3. Lie Derivatives And Lie Groups

3.1 Introduction: How A Vector Field Maps A Manifold Into Itself3.2 Lie Dragging A Function 3.3 Lie Dragging A Vector Field 3.4 Lie Derivatives 3.5 Lie Derivative Of A One-form 3.6 Submanifolds 3.7 Frobenius' Theorem (Vector Field Version) 3.8 Proof Of Frobenius' Theorem 3.9 An Example: The Generators Of S2

3.10 Invariance 3.11 Killing Vector Fields 3.12 Killing Vectors And Conserved Quantities In Particle Dynamics 3.13 Axial Symmetry 3.14 Abstract Lie Groups3.15 Examples Of Lie Groups3.16 Lie Algebras And Their Groups 3.17 Realizations And Representations 3.18 Spherical Symmetry, Spherical Harmonics And Representations

Of The Rotation Group

Congruence (flow):

Set of non-crossing curves that fill a part of M.

These curves are usually solutions of some vector field V.

Each point in M where V is non-singular (V0) is on 1 & only 1 curve.

dim(congruence) = dim(M) – 1

Lie derivative: Derivative along the congruence of a vector field V.

Lie dragging:

Moving each p in M by an amount Δλ along the congruence is an auto-diffeomorphism.

These draggings form a 1-parameter Lie group.

3.1. Introduction: How A Vector Field Maps A Manifold Into Itself

2. Pull-back (reciprocal image) of f :

*

kj

j k

yf V V

x y

*j

kj k

yf dx

x

Given : byf X Y x y f x one can define 2 related mappings:

* *: byP f Pf T X T Y V W f V 1. Push-forward (differential) of f :

Given a coord basis:

f* is also written as f , D f, or d f .

* * * *: byPf Pf T Y T X f

s.t. g : Y → , *f V g V g f

*

** *

fP f P

fP f P

f

g f g

T X T Y

T X T Y

X Y

R

s.t.

**x y

f V f V

Given a coord basis:

*f g g f

** y x

f V g V f g

See Choquet, pp.121 & 138.

,j k

j ky dx

jj

V Vx

jj dy

* y xV g ff V g

**

x yV ff V

*f g g f

** *gVf gV ff

1*f h h f

Let σ(t,x0) be the integral curve ofd

Vd t

that passes through x0 at t = 0.

i.e., ,

,i

id t xV t x

d t

with 0, x x

, , ,t s x t s x since both satisfy the same differential eqs & same I.C.

0, 0, 0,x x x

Define map , :t t X X by ,tx x t x

σt is called the local transformation generated by vector field V.

(c.f. Schutz’s Lie dragging. )

The set of all σt is a 1-parameter transformation group.

It’s local if the range of t is not all of .

In which case, it becomes a pseudo group since closure is violated.

tt s s , , ,t s x t s x →

See Choquet, p.144

3.2. Lie Dragging A Function

and vector field

Lie-dragging f along V by Δλ gives another function :f M R

s.t. f P f P

i.e. 1f x f x

dV T M

d

f is “Lie dragged” by V if fΔλ = f.

with integral curve σ(λ )

Given function f : M →

*f f

fΔλ is just the push-forward :

1f

or the pull-back :

*f f 1f

3.3. Lie Dragging A Vector Field

Given vector field dV T M

d with integral curve σ(λ )

Lie dragging a vector field W along V

= Operating on W by the push-forward of the local transformation.

* W ,

dV T M

d with integral curve σ(λ )

Lie derivative = Derivative along the congruence of a vector field.

3.4. Lie Derivatives

Definition: Lie derivative of a function

Given function f : M →

and vector field dV T M

d

*

0lim

V x

f x f xf

L

x

d f

d

xV f

with integral curves σ(λ, x )

0

limf x f x

0lim

f x f x

1*

0lim

f x f x

0

limf x f x

Vf V fL

Definition: Lie derivative of a vector field W along vector field V

1*

0

1lim t tV tx

W W x W xt

L

Setting ty x i i iy x t V x →

1*

j ji i

t ji i j

x xW y W y W y

y y x

i ji k j

i jk i

W x V xW x tV x t

x y

2j j

j k ijk i

W x V xW x t V x W x O t

x x

2W x t V x W x W x V x O t

,V

W VW WV V W L

j j

i i

V x V xO t

y x

where the multiplication

VW f V W fVW is defined by

Also VW V W

is not a vector field since the Leibniz rule is not obeyed.

C2 function f.

VW

1*

0

1lim t xt

W W xt

Properties of the Lie derivative (see Ex 3.1-3)

V V V A B A B A BL L L

, , , , , , 0X Y Z Y Z X Z X Y

L L L L L L L L L

,,

V W V W L L L

( Jacobi’s identity )

, ,i j i j

iV i jV W W VW L

( Leibniz rule: is a derivation )

( Coordinate basis )

i

i j i j i ji i jV j e

V e W W e V e V eWW L L ( General basis )

,i

ji jW W L

j

ji

W

x

( Coordinate-free partial )

iV i V L Lj

ji

V

x

i

V

x

i

W

x

WV VW L L L

3.5. Lie Derivative of a 1-form see Choquet, p.148.

Definition: Lie derivative of a 1-form field ω along vector field V

*

0

1lim t tV tx

x xt

L

Exercise: show that i

V

ij

j

Vdx dx

x

L c.f.

j

V i ji

V

x

L

iV Vidx L L V

iiV

ii dx dx L L i

ii j

i j

VV dx dx

x

ij i ji

j ji

VV dx dx

x x

i

ji ji ji

VV dx

x x

( c.f. Ex 3.14 )

The above can also be derived from V V VW W W L L L

Lie derivatives of tensors can be calculated using the Leibniz rule.

Basic formulae:

V Vd dL L

,i

V

i jjdx V dxL

,j

V i i jV L

d is an anti-derivation that changes a (mn) tensor into a (m

n+1 ) tensor.

is a derivation that leaves the rank of a tensor unchanged.

( proof: operate both sides in components on f )

Example:

i

V V iW W L L iiV i

i

VW W L L

, ,j i i j

j ii jV W W V

, ,j i j i

j ijV W W V

3.6. Submanifolds

See Frankel, § 1.3d.

A submanifold S of a manifold M is a subset of M that is also a manifold.

S is an s-D embedded submanifold of an m-D manifold M if there exists an atlas of M s.t. in every chart, the coordinates of S are of the form

1 1, , , 0, , 0s s mx x x x

or, more generally,

1 1, , , , ,s m sx x y y where the y j s are functions of the xk s.

Note: all solutions y to some set of equations involving x are of this form.

Consider a point pS.

A curve in S through p is also a curve in M.

→ A (tangent) vector in Tp(S) is also a vector in Tp(M).

Indeed, Tp(S) is a subspace of Tp(M) .

A vector in Tp(M) is not necessarily a vector in Tp(S).

A 1-form in Tp*(M) is also a 1-form in Tp

*(S) since it can map any vector in

Tp(S) to by treating it as a vector in Tp(M).

A 1-form in Tp*(S) is not necessarily a 1-form in Tp

*(M).

3.7. Frobenius' Theorem (Vector Field Version)

On a smooth manifold M, the order of partial derivatives is irrelevant.

,i j i j j if f f

Coordinate basis vector fields commute, i.e.,

, , 0j i i jf f f C M

, 0i j since

C(M) is the ring of all C functions on M.

It is not a field because f 1 may not exist.

The set (M) of vector fields on M is a vector space over the field because it is closed under linear combinations of constant coefficients.

If the coefficients of linear combinations are C functions, (M) becomes a module over the ring C(M).

If (M) is closed under the Lie bracket, it becomes a Lie algebra.

A set of vector fields are linearly independent if they are lin. indep. at each point.

A set of m linearly independent, mutually commuting, vector fields can be taken as a coordinate basis using the parametrization as coordinates.

By definition, their integral curves mesh to form a foliation (family of submanifolds). Each (m1)-D submanifold, called a leaf of the foliation, is specified by a fixed value of the mth coordinate.

Frobenius theorem generalize this to the case of a set of vector fields which spans a module over C(M) that is closed under the Lie bracket.

Lemma : The set of all linear combinations (non-constant coefficients allowed) of a set of mutually commuting vector fields is closed under the Lie bracket.

Alternative phrasing: the module over C(M) spanned by a set of mutually commuting vector fields is closed under the Lie bracket.

Mathematical terms: if , 0i jV V

i

a a ii

W W x V , 1, ,i j m

then ,a bi

ab ii

x VW W Proof: See Ex 3.5.

iiV Setting → vector fields on a submanifold are closed under the Lie bracket.

