Quantum Mechanics for Mathematicians

24.1 The metaplectic representation ford= 1 in terms ofaanda†

Poisson brackets of order two combinations ofzandzcan easily be computed
using the basic relation{z,z}=iand the Leibniz rule. On basis elements
z^2 ,z^2 ,zzthe non-zero brackets are

{zz,z^2 }=− 2 iz^2 , {zz,z^2 }= 2iz^2 , {z^2 ,z^2 }= 4izz

Recall from equation 16.8 that quadratic real combinations ofpandqcan be
identified with the Lie algebrasl(2,R) of traceless 2 by 2 real matrices with
basis

E=

(

0 1

0 0

)

, F=

(

0 0

1 0

)

, G=

(

1 0

0 − 1

)

Since we have complexified, allowing complex linear combinations of basis
elements, our quadratic combinations ofzandzare in the complexification of
sl(2,R). This is the Lie algebrasl(2,C) of traceless 2 by 2 complex matrices.
We can take as a basis ofsl(2,C) over the complex numbers

Z=E−F, X±=

1

2

(G±i(E+F))

which satisfy

[Z,X−] =− 2 iX−, [Z,X+] = 2iX+, [X+,X−] =−iZ

and then use as our isomorphism between quadratics inz,zandsl(2,C)

z^2

2

↔X+,

z^2

2

↔X−, zz↔Z

The element

zz=

1

2

(q^2 +p^2 )↔Z=

(

0 1

−1 0

)

exponentiates to give aSO(2) =U(1) subgroup ofSL(2,R) with elements of
the form

eθZ=

(

cosθ sinθ

−sinθ cosθ

)

(24.1)

Note thath=^12 (p^2 +q^2 ) =zzis the classical Hamiltonian function for the
harmonic oscillator.
We can now quantize quadratics inzandzusing annihilation and creation
operators acting on the Fock spaceF. There is no operator ordering ambiguity
for

z^2 →(a†)^2 =z^2 , z^2 →a^2 =

d^2

dz^2