Group theory for physicists

I'"

I'

p

",

(3 .143)

II, v mod 5.

Tttke n (1 ) = E to calculate 1>W. Then , taking R (2) in 1, MY 5 11 , which docs not appear ill 1>}}J, one calculates 1:> }?J. III this way, olle calculates
"'"

~~

=


=

<")~!

=

I'}


(E + I,- " To + 1,- 2WrJ + 1,21'Tf} + "I'T;}) / /5,

+ /,- "51 4 +1J- 2"SI2 + 11 21 '8 15 + 1,"5 13 ) 1V5, {(85 + ,,-I' H; + 11 - 21'TjI + ,,21'T4 + '11'11.1) + r/I'- "} (84 + ,,- I' R~ + 1} - 21'T~ + 1,21'T3 + J/I'R3) + 1/2{1' ,,) (5.1 + 1,- 1' R~ + I) 21'T:i + 1}21'T2 + ., /1' R2) + 1]- 2(,.- ,,) (82 + '1 - 1' R~ + J} - 2I'T; + .,}21'TI + 1)1' RJ) + 71 - (1' - ") (51 + IJ - I' R~ + '1 - 2"T24 + 1}2"Tr; + 1}"Rr;)} 15, { (810 + 'I- I'T? + 1} - 21' R~ + "/}21' fig + I}I'T;) + 1,(1' - ") (S!;I + ,,- I'TS1 + 1/_ 2" R?o + r?" Rg + li'r.n + 1/ 2{1'- "1 (S8 + ,,- I'T43 + ,} - 21' R~ + .,}21' R7 + ry'Tj) -t 71 - 2 (1' - ") (S7 I 1, - I'Tll,,- 2"R~ j ,P"R6 + ,"'Tl) + ,,- (1'- ") (St; + I,- "T? + 1,- 21' m+ 1/2" R IO + 71"T~)} 15, (5 11

(3.1 '1'1)

§3.7 1rrt:<J.tu:ible

B(j~e~

in Group Algebra

99

8(:'(:ond, calculate t he matrix of the class operator C ~ in (l>~~ and its cigCIl\'(:'(:lors. Due to Eq. (3. 1'1'1), only the coefficients of the te rms E , 5 11 , 8~, and SIO in C!.
n = T~Jl.J. ,

T3 = T6 RI , T4 = TC1 R-4 , T~= TJS1' = TJSl . TJ = ToJij , TJ = TJR2, Tt = ToRI.

Til = TJR4 ,

T;

S Il = S l2T6 = TOS l2TJ = TJSl2TJ = TJSl2 = TJSl2TO,

RJTo' = ToR'1 = TJR~TJ = TJ R2TJ = TOISI TO' SIO - R6TJ = ToRloTo = TJSsTJ = TJ R~ = 'l!: R1oT£:· Thcll , say, 8 11 :"": TJS12TJ = (TJ Ri)( R 2 S 12 n) = T 1(R iOn) = TI R~, 85 =

E = TjTj' = T}Tj,

TOS13 = TdS14 = T1R~ = Tt R~ = T2T'f = ~TJ T3Tr = TtT.~ = T4RS = T.tR6 = T5SS = rtS6 ,

8 11 S5

0 :5 j :5 5,

= ToR.I = nR~ = TITa = Tt T? = T 2S4 = TtSiO = T 3 S 9 = TiSI = T4Tg = Tt TJ = TsRs = ~Rl.

510 = ToTl = TilT? = TIRs = Ti'R~o = T2 S S = T~Sl 3 T3T? = TtTr = T4S L!. = T 44S l = TsR IO = TtRg. Thus, t he matrix of the class operator C~ in CP~,:! is obtained,

C5 (1)~~l = [(I}" + ,/- ") D + (17 2" + 17- 2,,) S5 + . . .J /.j5

IS (1/2" + '/ 2,,)
= (')"

C5 (l>;~ = =

+ ,/ ")(I) ;,~l +

Cs'ilW = [05"" ('12" + ,,- 2,,) E + (1/" + ,/- 1' + 1/" + ,/- " + '/" "-1-1/" ")85 -I- (I = 6,,,,../5 (1/2"

+ '/

")(1 + I()Sw + .. ·1/5

+ 1,- 2,,) (l» ,~l + ('/" + 1/- " + ,!" + 1/- "

+1/' V+ r( ")(l>},:~ + (J + '/" )( I + ln
C5(JI)~) = [05, ,(_ v) (1,- " -I- 1/- 2,,) 8 11 =

+ ( I + 1/" ) (I + 1]- ") S[, + (')" + ,/ " + I}" + '} " + ,/,d ·" + '/ I' ") 510 + .. .1/5 05, ,,)../5 ( ,,- /' + 1,- 2,,) 'I);~ + (1 + 'I") (1 + 1/- ") <[I~3J I(

+ (,/" + I) /' + 1( + '/ "+ 1/,'1" + 17 " ")
HlO

Chap. S Theory of Representations

It s cigcnvcctor~ an~ the irn,,"{iucible bases 11.,,, of I as listed in Table 2.1 5.