Frobenius theorem:

Let (U) be the set of all linear combinations ( with possibly non-constant coefficients ) of a set of vector fields on a region U of a manifold M.

[ i.e, (U) is a module of vector fields on UM over ring C(U). ]

If (U) is closed under the Lie bracket, then the integral curves of the vector fields mesh to form a foliation.

Proof: see §3.8.

The original version of the theorem was described in terms of differential forms (see §4.26) and dealed with the integrability conditions of Pfaffian systems (existence of solutions of system of partial differential equations).

Examples:

Fig.3.7

3.8 Proof of Frobenius' Theorem

Ancillary relations:

V Vd f d fL L , ,

i ji j

V f dx

,V V

d f W d f WL L , ,V V

d f W d f W L L , ,

i jj i

V f W

, ,V V

d f W d f W L L, , ,V

d f V W d f W L

Strategy of proof:

Since the theorem is satisfied if the vector fields commute, we need only show that a set of m linearly independent, mutually commuting, vector fields can be constructed out of m lin. indep. fields that are closed under the Lie bracket.

This will be done by induction.

The case m = 1 is trivial since foliation = curve = leaf.

Assuming case m1 is valid, we shall prove that case m also holds.

Let the m lin. indep. fields be ; 1, ,aV a m with 1

,m

caba b c

c

V V V

Let

mm

dV

d and

1

1

mj

i i jj

X V

for i = 1, … , m1

s.t. , 0 1, , 1m id X i m

1

,m

ai ji j a

a

X X V

→

1

1

mki j i jk m

k

X V

1

,m

am im i a

a

V X V

1

1

mkmi m jk m

k

X V

Next, we show that

, , 0m i jd X X

, ,m m jd V X

, , , ,V V

d f V W d f W d f W L LSince

we have , , , ,i im i j m j m jX X

d X X d X d X L L

i im mX Xd d L L

0m

i

dd

d

0

, 0m id X and

Similarly, , , , ,m m i m i md V X d X V

, ,i im m m mX X

d V d V L L 0

where

, ,m m m

m

dd V d

d

1m

m

d

d

→

, 1 0i im mX X

d V L L

1

1

,m

ki j i ji j k m

k

X X X V

1

1

,m

kmi m jm i k m

k

V X X V

→ 0 ,i j m md V i j

→ 0 ,m j m md V m j

i.e.,

1

1

,m

ki ji j k

k

X X X

1

1

,m

km im i k

k

V X X

{X(i)} is a set of m1 lin. indep. vector fields closed under the Lie bracket.Invoking the (m1) case of the theorem, {X(i)} can be transformed into a set of m1 of lin. indep. mutually commuting vector fields {Y(i)} whose integral curves mesh into a (m1)D foliation S.

Next, we extend {Y(i)} into the entire U by requiring it to be invariant under the group transformation generated by V(m) ( i.e., Lie dragging along V(m) ).

*

on

outsidem

i

ii

Y SZ

Y S

Calling the resultant fields {Z(i)}, this means

Since * * *, ,f V W f V f W we also have , 0i jZ Z

with , 0m i m iV

Z V Z L

QED

3.9. An Example: The Generators of S2

x ye y e xe

; , ,il i x y z

ji i j k kl x

x

zly xx y

~ ( ang mom op )

In general

, ,k mi j i k l jmn l nl l x x

ji j k kx

,k m m k k mik l jmn l n n l l nx x x x x x

k m m kik l jmn l n n lx x

k mik l j l n n ik l jmk lx x k

ik l j l n i l n j k l nx

ki j k n in k j i j nk i k n j nx

kin k j i k n j nx j i

i jx x i j k kl

Integral curves of form a foliation with spherical leaves.

( implied sum over repeated indices )

ji i j k k

rl r x

x

kj

i j k

xx

r

0

→ every li is tangent to the constant r surfaces.

Only 2 of them are linearly independent at each point.

Each leaf in the foliation is just S2 .

idr l

Comment: Angular momenta are generators of rotations.

Thus, they are antisymmetric tensors of rank 2.

Only in 3-D space are they equivalent to vectors.

3.10. Invariance

Principal use of Lie derivatives: Invariants of tensor fields

→ Symmetries, Lie groups …

A tensor T is invariant under a vector field V if 0V

TL

Let F = { T(1) , T(2) , … } be a set of tensor fields.

Then the set A of all vector fields under which F is invariant is a Lie algebra.

