Chapter 19

The Quantum Free Particle

as a Representation of the

Euclidean Group

The quantum theory of a free particle is intimately connected to the represen-
tation theory of the group of symmetries of space and time. This is well known
for relativistic theories, where it is the representation theory of the Poincar ́e
group that is relevant, a topic that will be discussed in chapter 42. It is less
well known that even in the non-relativistic case, the Euclidean groupE(3) of
symmetries of space plays a similar role, with irreducible representations ofE(3)
corresponding to free particle quantum theories for a fixed value of the energy.
In this chapter we’ll examine this phenomenon, for both two and three spatial
dimensions.
The Euclidean groupsE(2) andE(3) in two and three dimensions act on
phase space by a Hamiltonian group action. The corresponding moment maps
(momentapjfor translations, angular momentaljfor rotations) Poisson-com-
mute with the free particle Hamiltonian giving symmetries of the theory. The
quantum free particle theory then provides a construction of unitary representa-
tions of the Euclidean group, with the space of states of a fixed energy giving an
irreducible representation. The momentum operatorsPjgive the infinitesimal
action of translations on the state space, while angular momentum operatorsLj
give the infinitesimal rotation action (there will be only one angular momentum
operatorLin two dimensions since the dimension ofSO(2) is one, three in three
dimensions since the dimension ofSO(3) is three).
The Hamiltonian of the free particle is proportional to the operator|P|^2.
This is a quadratic operator that commutes with the action of all the elements
of the Lie algebra of the Euclidean group, and so is a Casimir operator play-
ing an analogous role to that of theSO(3) Casimir operator|L|^2 of section
8.4. Irreducible representations will be labeled by the eigenvalue of this oper-
ator, which in this case will be proportional to the energy. In the Schr ̈odinger