For example, when II = J) = 0, the matrix of ( 5 in ~) and its eigenvectors with the eigenvalues 12 , 4p-l, - 4p, a nd 0, res pectively, a rc

~ ~

(o

2J5 0

't' 2~) 6

"

2,;5 4

6

'

bob, b'{;o, blJ, a nd b{k, res pect ively.

Normalizing' t he eigenvectors, one obtains \\-llen It = 0 a ud

2 (

+ II'"

b'{;~) b;r~, and

""

_ 1/ Z" _

btL,

(I>&~ a ud ~~ is

0 olle has the ll ul.t ric of ( 5 ill

1 + 1}-v

_ l /Z" - '1]-21.'

I

/1

) .

Normalizing the eigeuvectors, olle obtains

1,-2"

res pectively.

T a ble 3.15

Ir r educible b ases

'1."

in the gr oup s p ace of I

1/ = exp( - i2ro /5), «=1

bA 00 -- ( '1>(1) 00

+ <}> DO (2) + v'51>(3) '51,(4)) a 110 + V'-' 00 / V'f2 L: ·

, ,, ", -,, , I'

e

0

-,0,

0 0 0

0 0 0

," _<,_I

'I-~

-TJ- 2 p

,1 _,,_ ' p_l

, -,

-'I

'1- 2

0 0' 0 0

0- '

_,,2

0 0 0

,

-0',

N

,

~

0

-, -, -, -, , 0

0 0

~

-,

.,

_'/p-I

2 4 2

_,,_2

'I

,

-,

_ p_l

4 2 4 2

DC,

"2 0

-2 2 0

-2

e

2 2 2 0 0 0

2 0

- 2

- 2

- 2

-2

,

0,

"0

0,

,"-

0 0 0

0

'1_ 2

_ '1- 1

-"

'Il, - I

0 0 0

- '7-'

"

'1-1 p-l

0 0

0' -,

, -, ,

0

,

,,'

-, ,,_2

'1-

1

1'

N 4 2 4

2 , , _,,_I , , -'I

2

,/2p

-'I _ p_l

2

§S ,7 Irreducible Bases in Group Algebra

0, ",, ,," , 0

0, 0 0 0

,, ,,, , , ,T ,, ,-' T 2

0 0 0

2

, ,, ,

T T T T

T

," ,"

T 2

",

,

0

T

",,

" ,,, , ,, 2

0,

v'

3

0

-,',

0

-'I-'P

_'1 2 p-l

_ '1- 2p-l

- 'I- ' P

,,'

,-' 0 0 0

"0

0

_'1 - '

0

- ,;5'I -~

_'1- 2 p- 2

- ,;',

0 0 0

,

T T

0

T

T

T T

2

",, ," ,T ,," 0

,"

",-'

0 0 0 0 0

,

,"

0 0 0

V5

V5 0 0 0 0

"

V5

- ,;5,f

0 0 0

,;5'1_' 0 0

"0

-'r

,

'II?

V5

3

_'I-~p-l

-,

- 'IP

,;5"

0

T

,-'

'1-'1'-'

0 0 0

-,

-,"

0 0 0

0 0 0 0 0

0 0 0

,

,

3 3

, ,T, ,, 0 0 0 0 0

-'IP

_ '1- 2

- OP

V5

0

,

-'1- ' p

0

,

_,, ~p -l

3 3 3 3 3 3 3

'1 _'

0

0 0 0

_ '1 2 p- 1

- 'I-Ip

_ '1- 2 _ ,,21'- 1 -'1- 2 p-'

0

0

N

- OP

0 0

0 0

,

,

0,

_ ,,-1 p -l

0

2

",

'"

,-' ,,2

l'

"

,,'

-" -,

,-'2

'1 '1- 2 1' _ ,fp- 2

.,

"

- 'I-'p- 1

3 3 3 3 3

N

-,11'

