6. One-Dimensional Continuous Groups

6.1 The Rotation Group SO(2) 6.2 The Generator of SO(2) 6.3 Irreducible Representations of SO(2) 6.4 Invariant Integration Measure, Orthonormality and

Completeness Relations 6.5 Multi-Valued Representations 6.6 Continuous Translational Group in One Dimension 6.7 Conjugate Basis Vectors

Introduction

• Lie Group, rough definition:

Infinite group that can be parametrized smoothly & analytically.

• Exact definition:

A differentiable manifold that is also a group.• Linear Lie groups = Classical Lie groups

= Matrix groups

E.g. O(n), SO(n), U(n), SU(n), E(n), SL(n), L, P, …• Generators, Lie algebra• Invariant measure• Global structure / Topology

6.1. The Rotation Group SO(2)

1 1 2cos sinR e e e

2 1 2sin cosR e e e

ji j iR e e R cos sin

sin cosR

1 2 1 2

cos sin

sin cosR e R e e e

2 1 2,span e e E2-D Euclidean space

Rotations about origin O by angle :

2 x Eiie xx

2 2:R E E

by R x x x iiR e x j i

j ie R x j

je x

jj i

ix R x

2 iix x x

2 jjx x x j ki

ki jR x R x

Rotation is length preserving:

j k kii j

R R TR R E

i.e., R() is special orthogonal.

2det det det 1TO O O det 1O

O n All n n orthogonal matrices

2SO R det 1R

If O is orthogonal,

T TO O E O O

Theorem 6.1:

There is a 1–1 correspondence between rotations in n & SO(n) matrices.

Proof: see Problem 6.1

Geometrically: 2 1 1 2 1 2R R R R R

and 2R n R n Z

Theorem 6.2: 2-D Rotational Group R2 = SO(2)

2 2R SO is an Abelian group under matrix multiplication with

0E R

and inverse

identity element

1 2R R R

Proof: Straightforward.

SO(2) group manifoldSO(2) is a Lie group of 1 (continuous) parameter

6.2. The Generator of SO(2)

Lie group: elements connected to E can be acquired by a few generators.

0R d E i d J

For SO(2), there is only 1 generator J defined by

d RR d R d

d

R() is continuous function of

R R d R i d R J

d Ri R J

d

with 0R E

J is a 22 matrix

Theorem 6.3: Generator J of SO(2)

i JR e

Comment:

• Structure of a Lie group ( the part that's connected to E ) is determined by a set of generators.

• These generators are determined by the local structure near E.

• Properties of the portions of the group not connected to E are determined by global topological properties.

cos sin

sin cos

d dR d

d d

1

1

d

d

E i d J

0

0

iJ

i

y Pauli matrix

J is traceless, Hermitian, & idempotent ( J2 = E ) i JR e

12 2 1

0 12 ! 2 1 !

j jj j

j j

E i Jj j

cos sinE i J cos sin

sin cos

6.3. IRs of SO(2)

Let U() be the realization of R() on V.

2 1 1 2U U U 1 2U U 2U n U

U d E i d J i JU e

U() unitary J Hermitian

SO(2) Abelian All of its IRs are 1-D

The basis | of a minimal invariant subspace under SO(2) can be chosen as

J iU e so that

2U n U 2i n ie e m Z

IR Um : J m m m m i mU m m e

m = 0: 0 1U Identity representation

m = 1: 1 iU e SO(2) mapped clockwise onto unit circle in C plane

m = 1: 1 iU e … counterclockwise …

m = n:

n i nU e SO(2) mapped n times around unit circle in C plane

Theorem 6.4: IRs of SO(2)

Single-valued IRs of SO(2) are given by m i mU e mZ

Only m = 1 are faithful

Representation cos sin

sin cosR

is reducible

1 1R U U

0

0

iJ

i

has eigenvalues 1 with eigenvectors

1 2

1

2e e i e

Problem 6.2

J e e iR e e e

6.4. Invariant Integration Measure, Orthonormality & Completeness Relations

Finite group g Continuous group dgIssue 1: Different parametrizations

d f g d f g

ξ

ξ ξ φ φφ

d f g φ φ

Remedy: Introduce weight :

gd d φ φ

d f g d f g ξ ξ ξ φ φ φso that &analytic f ξ φ

ξφ ξ φ

φ

Changing parametrization to = (), we have,

gd f g d f g φ φ

where = ( 1, …n ) & f is any complex-valued function of g.

