are rotations of the circle counterclockwise by an angleθ, or if we parametrize
the circle by an angleφ, just shifts
φ→φ+θ
By the same argument as in the caseG=R, we can use the representation on
functions given by equation 1.3 to get a representation onH
π(g(θ))ψ(φ) =ψ(φ−θ)
IfXis a basis of the Lie algebraso(2) (for instance taking the circle as the
unit circle inR^2 , rotations 2 by 2 matrices,X=
(
0 − 1
1 0
)
,g(θ) =eθX) then
the Lie algebra representation is given by taking the derivative
π′(aX)f(φ) =
d
dθ
f(φ−aθ)|θ=0=−a
d
dφ
f(φ)
so we have (as in theRcase, see equation 10.2)
π′(aX) =−a
d
dφ
This operator is defined on a dense subspace ofH=L^2 (S^1 ) and is skew-adjoint,
since (using integration by parts)
〈ψ 1 ,
d
dφ
ψ 2 〉=
1
2 π
∫ 2 π
0
ψ 1
d
dφ
ψ 2 dφ
=
1
2 π
∫ 2 π
0
(
d
dφ
(ψ 1 ψ 2 )−
(
d
dφ
ψ 1
)
ψ 2
)
dφ
=−〈
d
dφ
ψ 1 ,ψ 2 〉
The eigenfunctions ofπ′(X) are theeinφ, forn∈Z, which we will also write
as state vectors|n〉. These are orthonormal
〈n|m〉=δnm (11.1)
and provide a countable basis for the spaceL^2 (S^1 ). This basis corresponds to
the decomposition into irreducibles ofHas a representation ofSO(2) described
above. One has
(π,L^2 (S^1 )) =⊕n∈Z(πn,C) (11.2)
whereπnare the irreducible one dimensional representations given by the mul-
tiplication action
πn(g(θ)) =einθ
The theory of Fourier series for functions onS^1 says that any functionψ∈
L^2 (S^1 ) can be expanded in terms of this basis: