Introduction to Groups

8/14/2019 Introduction to Groups

1/164

Introduction

toGroups, Invariantsand

ParticlesFrank W. K. Firk, Emeritus Professor of Physics, Yale University

2000


2/164

CONTENTS 2

1. INTRODUCTION 3

2. GALOIS GROUPS 8

3. SOME ALGEBRAIC INVARIANTS 20

4. SOME INVARIANTS IN PHYSICS 29

5. GROUPS CONCRETE AND ABSTRACT 43

6. LIES DIFFERENTIAL EQUATION, INFINITESIMAL

ROTATIONS, AND ANGULAR MOMENTUM 57

7. LIES CONTINUOUS TRANSFORMATION GROUPS 68

8. PROPERTIES OF n-VARIABLE r-PARAMETER

LIE GROUPS 78

9. MATRIX REPRESENTATIONS OF GROUPS 83

10. SOME LIE GROUPS OF TRANSFORMATIONS 94

11. THE GROUP STRUCTURE OF LORENTZ

TRANSFORMATIONS 107

12. ISOSPIN 115

13. LIE GROUPS AND THE STRUCTURE OF MATTER 128

14. LIE GROUPS AND THE CONSERVATION LAWS

OF THE PHYSICAL UNIVERSE 159

15. BIBLIOGRAPHY 163


3/164

1 3

INT RODUC TION

The noti on of geom etric al symmetr y in Art a nd in Natu re is a

fam iliar one. In Moder n Phy sics, this noti on ha s evo lved to in clude

sym metri es of an a bstra ct ki nd. These new symme tries play an e ssential

par t in the t heori es of the micro struc ture of ma tter. The basic sym metri es

fou nd in Natu re se em to orig inate in t he ma thema tical stru cture of t he la ws

the mselvesla ws th at go vern the m otion s of the g alaxi es on the onehan d and the motio ns of quar ks in nucl eons on th e oth er.

In the N ewton ian e ra, t he la ws of Natu re we re de duced from a sm all

num ber o f imperfec t obs ervat ions by a small numb er of reno wned

sci entists an d mat hemat ician s. I t was not until the Einst einia n era ,

how ever, that the signi fican ce of the symme tries asso ciated wit h the laws

was full y app reciated. The disco very of sp ace-time symmet ries has led to

the wide ly-he ld be lief that the l aws o f Nat ure c an be deri ved f rom

sym metry , or invar iance , pri ncipl es. Our i ncomp lete knowledge of th e

fun damen tal intera ctions mea ns th at we are not y et in a po sitio n to confi rm

thi s belief. We t heref ore u se ar gumen ts ba sed o n emp irica lly established

laws and rest ricte d sym metry prin ciple s to guide us i n our sear ch fo r the

fun damen tal symmet ries. Fre quent ly, i t is impor tant to un derst and w hy

the symm etry of a system is obser ved t o be broke n.


4/164

In Geome try, an ob ject with a def inite shap e, si ze, location, a nd 4

ori entat ion c onstitutes a st ate w hose symme try p roper ties, or i nvari ants,

are to b e stu died. Any tran sform ation that leav es th e sta te un chang ed in

for m is calle d a s ymmet ry tr ansformati on. The g reate r the numb er of

sym metry tran sform ations tha t a s tate can u nderg o, th e hig her i ts

sym metry . If the numbe r of conditions that define th e sta te is redu ced

the n the symm etry of th e sta te is incr eased. Fo r exa mple, an o bject

cha racte rized by o blateness alone is s ymmet ric u nder all transformat ions

exc ept a chan ge of shap e.

In descr ibing the symme try o f a sta te of the most gener al kind (n ot

simply g eomet ric), the algebraic struc ture of th e set of s ymmet ry op erato rs

mus t be given; it is no t suf ficie nt to give the numbe r of opera tions, and

not hing else. The law of c ombin ation of the opera tors must be st ated. It

is the alg ebrai c gro up th at fu lly charac terizes th e sym metry of t he ge neral

sta te.

The The ory o f Gro ups ca me ab out u nexpe ctedly. G alois show ed

tha t an equat ion o f deg ree n , whe re n is an inte ger g reate r tha n or equal to

fiv e can not, in ge neral, be solve d by algebraic means . In the cours e of this

gre at wo rk, h e dev eloped the idea s of Lagra nge, Ruffi ni, and Ab el an d

int roduc ed th e con cept of a gro up. Galoi s discussed the func tional

rel ationships amon g the root s of an eq uation, an d sho wed t hat t he

rel ationships have symm etrie s associated wi th th em un der p ermut ations of

the root s.


5/164

The oper ators that tran sform one funct ional rela tions hip i nto 5

ano ther are e lemen ts of a se t tha t is chara cteri stic of th e equ ation ; the set

of opera tors is ca lled the G alois grou p of the e quati on.

In the 1 850s , Cay ley showed that ever y finite g roup is is omorp hic

to a cer tain permu tatio n gro up. The g eomet rical symm etries of cryst als

are desc ribed in t erms of fi nite group s. T hese symme tries are discu ssed in

man y sta ndard work s (see bib liography) and there fore, they will not be

discussed in this book.

In the b rief perio d bet ween 1924 and 1 928, Quant um Me chanics

was deve loped. Alm ost i mmedi ately, it was r ecogn ized by Weyl, and by

Wig ner, that certa in pa rts o f Gro up Th eory could be u sed a s a p owerf ul

ana lytic al to ol in Quan tum P hysic s. T heir ideas have been deve loped over

the deca des in man y are as th at ra nge f rom t he Th eory of So lids to Pa rticl e

Phy sics.

The esse ntial role play ed by grou ps th at ar e cha racte rized by

par amete rs th at va ry co ntinu ously in a give n ran ge wa s first em phasi zed

by Wigne r. T hese group s are know n as Lie Gro ups. They have becom e

increasingly impor tant in ma ny br anche s of conte mpora ry ph ysics ,

par ticul arly Nuclear an d Par ticle Phys ics. Fift y yea rs af ter G alois had

int roduc ed th e con cept of a group in t he Th eory of Eq uations, L ie

int roduc ed th e con cept of a continuous tran sform ation grou p in the T heory

of Diffe renti al Eq uations. Lies theo ry un ified many of t he di sconn ected

met hods of so lving diff erent ial e quations t hat h ad ev olved over a pe riod of


6/164

two hund red y ears. Inf inite simal unit ary t ransformat ions play a key role in 6

discussi ons o f the fund ament al co nserv ation laws of P hysic s.

In Class ical Dynam ics, the i nvari ance of th e equ ations of motio n of a

par ticle , or system of parti cles, unde r the Gali lean trans forma tion is a basic

par t of every day r elati vity. The sear ch fo r the tran sform ation that leav es

Max well s equ ations of Elect romag netism unc hange d in form (inva riant )

und er a linea r tra nsfor mation of the s pace-time coord inates, le d to the

discover y of the L orent z tra nsfor matio n. T he fu ndame ntal impor tance of

thi s tra nsfor matio n, an d its rela ted i nvari ants, cann ot be over state d.

This int roduc tion to Gr oup T heory , wit h its emph asis on Li e Gro ups

and thei r app licat ion t o the stud y of symme tries of t he fu ndame ntal

con stituents of ma tter, has its o rigin in a one- semester c ourse that I ta ught

at Yale University for more than ten y ears. The cour se wa s dev eloped for

Sen iors, and advan ced J unior s, ma joring in the P hysic al Sc ience s. T he

stu dents had gener ally completed the c ore c ourse s for thei r maj ors, and

had take n int ermed iate level cour ses in Lin ear A lgebr a, Re al an d Com plex

Ana lysis , Ord inary Line ar Di fferential Equa tions, and some of t he Sp ecial

Fun ctions of Physics. Group Theo ry wa s not a ma thema tical requ ireme nt

for a de gree in th e Phy sical Scie nces. The majo rity of ex istin g

und ergra duate text books on G roup Theor y and its appli cations in Phys ics

ten d to be ei ther highl y qua litat ive o r hig hly m athem atical. The pu rpose of

thi s int roduc tion is to stee r a m iddle cour se th at pr ovides the stud ent w ith

a s ound mathe matical ba sis f or st udying the symm etry prope rties of t he


7/164

fun damen tal p artic les. It i s not gene rally appr eciated by Phys icist s tha t 7

con tinuo us tr ansformati on gr oups originated in t he Th eory of Differential

Equ ations. T he in finit esima l gen erato rs of Lie Group s the refor e have

forms that involve differential operators and their commutators, and these

operators and their algebraic properties have found, and continue to find, a

natural place in the development of Quantum Physics.

Guilford, CT.

June, 2000


8/164

2 8

GALOIS GROUPS

In the early 19th - century, Abel proved that it is not possible to solve the

general polynomial equation of degree greater than four by algebraic means.

He attempted to characterize all equations that can be solved by radicals.

Abel did not solve this fundamental problem. The problem was taken up and

solved by one of the greatest innovators in Mathematics, namely, Galois.

2.1. Solving cubic equations

The main ideas of the Galois procedure in the Theory of Equations,

and their relationship to later developments in Mathematics and Physics, can

be introduced in a plausible way by considering the standard problem of

solving a cubic equation.

Consider solutions of the general cubic equation

Ax3 + 3Bx2 + 3Cx + D = 0, where A - D are rational constants.

If the substitution y = Ax + B is made, the equation becomes

y3 + 3Hy + G = 0

where

H = AC - B2

and

G = A2D - 3ABC + 2B3.


9/164

The cubic has three real roots if G2 + 4H3 < 0 and two imaginary roots if G2 9

+ 4H3 > 0. (See any standard work on the Theory of Equations).

If all the roots are real, a trigonometrical method can be used to obtain

the solutions, as follows:

the Fourier series of cos3u is

cos3u = (3/4)cosu + (1/4)cos3u.

Putting

y = scosu in the equation y3 + 3Hy + G = 0

(s > 0),

gives

cos3u + (3H/s2)cosu + G/s3 = 0.

Comparing the Fourier series with this equation leads to

s = 2 (-H)

and

cos3u = -4G/s3.

If v is any value of u satisfying cos3u = -4G/s3, the general solution is

3u = 2n 3v, where n is an integer.

Three different values of cosu are given by

u = v, and 2/3 v.

The three solutions of the given cubic equation are then

scosv, and scos(2/3 v).

Consider solutions of the equation


10/164

x3 - 3x + 1 = 0. 10

In this case,

H = -1 and G2 + 4H3 = -3.

All the roots are therefore real, and they are given by solving

cos3u = -4G/s3 = -4(1/8) = -1/2

or,

3u = cos-1(-1/2).