Proof:

0i i ia V b W V Wa b T T TL L L

V W V W L L LSince

we have

for a . and a V VaL L

i.e., A is a vector space.

From ,,

V WV W L L L ,

, 0i iV WV W T TL L Lwe have

i.e., A is a Lie algebra.

, & ,V W A a b R

aV bW A

,V W A

,V W A

Caution: A is a vector space only under combinations with constant coefficients.

Example:

il is linearly dependent in 3.

0iii

x lc can be satisfied with some ci (x) 0i.e.,

However, il is linearly independent if only constant ci are allowed.

→ Lie algebra generated by { li } is 3-D.

→ submanifold generated by { li } is 2-D.

3.11. Killing Vector Fields

V is a Killing vector field if the metric tensor g is invariant wrt it, i.e., 0V

gL

i ji jV V

g dx dx gL L k i j i k j j i kk i j i j k i j kV g dx dx g V dx dx g V dx dx

, , ,k k k i j

i j k k j i i k jV g g V g V dx dx

Formmx

V

we have ,i j

i j mVg dx dx gL

mi iV &

j is a Killing vector g is independent of xj

k k k i jk i j k j i i k jV g g V g V dx dx

3, Cartesian coordinates: ii j jg , ,x y z Killing vectors:

3, spherical coordinates: , ,r rdiag g 2 21 , , sindiag r r

Killing vectors: zl Also:

,x yl l

3.12. Killing Vectors & Conserved Quantities in Particle Dynamics

Classical mechanics H = T + Φ :

Φ indep of x → px conserved

Φ indep of φ → pφ conserved

Not so for other coordinate systems

Reason: Conserved momentum must be Killing vector field of configuration space.

Further discussions deferred to Chapter 5.

g is involved because the tensor form of the Newton’s equation is

i im x ij

j

xg

3.13. Axial Symmetry

Axial symmetry = invariance about an axis

Cylindrical symmetry = axial symmetry + translational symmetry along axis

Eq of motion: 0L L = linear operator invar under φ→ φ+ a

Caution: A solution is not necessarily axially symmetric, e.g., if is the path of a particle.

Fourier analysis: , j j imm

m

x x e

→ m im imm m mL e L e

( for a field )

0i mm m

m

L e

imm

m

L L e

Example:

22 2

2 2 2 2 2

1 1 1sin

sin sinL r

r r r r r

2 2~ ,i m i m i m i mm m m m m mL e L e f e f r e

22

2 2 2 2

1 1sin

sin sin m

mr f

r r r r r

= scalar axial harmonics

has axial eigenvalue m if ei m

L

e LIf is a scalar, then i mC e →

Let S be the submanifold (with boundary) φ= 0. Let be a 2-D basis for TP(S) at each pS. 1 2,je e e

is a basis for TP(M 3) at pS.→ , je e

around the axis of symmetry generates a basis pM.Lie dragging , je e

Let σ(φ) be an integral curve of e

0 0

0 0* *, ,j je e e e

The basis at a point with φ= φ0 is

Since , 0j jee e e

L

we have * 0j je ee e

L L

i.e., , je e are all axially symmetric.

Note: The Cartesian components of

are functions of φ.

* 0e e

e e

L L

For axis = z, S = x-z half-plane with x 0.

, je e

0je ee e

L L means

i mjm je e

e e e

L L

have axial eigenvalues 0. , je e

are vector axial harmonics with axial eigenvalues m.

, ,i m i mjm m je e e e e e

i m i mj je

e e e e

L

i mji m e e m ji m e

i.e.,

Thus, the solution to

jm j mV a e b e

where a1 , a2 and b are independent of φ.

eV i m V

L can be written as

The group of transformation (Lie draggings) { σφ } is the 1-parameter Lie group SO(2).

Spherical symmetry leads to SO(3).

e e

( sum over j implied )

3.14. Abstract Lie Groups

A Lie group G is a group that is also a differentiable manifold s.t. the group & differential structures are compatible.

1by ,G G G x y x y is a differentiable mapi.e.,

Choquet, pp.116, 153.

