Recently I've read in one article about very strange way to describe chiral anomaly on quasiclassical level (i.e., on the level of Boltzmann equation and distribution function).
Starting from Weyl hamiltonian $$ H= \sigma \cdot \mathbf p, $$ which describes massless chiral fermions, performing unitary transformation $$ |\psi\rangle \to V|\psi\rangle, $$ where $|\psi\rangle$ is fermion state and $V$ is $2\times 2$ matrix that diagonalizes $\sigma \cdot \mathbf p$, such that $$ V\sigma \cdot \mathbf pV^{\dagger} = |\mathbf p|\sigma_{3}, $$ we obtain the expression for the matrix element $$ \langle f|e^{iH(t_{f}-t_{i})}|i\rangle \equiv \left(V_{\mathbf p_{f}}\int Dx Dp \text{exp}\left[i\int dt(\mathbf p \cdot \mathbf x - |\mathbf p|\sigma_{3}-\hat{\mathbf{a}}\cdot \dot{\mathbf p})\right]V_{\mathbf p_{i}}^{\dagger}\right)_{fi}, \quad \hat{a}_{\mathbf p} = V_{p}\nabla_{\mathbf p}V^{\dagger}_{p} $$ By choosing $+1$ helicity and neglecting off-diagonal components of $\hat{a}_{\mathbf p}$ (which is called adiabaticity approximation; it is not valid near $\mathbf p = 0$), we obtain following quasiclassical action: $$ S = \int dt (\mathbf p \cdot \dot{\mathbf x} - |\mathbf p| - \mathbf a \dot{\mathbf p}) $$ The quantity $\mathbf a$ (which is called Berry phase) plays the role of gauge field in momentum representation, with curvature $$ \mathbf b = \nabla \times \mathbf a = \frac{\mathbf p}{|\mathbf p|^{3}} $$ Effect of this Berry phase is absent when $\dot{\mathbf p} = 0$.
If we, however, turn on external EM field, it becomes to be relevant. We obtain that the invariant phase volume element is $$ dV = \frac{d^{3}\mathbf x d^{3}\mathbf p}{(2\pi)^{3}}\Omega (\mathbf p), \quad \Omega (\mathbf p) = (1 + \mathbf b \cdot \mathbf B)^{2}, $$ where $\mathbf B$ is magnetic field. This captures chiral anomaly effect, $$ \tag 1 \dot{\rho} + \nabla_{\mathbf r}(\rho \dot{\mathbf r}) + \nabla_{\mathbf p}(\rho \dot{\mathbf p}) = 2\pi \mathbf E \cdot \mathbf B \delta^{3}(\mathbf p), \quad \rho = f\Omega , $$ with $\mathbf E$ being electric field and $f$ being distribution function.
I don't understand how this berry phase leads to description of chiral anomaly. They appear due to different reasons (anomaly arises because of non-trivial jacobian of chiral transformation, while berry phase arises because of formal manipulations), anomaly has topological nature connected with difference of number of zero modes of Dirac operator, while Berry phase hasn't such origin. Finally, the rhs of $(1)$ is non-zero only when adiabaticity approximation is violated (and hence the result isn't valid).
Could someone explain me the reason due to which berry phase somehow describes effects of anomaly on quasiclassical level?