unitary Lie algebra representation does not exist (due to extra scalar terms
in the commutation relations). Recall from section 15.3 that this potential
problem was already visible at the classical level, in the fact that givenL∈g,
the corresponding moment mapμLis only well-defined up to a constant. While
for the finite dimensional cases we studied, the constants could be chosen so as
to make the map
L→μL
a Lie algebra homomorphism, that turns out to no longer be true for the case
g=spres(orsores) acting on an infinite dimensional phase space. The potential
problem of the anomaly is thus already visible classically, but it is only when
one constructs the quantum theory and thus a representation on the state space
that one can see whether the problem cannot be removed by a constant shift
in the representation operators. This situation, despite its classical origin, is
sometimes characterized as a form of symmetry-breaking due to the quantization
procedure.
Note that this problem will not occur forGthat commute with the complex
structure, since for these the normal ordered Lie algebra representation opera-
tors will be a true representation ofu(∞)⊂spres. We will callU(∞)⊂Spres
the subgroup of elements that commute withJ 0 exactly, not just up to a Hilbert-
Schmidt operator. It turns out thatG⊂U(∞) for most of the cases we are
interested in, allowing construction of the Lie algebra representation by normal
ordered quadratic combinations of the annihilation and creation operators (as in
25.6). Also note that since normal ordering just shifts operators by something
proportional to a constant, when this constant is finite there will be no anomaly
since one can get operators with correct commutators by such a finite shift of the
normal ordered ones. The anomaly is an inherently infinite dimensional prob-
lem since it is only then that infinite shifts are necessary. When the anomaly
does appear, it will appear as a phase-ambiguity in the group representation
operators (not just a sign ambiguity as in finite dimensional case ofSp(2d,R)),
andHwill be a projective representation of the group (a representation up to
phase).
Such an undetermined phase factor only creates a problem for the action on
states, not for the action on operators. Recall that in the finite dimensional case
the action ofSp(2d,R) on operators (see 20.3) is independent of any constant
shift in the Lie algebra representation operators. Equivalently, if one has a
unitary projective representation on states, the phase ambiguity cancels out in
the action on operators, which is by conjugation.
39.4 Spontaneous symmetry breaking
In the standard Bargmann-Fock construction, there is a unique state| 0 〉, and
for the Hamiltonian of the free particle quantum field theory, this will be the
lowest energy state. In interacting quantum field theories, one may have state
spaces unitarily inequivalent to the standard Bargmann-Fock one. These can
have their own annihilation and creation operators, and thus a notion of particle