The set { σg } is a Lie group of transformations on manifold X if the map

: by , ,G X X g x g x

is differentiable and if the set of transformations

: ; ,g gX X x g x

is a group wrt composition, s.t.

g h g h ( Left action of G on X )

g h h g ( Right action of G on X )

σe is the identity transformation

or

with1

1g g

→

Left translation by g : : byg gL G G h L h g h

Right translation by g :

: byg gR G G h R h h g

Left translation by g near e

For X = G, the differential (push forward) of the translations are

* :g h g hL h T G T G

* :g h h gR h T G T G( Lg*(e) = Schutz’s Lg )

A vector field V on G is left invariant if it is invariant under left translations, i.e.,

*g gL V h V L h V gh ,g h G

Setting h = e gives *gL V g where V e

In local coordinates: *

i

jj

i

gL V hh

Vgh

h

*

i

g

e

i jj

h

gh

hL

Left invar.

iV gh

iV gFor h = e :

Note: V(h) V|h denotes the value of V at hG.

Let σ(t , g ) be the integral curve of V that passes through g for t = 0.

The transformations σt = σ( t, ) form an Abelian group A with group multiplication σt σs = σt+s .

Setting σ(t , e ) = g ( t ) turns the curve into a 1-parameter Abelian subgroup of G isomorphic to A with group multiplication

~t s t s g tL g s g t g s g t s g(0) = e.

The subgroup { g ( t ) =σ(t , e ) } is entirely specified by V e

The integral curve σ(t , h ) is the left coset h { g (t ) } of { g (t ) } wrt h.

By definition:

, expt h t h t V h

→ exptg t e t e

( Exponentiation always generates a left (or right) translation along σ(t) )

It is sometimes denoted by { gγ ( t ) }.

For each vector γ in Te(G), one can construct a left invariant field V & hence a unique 1-parameter subgroup { gγ ( t ) }.

Every element on the path connected submanifold of G containing e belongs to one of these 1-parameter subgroups.

Restriction to invariant fields guarantees manifold::group compatibility.

Since * * *, ,f V W f V f W

* * *, ,g g gL V W h L V L W h

,V W gh if V, W are left invar fields

→ Left (or right) invariant vector fields form a Lie algebra (G).

* * *g g gL aV bW h a L V h b L W h ,a b R

→ [ V, W ] is also a left-invariant field.

→ Left-invariant fields are closed under Lie bracket.

→ Left (or right) invariant vector fields form a vector space over .

aV bW gh if V, W are left invar fields

, , , , , , 0V U W U W V W V U automatically satisfied

Since each left invariant field is specified by its value at e, (G) is isomorphic to , the Lie algebra of G on Te(G).

Let

; 1, ,iV i n be a basis of or (G).

, ki ji j kV V c V

or (G) is specified by the structual constants cki j given by

ki jcC is a (1

2) tensor.

Every Lie group has a unique structure tensor but not vice versa.

3.15. Examples of Lie Groups

(i) n

It is a manifold under the usual (n-balls) topology.

It is an Abelian Lie group under vector addition.

The 1-parameter subgroups are rays ( straight lines through 0 ).

The congruence of a left invariant field consists of lines parallel to a given ray.

A left invariant field is therefore a constant vector field: V(x) = V(0) = γ.

→ All Lie brackets of left invariant fields vanish.

The Lie algebra is Abelian.

(ii) GL(n,) = Group of all nn invertible real matrices

Group operation = Matrix multiplication.

e = Unit matrix.

g1 = Inverse matrix.

It is a differentiable manifold (Lie group) because it is a submanifold of

Coordinates: n2 matrix elements.

2nR

Tangent space =2nR

Group manifold =

2

\ points giving singular matricesnR

1-parameter subgroup: exp 0A Ag t t A g

where generator 2

0

nA

t

d g tA

d t

R

is isomorphic to the space of all nn real matrices.

Other integral curves of the left-invariant vector field can be obtained by left translate (multiply) { gA(t) } by some group element h { gA(t) }.

exp t A e exp t A

Matrices with negative determinants don’t belong to any 1-paramter subgroup.

→ GL(n,) is a disconnected group.

Elements path-connected to e form the component of the identity, which is just the subgroup SL(n,).

The Lie bracket of the left-invariant fields is defined as

,A B A B B Ag t g t g t g t g t g t

where · is the group (matrix) multiplication.

the Lie bracket for the Lie algebra can be written as

, ~e

A B A B B A

where A is the matrix version of the vector A|

e.

(iii) O(n), SO(n) = Groups of all nn orthogonal real matrices ( S means det = +1)

O(n) is disconnected.

SO(n) = component of identity of O(n).

A GL(n,) can be taken as a (11) tensor on n.

A maps a (column) vector in n to another vector in n.