,-'

",

- '11'- '

3

'1 _ ~p _ 2

" ", ", , ",,

-,.,-' p

,-'

- ,;'

-'I-'P> 'IP-' " -,,-~

-, ,,'

'1 _ ' _ '1-'1'-' - ",' 'I - ~

p_2

,', ,

3

3

3

3

", " ", 3

'1- ' p2 _'1 - 2 1'

,fp-2 '1 - 'p - '

" ,,_2

11 - 2 1'

3

"

"

-'11'- '

3

Chap. S Theory of Representations

102

3.8

E xe rcises

1. Let C be a non-Abelian group, D(C) he a faithfnl repre;entatiOIl of the group C , and D(R) be the representation matrix of the element R. Tf the clement R in C corresponds to the following matrix in the set, please decide whether the following set forms Ii representation of the gl"Oup C. For example, in (a), if R <-----> D(R)t , plelL<;e decide whether the set D(G)t composed of D(R)t forms a representation of C. la) DIR)t; Ib) D I R )T; Ie) D IR - ' ); Id) DIR) ' ; Ie) DIR- ' )t; If) det DIR); Ig) T< DIR). 2. The homogeneous function space of degTee 2 spanned by the basis functions l/>J(X,y) = x'l, ·I .h(x,y) = xy, and W3(X,y) = y2 is invariant in the following rotations R in the two-dimensional coordinate space. Calculate the matrix form D( Il) of the cor n~po llding transformatiOlI operator Pn for scalar functions in the three-dimensional function space:

la) Ib)

R ~!. ( 2

- 1

J3

-V3) - 1

'

R ~ (10). 0-1 .

(e) R ~ (c~n -sinn). Sllln cosn

By rcplacing the basis function "l{Jz(x, y) = xy with ..j2xy, how will the matrix form of the operator PR change? 3. P rove that the module of any representatio n matrix ill a one-dimensional representation of a finite group is equal to 1. 4. Prove that any irreducible representation o f an infinite Abelian group is one-dimensional . 5. P rove that the similarity transformation matrix between two equivalent irr<:.-duciblc unitary representations of a finite group, if restricting its determinant to be 1, has to be unitary. 6 . Show that for a fi nit.e group G, the sum of the eharact.ers of all elements in any ilTeducible representation of G, excep t for the identical repres~ntation , is equal to zero. 7. Calculate t he similarity trans fo rmation matrix X between two equivalent regular representations D (R) and D(fl) , given in Eqs. (3.6) and (3.12), in the group space of the D3 group. How to generalize the res ult. to any other finile group?

'00

8. C" is a class in a finite group C .' and en is a vector in the group space composoo of the SlIlJI of all dements in Co.. Prove that the re presentation matrix of Co in an irreducible representat.ion is a constant mat.rix, and calculate this constant. 9. Prove that the number of self-reciprocal classes in a finite group G' is equal to the numbe r of inc<}uivaient. and irn.,
,,) ( y'

~

R

'

(x) y

'

where R is equal to the repr~entl;\tion matrix in the two-dimellsiollal representation DE(03) . For the generators D a nd A in 0 3 , one has

D

~

DEID)

~ ~2

(-'

J3

-v'3) - 1

'

A

~

DEIA)

~

(' 0 ) . 0- 1

The four-d imensioual functional space spanned by the following basis fUllctiolls is illvari(Hlt. in the grou p 03:

Calculate the representation matrices of the generators D and A of O:j in this re presentation . Then, rednce this representation into the

direct sum of the irreducible representations of 0 3 , a nd construct new basis funet.ions which arc the linear combinations of the original basis fundions and bdong to the irr(,.>{lucible representations. 12 , Calculate the characters and the representat.ion matrices of the proper symmetric gTOUp 0 of a cube with thc method of the quoticnt gTOUp and the method of coordinate transformations . 13. The lIlultipli{:at.ioll table for Il grou p G of o rder 12 is IJS follows.

