The following is from page 363 of Weinberg volume II.
We wish to evaluate the RHS of
\begin{align}\label{EQbbvbv} [d \psi][d \bar{\psi}] \rightarrow(\operatorname{Det} \mathscr{U} \operatorname{Det} \overline{\mathscr{U}})^{-1}[d \psi][d \bar{\psi}], \end{align}
in order to find the effect on the measure of a change of path integration variables corresponding to a local matrix transformation $\begin{align} \psi(x) \rightarrow U(x) \psi(x) \end{align}$. We define
\begin{align}\notag \mathscr{U}_{x n, y m} &:=U(x)_{n m} \delta^{4}(x-y), \text{ and}\\\label{EQdwdf22vfr} %% %% \bar{\mathscr{U}}_{x n, y m} &:=\left[\gamma_{4} U(x)^{\dagger} \gamma_{4}\right]_{n m} \delta^{4}(x-y), \end{align} where $\gamma_{4} := i \gamma^{0}$ is used in defining $\bar{\psi}=\psi^{\dagger} \gamma_{4}$.
Also note that the indices $n,m$ run over flavour labels and Dirac spin indices.
Let us consider $\alpha(x)$ to be an infinitesimal scalar function in the transformation \begin{align}\label{EQmnmmb4} U(x)=\exp \left[i \gamma_{5} \alpha(x) t\right] \end{align} where $t$ is a general hermitian matrix.
Note that Weinberg omits the working out from here, so the following 3 equations are my own work.
In this case, since the Taylor expansion of the exponential has negligible contributions from terms of order greater the one in $\alpha$, one obtains that \begin{align} [\mathscr{U}-1]_{n x, m y}=i \alpha(x)[\gamma _5 t]_{n m} \delta^{4}(x-y). \end{align} Therefore, \begin{align} \operatorname{Det} \mathscr{U} &= \exp{\text{Tr} \ln \{1+i \alpha(x)\left[\gamma_{5} t\right]_{n m} \delta^{4}(x-y)\}}\\ %% %% &= \exp{\, i \alpha(x)\text{Tr}\{\gamma_{5} t\} \delta^{4}(x-y)}, \end{align} where we have used the identity for the determinant of a matrix $M$, $\operatorname{Det} M=\exp \operatorname{Tr} \ln M$, and that $\ln (1+x)\rightarrow x$ as $x\rightarrow 0.$ But since $\mathscr U$ is pseudo-Hermitian, $(\begin{align} \overline{\mathscr{U}}=\mathscr{U} \end{align})$ we have
\begin{align} [d \psi][d \bar{\psi}] \rightarrow(\operatorname{Det} \mathscr{U})^{-2}[d \psi][d \bar{\psi}]. \end{align}
Weinberg now claims that the measure changes under this transformation as \begin{align}\label{EQnmnnghr3} [d \psi][d \bar{\psi}] \rightarrow \exp \left\{i \int d^{4} x \alpha(x) \mathscr{A}(x)\right\}[d \psi][d \bar{\psi}], \end{align} where we define the anomaly function \begin{align}\label{EQvccvezz33r43} \mathscr{A}(x)=-2 \operatorname{Tr}\left\{\gamma_{5} t\right\} \delta^{4}(x-x). \end{align} We use `Tr' to denote a trace to be taken over Dirac and species indices.
Question 1: Where does the $\delta^{4}(x-x)$ comes from? I can't see any reason why the argument of the delta function might change in the calculations I went through.
Question 2: Where does the integral over $d^4 x$ comes from in the second last equation? If we're working with the Jacobian as on the RHS of the first equation, I don't see how an integral could pop up.
Question 3: This is definitely a rather trivial question, but I've never encountered $\gamma^4$ before... I'm used to the definition $\bar{\psi}=\psi^{\dagger} \gamma_{0}$. This is probably just a different representation of the spinors. If so, could I get a name for it please? I can't seem to find it. Final trivial question: what is a species index?
