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?


1 Answer 1


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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