Quantum Mechanics for Mathematicians

one confusing point about these. Recall that in chapter 16 we found the moment
mapμL=−q·Apfor elementsL∈sp(2d,R) of the block-diagonal form
(
A 0
0 −AT

)

whereAis a realdbydmatrix and so ingl(d,R). That block decomposition
corresponded to the decomposition of basis vectors ofMinto the two setsqj
andpj. Here we have complexified, and are working with respect to a differ-
ent decomposition, that of basis vectorsM⊗Cinto the two setszjandzj.
The matricesAin this case are complex, skew-adjoint, and in a different non-
isomorphic Lie subalgebra,u(d) rather thangl(d,R). For the simplest example
of this,d= 1, the distinction is between theRLie subgroup ofSL(2,R) (see
section 20.3.4), for which the moment map is

−qp= Im(z^2 ) =

1

2 i

(z^2 −z^2 )

and theU(1) subgroup (see section 20.3.2), for which the moment map is

1

2

(q^2 +p^2 ) =zz

25.3 The metaplectic representation andU(d)⊂Sp(2d,R)

Turning to the quantization problem, we would like to extend the discussion of
quantization of quadratic combinations of complex coordinates on phase space
from thed= 1 case of chapter 24 to the general case. For anyj,kone can take

zjzk→−iajak, zjzk→−ia†ja†k

There is no ambiguity in the quantization of the two subalgebras given by pairs
of thezjcoordinates or pairs of thezj coordinates since creation operators
commute with each other, and annihilation operators commute with each other.
Ifj 6 =kone can quantize by taking

zjzk→−ia†jak=−iaka†j

and there is again no ordering ambiguity. Ifj=k, as in thed= 1 case there is
a choice to be made. One possibility is to take

zjzj→−i

1

2

(aja†j+a†jaj) =−i

(

a†jaj+

1

2

)

which will have the propersp(2d,R) commutation relations (in particular for
commutators ofa^2 jwith (a†j)^2 ), but require going to a double cover to get a true
representation of the group. The Bargmann-Fock construction thus gives us a