Chapter 25

The Metaplectic

Representation and

Annihilation and Creation

Operators, arbitrary d

In this chapter we’ll turn from thed= 1 case of chapter 24 to the general
case of arbitraryd. The choice ofdannihilation and creation operators picks
out a distinguished subgroupU(d)⊂Sp(2d,R) of transformations that do not
mix annihilation and creation operators, and the metaplectic representation
gives one a representation of a double cover of this group. We will see that
normal ordering the products of annihilation and creation operators turns this
into a representation ofU(d) itself (rather than the double cover). In this
way, aU(d) action on the finite dimensional phase space gives operators that
provide an infinite dimensional representation ofU(d) on the state space of the
ddimensional harmonic oscillator.
This method for turning unitary symmetries of the classical phase space
into unitary representations of the symmetry group on a quantum state space
is elaborated in great detail here not just because of its application to simple
quantum systems like theddimensional harmonic oscillator, but because it
will turn out to be fundamental in our later study of quantum field theories.
In such theories the observables of interest will be operators of a Lie algebra
representation, built out of quadratic combinations of annihilation and creation
operators. These arise from the construction in this chapter, applied to a unitary
group action on phase space (which in the quantum field theory case will be
infinite dimensional).
Studying theddimensional quantum harmonic oscillator using these meth-
ods, we will see in detail how in the cased= 2 the groupU(2)⊂Sp(4,R) com-
mutes with the Hamiltonian, so acts as symmetries preserving energy eigenspaces