8. The Group SU(2) and more about SO(3)

• SU(2) = Group of 22 unitary matrices with unit determinant.

• Simplest non-Abelian Lie group.

• Locally equivalent to SO(3); share same Lie algebra.

• Compact & simply connected All IRs are single-valued.

• Is universal covering group of SO(3).

Ref: Y.Choquet, et al, "Analysis, manifolds & physics"

• ( Y, f ) is a universal covering space for X if it is a covering space & Y is simply connected.

• A covering space for X is a pair ( Y, f ) where Y is connected & locally connected space & f : Y X is a homeomorphism ( bi-continuous bijection ) if restricted to each connected component of f –1(N(x)) neighborhood N(x) of every point xX.

• X is simply connected if every covering space (Y,f) is isomorphic to (X,Id)

8.1 The Relationship between SO(3) and SU(2) 8.2 Invariant Integration 8.3 Orthonormality and Completeness Relations of 8.4 Projection Operators and Their Physical Applications 8.5 Differential Equations Satisfied by the D j – Functions 8.6 Group Theoretical Interpretation of Spherical Harmonics 8.7 Multipole Radiation of the Electromagnetic Field

U(n):

Number of real components = 2 n2

Number of real constraints = n + 2 (n2–n)/2 = n2

Dimension = n2

Dimension of SU(n) = n2 –1

8.1. The Relationship between SO(3) and SU(2)

Proved in §7.3:

2 2 2 2

1/ 2

2 2 2 2

cos sin2 2, , ~ , ,

sin cos2 2

i i i i

i i i i

e e e eR D

e e e e

Converse is also true.

Proof ( of Theorem 8.1):

a bA

c d

Unitarity condition:

† * *

* *

a b a cA A

c d b d

* * * *

* * * *

a a b b a c b d

c a d b c c d d

Let

1 0

0 1

i.e.2 2

1a b 2 21c d * * 0a c b d

Ansatz:

cosaia e sinbib e sincic e cosdid e

cos sin sin cosa c b di ie e

0 ,2

0 , , , 2a b c d

cos sin sin cosa c b di ie e must hold ,

a c b di ie e cos sin sin cos 0 sin

2a c b d n m n, m = integers

0 ,2

( m = 0 only )

There's no loss of generality in setting n = 0.

a b

c d

a c b d 2a d b c or

Ansatz:

Theroem 8.1: U(2) matrices: 4-parameters

cos sin

sin cos

i ii

i i

e eU e

e e

0 , 2 0

02

Corollary: SU(2) matrices: 3-parameters

cos sin, ,

sin cos

i i

i i

e eU

e e

det 1U

1 1, ,

2 2 2

2 2 2 2

1/ 2

2 2 2 2

cos sin2 2, ,

sin cos2 2

i i i i

i i i i

e e e eD

e e e e

, 02

0 2 , 2

SU(2) matrices form a double-valued rep of SO(3)

2 , ,

However, this range of & covers twice the area covered by & .

One compromise, chosen by Tung, is to set 0 < < 2.

Cartesian parametrization of SU(2) matrices:

0 3 2 1

2 1 0 3

r i r r i rA

r i r r i r

with 2 2 2 20 1 2 3det 1A r r r r jr R

Group manifold = 4–D spherical surface of radius 1.

Compact & simply-connected.

Let 1X A X A 2A SU

Since X is hermitian & traceless, so is X'. iiX x

2 2 x x Rx x 3R SO

i.e.,

det detX X

Mapping 2 3SU SO A Rwith is 2-to-1 ( A to same R )

Let 1 2 3, ,x x xx &i

iX x

where i are the Pauli matrices

1

0 1

1 0

2

0

0

i

i

3

1 0

0 1

3 1 2

1 2 3det

x x i xX

x i x x

2 x

SU(2) SO(3)

Let ( r1, r2, r3 ) be the independent parameters in the Cartesian parametrization.

2 2 20 1 2 31r r r r &

0,0,0E A

1 2 3, ,A A r r ri.e.

Near E, we have kkr dr k = 1,2,3 20 1r O dr

3 2 1

2 1 3

1

1

i dr dr i drA

dr i dr i dr

k

kE i dr

i.e., { k } is a basis of the Lie algebra su(2).

Since , 2 k l mk l mi

su(2) & so(3) are the same if we set 1

~2 k kJ

Since SU(2) is simply-connected, all IRs of su(2) are also single-valued IRs of SU(2)

Higher dim rep's can be generated using tensor techniques of Chap 5:

• IRs are generated by irred tensors belonging to symm classes of Sn.

