are multiplication operators and, taking the Fourier transform of 19.2 gives the
differentiation operator
Γ ̃′S(l) =−(
p 1∂
∂p 2−p 2∂
∂p 1)
=−
∂
∂θ(use integration by parts to showqj=i∂p∂jand thus the first equality, then the
chain rule for functionsf(p 1 (θ),p 2 (θ)) for the second).
This construction of a representation ofE(2) starting with the Schr ̈odinger
representation gives the same result as starting with the action ofE(2) on
configuration space, and taking the induced action on functions onR^2 (the
wavefunctions). To see this, note thatE(2) has elements (a,R(φ)) which can
be written as a product (a,R(φ)) = (a, 1 )( 0 ,R(φ)) or, in terms of matrices
cosφ −sinφ a 1
sinφ cosφ a 2
0 0 1
=
1 0 a 1
0 1 a 2
0 0 1
cosφ −sinφ 0
sinφ cosφ 0
0 0 1
The group has a unitary representation
(a,R(φ))→u(a,R(φ))on the position space wavefunctionsψ(q), given by the induced action on func-
tions from the action ofE(2) on position spaceR^2
u(a,R(φ))ψ(q) =ψ((a,R(φ))−^1 ·q)
= =ψ((−R(−φ)a,R(−φ))·q)
=ψ(R(−φ)(q−a))This representation ofE(2) is the same as the exponentiated version of the
Schr ̈odinger representation Γ′Sof the Jacobi Lie algebragJ(2), restricted to the
Lie algebra ofE(2). This can be seen by considering the action of translations
as the exponential of the Lie algebra representation operators Γ′S(pj) =−iPj
u(a, 1 )ψ(q) =e−i(a^1 P^1 +a^2 P^2 )ψ(q) =ψ(q−a)and the action of rotations as the exponential of the Γ′S(l) =−iL
u( 0 ,R(φ))ψ(q) =e−iφLψ(q) =ψ(R(−φ)q)One also has a Fourier-transformed version ̃uof this representation, with
translations now acting by multiplication operators on theψ ̃E
̃u(a, 1 )ψ ̃E(θ) =e−i(a·p)ψ ̃E(θ) =e−i√
2 mE(a 1 cosθ+a 2 sinθ)ψ ̃E(θ) (19.3)and rotations acting by rotating the circle in momentum space
̃u( 0 ,R(φ))ψ ̃E(θ) =ψ ̃E(θ−φ) (19.4)