10·1

Chap. S Theory of Representations

E ,\

E

A

E

A

B B

A

H

E

B

A

B C

C

E K

D F I

D F I

N D M

J

J

K L M N

f(

L

L C

M

J

N

F

L

C

D

F

C

D L

F K M

N D

J E

B

F

E

C

N

K

M

J

L

.4 N

L I E D

M

H

C K I D

M A F

F

N

K

J

B

J C A

J J N C AI

D L

J E

M

F L

N J<

A D 8 K

E F A C

B

J<

L

AI

N

J(

L

N

M

J

F N

D F

At F K J

A L C E .1

B D I

C

A

H

I

K

C

A

J(

D C

J B

A I E N M

B N F L

F D M I. E

(a) Find out the inverse of eaeh element in C; (b) Point out which elemellts can commute wi t h any element in the group; (c) List the period and the order of each element;

(d) Find out the elements in each class of C; (e) Fiud out all iuvariant subgroups in C. For each invariant subgroup, list its cosels and point out onto which gTOUp its quo ticnt gTOUp is isomorphic;

(f) Establish the character table of C; (g) Decide whether C is isomorphic onto the symmetric group T of the tetrahedron, or the symmetric group 0 6 of the regular six-sided polygon. 14 . Calculate t.he character table of the group G given ill P i"Ob. 17 of Chap.

2. 15 . Calculate all inequivalent in·educible representations of the group

D2n+1 with t.he method of iuduced represent.ation. 16 . Calculate all inequivalent irreducible representations of the group 02" with the method of induced represe ntation.

17 . Calculate all inequivalcnt irreducible representations of the symmetric gi"OUp 0 of a cube with the method of induced representation. 18. The regula!" icosahedron is showll in Fig. 2.3. Calculate the character table of the proper symmetric group I of nil icoonhedrOIl with the method of induced representation. 19 . Calculate thc reduction of the subduced representation from each irredu(;ible representation of the gronp I with respect to the s ubgrollps C s , Os , and T .

005

20. Calculate the reduction of the subduccd I'cpn..-scntatioll from the regular representatio n of the improper symmetric grOllp h of an icooahedro n with respect to t.he subgroup C s ;, D"d, and T h. 21. Calculate the representation matrice; of Rij find SIZ o f J in its irreducible representat.ions by R6 = SITJS1 To- 1 and 5 12 = ToS t TJR 6 (sec Prob . 4 ill C ha p. 4), where t he representa tion matrices of the generators To and 5 1 are (see Prob. 12 ill C hap. 4):

DTl{ Tol DG(To)

= diag =

h, 1, 1]-I} ,

diag {r? , f/, 1,-1, 1,-2} ,

DT'{ Tol = diag hZ , I , II- 'l }

DH (Tol

,

= dias' {,, 2 ,1} , I , 11- 1" 1-2 } ,

, (1'/2-' /2 - "/2 )

DT,( S.) ~ -~ v5

-I "" p - VL-

P

_,

- p _ p- l

DG(Sd=

~5

(

-1 - I' _ p- l

1

I - I _p-l 2p - l

( 21'--, '

,,'

- I - I'

'-,

- p

- I' - I

)

,;6 - ,;6

_ p -2

,;6

1" ,;6

21'

- 2"

DH (Sd=~ ~

- ,;6

- I

21'

_ p-2

,;6

p'

_ 2p- l

p'

- 2p

J5

_ 2p-l

p-2

5

)

22. P lease use the projection operators to reduce the regular representation of ST OUp T. 23 . P lease use the proj('''CLioll operato rs to reOuce the reg ula!" representation of ST OUp O. 24. The lIlult ipli{;at.ioll table of the group G is givell

I!.S

folio"''!>.

HJ6

Chap. S Theory of R epres entati ons

~

E

A

B

C

F

K

M

N

F A B C F K AI N

"£

B M

C

F N

K

K

AI F

M B C

hi

N F A B C AI E K

left.

E A B C F K AI N

N

E

K

N /,

M C

F B

F A C

~ B

F

[(

E B C A

C P A B E N M

F A N {(

E

(a) Write the character table for the group C. (b) If we know that tlu;l represClltatiOIl matrices of the geIH~f1itors A and B in a two-dimensional irreducible representation Dare D (A ) =

(~ ~l) '

D(B) = (~~) ,

and two set.s of htlSi.<; functiOlls lPl' and .pI' transform according to this two-dimellSiollal representation of G, respectively:

PRJP!! =

L ",

t/!I"D1,,!,(R),

comhine the product fUllction tP1, t/J" such that the hllSi:;; function belongs to the irreducible representation of C. 25. Calculate tIle unitary similarity transformation matrix X for reducing the self-direct product of the three-dimensional irreducible unitary representation DT of the group T:

x -' (DT(R)

x DT ( R ) } X ~

L

" j D'(R ).

j

26. Calculate the Clebsch - Gordan series and the Clebsch- Gordan coefficients for the direct product representation of each pail' of two in"edllcible repre;;cntat.ions of the group O .

