Chapter 20

Representations of

Semi-direct Products

In this chapter we will examine some aspects of representations of semi-direct
products, in particular for the case of the Jacobi group and its Lie algebra, as
well as the case ofNoK, forN commutative. The latter case includes the
Euclidean groupsE(d), as well as the Poincar ́e group which will come into play
once we introduce special relativity.
The Schr ̈odinger representation provides a unitary representation of the
Heisenberg group, one that carries extra structure arising from the fact that
the symplectic group acts on the Heisenberg group by automorphisms. Each
such automorphism takes a given construction of the Schr ̈odinger representation
to a unitarily equivalent one, providing an operator on the state space called an
“intertwining operator”. These intertwining operators will give (up to a phase
factor), a representation of the symplectic group. Up to the problem of the
phase factor, the Schr ̈odinger representation in this way extends to a represen-
tation of the full Jacobi group. To explicitly find the phase factor, one can
start with the Lie algebra representation, where thesp(2d,R) action is given
by quantizing quadratic functions on phase space. It turns out that, for a finite
dimensional phase space, exponentiating the Lie algebra representation gives a
group representation up to sign, which can be turned into a true representation
by taking a double cover (calledMp(2d,R)) ofSp(2d,R).
In later chapters, we will find that many groups acting on quantum systems
can be understood as subgroups of thisMp(2d,R), with the corresponding
observables arising as the quadratic combinations of momentum and position
operators determined by the moment map.
The Euclidean groupE(d) is a subgroup of the Jacobi group, and we saw in
chapter 19 how some of its representations can be understood by restricting the
Schr ̈odinger representation to this subgroup. More generally, this is an example
of a semi-direct productNoKwithNcommutative. In such cases irreducible
representations can be characterized in terms of the action ofKon irreducible