Operator Product Expansion in Massless 2D QED

Question

In Peskin & Schroeder chapter 19 page 656, where the axial current anomaly of massless 2D QED is discussed, the authors go from: $$ \bar\psi(x+\varepsilon/2)\Gamma(x)\psi(x-\varepsilon/2)\tag{19.25} $$ (where $\Gamma(x)$ is some operator) to: $$ \bar\psi(x+\varepsilon/2)\Gamma(x)\psi(x-\varepsilon/2) \tag{19.27} $$ (where now the two Fermionic fields are contracted) to: $$ Tr\left[\Gamma(x)S_F(\varepsilon)\right] \tag{19.27} $$ (where $S_F(x)$ is the Fermion propagator between a spacetime interval $x$)

I really don't understand these transitions and would appreciate any help with how to do them. In particular:

1) Is it implicitly understood (from the very beginning of this derivation) that the axial current is in fact time-ordered (so that we can employ Wick's theorem) and always assume it operates on the vacuum (so that normal ordered terms vanish)?

2) Why does the contraction of the two Fermion fields over the $\Gamma$ operator lead to a trace, as if we had a loop?

Answer 1

For your second problem, the propagator can be written with its indices as

$$ (S_F)_{\alpha\beta}(x-y) = \langle T\{\psi_{\alpha}(x)\bar\psi_{\beta}(y)\} \rangle $$

Then we have

$$ \langle T\{\bar\psi(x)\Gamma\psi(y)\} \rangle = \Gamma_{\alpha\beta} \langle T\{\bar\psi_{\alpha}(x)\psi_{\beta}(y)\} \rangle = -\Gamma_{\alpha\beta} (S_F)_{\beta\alpha}(x-y) = -\mathrm{tr} [\Gamma S_F(x-y)] $$