Similarity transform B1AB is a basis transformation ei → B1 ei on A.

The canonical form of A SO(n) is block diagonal with blocks of

cos sin1, 1, or

sin cos

(see Ex.3.14)

where the number of 1 blocks must be even.

→ Every A SO(n) represents a rotation in some 2-D plane.

SO(n) = group of rotations.

The other component of O(n) represents inversions that change the handedness of basis.

The basis of so(3) is

1

0 0 0

0 0 1

0 1 0

L

2

0 0 1

0 0 0

1 0 0

L

3

0 1 0

1 0 0

0 0 0

L

or i i j kj kL

1 TT t B t BA A e e Lett BA e then

TB B

The Lie algebra o(n) of group O(n) consists of all anti-symmetric nn matrices.

The anti-symmetric condition gives n diagonal & n(n1)/2 off-diagonal relations.

→ Dimensions of both o(n) & so(n) are

2 1 11 1

2 2n n n n n n

so that i j imk jk nmnL L

i n m j i j mn

,i j i n m j jn mimnL L i j k k nm i j k k mn

L ,i j i j k kL L L

2

i i n mnm i i imnL 1 mni n

Convention: Algebras of subroups of GL(n,) are denoted by lower case letters.

(iv) SU(n) = Group of all nn unitary matrices with unit determinants

SU(n) is a subgroup of GL(n,).

SU(n) is the component of identity of U(n).

1 t A t AU U e e Let

t AU e then A A

u(n) consists of all nn anti-Hermitian matrices.

The (real) dimension of group U(n) is n2.

The anti-Hermitian condition imposes n(n1)/2 off-diagonal complex, & n imaginary diagonal, conditions.

→ (Real) dimension of u(n) is

2 22 1n n n n n

su(n) consists of all nn anti-Hermitian traceless matrices.

The (real) dimension of SU(n) is n21.

3.16. Lie Algebras & Their Groups

Every Lie group G has its Lie Algebra .

Every element g of G is on one of the integral curves of a left invariant field specified by a vector in Te(G).

If g is not a member of a 1-par subgroup, the integral curve that passes through it can be obtained by a left or right translation of some subgroup.

Hence, not all Lie groups can be obtained solely from their Lie algebras.

Definition: Lie Algebra

A Lie algebra is a vector space endowed a bilinear Lie bracket s.t.

, ,A B B A

, , , , , , 0A B C B C A C A B

Example:3,A B A B R

( M, π ) is a covering space of N if (see Choquet, p.19)

① M is connected & locally connected.

② π : M → N is onto & continuous.

③ π1 : N → M is a multi-valued homeomorphism.

Which means pN, neighborhood (p) s.t. the restriction of

π to each connected component Cα of π1 ((p)) is a homeomorphism.

Schutz’s notation: N is covered by M

M is disconnected if 2 disjoint open sets A1 & A2 s.t. A1 A2 = M.

M is connected & M are the only sets that are both open & closed.

M is locally connected if every neighborhood of every point contains a connected neighborhood.

M is simply connected if every closed curve can be shrunk continuously to a point.

Theorem:

1. Every Lie algebra is the Lie algebra of 1 & only 1 simply-connected Lie group.

2. Every Lie group with the same Lie algebra is covered by the simply-connected one.

Example: S1 is covered by

1: by cos , sinS x x x x R

2 , 2 1C

S1 is multiply-connected.

Sn is simply connected for n 2.

Example: is simply connected.

SU(2) is simply connected

Proof: Let set of all matrices of the form * *

a bH

b a

2 21a b SU(2) = H with

H is a 4-D real vector space with basis

→ 2 2det A a b A H

H\{O} is a Lie subgroup of GL(n,)0 0

0 0O

0 1 2 3

1 0 0 1 0 1 0

0 1 1 0 0 0 1

iI

i

Thus * *

a bA

b a

→ 3

2

0

2 1ii

SU A

SU(2) is simply connected

0 3 1 2

1 2 0 3

i

i

3

0i i

i

where

= S3 in 4-D α-space

αi

i.e., every element of SU(2) can be generated by exponentiation from e

3

1

, 2i j i j k kk

i

The Pauli matrices satisfy

, 1,2,3i j

( The structure constants should be real since the vector space & algebra are real )

2k k

iJ →

3

1

,i j i j k kk

J J J

Note: J2 in eq(3.65), Schutz, is wrong.