• Totally symmetric tensors of rank n form an (n+1)-D space for the j = n/2 IR of SU(2) [ See Example 2, §5.5 ]

• Explicit construction of mj

md

Let 1/ 2

c sr d

s c

where cos

2c

sin

2s

Spinor: 2; ,i i V

ii i j

jr Under rotation: i.e.

c s

s c

1 2

2; ,ni i ii nli V

Totally symmetric tensor in tensor space V2n:

k n ki 0 k n ( n+1 possible values )

j m j m

/ 2j n / 2m k n , 1, , 1,j j j j

n+1 independent 's in (convenient) normalized form:

! !

j m j m

m

j m j m

2

nj , 1, , 1,m j j j j

{ [m] } transforms as the canonical components of the j = n/2 IR of su(2):

mm r m mj

md

c.f. Problem 8.5

c s s c

2 2 22

0

! ! ! !cos sin

! ! ! ! 2 2

j m m k k m mjm kj

mk

j m j m j m j md

k j m k k m m j m k

Correctness of Eq(8.1-25)

Derivation: see Hamermesh, p.353-4

8.2. Invariant Integration

1A Ad f A d f B A BAd f A

, ,Ad d d d

A BAd d

, ,

, , , ,, ,

' , ,BA Ad d d d d

Specific method for SU(2) to find :Let A, A', & B be prarametrized by

0 1 2 3, , , , ,r r r r

, , &i i ir r s resp.

32

0

1ii

r

e.g.,

0 3 2 1

2 0 3

r i r r i rA

r i r r i r

with3

2

0

det 1ii

A r

3 3

2 2

0 0i i

i i

r r

{ ri } { r'i } is orthogonal 0 1 2 3

0 1 2 3

, , ,1

, , ,

r r r r

r r r r

0 1 2 3 0 1 2 3, , , , , ,r r r r r r r r 1set

Also holds for different parametrizations of same group element

( r r' is linear )

32

0 1 2 30

1A ii

d r dr dr dr dr

1 2 332

1

1

1A

ii

d dr dr dr

r

1

'j

j j

f x x xf x

3

20 0 0 0

0 0 0

1 11

2 2ii

r r r r rr r

32

01

1 ii

r r

0jf x Since where

Integrate over r0

where

Switching to { , , } parametrization

0 3 2 1

2 1 0 3

cos sin

sin cos

i i

i i

r i r r i re er

r i r r i re e

0 cos cosr r 3 cos sinr r 1 sin sinr r 2 sin cosr r

32

0i

i

r r

0 1 2 3

cos cos cos sin sin cos sin sin

sin cos sin sin cos cos cos sin, , ,

cos sin cos cos 0 0, ,

0 0 sin sin sin cos

r r r rr r r r

r rr

r r

3 cos sinr 31sin 2

2r

2 311 sin 2

2Ad r r dr d d d

3 2 2 21 11 1 1

2 2r r r r r r

1sin 22Ad d d d Integrate over r

1 1, ,

2 2 2

1sin 2

2Ad d d d 1sin8

d d d

1 10

2 2, , 1

0 0, , 2

1 10

2 2

1

4

Switching to { , , } parametrization

Theorem 8.2: Invariant Integration Measure

Let A() be a parametrization of a compact Lie group G & define

1

ii

AA J A

A G

A by

Then A A ii

d d with detA iA

where { J } are the generators of the Lie algebra g.

Proof: Let A() be another parametrization.

Consider any point under different parametrizations. We have

1 1 j

i j i

A AA A

j

ji

J A

i

J A

detA iA

det A

det detA

1

1

, ,

, ,

n

A

n

( as required )

A A

Let { i } be the local coordinates at A.

For a fixed element B, the coordinates at BA is BA B A

1 1 1

i i

ABA BA A B B

1

i

AA

i.e., BA A BA Ad d

QED

In case another parametrization { i } is used at BA, we have

detBA BA

1

1

, ,

, ,

n

A

n

i

J BA

i

J A

Another choice of generators { J' } can always be expressed as a linear combination of the old generators { J } , i.e.,

J J S where S is independent of coordinates.

det S constant

Example: SU(2) with Euler angle parametrization ( , , )

2 2

2 2

cos sin2 2

sin cos2 2

i i

i i

e eA

e e

3 32, , i J i Ji JA e e e 3 321 , , i J i Ji JA e e e , ,A

2 2

1

2 2

cos sin2 2

sin cos2 2

i i

i i

e eA

e e

2 2

2 2

cos sin2 2

2sin cos2 2

i i

i i

e eA i

e e

11 2 3sin cos sin sin cos

AA

Example: SU(2) with Euler angle parametrization ( , , )

