# Operator Product Expansion in Massless 2D QED

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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?

This post imported from StackExchange Physics at 2014-07-14 07:37 (UCT), posted by SE-user PPR
asked Jul 13, 2014
retagged Jul 14, 2014
Observe that the trace is invariant under cyclic permutation, this should give you the answer to your second question.

This post imported from StackExchange Physics at 2014-07-14 07:37 (UCT), posted by SE-user ACuriousMind

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