A vector in Te(SU(2)) can be written as3

1i i

i

t J

Elements of a 1-parameter subgroup of SU(2) are given by expg t t

E.g.

expkJ kg t t J

0 !

nn

kn

tJ

n

2k I

2 2 1

0 2 ! 2 2 1 ! 2

n nn n

kn

t tI i

n n

0

1

! 2

nn

kn

i t

n

cos sin2 2k

t tI i

→ left invar. fields

The generators Jk of a unitary matrix must be anti-hermitian.

The Pauli matrix σk are hermitian → Jk i σk

Setting

2 1i mni nmnL

The generators for SO(3) are (see §3.15):

i i j kj kL

so that ,i j i j k kL L L

3 1i mk ik ni kmnL imn 3

i iL L→

12 1 nni iL L

12 2nni iL L

with the , th element set to 0I i i

expkL kg t tL

1 !

nn

kn

tI L

n

1 1

2 1 2 2

1 2 1 ! 2 !

n n

n nk k

n

I t L t Ln n

n = 1,2,…

2 2sin cosk k kI L t L t L

E.g. 1

1 0 0

0 cos sin

0 sin cosLg t t t

t t

→ (SU(2) , π) is a covering space of SO(3).

Setting : 2 3SU SO by k k kJ J Lg t g t g t

The domain of kJg t is t [ 0, 4 π). That of

Restricting π to either t [ 0, 2 π) or t [ 2 π, 4 π) makes it a homeomorphism.

→ (SU(2) , π) is a double covering space of SO(3).

i.e., SO(3) is doubly covered by SU(2).

kLg t is t [ 0, 2 π).

The projection of S3 onto the 3-D space of coordinates ( α1 , α2 , α3 ) is the closed ball of radius 1.

( c.f. projection of S2 onto 2. )

2 20 1 r

Only the outermost sphere with = 1 can be taken to represent actual points on S3.

All other spheres with r < 1 are projections.

All points on a sphere have the same coordinate

Every point in the ball can be taken as e, different choices merely reflect different perspectives for the projection.

Every great circle C on the r = 1 sphere is also a great circle of S3.

By choosing e to lie on the r = 1 sphere, each C on it represents a 1-parameter subgroup of SU(2).

Going once around C changes t by 4π.

For SO(3), each pair of diametrically opposite points on C are identical. (The pair is also diametrically opposite on S3. )

This creates a 2nd kind of closed path that cannot be shrunk continuously to a point : any curve joining a diametrically opposite pair of points.

→ SO(3) is doubly connected.

SU(2) & SO(3) share the same Lie algebra → they are identical near e.

SU(2) is the only simply-connected Lie group associated with the Lie algebra.

A rotation in n-D space ( an element of SO(n) ) can be represented by a vector in the Lie algebra, which is also a vector space of dimension n(n1)/2.

Thus, the association of an n-D rotation with an n-D angular momentum vector is possible only if n = n(n1)/2, i.e. n = 3.

Objects whose rotations are elements of SU(2) are called spinors.

A vector in the Lie algebra ( internal space of the particle ) is called a spin.

The association of a SU(2) element with a 3-D vector allows the association of spin with an (angular momentum) vector in 3.The Lie algebra of the Lie group n is an n-D Abelian Lie algebra .

Since n is simply-connected, all other Lie groups G with Lie algebra must be covered by n.

Also n is Abelian & identical to G near e. → G must be Abelian.

All Lie groups of an Abelian algebra are Abelian.

3.17. Realizations & Representations

Abstract group : Group defined by manifold & group structure.

Group in physics : symmetry transformations (operations).

Realization of a group : A map T : G → L(M), where L(M) is the space of operators on M, that preserves the group structure, i.e.,

identity transform on T e I M

,T gh T g T h g h G

1 1T g T g

The realization is a representation if L(M) is the space of linear operators on a vector space M.

The realization / representation is faithful if T is 1-1.

Example (i):

The above still holds if G is applied to the 2-sphere S2.

However, since S2 is a manifold, G(S2) is a faithful realization of SO(3).

The abstract group SO(3) is defined by its Lie algebra & topology.

Consider the group G of all 33 orthogonal matrices with unit determinant.

Given any column matrix x corresponding to a point x3 and a matrix R G, Rx represents a rotation of x.

It’s straightforward to show that there is a 1-1 correspondence of G & SO(3) that preserves the group operations.