The values of u are therefore 2/9, 4/9, and 8/9, and the roots are

x1 = 2cos(2/9), x2 = 2cos(4/9), and x

3 = 2cos(8/9).

2.2. Symmetries of the roots

The roots x1, x2, and x3 exhibit a simple pattern. Relationships among

them can be readily found by writing them in the complex form 2cos = e i +

e-i where = 2/9 so that

x1 = ei + e -i ,

x2 = e2i + e-2i ,

and

x3 = e4i + e-4i .

Squaring these values gives

x12 = x2 + 2,

x22 = x3 + 2,

and


11/164

x32 = x1 + 2. 11

The relationships among the roots have the functional form f(x1,x2,x3) = 0.

Other relationships exist; for example, by considering the sum of the roots we

find

x1 + x22 + x2 - 2 = 0

x2 + x32 + x3 - 2 = 0,

and

x3 + x12 + x1 - 2 = 0.

Transformations from one root to another can be made by doubling-the-

angle, .

The functional relationships among the roots have an algebraic

symmetry associated with them under interchanges (substitutions) of the

roots. If is the operator that changes f(x1,x2,x3) into f(x2,x3,x1) then

f(x1,x2,x3) f(x2,x3,x1),

2f(x1,x2,x3) f(x3,x1,x2),

and

3f(x1,x2,x3) f(x1,x2,x3).

The operator 3 = I, is the identity.

In the present case,

(x12 - x2 - 2) = (x2

2 - x3 - 2) = 0,


12/164

and

2(x12 - x2 - 2) = (x3

2 - x1 - 2) = 0. 12

2.3. The Galois group of an equation.

The set of operators {I, , 2} introduced above, is called the Galois

group of the equation x3 - 3x + 1 = 0. (It will be shown later that it is

isomorphic to the cyclic group, C3).

The elements of a Galois group are operators that interchange the

roots of an equation in such a way that the transformed functional

relationships are true relationships. For example, if the equation

x1 + x22 + x2 - 2 = 0

is valid, then so is

(x1 + x22 + x2 - 2 ) = x2 + x3

2 + x3 - 2 = 0.

True functional relationships are polynomials with rational coefficients.

2.4. Algebraic fields

We now consider the Galois procedure in a more general way. An

algebraic solution of the general nth - degree polynomial

aoxn + a1x

n-1 + ... an = 0

is given in terms of the coefficients ai using a finite number of operations (+,-

,,,). The term "solution by radicals" is sometimes used because the

operation of extracting a square root is included in the process. If an infinite

number of operations is allowed, solutions of the general polynomial can be


13/164

obtained using transcendental functions. The coefficients ai necessarily belong 13

to afieldwhich is closed under the rational operations. If the field is the set

of rational numbers, Q, we need to know whether or not the solutions of a

given equation belong to Q. For example, if

x2 - 3 = 0

we see that the coefficient -3 belongs to Q, whereas the roots of the equation,

xi = 3, do not. It is therefore necessary to extendQ to Q', (say) by

adjoining numbers of the form a3 to Q, where a is in Q.

In discussing the cubic equation x3

- 3x + 1 = 0 in 2.2, we found

certain functions of the roots f(x1,x2,x3) = 0 that are symmetric under

permutations of the roots. The symmetry operators formed the Galois group

of the equation.

For a general polynomial:

xn + a1xn-1 + a2x

n-2 + .. an = 0,

functional relations of the roots are given in terms of the coefficients in the

standard way

x1 + x2 + x3 .. .. + xn = -a1

x1x2 + x1x3 + .. x2x3 + x2x4 + ..+ xn-1xn = a2

x1x2x3 + x2x3x4 + .. .. + xn-2xn-1xn = -a3

.

.

x1x2x3 .. .. xn-1xn = an.


14/164


15/164

It can be solved by writing x3 = y, y2 = 3 or 15

x = (3)1/3.

To solve the equation, it is necessary to calculate square and cube roots

not sixth roots. The equation x6 = 3 is said to be compound (it can be

broken down into simpler equations), whereas x2 = 3 is said to be atomic.

The atomic properties of the Galois group of an equation reveal

the atomic nature of the equation, itself. (In Chapter 5, it will be seen that a

group is atomic ("simple") if it contains no proper invariant subgroups).

The determination of the Galois groups associated with an arbitrary

polynomial with unknown roots is far from straightforward. We can gain

some insight into the Galois method, however, by studying the group

structure of the quartic

x4 + a2x2 + a4 = 0

with known roots

x1 = ((-a2 + )/2)1/2 , x2 = -x1,

x3 = ((-a2 - )/2)1/2 , x4 = -x3,

where

= (a 22 - 4a4)

1/2.

The field F of the quartic equation contains the rationals Q, and the

rational expressions formed from the coefficients a2 and a4.

The relations

x1 + x2 = x3 + x4 = 0


16/164

are in the field F. 16

Only eight of the 4! possible permutations of the roots leave these

relations invariant in F; they are

x1 x2 x3 x4 x1 x2 x3 x4 x1 x2 x3 x4{ P1 = , P2 = , P3 = ,

x1 x2 x3 x4 x1 x2 x4 x3 x2 x1 x3 x4x1 x2 x3 x4 x1 x2 x3 x4 x1 x2 x3 x4

P4 = , P5 = , P6 =x2 x1 x4 x3 x3 x4 x1 x2 x3 x4 x2 x1x1 x2 x3 x4 x1 x2 x3 x4

P7 = , P8 = }.x4 x3 x1 x2 x4 x3 x2 x1

The set {P1,...P8} is the Galois group of the quartic in F. It is a subgroup of

the full set of twentyfour permutations. We can form an infinite number of

true relations among the roots in F. If we extend the field F by adjoining

irrational expressions of the coefficients, other true relations among the roots

can be formed in the extended field, F'. Consider, for example, the extended

field formed by adjoining (= (a22 - 4a4)) to F so that the relation

x12 - x3

2 = is in F'.

We have met the relations

x1 = -x2 and x3 = -x4

so that

x12 = x2

2 and x32 = x4

2.

Another relation in F' is therefore


17/164

x22 - x4

2 = . 17

The permutations that leave these relations true in F' are then

{P1, P2, P3, P4}.

This set is the Galois group of the quartic in F'. It is a subgroup of the set

{P1,...P8}.

If we extend the field F' by adjoining the irrational expression

((-a2 - )/2)1/2 to form the field F'' then the relation

x3 - x4 = 2((-a2 - )/2)1/2

is in F''.

This relation is invariant under the two permutations

{P1, P3}.

This set is the Galois group of the quartic in F''. It is a subgroup of the set

{P1, P2, P3, P4}.

If, finally, we extend the field F'' by adjoining the irrational

((-a2 + )/2)1/2 to form the field F''' then the relation

x1 - x2 = 2((-a2 - )/2)1/2 is in F'''.

This relation is invariant under the identity transformation, P1 , alone; it is

the Galois group of the quartic in F''.

The full group, and the subgroups, associated with the quartic equation

are of order 24, 8, 4, 2, and 1. (The order of a group is the number of

distinct elements that it contains). In 5.4, we shall prove that the order of a

subgroup is always an integral divisor of the order of the full group. The


18/164

order of the full group divided by the order of a subgroup is called the index 18

of the subgroup.

Galois introduced the idea of a normal or invariant subgroup: if H is a

normal subgroup of G then

HG - GH = [H,G] = 0.

(H commutes with every element of G, see 5.5).

Normal subgroups are also called either invariant or self-conjugate subgroups.

A normal subgroup H is maximal if no other subgroup of G contains H.

2.5. Solvability of polynomial equations

Galois defined the group of a given polynomial equation to be either

the symmetric group, Sn, or a subgroup of Sn, (see 5.6). The Galois method

therefore involves the following steps:

1. The determination of the Galois group, Ga, of the equation.

2. The choice of a maximal subgroup of Hmax(1). In the above case, {P1, ...P8}

is a maximal subgroup of Ga = S4.

3. The choice of a maximal subgroup of Hmax(1) from step 2.

In the above case, {P1,..P4} = Hmax(2) is a maximal subgroup of Hmax(1).

The process is continued until Hmax = {P1} = {I}.

The groups Ga, Hmax(1), ..,Hmax(k) = I, form a composition series. The

composition indices are given by the ratios of the successive orders of the

groups:


19/164

gn/h(1), h(1)/h(2), ...h(k-1) /1.

The composition indices of the symmetric groups Sn for n = 2 to 7 are found

to be:

n Composition Indices

2 2

3 2, 3

4 2, 3, 2, 2

5 2, 60

6 2, 360

7 2, 2520

We shall state, without proof, Galois' theorem:

A polynomial equation can be solved algebraically if and only if its

group is solvable.

Galois defined a solvable group as one in which the composition indices are

all prime numbers. Furthermore, he showed that if n > 4, the sequence of

maximal normal subgroups is Sn, An, I, where An is the Alternating Group

with (n!)/2 elements. The composition indices are then 2 and (n!)/2. For n >

4, however, (n!)/2 is not prime, therefore the groups Sn are not solvable for n

> 4. Using Galois' Theorem, we see that it is therefore not possible to solve,

algebraically, a general polynomial equation of degree n > 4.


20/164

3 20

SOME ALGEBRAIC INVARIANTS

Although algebraic invariants first appeared in the works of Lagrange and

Gauss in connection with the Theory of Numbers, the study of algebraic

invariants as an independent branch of Mathematics did not begin until the

work of Boole in 1841. Before discussing this work, it will be convenient to

introduce matrix versions of real bilinear forms, B, defined by

m nB = aijxiyj

I=1 j=1

where

x = [x1,x2,...xm], an m-vector,

y = [y1,y2,...yn], an n-vector,

and aij are real coefficients. The square brackets denote a

column vector.

In matrix notation, the bilinear form is

B = xTAy

where

a11 . . . a1n

. . . .

A = . . . . .

. . . .

am1. . . amn

The scalar product of two n-vectors is seen to be a special case of a

bilinear form in which A = I.


21/164

Ifx = y, the bilinear form becomes a quadratic form, Q: 21

Q = xTAx.

3.1. Invariants of binary quadratic forms

Boole began by considering the properties of the binary

quadratic form

Q(x,y) = ax2 + 2hxy + by2

under a linear transformation of the coordinates

x' = Mx

where

x = [x,y],

x' = [x',y'],

and

| i j | M = .

| k l |

The matrix M transforms an orthogonal coordinate system into an

oblique coordinate system in which the new x'- axis has a slope (k/i), and the

new y'- axis has a slope (l/j), as shown:


22/164

y 22

y[i+j,k+l]

[j,l]x'

[0,1] [1,1]x

[i,k]

[0,0] [1,0] x

Fig. 1. The transformation of a unit square underM.

