I have questions in the section 19.1 of Peskin and Schroeder.
\begin{equation} \psi = \begin{pmatrix} \psi_+ \\ \psi_- \tag{19.7} \end{pmatrix} \end{equation}
The subscripts indicates the $\gamma^5$ eigenvalue.
Below we'll show that there is situtions where the axial current conservation law $$ \partial_{\mu} j^{\,\mu 5} =0 . $$ is violated, and the integrated version of $$ \partial_{\mu} j^{\,\mu 5} = \frac{e}{2\pi} \epsilon^{\mu\nu}F_{\mu\nu} \tag{19.18} $$ holds. where the totally antisymmetric symbol $\epsilon^{\mu \nu}$ is defined as $\epsilon^{01}=+1$ on page 653.
Let us analyze this problem by thinking about fermions in one space dimension in a background $A^1$ field that is constant in space and has a very slow time dependence. We will assume that the system has a finite length $L$, with the periodic boundary conditions.
So $A^1(t,0)=A^1(t,L) $, and also $ \psi(t,0)=\psi(t,L) $.
the single-particle eigen-states of $H$ have energies \begin{alignat}{2} \psi_+ : \quad \quad & E_n && =+(k_n -eA^1), \\ \psi_+ : \quad \quad & E_n && =-(k_n -eA^1). \tag{19.36} \end{alignat}
Now we consider the slow shift of $A^1$.
If $A^1$ changes by the finite amount
$$ \Delta A^1 = \frac{2\pi}{eL} \quad \quad \quad \quad \quad \quad \quad \quad (19.37)$$
Where $ \Delta A^1 \gt 0 $.
the spectrum of $H$ returns to its original form. In this process, each level of $\psi_+$ moves down to the next poston, and each level of $\psi_-$ moves up to the next position, The occupation numbers of levels should be maintained in this adiabatic process. Thus, remarkably, one right-moving fermion($\psi_+$) disappears from the vacuum and one extra left-moving fermion($\psi_-$) appears.
So $$ \quad \Delta N_R=-1 ,\quad \Delta N_L=1 $$.
At the same time, \begin{alignat}{2} \int d^2x \frac{e}{2\pi} \epsilon^{\mu\nu}F_{\mu\nu} &=&& \int dt dx \frac{e}{\pi} \partial_0 A^1 \\ &=&& \frac{e}{\pi} L(-\Delta A^1) \\ &=&& -2 \tag{19.38} \end{alignat}
where I've rectified a misprint on the LHS of the first line.
where we have inserted (19.37) in the last line. Thus the integrated form of the anomalous nonconservation equation (19.18) is indeed satisfied: $$ \Delta N_R - \Delta N_L = \int d^2x \frac{e}{2\pi} \epsilon^{\mu\nu}F_{\mu\nu} . \tag{19.39} $$
For $ \Delta N_R - \Delta N_L =-2 $.
From
$$ \int d^2x \partial_{\mu} j^{\,\mu 5} =\Delta N_R - \Delta N_L \tag{19.30} $$
we obtain
$$ \int d^2x \partial_{\mu} j^{\,\mu 5} = \int d^2x \frac{e}{2\pi} \epsilon^{\mu\nu}F_{\mu\nu}. $$
Question 1:
I've checked all the calculations leading to (19.38).
But I cannot understand the minus sign in the second anf the third line of (19.38).
Where does it come from?
Question 2:
In computing the changes in the separate fermion numbers, we have assumed that the vacuum cannot change the charge at large negative energies. This prescription is gauge invariant, but it leads to the nonconservation of the axial vector current.
It follws that if the vacuum changed the charge at large negative energies, this prescription would not be gauge invariant.
How come? Please would you explain?
