101

(b)

A

=

2l

+

1:

A(l+I

)(7i

l

+1)

=

El+l(7i

1

+1)

A(l+1)

( )

A

-

2l

+

2'

A(lH)

(1+1) - _

(l+I)A(l+1)

c - .

(72 -

10

1+1(72

(17.25)

(17.26)

(17.27)

(17.28)

(17.29)

(17.30)

Mukhi S., Mukunda N. Lectures on Advanced Mathematical Methods for Physicists

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