# The Berry phase and anomalous commutator

## The problem

Recently I've read an article "Adiabatic theorem and anomalous commutator", written by Iida and Kuratsuji. In this article the authors relate the Berry phase with anomalous commutator of canonical variables. Here is an idea of their considerations.

First, they assume the theory, in which some dynamical system (which they call external system) interacts with some internal system. They write the Hamiltonian of such system as $$H = H_{\text{ext}}(X) + H_{\text{int}}(X,q),$$ $X$ defines external system canonical variables, while $q$ defines internal system ones.

Second, they assume temporary adiabatic evolution of variables $X$ along the closed path $C$. With this assumption, the adiabaticity theorem is applicable, and for internal system we have instantaneous Hamiltonian $H_{\text{int}}(X(t),q)$ with the basis $$\tag 0 |n(X(t))\rangle : \quad H_{\text{int}}(X(t))|n(X(t))\rangle = \lambda_{n}(t)|n(X(t))\rangle$$ Third, they compute the trace of evolution operator $K(t) = \text{Tr}\left[\text{exp}^{-i\frac{H(t-t_{0})}{\hbar}}\right]$, during which the parameters evolve along the path $C$. It reads $$K(t) = \sum\int d\mu \text{exp}\left(\frac{1}{\hbar}\int dt S_{\text{eff}} \right),$$ where $S_{\text{eff}} = S_{\text{standard}} + \hbar \varphi^{(n)}_{B}$ and $$\tag 1 \varphi^{(n)}_{B} = \oint \limits_{C} dX_{i} \langle n| \partial_{X_{i}}|n\rangle$$ is called the Berry phase.

Fourth, they conclude that the Berry phase provides the modification of the symplectic structure of the $X$ variables phase space. This means, that the Berry phase modifies Poisson brackets $\{ X_{I},X_{J}\}$. In quantum theory, the modification of the Poisson brackets is nothing but modification of commutators, i.e., adding the anomalous commutators. It also modifies the Liouville measure of the phase space.

## My question

Suppose the case when the external system is EM field, and the internal system is Weyl fermions. In this case, the internal system quantity $n$ from $(0)$ is (here I "forget" about the two-dimensional subspace $\pm$ of helicities) $$\mathbf P = \mathbf p - \mathbf A,$$ where $\mathbf A$ is 3-vector potential and $\mathbf p$ is free fermion momenta. The canonical variables of EM field are $\{\mathbf A, \mathbf E\}$, where $\mathbf E$ is electric field strength.

With this, $(1)$ takes the form $$\tag 2 \varphi_{B}^{\pm} = \oint d\mathbf A \langle \mathbf P , \pm |\partial_{\mathbf A}|\mathbf P , \pm\rangle$$ From the other side, this theory has chiral anomaly, i.e., the chiral current isn't conserved: $$\tag 3 \partial_{\mu}J^{\mu}_{5}(x) = A(x),$$
I know that semiclassically $(3)$ be captured by the Berry phase: $$\tag 4 \varphi_{B}^{\pm} = \oint d\mathbf P \langle \mathbf P, \pm| \partial_{\mathbf P}|\mathbf P ,\pm\rangle$$ My question is: are the Berry phases $(2)$ and $(4)$ in fact the same? If yes, why the symplectic structure deformation in EM phase space is nothing but symplectic structure deformation in Weyl fermions phase space? If no, how the anomalous commutator of EM canonical variables (giving rise in fact the gauge anomaly), which is generated by $(2)$, implies the chiral anomaly equation $(3)$?