1 nd d d φ

1

1

, ,

, ,n

n

ξ

φ

Let G = { g() } & define

Issue 2: Rearrangement Theorem

1g gM g Md f g d f g g

gg g G ξ

Let g g gd d ξ ξ

g g g g g gd d ξ ξ

g g gd d g g g g g gd d ξ ξ ξ ξ

g gG Gd f g d f g g

M G Since

R.T. is satisfied by setting M = G if dg is (left) invariant, i.e.,

g g gd d g G

g g g gg g g ξ ξ ξ

e g gd d 0 ξ ξ ξ , eg e ξ 0

( Notation changed ! )

g gMd f g g

g g g gg g g ξ ξ ξFrom one can determine the (vector) function : ;g g g g ξ χ ξ ξ

g g ed J dξ ξ ξ

;g gξ χ 0 ξ

where deti

g g jJ Jξ ξ

;

g

ii g g

g jjg

J

ξ 0

ξ ξξ

eg g e

g

d

d ξξ 0

ξ e

gJ

0

ξ e(0) is arbitrary

g e

Theorem 6.5: SO(2) gd d

Proof: R R R ;

0

;J

1 Setting e(0) = 1 completes proof.

Theorem 6.6: Orthonormality & Completeness Relations for SO(2)

2

†

0 2m m

n n

dU U

Orthonormality

†nn

n

U U

Completeness

Proof: These are just the Fourier theorem since n i nU e

Comments:

• These relations are generalizations of the finite group results with g dg

• Cf. results for Td ( roles of continuous & discrete labels reversed )

6.5. Multi-Valued Representations

Consider representation / 21/ 2

iR U e

/ 21/ 2 1/ 22 iU e U

/ 21/ 2 1/ 24 iU e U 2-valued representation

m-valued representations :

//

i n mn mR U e ( if n,m has no common factor )

Comments:

• Multi-connected manifold multi-valued IRs:

• For SO(2): group manifold = circle Multi-connected because paths of different winding numbers cannot be continuously deformed into each other.

• Only single & double valued reps have physical correspondence in 3-D systems ( anyons can exist in 2-D systems ).

6.6. Continuous Translational Group in 1-D

R() ~ translation on unit circle by arc length

Similarity between reps of R(2) & Td

Let the translation by distance x be denoted by T(x)

Given a state | x0 localized at x0,

0 0T x x x x is localized at x0+x

0 0T x T x x T x x x 0x x x 0T x x x 0x

T x T x T x x

0 00 0T x x 0x 0E x 0x 0T E

T x T x T x x 0T E 1T x T x

1T T x x R is a 1-parameter Abelian Lie group

= Continuous Translational Group in 1-D

T dx E i dx P Generator P:

dTT x dx T x dx

d x

T x T dx T x i dx P

dT xi P T x

d x

i P xT x e

For a unitary representation T(x) Up(x), P is Hermitian with real eigenvalue p. Basis of Up(x) is the eigenvector | p of P:

P p p p p i p xU x p p e pR

Comments:

1. IRs of SO(2), Td & T1 are all exponentials: e–i m , e–i k n b & e–i p x, resp.

Cause: same group multiplication rules.

2. Group parameters arecontinuous & bounded for SO(2) = { R() }discrete & unbounded for Td = { T(n) }continuous & unbounded for T1 = { T(x) }

Invariant measure for T1: gd C dx

†

2p

p

d xU x U x p p

Orthonormality

†

2p

p

d pU x U x x x

Completeness

C = (2)–1 is determined by comparison with the Fourier theorem.

SO(2) Td T1

Orthonormality mn (k–k) (p–p)

Completeness (–) nn (x–x)

6.7. Conjugate Basis Vectors

Reminder: 2 kind of basis vectors for Td.

• | x localized state

• | E k extended normal mode

,E ku x x E k

T n x x nb

i k n bT n E k E k e

H E k E E k

For SO(2):

• | = localized state at ( r=const, )

• | m = eigenstate of J & R()

0 0U

0U m

m m

0m m U † 0U m

0 i mm e

i mU m m e

Setting 0 1m gives i mm e m

transfer matrix elements m | = representation function e–i m

i m

m

m e

i mm e

2 2

0 02 2i m mi m

m

d de m e

m mm

m

m

mm

m

2 ways to expand an arbitrary state | :

2

0 2

d

m

m m i mm

m

e

m m 2

0 2

dm

2

0 2i md

e

i m

m

J J m e

i m

m

m m e

i

1

Ji

in the x-representation

J J J is Hermitian:1

i

1

i

J = angular momentum component plane of rotation

For T1:

• | x = localized state at x

• | p = eigenstate of P & T(x)

0 0T x x x x

i p xT x p p e

2

d px p p x

0p x p T x 0i p xe p i p xe p | 0 set to 1

2i p xd p

p e

2i p p xi p x d p

d x e x p d x e

2i p xd p

x x x p e

2i p x xd pe

x x

d p p p p

p

i p xp p d x e p x

i p p xd x e

2 p p

T is unitary

2 ways to expand an arbitrary state | :

d x x x

2

d pp p

d x x x

2

d pp p

x x 2

d px p p

2

i p xd pe p

p p d x p x x

i p xd x e x

x P P x P+ = P :

2i p xd p

P x P p e

i xx

1x

i x

1

xi x

1P

i x

on V = span{ | x } P = linear momentum