39.3 The anomaly and the Schwinger term

39.3 The anomaly and the Schwinger term

Concept links · terms present in this machine draft; source roles are unverified: Lie algebra · Lie algebra representation · group representation · unitary representation

The argument above gives some motivation for the existence as d goes 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 algebra ${\mathfrak{sp}}_{res}$ of $S{p}_{res}$ . 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

$$-\frac{1}{2}tr(B{C}^{†}-C{B}^{†})1$$

This term is sometimes called the “Schwinger term”.

The Schwinger term causes a problem with the standard expectation that given some group $G$ acting on the phase space preserving the Poisson bracket, one should get a unitary representation of $G$ on the quantum state space $H$. This problem is sometimes called the “anomaly”, meaning that the expected 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 given $L\in \mathfrak{g}.$ the corresponding moment map ${\mu }_L$ is 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

$$L\to {\mu }_L$$

a Lie algebra homomorphism, that turns out to no longer be true for the case $\mathfrak{g}={\mathfrak{sp}}_{res}(\mathrm{or}{\mathfrak{so}}_{res})$ 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 for G that commute with the complex structure, since for these the normal ordered Lie algebra representation operators will be a true representation of $\mathfrak{u}(\infty )\subset {\mathfrak{sp}}_{res}$ . We will call $U(\infty )\subset S{p}_{res}$ the subgroup of elements that commute with $J_0$ exactly, not just up to a Hilbert-Schmidt operator. It turns out that $G\subset U(\infty )$ 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 problem 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 of Sp(2d, $R$)), and H will 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 of $Sp(2d,\mathbf{R})$ 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.

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