Since 3 is a vector space, G(3) is a faithful representation of SO(3).

Historically, G(3) is studied first & then “abstracted” to SO(3).

Most abstract groups are established in a similar manner.

Example (ii):

Every group has at least 2 faithful realizations :

The left- & right- translations of itself.

(h → gh) is called the progressive realizations & (h → hg1 ) the retrograde realizations.

Example (iii): Adjoint Representations

Given group G, the map

Ig : G → G by h g h g1 where gG

is called the adjoint realization of G on G.

Ig is an inner automorphism of G.

The group adjoint realization need not be faithful.

E.g., Ig = id g if G is Abelian.

Since 1gI e geg e

gG

The differential (push-forward) Ig* of Ig at e is called the adjoint map Adg.

* :gg g e eI eI Ad T T T

t XgI e

Thus, Ig maps a 1-par subgroup fX (t) to another 1-par subgroup gAd Xf t

i.e., g

g

t Ad

d

X

A Xe f t 1t Xg e g

See Aldrovandi, §8.4.

g XI f t

Taking t → 0 gives (see Ex 3.23)

sYg s e → Ys L

g sAd X e X

The map Ad: G → L(,) is called the adjoint representation of G on .

L(,) = space of all linear automorphisms of .

The adjoint rep is used in the definition of connection for fibre bundles.

3.18. Spherical Symmetry, Spherical Harmonics & Representations of the Rotation Group

A manifold M with metric g is spherically symmetric if the Lie algebra of its Killing vector fields has subalgebra isomorphic to so(3).

The above definition avoids the possible embarassment that the “center of spherical symmetry” does not exist because it lies outside M.

Reminder: A Killing vector field generates (isometric) group of transformations. I.e., g is invariant under translation along the integral curves of the field.

It was shown in §3.9 that the Killing vector fields on S2 are 1 2 3, ,l l l

with commutators , i j k kk

i jl l l

Reminder: lj are lin. indep. under linear combinations of constant coefficients.

On S2 of radius a : 2 2 2, , sindiag g g diag a a g

i.e., the Lie algebra of Killing fields on S2 the same as so(3).

Conversely, a foliation in a spherically symmetric M with spherical leaves.

, i j k kk

i j VV V

Switching to the basis iiV l gives commutators

Let L2(S2) be the Hilbert space of all square-integrable functions on S2.

The norm of f L2(S2) is defined as 2

1/ 22 2, sinS

f f d d

A realization of SO(3) on S2 is given by the map g σg , where σg is the group of transformation

σg (x) = σ( g, x ), x S2 , g SO(3)

s.t. σg h = σg σh .

This induces the pull back of σg s.t. *g gf x f x 2 2f L S

In physics, one usually defines the function transformed by R by

Rf Rx f x

For the transformation Rg corresponding to σg , this means 1

*g g

R

Hence the map R by g R(g) = Rg is a representation of SO(3) on L2(S2).

Note that Rg : L2(S2) → L2(S2) is an automorphism on linear space L2(S2).

A representation is irreducible if it contains no invariant subspaces.

One task of theoretical group theory is to find all irreducible representations of a group.

By definition, SO(3) has at least one faithful 3-D representation in terms of orthogonal matrices.

Since L2(S2) is an infinite dimensional space, the representation of SO(3) on it must be reducible.

Indeed, it can be decomposed into a countable set of finite dimensional irreducible representations (IR) each specified by an integer L = 0, 1, 2, ….

The Lth IR has dimension 2L+1 and basis { YLM , M = L, L+1, …, L}.

By definition:

2

* , ,Lg LM g LNM N S

R Y R Y d

gSO(3)

where

Lg LM g LNN

NM

R Y R Y

and Rg is some differential operator representing g.

Historically, YLM is chosen to be the pherical harmonics, which are eigenfunctions of lz and L2 2

klk

L

(See Ex.3.24-5 )

Since SU(2) doubly covers SO(3), the representation of SU(2) is double-valued.

Thus, a faithful representation of SU(2) consists of 2 disjoint parts, each of which is a faithful representation of SO(3).

IRs of SU(2) are labeled by half-integers J = 0, ½, 1, 3/2, ….

Every element is on some 1-par subgroup parametrized by t = [0, 4π) which can be splitted into 2 disjoint parts, [0, 2π) and [2π, 4π).

For J = L = integers, the representation on both parts are identical.

For J = odd half integers, it behaves differently.