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
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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.
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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