I would like to understand anomalies from the point of view of the canonical quantization. Noether's theorem claims that, given a continuous symmetry of the Lagrangian, there exists a function Q in the phase space that satisfies \begin{equation} \frac{dQ}{dt} = \{Q,H\}_{PB} = 0 \end{equation}

Under canonical quantization, it follows that

\begin{equation} i\hbar\frac{d\textbf{Q}}{dt} = [\textbf{Q},\textbf{H}] = 0 \end{equation}

holds as an operator equation where $\textbf{Q}$ and $\textbf{H}$ are operators as defined by the usual prescription. The statement of anomaly can be expressed as \begin{equation} \frac{d\textbf{Q}}{dt} \neq 0 \end{equation}

as operators in the quantum theory, which seems to be in disagreement with the canonical quantization prescription. Could this issue be clarified?


The replacement of the Poisson brackets by commutators per se is not yet a quantization. A quantization must include a prescription of how to map functions on the phase space into operators on the Hilbert space.

For example in the case of the flat phase space $\mathbb{R}^n$ with the canonical symplectic structure and coordinates $\{ q_i, p_i \}$, $i=1, ., ., ., n$. We have, for example, the Weyl quantization scheme in which we replace polynomials in $q$ and $p$ by a symmetrized version, for example $qp$ is quantized as $\frac{1}{2} (\hat{q}\hat{p}+\hat{p}\hat{q})$, etc.

On the same space, we also have the normal ordering which is a different quantization scheme :functions do not map into the same operators under the two schemes. (It is true however that for finite $n$, the two quantizations are unitarily equivalent, but this is not a general result).

The normal ordering scheme has an advantage that it makes the Hamiltonian bounded from below, thus it is favored in many cases.

Any quantization suffers from the Groenewold-Van Hove inconsistencies (please see for example the following review by: Curtright, Fairlie, and Zachos) so we can only demand that the Poisson brackets pass to commutators up to terms with higher powers of $\hbar$. Anomalies are just this type of corrections.

Once, a quantization scheme is selected, it is possible to obtain the anomalies within canonical quantization (in the Hamiltonian picture).

For example a 1+1 theory with $n$ species of chiral fermions: $$\mathcal{I} =\int dt \int dx \sum_i^n \bar{\psi_i}(x) \gamma^{\mu}\partial_{\mu} \psi_i(x)$$ has an $U(n)$ global Symmetry

If we assume the normal ordering prescription for the quantization of the Noether's charges of the $U(n)$ symmetry: $$\hat{T} = \int dx :\psi_i^{\dagger}(x)T_{ij}\psi_j(x):$$ then the current algebra gets the Schwinger term stemming from the chiral anomaly:

$$[\hat{T_1}, \hat{T_2}] = \widehat{[T_1, T_2]} + \frac{1}{2\pi} \mathrm{tr}(T_1, T_2)$$

Hamiltonian methods are not very popular in higher dimensions, but nevertheless, Dunne and Trugenberger were able to find a quantization scheme, they called kinetic normal ordering, in which the ordering is according the kinetic energy eigenvalues to obtain the chiral anomaly in 3+1 dimensions.

Thus in summary, there is no failure in canonical quantization with respect to anomalies.


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