Quantum Mechanics for Mathematicians

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: