Quantum Mechanics for Mathematicians

holds in the orthogonal group case). Elements ofsp(2d,R) corresponding to
Bogoliubov transformations (i.e., with non-zero commutator withJ 0 ) were of
the form
1
2

∑

jk

(Bjkzjzk+Bjkzjzk)

for symmetric complex matricesB. These acted on the metaplectic representa-
tion by

−

i

2

∑

jk

(Bjka†ja†k+Bjkajak) (39.1)

and commuting two of them gave a result (equation 25.9) corresponding to quan-
tization of an element of theu(d) subgroup, differing from its normal ordered
version by a term

−

1

2

tr(BC−CB) 1 =−

1

2

tr(BC†−CB†) 1

Ford=∞, this trace in general will be infinite and undefined. An alter-
nate characterization of Hilbert-Schmidt operators is that forBandCHilbert-
Schmidt operators, the traces

tr(BC†) and tr(CB†)

will be finite and well-defined. So, at least to the extent normal ordered op-
erators quadratic in annihilation and creation operators are well-defined, the
Hilbert-Schmidt condition on operators not commuting with the complex struc-
ture implies that they will have well-defined commutation relations with each
other.

39.3 The anomaly and the Schwinger term

The argument above gives some motivation for the existence asdgoes to∞of
well-defined commutators of operators of the form 39.1 and thus for the existence
of an analog of the metaplectic representation for the infinite dimensional Lie
algebraspresofSpres. There is one obvious problem though with this argument,
in that while it tells us that normal ordered operators will have well-defined
commutation relations, they are not quite the right commutation relations, due
to the occurrence of the extra scalar term

−

1

2

tr(BC†−CB†) 1

This term is sometimes called the “Schwinger term”.
The Schwinger term causes a problem with the standard expectation that
given some groupGacting on the phase space preserving the Poisson bracket,
one should get a unitary representation ofGon the quantum state spaceH.
This problem is sometimes called the “anomaly”, meaning that the expected