The transformation is linear, therefore the new function Q'(x',y') is a

binary quadratic:

Q'(x',y') = a'x'2 + 2h'x'y' + b'y'2.

The original function can be written

Q(x,y) = xTDx

where

| a h | D = ,

| h b |

and the determinant ofD is

detD = ab - h2, called the discriminant of Q.

The transformed function can be written


23/164

Q'(x',y') = x'TD'x' 23

where

a' h'

D' = ,h' b'

and

detD' = a'b' - h'2, the discriminant of Q'.

Now,

Q'(x',y') = (Mx)TD'Mx

= xTMTD'Mx

and this is equal to Q(x,y) if

MTD'M = D.

The invariance of the form Q(x,y) under the coordinate transformation M

therefore leads to the relation

(detM)2

detD' = detD

because

detMT = detM.

The explicit form of this equation involving determinants is

(il - jk)2(a'b' - h'2) = (ab - h2).

The discriminant (ab - h2) of Q is said to be an invariant

of the transformation because it is equal to the discriminant (a'b' - h'2) of Q',

apart from a factor (il - jk)2 that depends on the transformation itself, and not

on the arguments a,b,h of the function Q.


24/164

3.2. General algebraic invariants 24

The study of general algebraic invariants is an important branch of

Mathematics.

A binary form in two variables is

f(x1,x2) = aox1n + a1x1

n-1x2 + ...anx2n

= aix1n-ix2

i

If there are three or four variables, we speak of ternary forms or quaternary

forms.

A binary form is transformed under the linear transformation M as

follows

f(x1,x2) => f'(x1',x2') = ao'x1'n + a1'x1'

n-1x2' + ..

The coefficients

ao, a1, a2,.. ao', a1', a2' ..

and the roots of the equation

f(x1,x2) = 0

differ from the roots of the equation

f'(x1',x2') = 0.

Any function I(ao,a1,...an) of the coefficients of f that satisfies

rwI(ao',a1',...an') = I(ao,a1,...an)

is said to be an invariantof f if the quantity r depends only on the

transformation matrix M, and not on the coefficients ai of the function being

transformed. The degree of the invariant is the degree of the coefficients, and


25/164

the exponent w is called the weight. In the example discussed above, the 25

degree is two, and the weight is two.

Any function, C, of the coefficients andthe variables of a form f that is

invariant under the transformation M, except for a multiplicative factor that is

a power of the discriminant ofM, is said to be a covariant of f. For binary

forms, C therefore satisfies

rwC(ao',a1',...an'; x1',x2') = C(ao,a1,...an; x1,x2).

It is found that the Jacobian of two binary quadratic forms, f(x1,x2) and

g(x1,x2), namely the determinant

f/x1 f/x2

g/x1 g/x2

where f/x1 is the partial derivative of f with respect to x1 etc., is a

simultaneous covariant of weight one of the two forms.

The determinant

2f/x12 2f/x1x2

, 2g/x2x1

2g/x22

called the Hessian of the binary form f, is found to be a covariant of weight

two. A full discussion of the general problem of algebraic invariants is outside


26/164

the scope of this book. The following example will, however, illustrate the 26

method of finding an invariant in a particular case.

Example:

To show that

(aoa2 - a12)(a1a3 - a2

2) - (aoa3 - a1a2)2/4

is an invariant of the binary cubic

f(x,y) = aox3 + 3a1x

2y + 3a2xy2 + a3y

3

under a linear transformation of the coordinates.

The cubic may be written

f(x,y) = (aox2+2a1xy+a2y

2)x + (a1x2+2a2xy+a3y

2)y

= xTDx

where

x = [x,y],

and

aox + a1y a1x + a2y D = .

a1x + a2y a2x + a3y

Let x be transformed to x': x' = Mx, where

i j M =

k l


27/164

then 27

f(x,y) = f'(x',y')

if

D = MTD'M.

Taking determinants, we obtain

detD = (detM)2detD',

an invariant of f(x,y) under the transformation M.

In this case, D is a function of x and y. To emphasize this point, put

detD = (x,y)and

detD'= '(x',y')

so that

(x,y) = (detM)2'(x',y'

= (aox + a1y)(a2x + a3y) - (a1x + a2y)2

= (aoa2 - a12)x2 + (aoa3 - a1a2)xy + (a1a3 - a2

2)y2

= xTEx,

where

(aoa2 - a12 ) (aoa3 - a1a2)/2

E = .

(aoa3 - a1a2)/2 (a1a3 - a22 )

Also, we have

'(x',y') = x'TE'x'


28/164

= xTMTE'Mx 28

therefore

xTEx = (detM)2xTMTE'Mx

so that

E = (detM)2MTE'M.

Taking determinants, we obtain

detE = (detM)4detE'

= (aoa2 - a12)(a1a3 - a2

2) - (aoa3 -a1a2)2/4

= invariant of the binary cubic f(x,y) under the transformation

x' = Mx.


29/164

4 29

SOM E INV ARIAN TS IN PHYS ICS

4.1 . Gal ilean inva rianc e.

Eve nts o f fin ite e xtens ion a nd du ratio n are part of t he ph ysica l

wor ld. It will be conv enient to intro duce the n otion ofide al ev ents th at

hav e nei ther exten sion nor d urati on. Ideal even ts ma y be repre sente d as

mat hemat ical point s in a spa ce-ti me ge ometr y. A part icula r eve nt, E, i s

des cribed by the f our c ompon ents [t,x, y,z] where t is the time of th e eve nt,

and x,y, z, ar e its thre e spa tial coord inates. The tim e and spac e coo rdina tes

are refe rred to ar bitra rily chose n ori gins. The spat ial mesh n eed n ot be

Car tesia n.

Let an e vent E[t, x], r ecord ed by an o bserv er O at th e ori gin o f an x-

axis, be reco rded as th e eve nt E'[t ',x'] by a seco nd ob serve r O', movi ng at

con stant spee d V a long the x -axis . We supp ose t hat t heir clocks are

syn chron ized at t = t' = 0 w hen t hey c oinci de at a co mmon origin, x = x' =

0.

At time t, we writ e the plau sible equa tions

t' = t

and

x' = x - V t,

whe re Vt is t he di stanc e tra velle d by O' in a ti me t. The se eq uations ca n

be writt en

E' = GE


30/164

whe re 30

1 0 G = .

-V 1

G is the opera tor o f the Gali lean trans forma tion.

The inve rse e quations a re

t = t'

and

x = x ' + V t'

or

E = G-1E'

whe re G-1 is the inver se Ga lilea n ope rator . (I t und oes t he ef fect ofG).

If we mu ltipl y t a nd t' by t he co nstan ts k and k ', re spect ively , whe re

k a nd k' have dime nsions of velocity t hen a ll terms h ave d imensions of

length.

In space-spac e, we have the Pytha gorea n for m x 2 + y2 = r2, a n

inv ariant und er ro tatio ns. We ar e the refor e led to a sk th e que stion: is

(kt )2 + x2 in variant un der t he op erato r G in spac e-tim e? C alcul ation give s

(kt)2 + x2 = (k't ')2 + x'2 + 2Vx't ' + V 2t'2

= (k't' )2 + x'2 o nly if V = 0.

We see, there fore, that Gali lean space-time is n ot Py thago rean in its

algebraic for m. W e not e, ho wever , the key role played by acc elera tion in

Galilean -Newtonian phys ics:


31/164

The velo citie s of the e vents acco rding to O and O' ar e obt ained by 31

dif feren tiati ng th e equ ation x' = -Vt + x w ith r espect to time, giving

v' = -V + v,

a p lausi ble r esult, bas ed up on ou r exp erien ce.

Dif feren tiati ng v' with resp ect t o tim e giv es

d v'/dt ' = a ' = d v/dt = a

whe re a and a ' are the accel erati ons i n the two frame s of refer ence. The

classica l acceleration is inv arian tun der t he Ga lilea n tra nsfor matio n. I f the

rel ationship v' = v V is used to desc ribe the m otion of a puls e of light ,mov ing in emp ty sp ace a t v = c 3 x 108 m/ s, it does not fit t he fa cts. All

stu dies of ve ry hi gh sp eed p artic les t hat e mit e lectr omagn etic radiation

sho w tha t v' = c f or all va lues of th e rel ative spee d, V.

4.2 . Lor entz invar iance and Einst ein's spac e-tim e

sym metry .

It was E instein, above all others , who adv anced our under stand ing o f

the true natu re of spac e-tim e and rela tive motio n. W e sha ll se e tha t he

mad e use of a symm etry argum ent t o find the chan ges t hat m ust b e mad e

to the G alilean tr ansformati on if it i s to accou nt fo r the rela tive motio n of

rap idly movin g obj ects and o f bea ms of light. H e rec ognized an

inconsis tency in t he Ga lilea n-Newtonian equ ations, ba sed a s the y are , on

eve ryday expe rience. H ere, we sh all restrict th e discussi on to non-

acc elerating, or s o cal led inerti al, f rames


32/164

We have seen that the c lassi cal equations r elati ng th e eve nts E an d 32

E' a re E' = GE, a nd th e inv erse E = G-1E'

whe re

1 0 1 0 G = an d G-1 = . V 1 V 1

The se eq uations ar e con necte d by the s ubstitutio n V V; this is an

algebraic sta temen t of the N ewton ian p rinci ple o f rel ativi ty. Einst ein

incorpor ated this princ iple in his the ory. He a lso r etained th e lin earit y of

the classical equa tions in t he ab sence of a ny ev idence to the c ontra ry.

(Eq uispa ced i nterv als of tim e and dist ance in on e ine rtial fram e rem ain

equ ispac ed in any other iner tial frame ). H e the refor e sym metri zedth e

spa ce-ti me eq uations as foll ows:

t' 1 -V t= .

x' -V 1 x

Not e, ho wever , the inco nsist ency in th e dim ensio ns of the time- equat ion

tha t has now been intro duced :

t' = t Vx .

The term Vx h as dimensions o f [L] 2/[T], an d not [T]. Thi s can be

cor recte d by intro ducing the inva riant spee d of light , ca postu late in

Ein stein's th eory that is consist ent w ith e xperi ment:

ct' = ct Vx /c

so that all terms now h ave d imensions of length.


33/164

Ein stein went furt her, and i ntrod uced a dim ensio nless quan tity 33

instead of th e sca ling facto r of unity that appe ars i n the Gali lean equat ions

of space-time . Th is fa ctor must be co nsist ent w ith a ll observa tions. Th e

equ ations the n bec ome

ct ' = ct x

x ' = -ct + x, where =V/ c.

The se ca n be writt en

E' = LE,

whe re

- L = , and E = [ct,x ]

-

L is the opera tor o f the Lore ntz t ransformat ion.