2 2

2 2

cos sin2 2

sin cos2 2

i i

i i

e eA

e e

3 32, , i J i Ji JA e e e 3 321 , , i J i Ji JA e e e , ,A

2 2

1

2 2

cos sin2 2

sin cos2 2

i i

i i

e eA

e e

2 2

2 2

cos sin2 2

2sin cos2 2

i i

i i

e ei

e e

11 2 3sin cos sin sin cos

2

A iA

With the help of Mathematica, we get

11 2sin cos

2

A iA

132

A iA

ijAA

11 2 3sin cos sin sin cos

AA i J J J

11 2sin cos

AA i J J

13

AA i J

sin cos sin sin cos

sin cos 0

0 0 1

A i

detA A 3 sini sinAd C d d d C is an arbitrary constant

Group volume sinG AV d d d d

Normalized invariant measure: 1sinA

G

d d d dV

2 2

30 0 0

sinSOV d d d

28

2 4

20 0 0

sinSUV d d d

216

Rearrangement lemma for SU(2):

2 1 2 2 1 2

2 20 1 2 0 1 2

1 1cos , , cos , ,

16 16d d d f A d d d f BA

( Left invariant )

Left & right invariant measures coincide for compact groups.

See Gilmore or Miller for proof.

8.3. Orthonormality and Completeness Relations of D j

The existence of an invariant measure,

which is true for every compact Lie group,

establishes the validity of the rearrangement theorem,

which in turn guarantees that

1. Every IR is finite-dimensional.

2. Every IR is equivalent to some unitary representation.

3. A reducible representation is decomposable.

4. The set of all inequivalent IRs are orthogonal & complete.

Theorem 8.3: Orthonormality of IRs for SU(2)

† ' '2 1m nj j n m

A j j n mn mj d D A D A

2 1 2

'2

0 1 2

2 1cos

16n ni n j i m i n j i m

m m

jLHS d d d e d e e d e

In the Euler angle parametrization scheme [ dj() = real ]:

, ,m mj i m j i m

m mD e d e

2

1cos

16Ad d d d

• dA is normalized

• nj = 2j+1

2 1 2

'2

0 1 2

2 1cos

16n ni n n i m mj

j m m

jd e d d d d e

1

'2

1

2 12 cos 4

16n nn j m

n j mm m

jd d d

1

' '

1

12 1 cos

2n nj j

j jm mj d d d

RHS

( no sum over n, m )

Theorem 8.4: Completeness of D[R] (Peter-Weyl)

The IR D (A)mn form a complete basis in L2(G).

L2(G) = ( Hilbert ) space of (Lebesgue) square integrable functions defined on the group manifold of a compact Lie group G.

i.e., mnm n

f A f D A 2f A L G

For G = SU(2),

†2 1nn

jm A j mf j d D A f A

† †' '2 ' 1 2 ' 1

m m mn jA j A j jmn n n

j d D A f A j d D A f D A

'n j n m

A jm j n md f '

nj mf

†2 1 ' 'm nj

jn mj D A D A A A ( completeness )

2' 16 cos cosA A

1Ad A A

Comment:

• C.f. Fourier theorem in functional analysis.

• f(A) can be vector- or operator- valued.

mn jj m n

f A f D A †2 1n mj

A j m nj d D A f A D A

2 1 2

0 1 2

cos cos cos 1d d d

2 1 2

20 1 2

1cos

16d d d A A

Bosons

Fermions :

, , 2 , ,f f

0m

jmf

1

2j n

i.e. 0m

jmf j n

, , 2 , , 2mn j

j m nf f D 2, ,

m jn jj m nf D

, ,mn j

j m nf D

2 1j

j Bosons

Fermions :

2 1 2

'†2

0 1 0

2 1cos , , , ,

8mm

jm j m

jf d d d D f

0 2

' '† †

2 0

, , , , , , 2 , , 2m m

j jm md D f d D f

2

2

' 2†

0

, , , , 1m j

j md D f

2

'†

0

, , , ,m

j md D f

both cases

Often, , , , if f e with = n, or, n+ ½

For = 0

*

0

2 1, , 0 ,

4ml

lm

lD Y

i.e., the spherical harmonics { Ylm } forms an orthonormal basis for square integrable functions on the unit sphere.

