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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