The inve rse e quation is

E = L-1E'

whe re

L-1 = .

This is the i nvers e Lor entz trans forma tion, obta ined from L by chan ging

(o r ,V V); it h as th e effect o f und oing the t ransf ormat ion L.

We can t heref ore w rite

LL-1 = I

or


34/164

- 1 0 34= .

- 0 1

Equ ating elem ents gives

222 = 1

the refor e,

= 1/(1 2) ( taking the posi tive root) .

4.3 . The inva riant inte rval.

Pre viously, i t was show n tha t the spac e-tim e of Galileo an d New ton

is not P ythag orean in f orm. We n ow as k the ques tion: is E insteinian spac e-

tim e Pyt hagor ean i n for m? D irect calculati on le ads t o

( ct)2 + (x)2 = 2(1 + 2)(c t')2 + 42x'c t'

+2(1 + 2)x'2

(c t')2 + (x')2 if > 0.

Not e, ho wever , tha t the dif feren ce of squa res is an

inv ariant und er L:

(ct)2 (x )2 = (ct')2 (x ')2

bec ause

2(1 2) = 1.

Spa ce-ti me is said to b e pse udo-E uclid ean.

The nega tive sign that chara cteri zes L orent z inv ariance ca n be

included in t he th eory in a gener al wa y as follo ws.

We intro duce two k inds of 4- vecto rs


35/164

x = [x0, x1, x2, x3], a con trava riantve ctor, 35

and

x = [x0, x1, x2, x3], a cov arian tve ctor, wher e

x = [x0,-x1,-x2,-x3].

The scalar pr oduct of t he ve ctors is d efine d as

x Tx = (x0, x1, x2, x3)[x0,-x1,-x2,-x3]

= (x 0)2 (( x1)2 + (x2)2 + (x3)2)

The even t 4-v ector is

E

= [ct, x, y, z] a nd th e cov ariant for m is

E = [ct,- x,-y, -z]

so that the L orent z inv arian t sca lar p roduc t is

ETE = (ct)2 (x2 + y2 + z2).

The 4-ve ctor x tr ansforms a s follows:

x' = Lx

whe re

- 0 0- 0 0

L = .0 0 1 00 0 0 1

This is the o perat or of the Loren tz tr ansformati on if the motio n of O' is

along th e x-a xis o f O's fram e of refer ence.

Imp ortan t con seque nces of th e Lor entz trans forma tion are t hat

int ervals of time measu red i n two diff erent iner tial frame s are not the s ame

but are relat ed by the equat ion


36/164

t' = t 36

whe re t i s an inter val m easur ed on a cl ock a t res t in O's f rame, and

distance s are give n by

l' = l/

whe re l is a l ength meas ured on a ruler at r est i n O's fram e.

4.4 . The ener gy-mo mentu m inv arian t.

A d iffer ential time int erval, dt, cann ot be used in a Lore ntz-i nvari ant

way in k inematics. We must use t he pr oper time diffe renti al interva l, d,

def ined by

( cdt)2 - dx2 = (cdt' )2 - dx'2 (c d)2.

The Newt onian 3-ve locit y is

vN = [dx/d t, dy /dt, dz/dt],

and this must be r eplac ed by the 4-vel ocity

V = [d(ct )/d, d x/d, d y/d, d z/d]

= [ d(ct)/dt, dx/dt , dy/ dt, d z/dt] dt/d

= [c,vN] .

The scalar pr oduct is t hen

VV = (c)2 - (vN)

2

= (c)2(1 - (vN/c)2)

= c2.

(In form ing t he sc alar produ ct, t he tr anspo se is unde rstoo d).

The magn itude of t he 4- velocity is V = c, th e inv ariant spe ed of light.


37/164

In Class ical Mechanics, the conce pt of mome ntum is im porta nt be cause 37

of its r ole a s an invar iant in an isol ated system. W e the refor e int roduc e the

con cept of 4- momen tum i n Rel ativi stic Mechanics in or der t o find

pos sible Lore ntz i nvari ants invol ving this new q uantity. The c ontra variant

4-m oment um is defined a s:

P = mV

whe re m is th e mas s of the p artic le. ( It is a Lo rentz scalar, t he ma ss

mea sured in t he fr ame i n whi ch th e par ticle is a t res t).

The scalar pr oduct is

P P = (mc)2.

Now ,

P = [mc, mvN]

the refor e,

P P = (mc)2 (mvN)

2.

Wri ting

M = m, the r elati visti c mas s, we obta in

PP = (Mc)2 (M vN)

2 = (mc)2.

Multiply ing t hroug hout by c2 gi ves

M2c4 M2vN2c2 = m2c4.

The quan tity Mc2 ha s dim ensio ns of ener gy; w e the refor e wri te

E = Mc2

the tota l ene rgy o f a f reely movi ng pa rticl e.


38/164

This leads to the fun damen tal i nvari ant o f dyn amics 38

c2PP = E2 (p c)2 = Eo2

whe re

Eo = mc2 is the rest energ y of the p artic le, and

p is its rel ativi stic 3- momen tum.

The tota l ene rgy c an be writ ten:

E = Eo = Eo + T,

whe re

T = Eo

( 1) ,

the rela tivis tic k ineti c ene rgy.

The magn itude of t he 4- momen tum i s a L orent z inv ariant

P = mc.

The 4- m oment um tr ansforms a s follows:

P' = LP.

For rela tive motio n alo ng th e x-a xis, this equat ion is equ ivale nt to the

equ ations

E' = E cpx

and

cpx = -E + cpx .

Usi ng th e Pla nck-E instein eq uations E = h an d

E = pxc f or ph otons , the ener gy eq uation bec omes

' =


39/164

= (1 ) 39

= (1 )/( 1 2)1/2

= [(1 )/(1 + )]1/2 .

This is the r elati visti c Dop pler shift for the f reque ncy', measu red i n an

inertial fram e (pr imed) in t erms of th e fre quenc y me asure d in anoth er

inertial fram e (un prime d).

4.5 . The freq uency -wavenumbe r inv arian t

Par ticle -Wave dual ity, one o f the most prof ound

discover ies i n Phy sics, has its o rigins in Loren tz in variance. It w as

pro posed by d eBrog lie i n the earl y 192 0's. He u sed t he fo llowing

arg ument .

The disp lacement o f a w ave c an be writ ten

y(t ,r) = Acos (t - kr)

whe re = 2 (t he an gular freq uency ), k = 2/ (t he wa venum ber),

and r = [x, y , z] (the position v ector ). T he ph ase (t kr) c an be

wri tten ((/c) ct kr), and t his h as th e for m of a Lor entz invar iant

obt ained from the 4-vectors

E[ct , r], and K [/c, k]

whe re E is the event 4-ve ctor, and K is the "freq uency -wavenumbe r" 4-

vec tor.

deB roglie not ed th at th e 4-m oment um P is conn ected to t he ev ent 4 -

vec tor E th rough the 4-vel ocity V, a nd th e fre quenc y-wavenumb er 4-

vec tor K is conn ected to t he ev ent 4 -vector E th rough the Loren tz


40/164

inv ariant pha se of a wa ve ((/c) ct k r). He t heref ore p ropos ed th at a 40

dir ect c onnec tion must exist betw een P an d K; it is illu strat ed

in the f ollow ing d iagram:

E[ct ,r]

( Einst ein) PP=in v. EK=in v. (d eBrog lie)

P [E/ c,p] K[/c,k]

(deB roglie)

Fig . 2. The coup ling betwe en P an d K vi a E.

deB roglie pro posed that the conn ection is the

simplest poss ible, name ly, P an d K ar e pro porti onal to ea ch ot her. He

rea lized that ther e cou ld be only one value for the c onsta nt of

pro porti onali ty if the Planck-Einstein resu lt for pho tons E = h/2 is but a

spe cial case of a gener al re sultit must be h /2, w here h is Plancks

con stant . Th erefo re, d eBrog lie propos ed th at

P K

or

P = (h/2)K.

Equ ating the elements o f the 4-ve ctors give s

E = (h/2)

and

p = (h/2)k .


41/164

In these rema rkabl e equ ations, ou r not ions of pa rticl es an d wav es ar e 41

for ever merge d. T he sm allne ss of the value of P lanck's co nstan t pre vents

us from obser ving the d ualit y dir ectly; how ever, it i s clearly obser ved a t

the mole cular, ato mic, nuclear, a nd pa rticl e lev el.

4.6 . deB rogli e's i nvari ant.

The inva riant form ed fr om th e fre quenc y-wavenumb er 4- vecto r is

KK = (/c, k)[/c, - k]

= (/c)2 k2 = (o/c)2, w here o is the prope r

ang ular frequ ency.

Thi s inv arian t is the w ave v ersion of Einst ein's

ene rgy-m oment um in varia nt; it giv es th e dispersion re latio n

o2 = 2 (k c)2.

The rati o /k is th e pha se ve locit y of the w ave, v.

For a wa ve-packet, the group velo city vG is d/dk ; it can b e obt ained by

dif feren tiati ng th e dispersion eq uation as follo ws:

d kc2dk = 0

the refor e,

vG = d/dk = kc2/.

The deBr oglie inva riant invo lving the produ ct of the phase and group

velocity is t heref ore

vvG = (/k) .(kc2/) = c2.

Thi s is the w ave-equivalent of Ei nstein's f amous


42/164

E = M c2. 42

We see t hat

vvG = c2 = E/M

or,

vG = E/Mv = Ek/M = p/M = vN, t he pa rticl e

velocity.

This res ult p layed an i mport ant p art i n the deve lopment of Wave

Mec hanics.

We shall find in l ater chapt ers, that Loren tz tr ansformati ons f orm a

gro up, a nd th at in variance p rinci ples are r elated dir ectly to s ymmet ry

tra nsfor matio ns an d the ir as socia ted g roups .


43/164

5 43

GROUPS CONCRETE AND ABSTRACT

5.1 Some concrete examples

The elements of the set {1, i}, where i = -1, are the roots of the

equation x4 = 1, the fourth roots of unity. They have the following special

properties:

1. The product of any two elements of the set (including the same two

elements) is always an element of the set. (The elements obey closure).

2. The order of combining pairs in the triple product of any elements

of the set does not matter. (The elements obey associativity).

3. A unique element of the set exists such that the product of any

element of the set and the unique element (called the identity) is equal to the

element itself. (An identity element exists).

4. For each element of the set, a corresponding element exists such

that the product of the element and its corresponding element (called the

inverse) is equal to the identity. (An inverse element exists).

The set of elements {1, i} with these four properties is said to form

a GROUP.

In this example, the law of composition of the group is multiplication; this

need not be the case. For example, the set of integers Z = {.., -2, -1, 0, 1, 2,

...} forms a group if the law of composition is addition. In this group, the

identity element is zero, and the inverse of each integer is the integer with the

same magnitude but with opposite sign.