2 1 2

'†2

0 1 0

2 1cos , ,0 ,

8mm i m i

jm j m

jf d d d D e f e

2 1

†

0 1

2 1cos , ,0 ,

4m

j m

jd d D f

Peter-Weyl: 0

0, , , 0

mllmf f D

2 1

0 1

cos , ,l m lmc d d Y f

* ,lm lml m

c Y 04

2 1l m lmc fl

, , , ,mj

j mf f D

0

0, , , , ,0

mjj mf f f D

Setting (,) → (,) gives

, ,0mj i

j mf D e

8.4. Projection Operators & their Physical Applications

12 1nn

jm A j mP j d D A U A Transfer operator: c.f. Chap 4

, ,njmP x m j j if non-vanishing, transforms like the IBVs

{ | j m } under SU(2) / SO(3)

i.e., kjn njm jk m

U A P x P x D A

njmP j m j n

' 'njmP j m j m j n j m '

j nj mj m

Henceforth, indices within | or | are exempted from summation rules

(error in eq8.4-2)

8.4.1. Single Particle State with Spin

Intrinsic spin = s states of particle in rest frame are eigenstates of J2 with eigenvalue s(s+1) .

Denote these states by 0, , ,s s p

with 2 , 1 ,J s s p 0 p 0

3 , ,J p 0 p 0

Task: Find | p 0,

, , J P P L S P P , L P P

,i j k j k i mx p p p ,i j k j m k ix p p p

i j k jm k ii p p im k k ii p p

0

k i

km i i ki p p

im k i ki p p 1

23, , , 0J X P X J Xi.e., find X

Let ˆ,p z be the "standard state" . Then

1 2ˆ ˆ, , 0P p P p z z 3 ˆ ˆ, ,P p p p z z

3 3ˆ ˆ, ,J P

p pp p

J Pz z 3 ˆ,J p z

( Helicity = )

ˆ,set

p z

L3 = 0 since motion is along z

3 ˆ,S p z

, 0 J P P

Alternatively, treating J & P as the generators of rotation & translation, resp,

, mk l k l mP J i P (eq 9.6-5)

( Theorem 9.12 )

J·P , P , J2, J3 share the same eigenstates

, 0 J P J Prove it !Similarly,

Let ˆpp n ˆ ˆ , n nwhere

and , , , ,p p ˆ, , 0 ,U p z

, , P p p p ( Problem 8.1 )

ˆ, , , 0 ,U R pp p

J P J Pp z

1 ˆ,U R U R U R pp

J P

z

ˆ,U R pp

J Pz

ˆ,U R p z

, p

i.e., helicity of a particle is the same in all inertial frames.

, 0 J P J

States with definite angular momentum (J, M) :

ˆ,J Mp J M P p z ˆ,J M J p z

1 ˆ2 1 ,A J MJ d D A U A p

z

2 1 2

†2

0 1 0

2 1ˆcos , , , , ,

8 J M

Jd d d U p D

z

| & | excluded from summation

3ˆ ˆ, , , , , 0 ,i JU p U e p z z ˆ, , 0 , iU p e z

† †, , , , 0 iJ JM M

D D e

2 1

†

0 1

2 1ˆcos , , 0 , , , 0

4 J M

Jp J M d d U p D

z

†2 1, , , , , 0

4 J M

Jd p D

' '', ,m mj i m j i m

m mD e d e

c.f. Peter-Weyl Thm, eq(8.3-4)

For a spinless particle, s = = 0:

2 1

0†

0 1

2 1cos , , , , 0

4 l m

lp l m d d p D

2 1

0 1

2 1cos , , ,

4 l m

ld d p Y

where *

0

2 1, , 0 ,

4ml

l m

lD Y

c.f. § 7.5.2

2 1

0 1

2 1cos , , ,

4

lp l m d d p l m

{ | p J M ; fixed } is complete for 1-particle states

, , , , , 0MJ

J M

p p J M D

2 1

†

0 1

2 1cos , , , , , 0

4 J M

Jp J M d d p D

can be inverted using

Standard state :

ˆ, , 0, 0,p p z 0, 0, 0MJ

J M

p J M D

J

p J

Traditional description: eigenstates of P2, L2, L3, S3 : , , ,p l m

with 2 1

0 1

, , , cos , , , ,l mp l m d d p Y

Difficulty: L3, S3 not conserved

Partial remedy: ,m

p J M l p l m m l s J M

Helicity is preferred

†2 1 ' 'm nj

jn mj D A D A A A

to give

D diagonal

8.4.2. Two Particle States with Spin

Group theoretical methods essential to avoid complications such as the L–S & j–j coupling schemes.

