Transcript 6._1DContinuousGroups

6. One-Dimensional Continuous Groups
6.1
6.2
6.3
6.4
6.5
6.6
6.7
The Rotation Group SO(2)
The Generator of SO(2)
Irreducible Representations of SO(2)
Invariant Integration Measure, Orthonormality and
Completeness Relations
Multi-Valued Representations
Continuous Translational Group in One Dimension
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)
2-D Euclidean space
E2  span  e1 , e2
Rotations about origin O by angle  :
R   e1  e1 cos  e2 sin 
R   e2  e1 sin   e2 cos 
R   ei  e j R  
 R   e
1
j
i
R   e2    e1
 cos 
R    
 sin 
 cos 
e2  
 sin 
 sin  
cos  
 sin  
cos  

 x  E2
x  ei x i
R   : E2  E2
j
x  R   x  R   ei x i  e j R   i x i  e j x j
x
by
x  j  R  

j
i
xi
Rotation is length preserving:
x
2
 x j xj  R  
j
x i R   j xk  x
k
i
R   i R   j   i k
j

If O is orthogonal,

 x i xi
R   RT    E
k
det R     1
2
i.e., R() is special orthogonal.

SO  2    R  
O OT  E  OT O
det O det OT  det O   1
2
O  n    All n  n orthogonal matrices
det O   1


Theorem 6.1:
There is a 1–1 correspondence between rotations in En & SO(n) matrices.
Proof: see Problem 6.1
R 2  R 1   R 1  2   R 1  R 2 
Geometrically:
R  2 n     R  
and
Theorem 6.2:
 nZ
2-D Rotational Group R2 = SO(2)
R2  SO  2
is an Abelian group under matrix multiplication with
identity element
E  R   0
and inverse
R1    R     R  2   
Proof: Straightforward.
SO(2) is a Lie group of 1 (continuous) parameter 
SO(2) group manifold
6.2. The Generator of SO(2)
Lie group: elements connected to E can be acquired by a few generators.
For SO(2), there is only 1 generator J defined by
R  d  0   E  i d J
J is a 22 matrix
R() is continuous function of  
R   d  
d R  
R    d
d
 R   R  d   R    i d R   J

d R  
 i R    J
d
Theorem 6.3:
with
Generator J of SO(2)
R    ei  J
R  0  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 d
R  d   
 sin d

 0 i 
J 

i
0


 sin d 
cos d 
y
 1
 d

d 
 E  i d J

1 
Pauli matrix
J is traceless, Hermitian, & idempotent ( J2 = E )
R    ei  J
  2 j


j  0  2 j !

j
 E cos   i J sin 
   2 j 1

E i 
j  1  2 j  1 !
 cos 

 sin 

j 1
 sin  
cos  
J
6.3. IRs of SO(2)
Let U() be the realization of R() on V.

U 2 U 1   U 1  2   U 1  U 2 
U  d   E  i d  J
U() unitary

SO(2) Abelian 
U  2 n     U  
U    e i  J
J Hermitian
All of its IRs are 1-D
The basis |   of a minimal invariant subspace under SO(2) can
be chosen as
J    
U  2 n     U  
so that

U      e i  
e i  2 n     e i  
IR Um :
J m  m m
m = 0:
U 0    1

  m Z
U m   m  m e i  m

Identity representation
m = 1:
U 1    ei 
SO(2) mapped clockwise onto unit circle in C plane
m = 1: U 1    e i 
m = n: U  n    e
Theorem 6.4:
… counterclockwise …
i n
SO(2) mapped n times around unit circle in C plane
IRs of SO(2)
U m    e i m 
Single-valued IRs of SO(2) are given by
m Z
Only m = 1 are faithful
Representation
 0 i 
J 

i
0



 cos 
R    
 sin 
 sin  
is reducible

cos  
has eigenvalues 1 with eigenvectors
J e   e
R   e   e e
i
e 
1
 e1  i e2 
2
R  U 1  U 1
Problem 6.2
6.4. Invariant Integration Measure,
Orthonormality & Completeness Relations
Finite group g

Continuous group  dg
Issue 1: Different parametrizations
Let G = { g() } & define
 d
g
f  g    dφ f  g  φ 
dφ  d1
d n
where  = ( 1, …n ) & f is any complex-valued function of g.
Changing parametrization to  = (), we have,
ξ
 d ξ f  g  ξ    dφ  φ f  g  φ
  dφ f  g  φ 
Remedy:
so that

Introduce weight :
 ξ  1 ,

 φ  1 ,
d g    φ dφ
 d ξ   ξ  f  g  ξ    dφ  φ f  g φ
  φ 
ξ
 ξ  φ  
φ
, n 
,n 
 analytic ξ  φ & f
( Notation changed ! )
Issue 2: Rearrangement Theorem

G
Since
d g f  g    d g f  g g 
G

M
d g f  g   
g  1M
 g   g  ξ g   G
d g f  g g    d gg f  g g 
M G
M
R.T. is satisfied by setting M = G if dg is (left) invariant, i.e.,
 g  G
d g  d g  g
Let

d g  d g g
d g    ξ g  d ξ g
d g g    ξ g g  d ξ g g

g  ξ g g   g  ξ g   g  ξ g 
  ξ g  d ξ g    ξ g g  d ξ g g
  0  d ξ e    ξ g  d ξ g
 g  e,
ξe  0