44/164

In a different vein, we consider the set of 44 matrices: 44

1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0{M} = 0 1 0 0 , 1 0 0 0 , 0 0 0 1 , 0 0 1 0 .

0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1

0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0

If the law of composition is matrix multiplication , then {M} is found to obey:

1 --closure

and

2 --associativity,

and to contain:

3 --an identity, diag(1, 1, 1, 1),

and

4 --inverses.

The set {M} forms a group under matrix multilication.

5.2. Abstract groups

The examples given above illustrate the generality of the group

concept. In the first example, the group elements are real and imaginary

numbers, in the second, they are positive and negative integers, and in the

third, they are matrices that represent linear operators (see later discussion).

Cayley, in the mid-19th century, first emphasized this generality, and he

introduced the concept of an abstract group, Gn which is a collection of n

distinct elements (...gi...) for which a law of composition is given. If n is finite,

the group is said to be a group of order n. The collection of elements must

obey the four rules:


45/164

1. If gi, gj G then gn = gjgi G gi, gj G (closure) 45

2. gk(gjgi) = (gkgj)gi [leaving out the composition symbol] (associativity)

3. e G such that gie = egi = gi gi G (an identity exists)

4. If gi G then gi-1 G such that gi

-1gi = gigi-1 = e (an inverse exists).

For finite groups, the group structure is given by listing all

compositions of pairs of elements in a group table, as follows:

e . gi gj . (1st symbol, or operation, in pair)e . . . .. . . . .

gi . . g igi g igj .

gj . gjgi gjgj .

gk . gkgi gkgj ...

If gjgi = gigj gi, gj G, then G is said to be a commutative or abelian

group. The group table of an abelian group is symmetric under reflection in

the diagonal.

A group of elements that has the same structure as an abstract group is

a realization of the group.

5.3 The dihedral group, D3

The set of operations that leaves an equilateral triangle invariant under

rotations in the plane about its center, and under reflections in the three

planes through the vertices, perpendicular to the opposite sides, forms a


46/164

group of six elements. A study of the structure of this group (called the 46

dihedral group, D3) illustrates the typical group-theoretical approach.

The geometric operations that leave the triangle invariant are:

Rotations about the z-axis (anticlockwise rotations are positive)

Rz(0) (123) (123) = e, the identity

Rz(2/3)(123) (312) = a

Rz(4/3)(123) (231) = a2

and reflections in the planes I, II, and III:

RI (123) (123) = b

RII (123) (321) = c

RIII (123) (213) = d

This set of operators is D3 = {e, a, a2, b, c, d}.

Positive rotations are in an anticlockwise sense and the inverse rotations are in

a clockwise sense., so that the inverse of e, a, a2 are

e-1 = e, a-1 = a2, and (a2)-1 = a.

The inverses of the reflection operators are the operators themselves:

b-1 = b, c -1 = c, and d-1 = d.

We therefore see that the set D3 forms a group. The group

multiplication table is:

e a a2 b c de e a a2 b c da a a2 e d b ca2 a2 e a c d bb b c d e a a2

c c d b a2 e ad d b c a a2 e


47/164

In reading the table, we follow the rule that the first operation is written on 47

the right: for example, ca2 = b. A feature of the group D3 is that it can be

subdivided into sets of either rotations involving {e, a, a2

} or reflections

involving {b, c, d}. The set {e, a, a2} forms a group called the cyclic group

of order three, C3. A group is cyclic if all the elements of the group are

powers of a single element. The cyclic group of order n, Cn, is

Cn = {e, a, a2, a3, .....,a

n-1},

where n is the smallest integer such that an = e, the identity. Since

aka

n-k= a

n= e,

an inverse an-k

exists. All cyclic groups are abelian.

The group D3 can be broken down into a part that is a group C3, and a

part that is the product of one of the remaining elements and the elements of

C3. For example, we can write

D3 = C3 + bC3 , b C3

= {e, a, a2} + {b, ba, ba2}

= {e, a, a2} + {b, c, d}

= cC3 = dC3.

This decomposition is a special case of an important theorem known as

Lagranges theorem. (Lagrange had considered permutations of roots of

equations before Cauchy and Galois).

5.4 Lagranges theorem


48/164

The order m of a subgroup Hm of a finite group Gn of order n is a 48

factor (an integral divisor) of n.

Let

Gn = {g1=e, g2, g3, ...gn} be a group of order n, and let

Hm = {h1=e, h2, h3, ...hm} be a subgroup of Gn of order m.

If we take any element gkof Gn which is not in Hm, we can form the set of

elements

{gkh1, gkh2, gkh3, ...gkhm} gkHm.

This is called the left cosetof Hm with respect to gk. We note the important

facts that all the elements of gkhj, j=1 to m are distinct, and that none of the

elements gkhjbelongs to Hm.

Every element gkthat belongs to Gn but does not belong to Hm

belongs to some coset gkHm so that Gn forms the union of Hm and a number

of distinct (non-overlapping) cosets. (There are (n - m) such distinct cosets).

Each coset has m different elements and therefore the order n of Gn is

divisible by m, hence n = Km, where the integer K is called the index of the

subgroup Hm under the group Gn. We therefore write

Gn = g1Hm + gj2Hm + gk3Hm + ....gKHm

where


49/164

gj2 Gn Hm, 49

gk3 Gn Hm, gj2Hm

.

gnK Gn Hm, gj2Hm, gk3Hm, ...gn-1, K-1Hm.

The subscripts 2, 3, 4, ..K are the indices of the group.

As an example, consider the permutations of three objects 1, 2, 3 ( the

group S3) and let Hm = C3 = {123, 312, 231}, the cyclic group of order

three. The elements of S3 that are not in H3 are {132, 213, 321}. Choosing

gk= 132, we obtain

gkH3 = {132, 321, 213},

and therefore

S3 = C3 + gk2C3 ,K = 2.

This is the result obtained in the decomposition of the group D3, if we make

the substitutions e = 123, a = 312, a2 = 231, b = 132, c = 321, and d = 213.

The groups D3 and S3 are said to be isomorphic. Isomorphic groups have

the same group multiplication table. Isomorphism is a special case of

homomorphism that involves a many-to-one correspondence.

5.5 Conjugate classes and invariant subgroups


50/164

If there exists an element v Gn such that two elements a, b Gn are 50

related by vav-1 = b, then b is said to be conjugate to a. A finite group can

be separated into sets that are conjugate to each other.

The class of Gn is defined as the set of conjugates of an element a

Gn. The element itself belongs to this set. If a is conjugate to b, the class

conjugate to a and the class conjugate to b are the same. If a is not conjugate

to b, these classes have no common elements. Gn can be decomposed into

classes because each element of Gn belongs to a class.

An element of Gn that commutes with all elements of Gn forms a class

by itself.

The elements of an abelian group are such that

bab -1 = a for all a, b Gn,

so that

ba = ab.

If Hm is a subgroup of Gn, we can form the set

{aea-1, ah2a-1, ....ahma

-1} = aHma-1 where a Gn .

Now, aHma-1 is another subgroup of Hm in Gn. Different subgroups may be

found by choosing different elements a of Gn. If, for all values of a Gn

aHma-1 = Hm


51/164

(all conjugate subgroups of Hm in Gn are identical to Hm), 51

then Hm is said to be an invariant subgroup in Gn.

Alternatively, Hm is an invariant in Gn if the left- and right-cosets

formed with any a Gn are equal, i. e. ahi = hka.

An invariant subgroup Hm of Gn commutes with all elements of Gn.

Furthermore, if hi Hm then all elements ahia-1 Hm so that Hm is an

invariant subgroup of Gn

if it contains elements of Gn

in complete classes.

Every group Gn contains two trivial invariant subgroups, Hm = Gn and

Hm = e. A group with no proper (non-trivail) invariant subgroups is said to

be simple (atomic). If none of the proper invariant subgroups of a group is

abelian, the group is said to be semisimple.

An invariant subgroup Hm and its cosets form a group under

multiplication called thefactor group (written Gn/Hm) of Hm in Gn.

These formal aspects of Group Theory can be illustrated by considering

the following example:

The group D3 = {e, a, a2, b, c, d} ~ S3 = {123, 312, 231, 132, 321, 213}.

C3 is a subgroup of S3: C3 = H3 = {e, a, a2} = {123, 312, 231}.

Now,

bH3 = {132, 321, 213} = H3b


52/164


53/164

If we now consider Hm = {e, b} an abelian subgroup, we find 53

aHm = {a,d}, Hma = {a.c},

a2Hm = {a2,c}, Hma2 = {a2, d}, etc.

All left and right cosets are not equal: Hm = {e, b} is therefore not an

invariant subgroup of S3. We can therefore write

S3 = {e, b} + {a, d} + {a2, c}

= Hm + aHm + a2Hm.

Applying Lagranges theorem to S3 gives the orders of the possible

subgroups: they are

order 1: {e}

order 2: {e, d}; {e, c}: {e, d}

order 3: {e, a, a2} (abelian and invariant)

order 6: S3.

5.6 Permutations

A permutation of the set {1, 2, 3, ....,n} of n distinct elements is an

ordered arrangement of the n elements. If the order is changed then the

permutation is changed. the number of permutations of n distinct elements is

n!

We begin with a familiar example: the permutations of three distinct

objects labelled 1, 2, 3. There are six possible arrangements; they are

123, 312, 231, 132, 321, 213.


54/164

These arrangements can be written conveniently in matrix form: 54

1 2 3 1 2 3 1 2 31 = , 2 = , 3 = ,

1 2 3 3 1 2 2 3 1

1 2 3 1 2 3 1 2 34 = , 5 = , 6 = .

1 3 2 3 2 1 2 1 3

The product of two permutations is the result of performing one arrangement

after another. We then find

23 = 1

and

32 = 1

whereas

45 = 3

and

54 = 2.

The permutations 1, 2, 3 commute in pairs (they correspond to the

rotations of the dihedral group) whereas the permutations do not commute

(they correspond to the reflections).

A general product of permutations can be written

s1 s2 . . .sn 1 2 . . n 1 2 . . n= .

t1 t2 . . .tn s1 s2 . . sn t1 t2 . . tn

The permutations are found to have the following properties:

1. The product of two permutations of the set {1, 2, 3, ...} is itself a

permutation of the set. (Closure)


55/164

2. The product obeys associativity: 55

(kj)i = k(ji), (not generally commutative).

3. An identity permutation exists.

4. An inverse permutation exists:

s1 s2 . . . sn

-1 = | |1 2 . . . n

such that -1 = -1 = identity permutation.

The set of permutations therefore forms a group

5.7 Cayleys theorem:

Every finite group is isomorphic to a certain permutation group.

Let Gn ={g1, g2, g3, . . .gn} be a finite group of order n. We choose any

element gi in Gn, and we form the products that belong to Gn:

gig1, gig2, gig3, . . . gign.

These products are the n-elements of Gn

rearranged. The permutation i,

associated with gi is therefore

g1 g2 . . gn i = .

gig1 gig2 . . gign

If the permutation j associated with gj is

g1 g2 . . gn

j = ,gjg1 gjg2 . . gjgn

where gi gj, then

g1 g2 . . gn ji = .

(gjgi)gi (gjgi)g2 . . (gjgi)gn


56/164

This is the permutation that corresponds to the element gjgi of Gn. 56

There is a direct correspondence between the elements of Gn and the n-

permutations {1,

2, . . .n}. The group of permutations is a subgroup of

the full symmetric group of order n! that contains all the permutations of the

elements g1, g2, . . gn.

Cayleys theorem is important not only in the theory of finite groups

but also in those quantum systems in which the indistinguishability of the

fundamental particles means that certain quantities must be invariant under

the exchange or permutation of the particles.


57/164

6 57

LIES DIFFERENTIAL EQUATION, INFINITESIMAL ROTATIONS

AND ANGULAR MOMENTUM OPERATORS

Although the field of continuous transformation groups (Lie groups)

has its origin in the theory of differential equations, we shall introduce the

subject using geometrical ideas.

6.1 Coordinate and vector rotations

A 3-vector v = [vx, vy, vz] transforms into v = [vx, vy, vz] under a

general coordinate rotation R about the origin of an orthogonal coordinatesystem as follows:

v = R v,

where

i.i j.i k.i

R

=i.j j.j k.j i.k j.k k.kcosii . .

= cosij . . cosik . coskk

where i,j, k, i,j, k are orthogonal unit vectors, along the axes, before and

after the transformation, and the cosiis are direction cosines.

The simplest case involves rotations in the x-y plane:

vx = cosii cosjivxvy cosij cosjj vy


58/164

= cos sin = R c()v 58-sin cos

where R c() is the coordinate rotation operator. If the vector is rotated in a

fixed coordinate system, we have so that

v = R v()v,where

R v() = cos -sin .sin cos

6.2 Lies differential equation

The main features of Lies Theory of Continuous Transformation

Groups can best be introduced by discussing the properties of the rotation

operator R v() when the angle of rotation is an infinitesimal. In general,

R v() transforms a point P[x, y] in the plane into a new point P[x, y]:

P = R v()P. Let the angle of rotation be sufficiently small for us to put

cos() 1 and sin() , in which case, we have

R v() = 1 1

and

x = x.1 - y = x - y

y = x + y.1 = x + y


59/164

Let the corresponding changes x x and y y be written 59

x = x + x and y = y +y

so that

x = -y and y = x.

We note that

R v() = 1 0 + 0 -1 0 1 1 0

= I + iwhere

i = 0 -1 = Rv(/2).

1 0

Lie introduced another important way to interpret the operator

i = R v(/2), that involves the derivative ofR v() evaluated at the identity

value of the parameter, = 0:

dRv

()/d

=

sin cos

=

0 -1

= i

=0 cos sin 1 0 = 0

so that

R v() = I + dR v()/d ., = 0

a quantity that differs from the identity I by a term that involves the

infinitesimal, : this is an infinitesimal transformation.

Lie was concerned with Differential Equations and not Geometry. He

was therefore motivated to discover the key equation


60/164

dR v()/d = 0 -1cos -sin 601 0 sin cos

= iR v() .This isLies differential equation .

Integrating between = 0 and = , we obtain

R v()

dR v()/R v() = id I 0

so that

ln(R v()/I) = i,or

R v() = Iei

, the solution of Lies equation.

Previously, we obtained

R v() = Icos + isin.

We have, therefore

Iei

= Icos + isin .This is an independent proof of the famous Cotes-Euler equation.

We introduce an operator of the form

O = g(x, y, /x, /y),

and ask the question: does

x = Of(x, y; ) ?

Lie answered the question in the affirmative; he found

x = O(x) = (x/y y/x)x = y

and


61/164

y = O(y) = (x/y y/x)y = x . 61

Putting x = x1 and y = x2, we obtain

xi = Xxi , i = 1, 2

where

X = O = (x1/x2 x2/x1), the generator of rotations in the plane.

6.3 Exponentiation of infinitesimal rotations

We have seen that

R v() = ei,

and therefore

R v() = I + i, for an infinitesimal rotation,

Performing two infinitesimalrotations in succession, we have

R v2() = (I + i)2

= I + 2i to first order,

= R v(2).

ApplyingR v() n-times gives

R vn() = R v(n) = e

in= e

i

= R v() (as n and 0, the

product n).

This result agrees, as it should, with the exact solution of Lies differential

equation.


62/164

A finite rotation can be built up by exponentiation of infinitesimal 62

rotations, each one being close to the identity. In general, this approach has

the advantage that the infinitesimal form of a transformation can often be

found in a straightforward way, whereas the finite form is often intractable.

6.4 Infinitesimal rotations and angular momentum operators

In Classical Mechanics, the angular momentum of a mass m, moving in

the plane about the origin of a cartesian reference frame with a momentum p

is

Lz = rp = rpsinnz

where nz is a unit vector normal to the plane, and is the angle between r

and p.

In component form, we have

Lzcl = xpy ypx, where px and py are the cartesian

components ofp.

The transition between Classical and Quantum Mechanics is made by

replacing

px by i(h/2)/x (a differential operator)

and py by i(h/2)/y (a differential operator),where h

is Plancks constant.

We can therefore write the quantum operator as

LzQ = i(h/2)(x/y y/x) = i(h/2)X

and therefore


63/164

X = iLzQ/(h/2), 63

and

xi = Xxi = (2iLzQ/h)xi, i = 1,2.

Let an arbitrary, continuous, differentiable function f(x, y) be

transformed under the infinitesimal changes

x = x - y

and

y = y + x .

Using Taylors theorem, we can write

f(x, y) = f(x + x, y + y)

= f(x y, y + x)

= f(x, y) + ((f/x)x + ((f/y)y)

= f(x, y) + (y(/x) + x(/y))f(x, y)

= I + 2iLz

/h)f(x, y)

= e2iLz/h

f(x, y)

= R v(2Lz/h) f(x, y).

The invatriance of length under rotations follows at once from this result:

If f(x, y) = x2 + y2 then

f/x = 2x and f/y = 2y, and thereforef(x, y) = f(x, y) + 2xx + 2yy

= f(x, y) - 2x(y) + 2y(x)

= f(x, y) = x2 + y2 = invariant.


64/164

This is the only form that leads to the invariance of length under rotations. 64

6.5 3-dimensional rotations

Consider three successive counterclockwise rotations about the x, y,

and z axes through angles , , and , respectively:

zz y

about xy y

x x, x

z y z y, y about y

x x xz z

y y about z

x x xThe total transformation is

R c(, , ) = R c()R c()R c()

coscos cossinsin + sincos cossincos + sinsin= sincos sinsinsin + coscos sinsincos + sinsin

sin cossin coscos

For infinitesimal rotations, the total rotation matrix is, to 1st-order in the s:

1 R c(, , ) = 1 . 1


65/164

The infinitesimal form can be written as follows: 65

1 0 1 0 1 0 0 R c(, , ) = 1 0 0 1 0 0 1

0 0 1 0 1 0 1

= I + Y3I + Y2I + Y1

where

0 0 0 0 0 -1 0 1 0

Y1 = 0 0 1 , Y2 = 0 0 0 , Y3 = -1 0 0 .0 -1 0 1 0 0 0 0 0

To 1st-order in the s, we have

R c(, , ) = I + Y1 + Y2 + Y3 .

6.6 Algebra of the angular momentum operators

The algebraic properties of the Ys are important. For example, we find

that their commutators are:

0 0 0 0 0 -1 0 0 -1 0 0 0[Y1, Y2] = | 0 0 1|| 0 0 0| | 0 0 0|| 0 0 1|

0 -1 0 1 0 0 1 0 0 0 -1 0

= Y3 ,

[Y1, Y3] = Y2 ,

and

[Y2

, Y3

] = -Y1

.

These relations define the algebra of the Ys. In general, we have

[Yj, Yk] = Yl = jklYl ,


66/164


67/164

x = x - x = y - z 67

y = y - y = -x + z

z = z - z = x - y .

Introducing the classical angular momentum operators: Licl, we find that

these small changes can be written 3

xi = kXkxi k = 1

For example, if i = 1

x1 = x = (z/y - y/z)x

+ (-z/x + x/z)x

+ (y/x - x/y)x = -z + y .

Extending Lies method to three dimensions, the infinitesimal form

of the rotation operator is readily shown to be

3R c(, , ) = I + (R c/i)| i .

i= 1 All is = 0


68/164

7 68

LIES CO NTINUOUS T RANSF ORMAT ION G ROUPS

In the p revio us ch apter , we discu ssed the p roper ties of infinit esima l

rot ations in 2- an d 3-d imensions, and we fo und t hat t hey a re re lated

dir ectly to t he an gular mome ntum opera tors of Qu antum Mech anics .

Imp ortan t al gebra ic pr opert ies of the matr ix re prese ntati ons o f the

ope rator s also wer e int roduc ed. In th is chapter , we shall cons ider the

sub ject in ge neral term s.

Let xi, i = 1 to n be a set o f n v ariables. They may be co nside red t o

be the c oordi nates of a poin t in an n- dimensiona l vec tor s pace, Vn. A set

of equat ions invol ving the x is is ob tained by the t ransformat ions

x i = fi(x1, x2, . ..xn: a1, a2, . ...ar), i = 1 to n

in which the set a1, a2, . ..ar co ntains r-indepe ndent para meter s. T he se t Ta,

of trans forma tions maps x x . We shall write

x = f(x; a) or x = Tax

for the set o f fun ctions.

It is assumed that the funct ions fi ar e differen tiabl e wit h res pect to

the xs and t he a s to any r equir ed or der. Thes e fun ctions nec essarily

dep end o n the esse ntial para meter s, a. Thi s mea ns th at no two

tra nsfor matio ns with di fferent nu mbers of p arame ters are t he sa me. r is

the smallest numbe r req uired to c harac teriz e the tran sform ation,

com pletely.