Standard state: C.M. frame,

1 2ˆp p z p

1 2 1 2ˆ ˆ ˆ, , 0, , 0 ,p p U p z z z

3 1 2 1 2 1 2ˆ ˆ, ,J p p z z

2 2 2 p j p1 1 1 p j p, ,

1 2 J j 1 1 j 3 1 23 31 1J j j

3 1 2 1 1 2 1 2 23 3ˆ ˆ ˆ ˆ ˆ, , , , ,J p p U p p U p z j z z z j z

1 1 1 13ˆ ˆ, ,p p j z z

12 2 2 23 3

ˆ ˆ, ,U p U U U p j z j z 2 2ˆ,U p z

General plane-wave states with

1 2 1 2ˆ, , , , , 0 ,p U p z

1 2ˆ ,p p n p

1 2

1 2 1 2ˆ,J Mp J M P p z

1 2

2 1 2†

1 220 1 0

2 1ˆcos , , , , ,

8 J M

Jd d d U p D

z

31 2 1 2ˆ ˆ, , , , , 0 ,i JU p U e p z z

1 2

1 2ˆ, , 0 ,i

U p e z

1 2 1 2 1 2† †, , , , 0i

J JM MD D e

1 2

2 1†

1 2 1 2

0 1

2 1ˆcos , , 0 , , , 0

4 J M

Jp J M d d U p D

z

1 2†1 2

2 1, , , , , 0

4 J M

Jd p D

{ | p J M 1, 2 ; j fixed } is complete for 2-particle states

1 2

1 2 1 2, , , , , 0MJ

J M

p p J M D

See Jacob & Wick, Annals of Physics (NY) 7, 401 (59)

Advantages of the helicity states:

• All quantum numbers are measurables.

• Relation between linear- & angular- momentum states is direct: there is no need for the coupling-schemes.

• Well-behaved under discrete symmetries.

• Applicable to zero-mass particles.

• Simplifies application to scattering & decay processes.

8.4.3. Partial Wave Expansion for 2-Particle Scattering with Spin

Initial state:, ,i a b i a b a b

J

p J M p

Final state:

, , , ,0c d

MJf c d f c d

J M

p J M D

p

All known interactions are invariant under SO(3).

Scattering matrix preserves J.

Wigner–Eckart theorem: , ,f f f c d i i i a bp J M T p J M

f f

i i

J M Jc d J a b J M JT E

, ,f c d i a bT p p

*

'

, , , ,0c d

M

f c d i a b a b JJ J M

p J M T p J D

a b a b

c d

iJc d J a b

J

T E d e

, ,f c d i a bT p p

*

'

, , , ,0c d

M

f c d i a b a b JJ J M

p J M T p J D

'' *'' '' '

'

, ,0a b c d

MJ M Jc d J a b J J J

J J M

T E D

* , ,0 a b

c dc d J a b J

J

T E D

( General partial wave expansion for 2-particle scattering )

c.f. §§ 7.5.3, 11.4, 12.7

Static spin version would involve multiple C–GCs.

8.5. Differential Equations Satisfied by the D j – Functions

1-D translation (§6.6):

T dx E i dx P d Ti P T

d x

i x PT e

P p p p i x pT p p e

Functions { e– i x p } are IRs of Lie group T1.

3 32, , i J i Ji JR e e e

3 323, , i J i Ji Ji R J e e e

3J R

13R J R R

3 322, , i J i Ji Ji R e e J e

3 323, , i J i Ji Ji R e e e J

3R J

13 3

kkR J R J R

cos cos cos sin sin cos cos sin sin cos cos sin

, , sin cos cos cos sin sin cos sin cos cos sin sin

sin cos sin sin cos

R

1 , , , ,TR R

1 2 3cos sin sin cosJ J J

The following derivations are Mathematica assisted. See R_New.nb

3 3 12

i J i JR e J e R R

Tung's version is described in SU(2).ppt & R.nb

3

1sin cos

2i iJ e J e J

3

cos sin 0

sin cos 0

0 0 1

R

Using

13 1 2 3cos sin sin cosR J R J J J

3

1 1cos sin sin cos

2 2J J J J J

i

3 3 1 12 1 2sin cosi J i JR e J e R R J J R 1 2sin cosk k

kJ R R

2 1cos sinJ J 1

2i ii J e J e

3i R J R

1

2i ii R i J e J e R

3

1sin cos

2i ii R J e J e J R

, ,R R

(1)(2)

(3)

(3) + i sin (2) – cos (1) :

sin cos sinii R R i R J R e

cossin

i ie R J R

(3) – i sin (2) – cos (1) :

cossin

i ie R J R

sin cos sinii R R i R J R e

RJ

RJ

R A RJ U AUJ

j m j jU m j A mUm J

m

mj

mD R j m A j m j m U j m

J

( Differential equation for D j )

mj

mm

j m A j m D R

3 32, , i J i Ji Jj m U j m j m e e e j m

'mi m i m

me d e

3 32

'', '''

