Although we won’t prove it here, the representations constructed in this way
provide essentially all the unitary irreducible representations ofE(2), parame-
trized by a real numberE >0. The only other ones are those on which the
translations act trivially, corresponding toE = 0, withSO(2) acting as an
irreducible representation. We have seen that suchSO(2) representations are
one dimensional, and characterized by an integer, the weight. We thus get
another class ofE(2) irreducible representations, labeled by an integer, but
they are just one dimensional representations onC.
19.2 The case ofE(3)
In the physical case of three spatial dimensions, the state space of the theory of a
quantum free particle is again a Euclidean group representation, with the same
relationship to the Schr ̈odinger representation as in two spatial dimensions. The
main difference is that the rotation group is now three dimensional and non-
commutative, so instead of the single Lie algebra basis elementlwe have three
of them, satisfying Poisson bracket relations that are the Lie algebra relations
ofso(3)
{l 1 ,l 2 }=l 3 ,{l 2 ,l 3 }=l 1 ,{l 3 ,l 1 }=l 2
Thepjgive the other three basis elements of the Lie algebra ofE(3). They
commute amongst themselves and the action of rotations on vectors provides
the rest of the non-trivial Poisson bracket relations
{l 1 ,p 2 }=p 3 , {l 1 ,p 3 }=−p 2{l 2 ,p 1 }=−p 3 , {l 2 ,p 3 }=p 1
{l 3 ,p 1 }=p 2 , {l 3 ,p 2 }=−p 1
An isomorphism of this Lie algebra with a Lie algebra of matrices is given
by
l 1 ↔
0 0 0 0
0 0 −1 0
0 1 0 0
0 0 0 0
l^2 ↔
0 0 1 0
0 0 0 0
−1 0 0 0
0 0 0 0
l^3 ↔
0 −1 0 0
1 0 0 0
0 0 0 0
0 0 0 0
p 1 ↔
0 0 0 1
0 0 0 0
0 0 0 0
0 0 0 0
p^2 ↔
0 0 0 0
0 0 0 1
0 0 0 0
0 0 0 0
p^3 ↔
0 0 0 0
0 0 0 0
0 0 0 1
0 0 0 0
Theljare quadratic functions in theqj,pj, given by the classical mechanical
expression for the angular momentum
l=q×p