The set of fu nctio ns fi fo rms a fini te co ntinu ous g roup if:


69/164

1. The result of two s uccessive trans forma tions x x x is equivalent 69

to a single t ransf ormat ion x x :

x = f( x; b) = f(f( x; a); b)

= f(x ; c)

= f(x ; (a; b))

whe re c is th e set of p arame ters

c = (a ; b) , = 1 to r,

and

2. To e very trans forma tion there corr espon ds a uniqu e inv erse that

belongs to th e set :

a such that x = f (x; a) = f(x; a)

We have

TaTa-1 = Ta

-1Ta = I, the i dentity.

We shall see that 1) is a hi ghly restr ictiv e req uirem ent.

The tran sform ation x = f(x; a0) is the iden tity. Without loss of

gen erali ty, w e can take a0 = 0. T he es sential po int o f Lie s th eory of

con tinuo us tr ansformati on gr oups is to cons ider that part of th e gro up th at

is close to t he id entit y, an d not to c onsider th e gro up as a wh ole.

Suc cessi ve in finit esima l cha nges can b e use d to build up t he finite chang e.

7.1 One- param eter group s

Con sider the trans forma tion x x unde r a f inite chan ge in a si ngle

par amete r a, and t hen a chan ge x + dx . T here are t wo pa ths f rom x

x + dx ; the y are as s hown:


70/164

x 70an infi nitesimal

aa , a f inite para meter chan ge

x + dx a + daa di fferential

x (a = 0)

We have

x + d x = f(x; a + d a)

= f (f(x; a); a) = f(x ; a)

The 1st- order Taylor ex pansi on is

dx = f (x; a)/a a u( x) a a = 0

The lie group cond ition s the n dem and

a + da = (a; a).

But

(a; 0) = a, (b = 0)

the refor e

a + da = a + (a; b)/b a b = 0

so thatd a = (a; b)/b a

b = 0

or

a = A(a)da.

The refor e

dx = u (x)A (a)da ,

leading to


71/164


72/164

and

x2 = 2x2a. 72

In the l imit, thes e equ ations giv e

dx1/x1 = dx2/2x2 = da.

The se ar e the diff erent ial e quations t hat c orres pond to th e infinite simal

equ ations abo ve.

Int egrat ing, we ha ve

x1 a x2 a

dx 1/x1 =

da and

dx 2/2x2 = da ,

x1 0 x2 0

so that

lnx1 - ln x1 = a = ln (x1/x1)

and

ln (x2/x2) = 2a = 2ln(x1/x1)

or

U = (x2/x12) = U = ( x2/x1

2) .

Put ting V = lnx1, w e obt ain

V = V + a and U = U, th e tra nslat ion g roup.

7.2 Det ermin ation of t he fi nite equat ions from the i nfini tesim al

for ms

Let the finit e equ ations of a one -para meter grou p G(1) be

x1 = (x1, x2 ; a)

and

x2 = (x1, x2 ; a),


73/164

and let the i dentity co rresp ond t o a = 0. 73

We consi der t he tr ansformati on of f(x1, x2) t o f(x1, x2). We expan d

f(x1, x2) in a Macla urin series in the p arame ter a (at defin ite v alues of x 1

and x2):

f(x1, x2) = f (0) + f (0)a + f (0) a2 /2! + ...

whe re

f(0) = f(x1, x2)| a=0 = f(x1, x2),

and

f(0) = (df(x1, x2)/ da| a=0

= {(f/x1)( dx1/d a) + (f /x2)( dx2/d a)}| a= 0

= {(f/x1)u(x1, x2) + (f/x2)v(x1, x2)}|a=0

the refor e

f (0) = {( u(/x1) + v(/x2))f }|a=0

= Xf(x1, x2).Con tinui ng in this way, we h ave

f (0) = {d2f(x1, x2)/ da2}|a=0 = X

2f(x1, x2), etc....

The func tion f(x1, x2) can b e exp anded in t he se ries

f(x1, x2) = f (0) + af (0) + ( a2/2! )f( 0) + ...

= f(x1, x2) + aXf + (a2/2! ) X2f + .. .

Xn

f is the symb ol for ope ratin g n-t imes in su ccess ion o f f w ith X.

The fini te eq uations of the group are there fore

x1 = x1 + aXx1 + (a2/2! ) X2x1 + ...

andx2 = x2 + aXx2 + (a

2/2! ) X2x2 + = ...


74/164

If x1 a nd x2 ar e def inite valu es to whic h x1an d x2 r educe for the i dentity 74

a=0 , the n the se eq uations ar e the seri es so lutio ns of the diffe renti al

equ ations

dx1/u (x1, x2) = d x2/v (x1, x2) = d a.

The grou p is referred t o as the g roupXf.

For exam ple, let

Xf = (x1/x1 + x2/x2)f

the n

x1 = x1 + aXx1 + (a2/2! ) X2f . ..

= x1 + a(x1/x1 + x2/x2)x1 + ...

= x1 + ax1 + (a2/2! )(x 1/x1 + x2/x2)x1 +

= x1 + ax1 + (a2/2! )x 1 + ...

= x1(1 + a + a2 /2! + ...)

= x1ea

.

Also, we find

x2 = x2ea.

Put ting b = ea, w e hav e

x1 = bx 1, a nd x2 = bx2.

The fini te gr oup i s the grou p of magni ficat ions.

IfX = (x/y - y/x ) we find, for examp le, that t he finite group is t he

gro up of 2-dimensional rotat ions.


75/164

7.3 Inv arian t fun ction s of a gro up 75

Let

Xf = (u/x1 + v/x2)f defin e a o ne-paramet er

gro up, a nd le t a=0 give the identity. A fu nction F(x1, x2) is ter med a n

inv arian tun der t he tr ansformati on gr oup G (1) if

F( x1, x2) = F (x1, x2)

for all values of the p arame ter, a.

The func tion F(x1, x2) can b e exp anded as a seri es in a:

F(x1, x2) = F(x 1, x2) + aXF + (a2

/2! ) X(XF) + . ..

If

F(x1, x2) = F (x1, x2) = in variant fo r all valu es of a,

it is necessa ry fo r

XF = 0,

and this mean s tha t

{u( x1, x2)/x1 + v(x 1, x2)/x2}F = 0 .

Con seque ntly,

F(x1, x2) = co nstan t

is a solution of

dx1/u( x 1, x2) = dx2/v( x 1, x2) .

This equ ation has one s oluti on th at de pends on o ne ar bitra ry co nstan t, an d

the refor e G(1) ha s onl y one basic inv arian t, an d all othe r pos sible inva riant s

can be g iven in te rms o f the basic inv ariant.

For exam ple, we no w rec onsider th e the inva riant s of rotat ions:


76/164

The infi nitesimal trans forma tions are given by 76

Xf = (x 1/x2 - x2/x1),

and the diffe renti al eq uation tha t giv es th e inv ariant fun ction F of the

gro up is obta ined by so lving the chara cteri stic diffe renti al eq uations

dx1/x2 = d, a nd dx 2/x1 = -d,

so that

dx1/x2 + dx2/x1 = 0.

The solu tion of th is equation is

x12

+ x22

= cons tant,

and ther efore the invar iant funct ion is

F (x1, x2) = x12 + x2

2.

All func tions of x 12 + x2

2 ar e the refor e inv ariants of the 2-dimensio nal

rot ation grou p.

This met hod c an be gene raliz ed. A gro up G (1) in n-va riabl es de fined

by the e quation

xi = (x1, x2, x3, . ..xn; a ), i = 1 to n ,

is equivalent to a uniq ue in finit esima l tra nsfor matio n

Xf = u1(x1, x2, x3, . ..xn)f /x1 + ... un(x1, x2, x3, . ..xn)f /xn .

If a is the g roup param eter then the i nfini tesimal tr ansformati on is

x i = xi + u i(x1, x2, . ..xn)a (i = 1 t o n),

the n, if E(x1, x2, . ..xn) is a f unction th at ca n be diffe renti ated n-times with

res pect to it s arg ument s, we have


77/164

E (x1, x2, ...xn) = E (x1, x2, . ..xn) + aXE + (a2/2! ) X2E + . 77

Let (x1, x2, . ..xn) b e the coor dinates of a po int in n-s pace and l et a be a

par amete r, in depen dent of th e xis. As a var ies, the p oint (x1, x2, . ..xn) w ill

des cribe a tr ajectory, start ing from t he in itial poin t (x1, x2, . ..xn). A

nec essary and sufficien t con ditio n tha t F(x 1, x2, . ..xn) b e an invar iant

fun ction is t hat XF = 0. A cur ve F = 0 i s a t rajectory and t heref ore a n

inv ariant cur ve if

XF(x1, x2, x3, . ..xn) = 0.


78/164

8 78

PROPERTIES OF n-VARIABLE, r-PARAMETER LIE GROUPS

The change of an n-variable function F(x) produced by the

infinitesimal transformations associated with r-essential parameters is:n

dF = (F/xi)dxii = 1

wherer

dxi = ui(x)a , the Lie form. = 1

The parameters are independent of the xis therefore we can write r n

dF = a{ ui(x)(/xi)F} =1 i = 1 r

= aX F = 1

where the infinitesimal generators of the group are n

X ui(x)(/xi) , = 1 to r. i = 1

The operator

r I + Xa = 1

differs infinitesimally from the identity.

The generators X have algebraic properties of basic importance in the

Theory of Lie Groups. The Xs are differential operators. The problem is

therefore one of obtaining the algebraic structure of differential operators.

This problem has its origin in the work of Poisson (1807); he

introduced the following ideas:

The two expressions

X1f = (u11/x1 + u12/x2)fand


79/164

X2f = (u21/x1 + u22/x2)f 79

where the coefficients ui are functions of the variables x1, x2, and f(x1, x2)

is an arbitrary differentiable function of the two variables, are termed

linear differential operators.

The product in the order X2 followed by X1 is defined as

X1X2f = (u11/x1 + u12/x2)(u21f/x1 + u22f/x2)

The product in the reverse order is defined as

X2X1f = (u21/x1 + u22/x2)(u11f/x1 + u12f/x2).

The difference is

X1X2f - X2X1f = X1u21f/x1 + X1u22f/x2

- X2u11f/x1 - X2u12f/x2.

= (X1u21 - X2u11)f/x1 + (X1u22 - X2u12)f/x2

[X1, X2]f.

This quantity is called the Poisson operatoror the commutatorof the

operators X1f and X2f.

The method can be generalized to include = 1 to r essential parameters

and i = 1 to n variables. The ath-linear operator is then

Xa = uiaf/xi n

= uiaf/xi , ( a sum over repeated indices). i = 1

Lies differential equations have the formxi/a = uik(x)Ak(a) , i = 1 to n, = 1 to r.

Lie showed that

(ck/a)uik = 0


80/164

in which 80

ujui/xj - ujui/xj = ck(a)uik(x),

so that the cks are constants. Furthermore, the commutators can be

written

[X, X] = ( ckujk)/xj

= ckXk.