'' '' ''' '''i J i Ji J

m m

j m e j m j m e j m j m e j m

'''' ' ''' ''''' '''

'', '''

mi m m i m mm mm

m m

e d e

', ,

mi m i m

mj m U j m i m e d e

', ,

mi m i m

m

dj m U j m e d e

d

', ,

mi m i m

mj m U j m i m e d e

'' 1 1' cos

sinmi m i m

m

de m m d

d

, ,

cos , ,sin

i

j m U j m

ij m e U j m

J

''

, , , ,m

j m J U j m j m J j m j m U j m

''' 1

''

1 1 , ,mm

m

j j m m j m U j m

''' 1

''

1 1mm i m i m

m mm

j j m m e d e

' 111 1mi m i m

mj j m m e d e

3 3''

, , , ,m

j m J U j m j m J j m j m U j m

'

''

, ,mm

m

m j m U j m

'''

''

mm i m i mm m

m

m e d e

'mi m i m

mm e d e

' ' 11' cos 1 1

sinm m

m m

dm m d j j m m d

d

The J3 equation is the identity:

The J eqs give the recurrence relations

2 23 3J J J 1 2 1 2J J J i J J i J 2 2

1 2 2 1 1 2J J i J J J J

2 23 3J R J J J J R

Since the J's are independent of ,, & , we have

R J R J 3 3R J RJ

R J R J J J J R J J J R

3 3 3 3R J RJ J J 3 3J R J 23J R

22

2cos cos

sin sini ii i

J R e e i R

Note reversed order

i m i mi e m e

2 2 2 2

2 2 2 2

1cot 2 cos

sinR

2 2 2

2 2 2

1 1sin 2cos

sin sinR

'2 , , 1 , ,mj

mj m J U j m j j D

2 2 2

'

2 2 2

1 1sin 2cos 1 , , 0

sin sinmj

mj j D

(Mathematica R_New.nb )

' ', ,

m mj i m i m

m mD e d e

'2 22

1 1sin ' 2 ' cos 1 0

sin sinmj

m

d dm m m m j j d

d d

22

2cos cos

sin sini ii i

J R e e i R

2 2 2

'

2 2 2

1 1sin 2cos 1 , , 0

sin sinmj

mj j D

For m = 0, j must be an integer & D j is independent of .

Let ( j, m') = ( l, m ) & ( ,) = (,), we have

2

2 2 0

1 1sin 1 , , 0 0

sin sinmll l D

*

0

2 1, , 0 ,

4ml

l m

lD Y

0

!2 1 2 1cos

4 4 !m ml i m i m

l m

l ml ld e P e

l m

0

!cos

!m ml

l m

l md P

l m

d j is related to the Jacobi polynomials by [ Eq(8.5-13) is wrong ]:

,! !

cos sin cos! ! 2 2

n m n mnj n m n m

j nm

j n j nd P

j m j m

From (Mathematica R_New.nb) we have

2

2 ,2

1 2 1 0l

d dz z l l P z

d z d z

In particular, setting ( j,n,m ) ( l,m,0 ) gives

,

0

! !cos sin cos

! ! 2 2

m mml m m

l m

l m l md P

l l

/ 22 ,1! ! 2 1

!

mm m ml ml m l m z P z

l

!

!m

l m

l mP z

l m

/ 2, 2!2 1

!

mmm ml m lm

lP z z P z

l m

For n = m = 0: 00, 00 0

ll l lP z P z P z d

, ,

cos ,

n m n m

z l j n

8.6. Group Theoretical Interpretation of Spherical Harmonics

Special functions ~ Group representation functions

*

0

2 1, , , 0

4ml

l m

lY D

, l m

0

2 1, , , 0

4ml

l m

ll m D

Roles played by Ylm(,) :

• They are matrix elements of the IRs of SO(3).

• They are transformation coefficients between bases | & | l m .

