Quantum Mechanics for Mathematicians

Quadratic combinations of the creation and annihilation operators give rep-
resentations onHof three subalgebras of the complexificationsp(4,C) ofsp(4,R):

• A three dimensional commutative Lie sub-algebra spanned byz 1 z 2 ,z^21 ,z^22 ,
  with quantization

Γ′BF(z 1 z 2 ) =−ia 1 a 2 , Γ′BF(z^21 ) =−ia^21 , Γ′BF(z^22 ) =−ia^22

• A three dimensional commutative Lie sub-algebra spanned byz 1 z 2 ,z^21 ,z^22 ,
  with quantization

Γ′BF(z 1 z 2 ) =−ia† 1 a† 2 , Γ′BF(z 12 ) =−i(a† 1 )^2 , Γ′BF(z 22 ) =−i(a† 2 )^2

• A four dimensional Lie subalgebra isomorphic togl(2,C) with basis

z 1 z 1 ,z 2 z 2 ,z 2 z 1 ,z 1 z 2

and quantization

Γ′BF(z 1 z 1 ) =−

i

2

(a† 1 a 1 +a 1 a† 1 ), Γ′BF(z 2 z 2 ) =−

i

2

(a† 2 a 2 +a 2 a† 2 )

Γ′BF(z 2 z 1 ) =−ia† 2 a 1 , Γ′BF(z 1 z 2 ) =−ia† 1 a 2

Real linear combinations of

z 1 z 1 , z 2 z 2 , z 1 z 2 +z 2 z 1 , i(z 1 z 2 −z 2 z 1 )

span the Lie algebrau(2)⊂sp(4,R), and Γ′BF applied to these gives a

unitary Lie algebra representation by skew-adjoint operators.

Inside this last subalgebra, there is a distinguished elementh=z 1 z 1 +z 2 z 2
that Poisson-commutes with the rest of the subalgebra (but not with elements
in the first two subalgebras). Quantization ofhgives the Hamiltonian operator

H=

1

2

(a 1 a† 1 +a† 1 a 1 +a 2 a† 2 +a† 2 a 2 ) =N 1 +

1

2

+N 2 +

1

2

=z 1

∂

∂z 1

+z 2

∂

∂z 2

+ 1

This operator will multiply a homogeneous polynomial by its degree plus one,
so it acts by multiplication byn+ 1 onHn. Exponentiating this operator (mul-
tiplied by−i) one gets a representation of aU(1) subgroup of the metaplectic
coverMp(4,R). Taking instead the normal ordered version

:H: =a† 1 a 1 +a† 2 a 2 =N 1 +N 2 =z 1

∂

∂z 1

+z 2

∂

∂z 2

one gets a representation of aU(1) subgroup ofSp(4,R). NeitherHnor :H:
commutes with operators coming from quantization of the first two subalgebras.