The commutators are linear combinations of the Xks. (Recall the earlier

discussion of the angular momentum operators and their commutators).

The cks are called the structure constants of the group. They have the

properties

ck = -ck ,

cc + cc + cc = 0.

Lie made the remarkable discovery that, given these structure constants,

the functions that satisfy

xi/a = uikAk(a) can be found.

(Proofs of all the above important statements, together with proofs of

Lies three fundamental theorems, are given in Eisenharts

standard workContinuous Groups of Transformations, Dover Publications,

1961).

8.1 The rank of a groupLet A be an operator that is a linear combination of the generators

of a group, Xi:

A = iXi (sum over i),


81/164

and let 81 X = xjXj .

The rank of the group is defined as the minimum number of commuting,

linearly independent operators of the form A.

We therefore require all solutions of

[A, X] = 0.For example, consider the orthogonal group, O+(3); here

A = iXi i = 1 to 3,and

X = xjXj j = 1 to 3

so that[A, X] = ixj[Xi, Xj] i, j = 1 to 3

= ixjijkXk .

The elements of the sets of generators are linearly independent, therefore

ixjijk = 0 (sum over i, j,, k = 1, 2, 3)

This equation represents the equations

-2 1 0 x1 03 0 -2x2 = 0 .0 -3 2 x3 0

The determinant of is zero, therefore a non-trivial solution of the xjs

exists. The solution is given by

xj = j (j = 1, 2, 3)

so that

A = X .

O+(3) is a group of rank one.

8.2 The Casimir operator of O+(3)


82/164

The generators of the rotation group O+(3) are the operators. Yks, 82

discussed previously. They are directly related to the angular momentum

operators, Jk:

Jk = -i(h/2)Yk(k = 1, 2, 3).

The matrix representations of the Yks are

0 0 0 0 0 -1 0 1 0

Y1 = 0 0 1, Y2 = 0 0 0 , Y3 = -1 0 0 .0 -1 0 1 0 0 0 0 0

The square of the total angular momentum, J is

3J2 = Ji2

1

= (h/2)2 (Y12 + Y2

2 + Y32)

= (h/2)2(-2I).

Schurs lemma states that an operator that is a constant multiple ofI

commutes with all matrix irreps of a group, so that

[Jk, J2] = 0 , k = 1,2 ,3.

The operator J2 with this property is called the Casimir operator of the

group O+(3).

In general, the set of operators {Ci} in which the elements commute

with the elements of the set of irreps of a given group, forms the set of

Casimir operators of the group. All Casimir operators are constant multiples

of the unit matrix:

Ci = aiI.

The constants ai are characteristic of a particular representation of a group.


83/164

9 83

MAT RIX R EPRES ENTATIONS OF GR OUPS

Mat rix r epres entat ions of li near opera tors are i mport ant i n Lin ear

Algebra; we s hall see t hat t hey a re eq ually impo rtant in G roup Theor y.

If a gro up of m m matri ces

Dn(m) = {D1

(m)(g1),. ..Dk(m)(gk), ...Dn

(m)(gn)}

can be f ound in wh ich e ach e lement is associated with the corre spond ing

element gkof a gr oup o f ord er n

Gn = {g1,...gk,....gn},

and the matri ces o bey

Dj(m)(gj)Di

(m)(gi) = Dji(m)(gjgi),

and

D1(m)(g1) = I, t he id entit y,

the n the matr ices Dk(m)(gk) a re sa id to form an m -dime nsional

rep resen tatio n of Gn. If th e ass ociation is one -to-one we have an

isomorph ism and th e rep resen tatio n is said to be fai thful .

The subj ect o f Gro up Re prese ntations f orms a ver y lar ge bra nch o f

Gro up Th eory. The re ar e man y sta ndard work s on this topic (see the

bib liography), eac h one cont ainin g num erous definitio ns, lemmas and

the orems . He re, a rath er br ief accoun t is given of s ome o f the more

imp ortan t res ults. The read er sh ould delve into the deepe r asp ects of th e

sub ject as th e nee d ari ses. The subje ct wi ll be intr oduce d by consi derin g


84/164

rep resen tatio ns of the rotat ion g roups , and thei r cor respo nding cycl ic 84

gro ups.

9.1 The 3-di mensi onal repre senta tion of ro tatio ns in the plane

The rota tion of a vecto r thr ough an an gle in the plane is

cha racte rized by t he 2 x 2 m atrix

c os -sin R v() = .

s in cos

The grou p of symme try t ransformat ions that leave s an equil atera l

tri angle inva riant unde r rot ations in the p lane is of orde r thr ee, a nd ea ch

element of th e gro up is of d imension t wo

Gn ~ R 3(2) = {R (0), R (2 /3), R (4 /3)}

= 1 0 , -1 /2 -3/2 , -1/2 3 /2 .0 1 3/2 -1/ 2 , -3 /2 -1 /2

{ 123, 312, 231} = C3.

The se ma trice s for m a 2 -dime nsional re prese ntati on of C3 .

A 3 -dime nsional re prese ntati on of C3 ca n be obtained a s follows:

Con sider an e quila teral tria ngle located in the plane and let t he

coo rdina tes o f the thre e ver tices P1[x, y], P2[x , y] , and P3[x , y] be

wri tten as a 3-vector P13 = [P1, P2, P3], in no rmal order . We intr oduce

3 3 matri x ope rator s Di(3) th at ch ange the o rder of th e elements ofP13,

cyc lically. The identity is

P13 = D1(3) P13, w here D1

(3) = diag (1, 1 , 1).


85/164

The rear range ment 85

P13 P23[P3, P1, P2] i s giv en by

P23 = D2(3) P13,

whe re

0 0 1 D2

(3) = 1 0 0 ,0 1 0

and the rearr angem ent

P13P33[P2, P3, P1] i s giv en by

P33 = D3(3) P13

whe re

0 1 0 D3

(3) = 0 0 1 .1 0 0

The set of ma trice s {Di(3) } = {D1

(3) , D2(3) , D3

(3) } i s said to form a 3-

dim ensio nal r epresentat ion of the original 2 -dime nsional re prese ntati on

{R 3(2) }. The elements Di

(3) ha ve th e sam e gro up mu ltipl icati on ta ble a s

tha t ass ociated wi th C3.

9.2 The m-di mensi onal repre senta tion of sy mmetr y

tra nsfor matio ns in d-di mensi ons

Con sider the case in wh ich a grou p of order n

Gn = {g1, g2, . ..gk, . ..gn}

is repre sente d by


86/164

R n(m) = {R 1

(m) , R 2(m) , . ....R n

(m) 86

whe re

R

n

(m)

~ Gn,

and R k(m) is an m m matri x rep resen tatio n of gk. Let P1d be a ve ctor in

d-d imensional spac e, wr itten in n ormal orde r:

P1d = [P1, P2, . ..Pd],

and let

P 1m = [P1d, P2d, . ...Pmd]

be an m- vecto r, wr itten in n ormal orde r, in whic h the comp onent s are each

d-v ector s. I ntrod uce t he m m matri x ope rator Dk(m)(gk) s uch t hat

P 1m = D1(m)(g1)P 1m

P 2m = D2(m)(g2)P 1m

.

.

P km = Dk(m)(gk)P 1m , k = 1 to m , the numb er of

symme try o perat ions,

whe re P km is the kth ( cycli c) pe rmuta tion ofP 1m , and Dk(m)(gk)

is call ed

the m-d imensional repr esent ation of g k.


87/164

Infinite ly ma ny re prese ntati ons o f a g iven repre senta tion can b e 87

fou nd, f or, i fS is a ma trix repre senta tion, and M is any defin ite m atrix

wit h an inver se, w e can form T(x) = MS(x)M-1, x G. Sin ce

T(xy ) = MS(xy )M-1 = MS(x)S(y)M-1 = MS(x)M-1MS(y)M-1

= T(x)T(y),

T is a re prese ntation of G. The n ew re prese ntation si mply invol ves a

cha nge o f var iable in t he co rresp ondin g sub stitutions. Re prese ntati ons

rel ated in th e man ner o fS an d T ar e equ ivale nt, and a re no t reg arded as

dif feren t rep resen tatio ns. All repres entat ions that are e quivalent to S ar e

equ ivale nt to each othe r, an d the y for m an infin ite c lass. Two equi valent

rep resen tatio ns wi ll be writ ten S ~ T.

9.3 Dir ect s ums

IfS is a re prese ntati on of dime nsion s, a nd T is a re prese ntation of

dim ensio n t o f a g roup G, th e mat rix

S(g) 0 P = , (g G)

0 T(g)

of dimension s + t is c alled the dir ect s um of the matri ces S(g) and T(g),

wri tten P = ST. There fore, give n two repr esent ations (th ey ca n be the

sam e), we can obta in a third by a dding them dire ctly. Alt ernat ively , let P

be a rep resen tatio n of dimension s + t ; we suppo se th at, f or al l x G, the

mat rix P(x) is o f the form

A(x) 0

0 B(x)


88/164

whe re A(x) and B(x) are s s and t t matri ces, respe ctively. (The 0s 88

are s t and t s zero matri ces). Def ine t he ma trice s S an d T as foll ows:

S(x) A(x) and T(x) B(x), x G.

Sin ce, b y the grou p pro perty , P(xy ) = P(x)P(y),

A(xy ) 0 A(x) 0 A(y) 0=

0 B(xy ) 0 B(x) 0 B(y)

A(x)A(y) 0= .

0 B(x)B(y)

The refor e, S(xy ) = S(x)S(y) and T(xy ) = T(x)T(y), so that S an d T ar e

rep resen tatio ns. The r epres entat ion P is said to b e dec ompos able, w ith

com ponen ts S an d T. A rep resen tation is indecompos able if it cann ot be

dec ompos ed.

If a com ponen t of a dec ompos able repre senta tion is it self

dec ompos able, we c an co ntinu e in this manne r to decom pose any

rep resen tatio n int o a f inite numb er of inde compo sable comp onent s. ( It

sho uld b e not ed th at th e pro perty of i ndeco mposa blity depe nds o n the fiel d

of the r epres entat ion; the r eal field must somet imes be ex tende d to the

com plex field to c heck for i ndeco mposa bilit y).

A w eaker form of d ecomp osabi lity arises whe n we consi der a

mat rix o f the form

A(x) 0 P(x) =

E(x) B(x)


89/164

whe re A(x), and B(x) are matri ces o f dimensio ns s s and t t 89

res pectively and E(x) is a matr ix th at de pends on x , and 0 is the s t zero

mat rix. The matri x P, a nd an y equ ivale nt fo rm, i s said to be red ucibl e.

An irr educi blerep resen tatio n is one t ha

Documents

Introduction to Groups