8.6.1. Transformation under Rotation

Let ˆ ˆ ˆ, ,ˆ ,

x y z u

, , ˆ ˆ ˆˆ ˆ ˆ, , , ,R

x y z X Y Z

ˆ ˆ ˆˆ ˆ ˆ, , , ,ˆ , ,

x y z X Y Z v

, , , ,U

, , , ,l m U l m , , ,ml

ml m D

, , , ,ml

l m l m mY Y D

ˆ ˆml

l m l m mY Y R D R

u u

ˆlmY R u=U c.f. eq(7.6-5)

ˆ ˆRv u

8.6.2. Addition Theorem

, , , ,ml

l m l m mY Y D

For m = 0:

0

2 1, cos

4l l

lY P

0

, , ,ml

l mY D

* 4, ,

2 1l m l mm

Y Yl

*4cos , ,

2 1l l m l mm

P Y Yl

( Addition

Theorem )

Note: , ,ˆ ˆ,

x y zZ R z so that , , , ,

ˆˆ, , cosx y z x y z

v Z

*4ˆ ˆ ˆ ˆ

2 1l l m l mm

P Y R Y Rl

u z u z *4 ˆˆ2 1 l m l m

m

Y Yl

v Z ˆˆlP v Z

8.6.3. Decomposition of Products of Ylm with the Same Arguments

From §7.7:

,

2 1 2 ' 1, , , ' 0 ' 0,0 ,

4 2 1l m l m L m mL

l lY Y m m l l L m m L l l Y

L

' , ' ' ,m m Mj j J

n n NJ M N

D R D R m m j j J M D R J N j j n n

'

0 0, ,0 , ,0 , ' , ,0 ' 0,0

m m Ml l L

NL M N

D D m m l l L M D L N l l

0

, ' , ,0 0 ' 0,0m mL

L

m m l l L m m D L l l

*

0

2 1, , , 0

4ml

l m

lY D

eq (8.6-4) is wrong

ˆ ˆ, , , 0R u z

8.6.4. Recursion Formulas

, , , , 0

, , 0 0 , , 0 0m

l m J U l l m J l m l m U l

1 0

1 1 , , 0mm l

mm

l l m m D

0, , 0 0 , , 0

mll m U l i m D

cossin

i ie R J R

cot , , 0 , , 0ie i U J U

*1

41 1 ,

2 1l ml l m m Yl

* 4,

2 1l mi m Yl

* *1cot , 1 1 ,i

l m l me m Y l l m m Y

1cot , 1 1 ,il m l me m Y l l m m Y

1cot , 1 1 ,il m l me i Y l l m m Y

eqs(8.8.6-5,6 ) are wrong (see Edmonds)

Recursions involving different l's can be done using direct product reps.

E.g., setting

,

2 1 2 ' 1, , , ' 0 ' 0,0 ,

4 2 1l m l m L m mL

l lY Y m m l l L m m L l l Y

L

' 1, 0l m in

we have

10

3, , , cos

4l m l mY Y Y

,

2 1 3, 0 1 0 1 0,0 ,

4 2 1L mL

lm l L m L l Y

L

1,

2 1 3, 0 1 1 1 0 1 0,0 ,

4 2 3l m

lm l l m l l Y

l

,

2 1 3, 0 1 0 1 0,0 ,

4 2 1l m

lm l l m l l Y

l

1,

2 1 3, 0 1 1 1 0 1 0,0 ,

4 2 1l m

lm l l m l l Y

l

1,

1 1 1 1 2 1 33, cos ,

4 2 1 1 2 1 1 4 2 3l m l m

l m l m l l lY Y

l l l l l

,

2 1 30,

4 2 11 1l m

lmY

ll l l l

1,

2 1 3,

2 1 2 1 4 2 1l m

l m l m ll lY

l l l l l

1, 1,

1 12 1 cos , , ,

2 3 2 1l m l m l m

l m l m l m l ml Y Y Y

l l

Using the CGCs in App V, we have

8.6.5. Symmetry in m

*

0

2 1, , , 0

4ml

l m

lY D

*

0

2 1

4mi m ll

e d

0

2 1

4mi m ll

e d

From §7.4: m m m m mj j j

m m md d d

0

2 1,

4mi m l

l m

lY e d

0

2 1

4m mi m ll

e d

*, ,m

l m l mY Y

( d j is real )

8.6.6. Orthonormality and Completeness

Theorem 8.3: † ' '2 1m nj j n m

A j j n mn mj d D A D A

*' '2 1

n nj j j j n nA m mm m

j d D A D A

*

' '

0 02 1 , ,0 , , 0

4m ml l l l m md

l D D

*', ,l m l m l l m md Y Y

Orthonormality

Theorem 8.4 (Peter-Weyl, for j = integer l):

* , , cos cosl m l ml m

Y Y

*, , , ,l m l ml m

f Y d Y f c.f. eqs(8.3–14,15)

8.6.7. Summary Remarks

• Geometric interpretations were given for– Differential eqs– Recursion formulae– Addition theorem– Orthonormality & completeness relations– …

• Further development: generalization of Fourier analysis to functions on manifold of any compact Lie group (for which the Peter-Weyl theorem holds).

