Chapter 13

The Heisenberg group and

the Schr ̈odinger

Representation

In our discussion of the free particle, we used just the actions of the groups
R^3 of spatial translations and the groupRof time translations, finding corre-
sponding observables, the self-adjoint momentum and Hamiltonian operatorsP
andH. We’ve seen though that the Fourier transform allows a perfectly sym-
metrical treatment of position and momentum variables and the corresponding
non-commuting position and momentum operatorsQjandPj.
ThePjandQj operators satisfy relations known as the Heisenberg com-
mutation relations, which first appeared in the earliest work of Heisenberg and
collaborators on a full quantum-mechanical formalism in 1925. These were
quickly recognized by Hermann Weyl as the operator relations of a Lie algebra
representation, for a Lie algebra now known as the Heisenberg Lie algebra. The
corresponding group is called the Heisenberg group by mathematicians, with
physicists sometimes using the terminology “Weyl group” (which means some-
thing else to mathematicians). The state space of a quantum particle, either
free or moving in a potential, will be a unitary representation of this group,
with the group of spatial translations a subgroup.
Note that this particular use of a group and its representation theory in
quantum mechanics is both at the core of the standard axioms and much more
general than the usual characterization of the significance of groups as “symme-
try groups”. The Heisenberg group does not in any sense correspond to a group
of invariances of the physical situation (there are no states invariant under the
group), and its action does not commute with any non-zero Hamiltonian opera-
tor. Instead it plays a much deeper role, with its unique unitary representation
determining much of the structure of quantum mechanics.