From
g  ξ g g   g  ξ g   g  ξ g 
ξ g g  χ  ξ g  ; ξ g 

ξ g  χ 0 ; ξ g 
d ξ g  J ξ g  d ξe
J ξ g 
i
j

Proof:
 g  e
J  ξ g   det J  ξ g 
where
i
j
  i  ξ g ; ξ g 
  gj
 g  ξ g   e  0
Theorem 6.5:
one can determine the (vector) function  :
SO(2)
ξg  0
d ξe
 0
 e
d ξg
J ξ g 
e(0) is arbitrary
d g  d 
R   R   R   
    ;  
1
J   
 
0

  ;     
Setting e(0) = 1 completes proof.
Theorem 6.6:
2

0
Orthonormality & Completeness Relations for SO(2)
d †
U n   U m     nm
2


n  
U n   U n†        
Orthonormality
Completeness
Proof: These are just the Fourier theorem since
U n    ei n 
Comments:
• These relations are generalizations of the finite group results with
g   dg
• Cf. results for Td ( roles of continuous & discrete labels reversed )
6.5. Multi-Valued Representations
R  
Consider representation
U1/ 2    ei  / 2
U1/ 2  2     ei  / 2  U1/ 2  
U1/ 2  4     e
i  / 2
 U1/ 2  
2-valued representation
m-valued representations :
R  
Un / m    ei n / m
( 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,
T  x  x0  x0  x
T  x T  x  x0  T  x x0  x
 T  x  x x0
T  x T  x   T  x  x 

T  0 x0  x0  0
 x0  E x0
T xT  x  T x  x

 x0  x  x
is localized at x0+x

T1  T  x  x  R
 T  0  E

 x0


T  0  E
T  x  T x
1
is a 1-parameter Abelian Lie group
= Continuous Translational Group in 1-D
 x0
T  dx   E  i dx P
dT
T  x  dx   T  x   dx
dx

 T  x  T  dx   T  x   i dx P
Generator P:

dT  x 
 i P T  x 
dx
T  x   ei P x
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
U p  x  p  p e i p x
pR
Comments:
1.
IRs of SO(2), Td & T1 are all exponentials: e–i m , e–i
knb
& e–i p x, resp.
Cause: same group multiplication rules.
2.
Group parameters are
continuous &
bounded for
discrete
& unbounded for
continuous & unbounded for
SO(2) = { R() }
Td
= { T(n) }
T1
= { T(x) }
Invariant measure for T1:
d g  C dx


dx †
p
U
x
U


 x     p  p
 2 p
Orthonormality

dp p
†


U
x
U


p x    x  x 
 2
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
T  n  x  x  nb
•
|Ek
extended normal mode
T  n  E k  E k e i k n b
uE , k  x   x
H Ek E Ek
Ek
For SO(2):
U   0    0
• |   = localized state at ( r=const,  )
U   m  m ei m 
• | m  = eigenstate of J & R()
  U   0
 


m 
m
m  
m   m U   0
Setting
m 0 1
 U   m
†
gives


0
 m 0 ei m 
m   ei m 
m
 transfer matrix elements  m |   = representation function e–i m 
 


m   ei m 
m ei m 
m  
2

0
2

d i m 
e
   m
2
m  

0

d i  mm   
m  m m 
e

m  
2
 m
2 ways to expand an arbitrary state |   :
 


m  
m m 

0
 
     
m  m 
2
2


0
d
2
   
m
m 
  ei m m
m
m
d
2
2
m 
 


0
d  i m 
e
  
2
 
J


J m e i m 
m  
J is Hermitian:

J



m  

1 
i 
m m ei m   i  

J   J 
1 

 
i 
in the x-representation
 J = angular momentum component  plane of rotation
 
1 

  
i 
For T1:
• | x  = localized state at x
T  x  x0  x  x0
• | p  = eigenstate of P & T(x)
T  x  p  p e i p x
p
x  p T  x 0

x 


dp
2



d x ei p x x 



x

x 


p
p


dp
2


d p
p
2
dp
2
dx e


dx e
i  p  p  x

x p e  i p x
ipx
p x
p ei p x



d p
p   p  p   p






p 

x
T is unitary
 p | 0  set to 1

p
 ei p x
 ei p x p 0




d p i p  x  x 
e
   x  x 
2
d x ei  p  p  x  2   p  p 

2 ways to expand an arbitrary state |   :

 

dx

dp
2




x 
x

P+
dx
x   x
dp
2

p   p



dp
2
x
p

dx
p
x
x 

x
1 
i x






p 
P   Px 
on V = span{ | x  }



d p ipx
e   p
2
d x e i p x   x 


dp
i p x

i
x
P p e
x
2
=P:
P


  p  p 



p 
p

P x 


  x  x 



1 
i x
x 

1 
  x
i x
 P = linear momentum