27 . Calculate the Clebsch- Gordan series and Clebscb - Gordall coefficients in the reduction of direct product re presentation in terms of the character table of the group I given ill P rob. 18: ( 1) D T ] x D T.; (2 ) DT! x D T,; (3) D T, x D T,; (4) D T, x DC;

(5) DTo x DC;

(6) DT, x DlI ; (7) DTo x Dllj

(9) DC x Dll ;

(10) DH x DlI.

(8) DC x DC;

C hapte r 4

THREE-DIMENSIONAL ROTATION GROUP

A system with the spherical symmetry has a symmetric center, chosen as the origin, and the system is isotropic with respect to the origin. T he system is invariant. under any rotation around any axis ac roos the origin thro ugh any angle. T he symmetric gronp of the system is three-dimensional rotation group 80 (3). The s phe rical sYlllmetry is the llloot important symmet ry studied in physics, because a system with the spherical symmetry is easier to deal with and most systems in physics have this approximative symmetry. The group 80(3) is a Lie group, where lts elements can be characterized by a set of continuous parameters, and it can be dealt with by mathematical analysis. T herefore, t.he study on three-dimensional rotation gronp is very important bot.h in physics and in mathematics .

4.1

Three-dimen sio n a l Rotations

In a real t.hree-dimensional space, a spat.ial rotation keeps both the position of the origin and the distance of any two points invaria nt . T here are two viewpoiuts in de;;cribing a rotation in a titret:l-dimensional sp ace. [n one viewpoint the system rotates in a· laboratory frame , and in another viewpoint the coordinate frame rotates but the system is fixed . III this book we adopt the first viewpoint. Let K be a perpendicular coordinate frame which is laboratory-fixed. All arbitrary poiu!. P is described by a position vector r which is a veetor pointing· from the origin 0 to P. T he coordinate basis veetors in the laboratory frame are denoted bye", a = 1, 2, 3,

,

r =

L

(4.1)

e"x"

a=l

There is a one-to-one {;orrespolldence between the position vector r and

'" I).

pr

~ido

ocr

.08

Chap.

4 Three-dimensional Rotatim. Group

three coordinates Xu when e o arc fixed . In a spatial rotation R the point P transforms to pi wbose position vector is denoted by r ' = L:u e ax~ . Since the position of the origin and the dist.ance of any two points keep invariant in R , R can be denoted by a three-dimensional real orthogonal matrix

E= (;£') T

l

R:E.,

R* = R .

= ,£T,!.,

(4.2) (4 .3)

T he set of all three-dimensional real orthogonal matrices, in the multiplication rule of matrices, satisfies four axioms of a group and forms a group 0(3). Let K' be a body-fixed frame wit h the coordinate basis vectors e~. The components of r' in the frame f{' are invariant in the rota tion R,

r' =

, L

, eax~ = L

«= 1

e~xa.

(4.4)

b= L

Suhstituting EC]. (4 .2) into Efj. (4.4) , one ohtains

, e~ = L

c aR,,!,_

(4.5)

,-.=\

The chirality of Ii coordinate frame is determined by the mixed product of coordinate basis vectors. /( is a right-handerl coordinate frame,

(4 .6) The chirality of /{' depends

Oil

det R

(4.7)

K' is righ t- handed if R is a proper rotation, det R = 1. /{' is left- handed if R is an improper rotat.ion, det. R = -1. The set of all proper rotations forms an invariant subgroup of 0(3), called the three-dimensional rotation group SO(3). The set of all improper rotations forms the only coset of 50(3) in 0 (3). The spatial inversion q is the representl:l.tive element ill the coset. Every improper rotat.ion R' is equal to the product of (1 and a proper rotation R. Study some special rotations . A rotation around the z-axi>; through an angle w is deuoted by R(e3Jw) :

§4.1

Three -dimensional Rotations

.09

x~ = XICOSW-X2sinw,

- sin,,",'

Xt = Xl sinw -I- X"l COSW,

cosw

(4.8)

o

x~ = X~, y

P'

Fig. 4 . I A rotation around the z-axis.

In terms of the Pauli matrices. 0,

~

(1a(1b = Oab 1

, -I- i'L

=

(12

(: :)

(0 -i0 )' OF (:~1) ' i

Tr

(ab<;(1c ,

(1a =

0,

Tr ((1a(1b)

(4.9)

= 26"b.

c=l ~xp

. }= L {- lW0"2 "

= 1

1 "( - UJ"l . )" -w

n!

