representations ofN, together with the irreducible representations of certain
subgroups ofK.
The reader should be warned that much of the material included in this chap-
ter is motivated not by its applications to non-relativistic quantum mechanics, a
context in which such an abstract point of view is not particularly helpful. The
motivation for this material is provided by more complicated cases in relativistic
quantum field theory, but it seems worthwhile to first see how these ideas work
in a simpler context. In particular, the discussion of representations ofNoK
forNcommutative is motivated by the case of the Poincar ́e group (see chapter
42). The treatment of intertwining operators is motivated by the way symmetry
groups act on quantum fields (a topic which will first appear in chapter 38).
20.1 Intertwining operators and the metaplectic representation
For a general semi-direct productNoKwith non-commutativeN, the repre-
sentation theory can be quite complicated. For the Jacobi group case though, it
turns out that things simplify dramatically because of the Stone-von Neumann
theorem which says that, up to unitary equivalence, we only have one irreducible
representation ofN=H 2 d+1.
In the general case, recall that for eachk∈Kthe definition of the semi-direct
product comes with an automorphism Φk:N→Nsatisfying Φk 1 k 2 = Φk 1 Φk 2.
Given a representationπofN, for eachkwe can define a new representation
πkofNby first acting with Φk:
πk(n) =π(Φk(n))In the special case of the Heisenberg group and Schr ̈odinger representation ΓS,
we can do this for eachk∈K=Sp(2d,R), defining a new representation by
ΓS,k(n) = ΓS(Φk(n))The Stone-von Neumann theorem assures us that these must all be unitarily
equivalent, so there must exist unitary operatorsUksatisfying
ΓS,k=UkΓSUk−^1 = ΓS(Φk(n))We will generally work with the Lie algebra version Γ′Sof the Schr ̈odinger rep-
resentation, for which the same argument applies: we expect to be able to find
unitary operatorsUkrelating Lie algebra representations Γ′Sand Γ′S,kby
Γ′S,k(X) =UkΓ′S(X)Uk−^1 = Γ′S(Φk(X)) (20.1)whereXis in the Heisenberg Lie algebra, andkacts by automorphism Φkon
this Lie algebra.
Operators likeUkthat relate two representations are called “intertwining
operators”: