Quantum Mechanics for Mathematicians

For the case ofzz(which is real), in order to get thesl(2,R) commutation
relations to come out right (in particular, the Poisson bracket{z^2 ,z^2 }= 4izz),
we must take the symmetric combination

zz→

1

2

(aa†+a†a) =a†a+

1

2

=z

d

dz

+

1

2

(which of course is the standard Hamiltonian for the quantum harmonic oscil-
lator).
Multiplying as usual by−i(to get a unitary representation of the real Lie
algebrasl(2,R)), an extension of the Bargmann-Fock representation Γ′BF of
h 3 ⊗C(see section 22.4) to ansl(2,C) representation can be defined by taking

Γ′BF(X+) =−

i

2

a^2 , Γ′BF(X−) =−

i

2

(a†)^2 , Γ′BF(Z) =−i

1

2

(a†a+aa†)

This is the right choice of Γ′BF(Z) to get ansl(2,C) representation since

[Γ′BF(X+),Γ′BF(X−)] =

[

−

i

2

a^2 ,−

i

2

(a†)^2

]

=−

1

2

(aa†+a†a)

=−iΓ′BF(Z) = Γ′BF([X+,X−])

As a representation of the real sub-Lie algebrasl(2,R) ofsl(2,C), one has
(using the fact thatG,E+F,E−Fis a real basis ofsl(2,R)):

Definition(Metaplectic representation ofsl(2,R)). The representationΓ′BF
onFgiven by

Γ′BF(G) = Γ′BF(X++X−) =−

i

2

((a†)^2 +a^2 )

Γ′BF(E+F) = Γ′BF(−i(X+−X−)) =−

1

2

((a†)^2 −a^2 ) (24.2)

Γ′BF(E−F) = Γ′BF(Z) =−i

1

2

(a†a+aa†)

is a representation ofsl(2,R), called the metaplectic representation.

Note that this is clearly a unitary representation, since all the operators are
skew-adjoint (using the fact thataanda†are each other’s adjoints).
This representation Γ′BFonFwill be unitarily equivalent using the Bargmann
transform (see section 23.5) to the Schr ̈odinger representation Γ′Sfound earlier
when quantizingq^2 ,p^2 ,pqas operators onH=L^2 (R). For many purposes it is
however much easier to work with since it can be studied as the state space of
the quantum harmonic oscillator, which comes with a basis of eigenvectors of
the number operatora†a. The Lie algebra acts simply on such eigenvectors by
quadratic expressions in the annihilation and creation operators.
One thing that can now easily be seen is that this representation Γ′BFdoes
not integrate to give a representation of the groupSL(2,R). If the Lie algebra