(1 - ~!,,",,2 +

:!",,4 _

..

+.,.)

(COSW -Sinw) . . SIIlW COSW

= l COS,,",' - ·lO"ZSlIl W =

(4.10) R ( e 3,w) can be expresseO as a n exponential function of mat rix

R.(e3 ,w) = exp{ - iwT3} =

(

COSW

-sinw

-i

Si~w

co~w

o

o

From t.he cyclic of three axes, one has

R (el,W) = exp{ - lwTI } =

(~ ~w o

SlllW

IIII

(:hll7!.

4 Three-dimensional Rotation Group

cosw (

s~nw

-

o

SinW)

10, o cosw

The matrix entries of Ta are

(4

Another useful rotation is S(cp,B), which transforms e3 to the direc n(B, cp), where 8 and cp are the polar angle and the azimuthal angle of

cp cos B - sin cp cos cp sin B) sincpcosB coscp sincpsinB , cosB - sinB 0

COS

S(cp,B)

= R(e3,cp)R(e2,B) = S(cp,B)

(

(~) = (~~~;::~;) = (~~) . cos B

1

(4

n3

It is easy to check (see Prob. 9 of Chap. 1 in [Ma and Gu (2004)])

S(cp,B)T3S(cp,B)-1 = nlT! + n2T2 + n3T3 = n· T, T = e1T1 + e2T2

(4

+ e3T3.

Thus, the rotation R( n, w) around the direction n through an angle w be expressed as an exponential function of matrix ,

R(n,w)

= S(cp , B)R(e3,w)S(cp,B)-1

= exp{-iwST3 S- I

= exp

{-i t

= w sin

Bsin cp,

= exp{ -iwn . T}

waTa} .

}

(4

a= l Wj

=wsinBcoscp,

W2

(4

Note that R(n,w)n = n, TrR(n,w) = 1 + 2cosw, and

R(n,w+27T)

= R(n,w)

= R( -n, 27T-W) ,

R(n,7T) = R( -n, 7T) . (4

On the other hand, any element R in SO(3) is a rotation aroun direction n through an angle w where n is the eigenvector of R to eigenvalue 1 and w is determined by Tr R = 1 + 2 cosw. Thus, the elem R(n, w) E 80(3) can be described by w through its spherical coordin (w, B, cp) or its rectangular coordinates (WI, W2, W3). The variation re of w is a spheroid with radius 7T, where in the sphere two end points diameter describe the same element (rotation). Equation (4.14) shows the elements R(n,w) with the same w form a class in 80(3).

§4·2 Fundamental Concepts of a Lie Group

4.2

4.2.1

111

Fundamental Concept of a Lie Group

The Composition Functions of a Lie Group

A group G is called a continuous group if its element R can be characterized by 9 independent and continuous real parameters rp, 1 :::; P :::; g, varying in a g-dimensional region. The region is called the group space of G and 9 is the order of G. It is required that there is a one-to-one correspondenc between the group element R and the set of parameters r p, at least in the region where the measure is not vanishing. For example, the group space of 80(3) is a spheroid with radius Jr. Each point inside the spher corresponds to one and only one element of 80(3), but in the sphere wher the measure is vanishing, two end points of a diameter correspond to on element. Hereafter, a point in the group space which corresponds to an element R of G will be said as a point R in the group space for convenience The parameters tp of the product T = RS of two elements are th functions of the parameters of the factors Rand S

(4.17

Jp(r; s) are called the composition functions, which describe the multipli cation rule of elements in G completely. A continuous group is called a Lie group if its composition functions are the analytic functions or at leas piecewise continuously differentiable up to the second order. The mathe matical analysis can be used in studying a Lie group so that the theory o Lie groups has been studied thoroughly. The domain of the definition of Jp(r; s) is a square of the group space and the domain of function is the group space. In order to meet four axiom for a group, the composition functions have to satisfy the conditions: Jp(f(r; s); t)

= Jp(r; J(8; t)),

where Tp are the parameters of the inverse R- 1 , and ep are the parameter of the identical element E. e p are usually taken to be zero for convenience As a matter of fact, the composition functions are mainly used in the the oretical analysis, but rarely in the real calculations. Even for a very simpl Lie group, such as 80(3), the composition functions are quite complicated and their forms do not write evidently. Many fundamental concepts of a group are also suitable for a Lie group such as the concepts of an Abelian group, a subgroup, a coset, the conju gate elements, a class, an invariant subgroup, the quotient group, the iso

