Chapter 24

The Metaplectic

Representation and

Annihilation and Creation

Operators, d = 1

In section 22.4 we saw that annihilation and creation operators quantize com-
plexified coordinate functionsz,zon phase space, giving a representation of
the complexified Heisenberg Lie algebrah 3 ⊗C. In this chapter we’ll see what
happens for quadratic combinations of thez,z, which after quantization give
quadratic combinations of the annihilation and creation operators. These pro-
vide a Bargmann-Fock realization of the metaplectic representation ofsl(2,R),
the representation which was studied in section 17.1 using the Schr ̈odinger real-
ization. Using annihilation and creation operators, the fact that the exponenti-
ated quadratic operators act with a sign ambiguity (requiring the introduction
of a double cover ofSL(2,R)) is easily seen.
The metaplectic representation gives intertwining operators for theSL(2,R)
action by automorphisms of the Heisenberg group. The use of annihilation and
creation operators to construct these operators introduces an extra piece of
structure, in particular picking out a distinguished subgroupU(1)⊂SL(2,R).
Linear transformations of thea,a†preserving the commutation relations (and
thus acting as automorphisms of the Heisenberg Lie algebra structure) are
known to physicists as “Bogoliubov transformations”. They are naturally de-
scribed using a different, isomorphic, form of the groupSL(2,R), a group of
complex matrices denotedSU(1,1).