Quantum Mechanics for Mathematicians

for a complex symmetric matrixBwith matrix entriesBjk. There is no normal
ordering ambiguity here, and quantization will give the unitary Lie algebra
representation operators

−

i

2

∑

jk

(Bjka†ja†k+Bjkajak)

Exponentiating such operators will give operators which take the state| 0 〉to a
distinct state (one not proportional to| 0 〉).
Using the canonical commutation relations one can show

[a†ja†k,alam] =−ala†jδkm−ala†kδjm−a†jamδkl−a†kamδjl

and these relations can in turn be used to compute the commutator of two such
Lie algebra representation operators, with the result

[−

i

2

∑

jk

(Bjka†ja†k+Bjkajak),−

i

2

∑

lm

(Clma†la†m+Clmalam)]

=

1

2

∑

jk

(BC−CB)jk(a†jak+aka†j) =U ̃B′C−CB

Note that normal ordering of these operators just shifts them by a constant,
in particular

UB′C−CB= :U ̃B′C−CB: =U ̃B′C−CB−

1

2

tr(BC−CB) 1 (25.9)

The normal ordered operators fail to give a Lie algebra homomorphism when
extended tosp(2d,R), but this failure is just by a constant term. Recall from
section 15.3 that even at the classical level, there was an ambiguity of a constant
in the choice of a moment map which in principle could lead to an “anomaly”, a
situation where the moment map failed to be a Lie algebra homomorphism by a
constant term. The situation here is that this potential anomaly is removable,
by the shift

UA′ →U ̃A′ =UA′ +

1

2

tr(A) 1

which gives representation operators that satisfy the Lie algebra homomorphism
property. We will see in chapter 39 that for an infinite number of degrees of
freedom, the anomaly may not be removable, since the trace of the operatorA
in that case may be divergent.

25.6 For further reading

The references from chapter 26 ([26], [94]) also contain the general case discussed
here. Given aU(d)⊂Sp(2d,R) action of phase space, the construction of
corresponding metaplectic representation operators using quadratic expressions