• The D-functions, e.g.,{ Ylm}, are also natural bases for Hilbert space vectors & (tensor) operators (see §§7.5, 8.4 & 8.7).

8.7. Multipole Radiation of the Electromagnetic Field

Plane wave photon state of helicity : , , , ,k k ˆ, , 0 ,U k z

Photon state with angular momentum specified by J,M (c.f. §8.4.1) :

*2 1

, , , , , 04

MJJk J M d k D

The creation operators a†(k,) & a†( k, J, M, ) are defined by

†, , 0a k k † 0k J M a k J M

where | 0 is the (vacuum) state of no photons.

*

† †2 1, , , , , 0

4MJJ

a k J M d a k D

Using the half-integer case of Peter-Weyl theorem (see eqs(8.3–11,12):

*2 1

, , 0 , , 0 cos cos4

m mj j

j m

jD D

we get † †, , , , , 0MJ

J M

a k a k J M D

Annihilation operators:

*

, , , , , 0MJ

J M

a k a k J M D

Vector potential in a source–free region is given by :

3 *, , ,i t i tt d k a e a e

k kA x k A x k A x

where ˆ , ie k xkA x e k ˆ , 0 k e k

Electromagnetic fields ( potential = 0 ) :

ic

k kE x A x

0 A

,,

tt

c t

A x

E x i k kA x

, ,t tB x A x i k kB x k A x

*

3, , , 0 . .MJ i t

J M

t d k a k J M D e h c

kA x A x

2 . .k i tJ M

J M

k dk a k J M e h c

A x

where *

, , 0Mk J

J MA d D

k k k kx A x are the multipole wave functions

Evaluation of AJMk(x)

ˆ ˆ ˆ, , , 0 ,R e k e z 1ˆ ˆ, , , 0D

e z

1ˆ ˆ 1

ˆ ˆ, 2ˆ 0

ifor

x ye z

z

where

exp cosik xe i k x k x

2 1 cosll l k x

l

i l j k x P

*4 , ,ll l m l m

l m

i j k x Y Y k k x x

( Addition theorem )

See Jackson §16.8

ˆ , ie k xkA x e k

* 1ˆ ˆ4 , , , , , 0ll l m l m

l m

i j k x Y Y D

x x k k k ke z

See §7.8

* 1 1

0

2 1, , , 0 , , 0 , , 0

4

mll m

lY D D D

k k k k k k k k

2 1, 1 , ,0 1 0,

4

MJ

NJ M N

lm l J M D J N l

k k

From § 7.7 : ' , ' ' ,m m Mj j J

n n ND R D R m m j j J M D R J N j j n n

2 1, 1 , ,0 1 0,

4

MJ

J M

lm l J M D J l

k k

2 1ˆ ˆ4 , ,

4

, 1 , ,0 1 0,

ll l m

l m

MJ

J M

li j k x Y

m l J M D J l

k x x

k k

A x e z

*

, , 0Mk J

J MA d D

k k k kx A xComparing with the inverse of

i.e.,

2 1, , 0

4

MJ kJ M

J M

JD A

k k kA x x we have

216 2 1

ˆ ˆ, , , 1 1 0,2 1 4

k lJ M l l m

l m

lA i j k x Y m l J M J l

J

x xx e z

216 2 1

1 0,2 1 4

l ll J M

l

li j k x J l

J

T x

ˆ ˆ, , , 1lJ M l m

m

Y m l J M

x xT x e z

216 2 1

ˆ ˆ, , , 1 1 0,2 1 4

k lJ M l l m

l m

lA i j k x Y m l J M J l

J

x xx e z

where

= Vector spherical harmonics c.f. Prob 8.10

Electric and magnetic multipoles ( of definite parities ) :

k kJ M J M A x A x See Chap 11

Note: The above results are derived with no explicit reference to the Maxwell eqs.

c.f.

1,

4,

2 1lm

lmll m

qY

l r

x

error in eq(8.7-15)

Example: Photo-Absorption

, , ,i i i ii E j m k J M , , ,f f f ff E j m

31H e d x J x A x

1st order perturbation transition probability amplitude:

ki f J M i fT e f H i E E

3k kJ M J MH e d x J x A x

Using the Wigner-Eckart theorem, we have

, , , , , ,J M kf f f f J M i i i if H i E j m H E j m

,f f i i f f J i ij m J j M m j H E j

Final Exam

Problems

7.7, 8.6, 8.7 & 8.10