112

Chap. 4 Three-dimensional Rotation Group

morphism, the homomorphism, a linear representation, the character, equivalent representations, an irreducible representation, a self-conju representation, and so on. The matrix entries and the character of a re sentation of a Lie group are the single-valued analytic functions of the g parameters in the group space, at least in the region where the measu not vanishing. 4.2.2

The Local Property of a Lie Group

The group elements in a Lie group G are said to be adjacent if their rameters differ only slightly from one another . An element A(a) is sai be infinitesimal if it is adjacent to the identical element E . The parame e p of E are chosen to be zero. Thus, the paTameters a p of an infinites element A(a) are infinitesimal. Evidently, RA and AR are both adja to the element R, and the product of R- 1 and an element adjacent to an infinitesimal element. To speak roughly, the point R moves continuo in the group space if R is multiplied with infinite number of infinites elements continually. Conversely, if the points Rand E can be conne by a continuous curve embedded completely in the group space, R is product of infinite number of infinitesimal elements. The infinitesimal ments are related to the differential calculus of the group elements so they describe the local property of a Lie group. The precise version of statement will be discussed later. The product of two infinitesimal elements A(a) and B(j3) is an infin imal element. The parameters of AB can be expanded as a Taylor s with respect to the infinitesimal parameters a v and j3v

(4 where e p = 0, AE = A, EB = B , and the infinitesimal quantities of second order have been omitted. Therefore, the product of two infinites elements are commutable, and the parameters of the product are the su those of two elements. It does not m ean that the group is Abelian. Deno by Ci p the parameters of the inverse A-J of an infinitesimal element A one has

(4

§4.2 Fundamental Concepts oj a Lie Group

1.2.3

11

Generators and Differential Operators

There are infinite number of infinitesimal elements in a Lie group C. How t study their property in the transformation group Pc and in a representatio D(G)? PR is the transformation operator corresponding to the element R o G, PR'Ij!(x) = 'Ij!(R-1x), where x denotes the coordinates of all degrees o freedom in the system. Let R be an infinitesimal element A(a),

(4.21

The differential operators 1~O), whose number is g, characterize the actio of every infinitesimal element A(a) on a scalar function 'Ij!(x). For the group SO(3), its element R is a rotation in a real three dimensional space. If the system is a mass point and x denotes thre coordinates of the mass point, one obtains from Eq. (4.11), (Ax)"

=

L

{Jab -

i

b

L

ad(Td)ab} Xb

= Xa

-

L

d

adCdabXb·

(4.22

bd

Substituting Eq. (4.22) into Eq. (4.21), one has 1d(0)

__ . ' " ' 2L CdbaXb ab

8a = L d,

(4.23

Xa

where the natural units are used, Ii = c = 1. The differential operators 1~O of SO(3) are nothing but the orbital angular momentum operators. For a n-body system, 1~O) is the total orbital angular momentum operators (se Eq. (4.220)). If an m-dimensional functional space £ spanned by the basis function 1/JJ1.(x) is invariant with respect to the transformation group Pc and corre sponds to a representation D( G) of a Lie group G, m

PR 1/JI'(x) =

L v=l

1/Jv(x)DvJ1.(R).

(4.24

Chap. " Three-dimensional Rotation Group

Expand the representation matrix D(A) of an infinitesimal element, 9

D(A)

=1-

i

L

I

o.pIp,

_ . oD(A) p -

00.

Z

p

p=l

I

(4

.

0'=0

Ip is nothing but the matrix forms of the differential operators I~O) in basis functions 'l/JJ.L(x) of L, m

I~O)'l/JJ.L(x)

=

L

(4

'l/Jy(x) (Jp)yJ.L .

y=l

I p , 1 :::; P :::; g, are called the generators in a representation D(G) o which characterize the transformation of the basis functions 'l/JJ.L(x) as as all functions in L under the action of every infinitesimal element A If D(G) is a unitary representation, Ip are Hermitian because we introd a factor -i in front of Ip in Eq. (4.25).

4.2.4

The Adjoint Representation of a Lie Group

Let RSR- 1 = T, where the parameters ofT are the functions of parame of Sand R, (4

For a faithful representation D(G), one calculates the derivative D(R)D(S)D(R)-l = D(T) with respect to the parameter Sj, and th takes iij = 0, D(n) oDeS) D(R) - ll = ""' os J· ~ s=O k D(R)IjD(R)-1

=

L

h D t1(R),

oD(T) otk

Dt1(R)

I [=0

PRI?) p RI

I '

OSj

s=O

= o'I/J~(s;r) I sJ

k

Similarly,

o'I/Jds; r)

=L

IkO) Dt1(R).

.

(4

s=O

(4

k

Equation (4.28) (or Eq. (4.29)) gives a one-to-one or many-to-one co spondence between the group element R and the matrix Dad(R), and correspondence is invariant in the multiplication of group elements. Th D"c\ (n) is a representation of a Lie group, called the adjoint representat The dimension of the adjoint representation is the order 9 of G. Every )!;roup has its adjoint representation.

§4.2 Fundamental Concepts of a Lie Group

4.2.5

115

The Global Property of a Lie Group

[n fact, the global property of a Lie group is the topological property of the group space of the Lie group. We will explain the global property of a Lie group with the language familiar to physicists. First is the continuity of the group space. A Lie group is called mixed if its group space falls into several disjoint pieces. A mixed Lie group contains an invariant Lie subgroup G whose group space is a connected piece in which the identical element lies. G is called a connected Lie group The set of elements related to the other connected piece is the coset of G. The property of the mixed Lie group can be characterized completely by G and the representative elements belonging to each coset, respectively. For example, the group space of 0(3) falls into two pieces where det R = 1 and det R = -1, respectively. An invariant subgroup 80(3) of 0(3) has a connected group space with det R = 1. A representative element in the coset with det R = -1 is usually taken to be the spatial inversion a. The 80(3) group and a completely characterize the group 0(3) because any improper rotation in 0(3) is a product of a and a proper rotat.ion belonging to 80(3). Hereafter, we will mainly study the connected Lie groups . 8econd is the degree of continuity of t.he group space. The group space of a connected Lie group G can be simply or multiply connected . The Lie group G is called a multiply connected with the degree of continuity n if the connected curves of any two points in the group space are separated into n classes where any two curves in each class can be changed continuously from one to another in the group spa.ce, but two curves in different cla.sses cannot, where "continuous change of two curves in the group space" means that there exist a set of curves ](t) that depends on a continuous pa.ramete t, 0 ~ t ~ 1, such that each ](t) embeds completely in the group space, and ](0) and ](1) coincide with the two curves, respectively. The Lie group with the degree of continuity n = 1 is called simply-connected. As proved in mathematics, for a connected Lie group G with the degree of continuity n, there exists a simply-connected Lie group G' homomorphic onto G with an n-to-one correspondence . G' is called the covering group of G . Consider the group 80(2) composed of all rotations R( e3, if!) around the z-axis. Every rotation is characterized by its rotation angle if!. if! varies in a unit circumference, which is the group space of 80(2). Any two points in the circumference can be connected by a curve in the circumference through several loops around the circumference. The loops contained in the curve is enumerated by an integer n: + 1 for each clockwise loop and -1 for each

116

Chap. 4 Three-dimensional Rotation Group

counter-clockwise loop. The curves connecting two points are sepa into classes signed by n. The degree of continuity of SO(2) is infinite SO(2) is an Abelian Lie group, whose representation has to satisf condition D(rp + 2'll') = D(rp). There are infinite number of inequiv representations denoted by an integer m,

e- i2mrr = l.

The covering group of SO(2) is composed of all real numbers wher multiplication rule of elements is defined with the addition of digits. the elements with the parameters rp + 2'll' and rp in the covering grou different, its representation is e-ir
,..

ll7

§4·3 The Covering Croup of 50(3)

variation region is open because the velocity can tend to the velocity c o light, but cannot be equal to c. In order to generalize the formulas for the finite group given in Chap. 3 to a Lie group, the key is to define the average of a group function over the group elements. For a Lie group the average becomes an integral over the group parameters, instead of a sum over the group elements. As proved in mathematics, the integral can be defined only for a compact Lie group.

4.3

4.3.1

The Covering Group of SO(3)

The Group SU(2)

The set of all two-dimensional unimodular (det u = 1) unitary matrices u in the multiplication rule of matrices , forms a Lie group, called SU(2). An arbitrary element u in SU (2)

u

=

(~

!)

E SU(2)

satisfies aa' + ec' = bb* + dd* = ad - be = 1 and abo + cd* = O. The solution is a = d', b = -c', and Icl 2 + Idl 2 = 1 (see Prob. 7 of Chap 1 in [Ma and Gu (2004)]). Letting d = cos(w/2) + isin(w/2) cos() and c = sin(w /2) sin () (sin